Class Notes 7008 Macroeconomics Spring 2003 Hans G. Ehrbarcontent.csbs.utah.edu/~ehrbar/macrec.pdf · Class Notes 7008 Macroeconomics Spring 2003 Hans G. Ehrbar Economics Department,

Class Notes 7008 Macroeconomics Spring 2003

Hans G. Ehrbar

Economics Department, University of Utah, 1645 Campus CenterDrive, Salt Lake City UT 84112-9300, U.S.A.

URL: www.econ.utah.edu/ehrbar/macrec.pdfE-mail address: [email protected]

http://www.econ.utah.edu/ehrbar/macrec.pdf

mailto:[email protected]

Contents

Chapter 1. Syllabus for Econ 7008, Spring 2003 7

Chapter 2. Who is Who in Macroeconomics 111. Demand and Supply of Money 172. Who are the Neo-Conservatives? 23

Chapter 3. Dynamics in Aggregate Supply and Demand 271. Adaptive Expectations and Stability 272. Hysteresis 32

Chapter 4. Rational Expectations and Economic Policy 41

3

4 CONTENTS

Chapter 5. Anticipation Effects and Economic Policy 53

Chapter 6. The Macroeconomics of Quantity Rationing 651. Malinvaud’s Prototype Model 682. Keynesian Unemployment 773. Classical Unemployment 834. Repressed Inflation 86

Chapter 7. Government Budget Deficit 971. Ricardian Equivalence 972. Tax Smoothing 104

Chapter 8. Labor Market 1051. Unemployment Stylized Facts 1052. Minimum Wage too High? 1063. Efficiency Wages 110

Chapter 9. Trade Unions 115

Chapter 10. Search Theory 1191. Relationship between Unemployment Rate and Labor Market Tightness 1212. Firms and Workers 126

CONTENTS 5

3. Policy Implications 141

Chapter 11. Dynamic Inconsistency and Credibility 1431. A Simple Model with Dynamic Inconsistency 1442. Modelling Reputation 149

Chapter 12. Open Economy Macro 1531. Which Measure of National Product? 1532. Accounting Identities 1543. The Mundell-Fleming Model 1574. Dornbusch Model 166

Chapter 13. Money 1691. Errors of Neoclassical Theory 1692. Functions of Money 1713. Overlapping Generations Model of Money 1734. Optimal Quantity of Money 177

Chapter 14. New Keynesian Economics 185

Bibliography 195

CHAPTER 1

Syllabus for Econ 7008, Spring 2003

Textbook is [HVdP02]. Class notes are available at www.econ.utah.edu/ehrbar/macrec.pdf for screen reading, and www.econ.utah.edu/ehrbar/macrel.pdfin letter-page format. Hans’s office hours are Tuesday Thursday 1–2 pm, tel 581-7797, email [email protected]. The class is in the Rasmussen room, 2–4:30pm.

February 27: Chapter 1March 4: Chapter 2March 6: Chapter 3March 11: Chapter 4March 13: Chapter 5March 18: Spring break

7



http://www.econ.utah.edu/ehrbar/macrel.pdf

mailto:[email protected]

8 1. SYLLABUS FOR ECON 7008, SPRING 2003

March 20: Spring breakMarch 25: Chapter 6March 27: Midterm: questions 1, 8, 11, 12, 13, 17, and 22.April 1: Chapter 7April 3: Chapter 8April 8: Chapter 9April 10: Chapter 10April 15: Chapter 11April 17: Chapter 12April 22: Chapter 13The final exam is a takehome exam. The first four questions give some basic

little details which were taken for granted in the class itself. 2 justifies the use of anaggregate production function although the economy consists of many independentfirms; 3 is a standard model of money demand, 4 is a standard model of moneysupply, 7 shows how the real rate of interest can be defined in continuous time.The other four questions refer more specifically to the material covered since themidterm: 24 computes the elasticities of the Matching function, 27 is a simple rationalexpectations model. 28 is an essay question about time inconsistency, and 31 refersto the open economy.

1. SYLLABUS FOR ECON 7008, SPRING 2003 9

In the qualifying exam in macro in 2002, students had to answer the followingfour questions. (They had choices for each question.)

• As your first question please do either question 2 about the aggregationproblem, or question 11 about rational expectations in an econometricmodel.

• The second question is either a simple IS-LM model 5 or a simple maxi-mization problem 26.

• The third question is an essay question: either 29 or 33.• The fourth question goes through one of the models discussed in class in

some detail. Either 19 or 25.

CHAPTER 2

Who is Who in Macroeconomics

Chapter 1 in [HVdP02]. They put a bar on K because the capital stock isassumed given.

Here is, in a little more mathematical detail than in the book, the algebraicderivation of the derivatives of labor demand on pp. 2–4, and the elasticity on p. 10.Profits are

Π = PY −WN(1)

Π = PF (N, K)−WN(2)

11

12 2. WHO IS WHO IN MACROECONOMICS

This is the objective function to maximize. Partial derivative with respect to N is

∂Π∂N

= PF1(N, K)−W(3)

Set this zero and put superscript D on N at the same time (the book did not do thisright; (1.4) should have ND):

PF1(ND, K)−W = 0(4)

F1(ND, K)−W/P = 0(5)

This is an implicit function defining the labor demand ND. If we solve this for ND

we get the function the book shows as first expression in (1.7):

ND = ND(W/P, K)(6)

In order to get the partial derivatives of this function one does not need to know itsexplicit functional form. One can get these partials already from the implicit form(5): simply take the total differential

F11 dND + F12 dK − d(W/P ) = 0.(7)

2. WHO IS WHO IN MACROECONOMICS 13

I.e., if dK = 0 then the relationship between dND and d(W/P ) is

F11 dND = d(W/P ),(8)

and now simply divide

dND

d(W/P )=

1F11

.(9)

Since this only holds if dK = 0 it is called the partial derivative

∂ND

∂(W/P )

∣∣∣K=const

=1

F11.(10)

Whenever one writes down a partial derivative, one should also indicate which othervariables are kept constant; this is usually not done because it is clear from thecontext.

Problem 1. (4 points) The first-order condition for profit maximization is

(11) F1(ND, K)−W/P = 0


The elasticity of labor demand with respect to the real wage w = W/P is defined as

ε = − w

ND

∂ND

∂w(12)

i.e., it indicates by how many percent labor demand increases if real wages fall by 1percent. Show that

ε = − F1

NDF11.(13)

Answer. The total differential of (11) is

F11 dND + F12 dK − d(W/P ) = 0.(14)

This can also be written as

F11ND dND

ND+ F12K

dK

K−

W

P

d(W/P )

W/P= 0;(15)

but since NDF11 = −KF12 by (??), and W/P = F1 because of (11), andd(W/P )

W/P= dW

W− dP

P,

this gives (1.16):

dND

ND=

dK

K+

F1

NDF11

[dW

W−

dP

P

](16)

Setting dK = 0 and using the negative of this coefficient gives the required number.

2. WHO IS WHO IN MACROECONOMICS 15

Problem 2. This exercise, derived from [Sar87, pages 7, 8, 9], shows how anaggregate production function can come about as the aggregation of many individualproduction functions.

Assume the economy has n firms, which produce the same good and sell it atthe same price p. The ith firm, which has inherited from the past the amount ofcapital Ki, hires that amount of labor Ni which maximizes its profit. All firms haveidentical production functions Yi = F (Ki, Ni) which exhibit constant returns to scale,i.e., which satisfy F (λK, λN) = λF (K, N), with F2 > 0, and F22 < 0. All firms selltheir product at the same price p and hire labor at the same wage w.

• a. (3 points) Show that in this case Ki/Ni is the same for all i (call it Ki/Ni =k).

Answer. Since F is homogeneous of degree one, F2 is homogeneous of degree zero. Proof

by differentiating F (λK, λN) = λF (K, N) with respect to N , (when we wanted to prove Euler’sidentity we differentiated with respect to λ) which gives λF2(λK, λN) = λF2(K, N), and then

divide both sides by λ. Setting λ = 1/N gives F2(K, N) = F2(KN

, 1). The first-order conditionwp

= F2(Ki, Ni) is therefore equivalent to wp

= F2(KiNi

, 1), i.e., KiNi

has equal values for all firms.

• b. (2 points) Show furthermore that the real wage satisfies wp = F2(K, N),

where K =∑

i Ki and N =∑

i Ni.


Answer. Call Ki/Ni = k, then Kj = kNj for all j, therefore

(17)K

N=

∑Ki∑Ni

=

∑kNi∑Ni

=k

∑Ni∑

Ni= k.

From KiNi

= KN

and F2 homogeneous of degree zero follows wp

= F2(K, N).

• c. (4 points) Finally show that the total product of the economy is Y =F (K, N), where Y =

∑Yi.

Answer. In the same way one sees that F1(Ki, Ni) is the same for all i. Therefore by Euler’stheorem

Fi(Ki, Ni) = F1(Ki, Ni)Ki + F2(Ki, Ni)Ni = F1(K, N)Ki + F2(K, N)Ni(18)

Y =∑

i

Fi(Ki, Ni) = F1(K, N)∑

Ki + F2(K, N)∑

Ni =

= F1(K, N)K + F2(K, N)N = F (K, N).(19)

• d. (4 points) The production function Y = F (K, N) is constant returns toscale, i.e., F (λK, λN) = λF (K, N) for all λ > 0. Does this imply that F (

∑Ki,

∑Li) =∑

F (Ki, Li)? If yes, give a proof; if no, give a counterexample.

1. DEMAND AND SUPPLY OF MONEY 17

Answer. This does not hold. For a counterexample choose K1, K2, L1, and L2 such that

K1/L1 6= K2/L2. Say F (K, N) =√

KN . Then

K1 = 2 N1 = 2√

K1N1 = 2(20)

K2 = 3 N2 = 12√

K2N2 = 6(21)

K = 5 N = 14√

KN =√

70 > 8(22)

1. Demand and Supply of Money

Problem 3. The following question illustrates the inventory theory of the de-mand for money (Baumol, Tobin), see the intermezzo in the textbook [HVdP02, p.13/14]. Assume a family earns $3000 every month, and spends all of its income overthe month, at a uniform rate. If they receive their full monthly salary in cash, anddraw down their cash just before the next payday, then their cash balance starts at$3000 and declines linearly to zero over the month. Their average cash holding is inthis case $1500. One can say, this family’s money demand is Y/2, where Y is theirmonthly income.

An alternative option for this family would be to have the pay check deposited intheir savings account at the beginning of the month, and make more than one cashwithdrawal during the month. The family could, e.g., withdraw half of its salary at


the beginning of the month and the second half at the 15th day of the month; orit could withdraw one third at the beginning, one third on the tenth, and one thirdon the 20th. (For simplicity assume every month has 30 days.) Assume that theirbalance in the savings account earns interest at a rate of 1% per month. However thewithdrawals are costly; there may be bank service charges, or one might just considerthe trips to the bank a cost. Assume these costs have a monetary equivalent of $2.–per withdrawal.

• a. (4 points) Which number of withdrawals would be optimal in the sense thatinterest income minus withdrawal costs is maximized? (Hint: simply determine itby computing transaction costs and interest income for 2, 3, and 4 transactions permonth and choose the best number among these three).

Answer. If there are two withdrawals, then $1500 dollars earn interest for half a month, whichis $7.50, less the transfer costs leaves $3.50.

If there are three withdrawals, then the savings account has a balance of $2000 dollars duringthe first 10 days, and $1000 in the next ten days. This gives $10 interest, minus transfer costs thisis $4.

Four withdrawals mean: the balance of the saving account is $2250 in the first quarter of themonth, $1500 in the second quarter, $750 in the third quarter. In the average this is $1500 for threequarters of a month, which gives $11.25 interest. Less transaction costs this is $3.25.

From then on it goes down, the optimal number of withdrawals is therefore 3.


• b. (2 points) What would this family’s demand for average cash holdings be(as a function of monthly income)?

Answer. If they make three withdrawals of $1000 each, their average cash holdings is $500,therefore the money demand is Y/6.

• c. (2 points) Now let us solve the same problem analytically. The only differ-ence between the following and the above is that the withdrawal amounts are arbitrary.If the monthly income if Y and the amount of each withdrawal is W , then the averagenumber of withdrawals per month is k = Y/W . Call the cost of each withdrawal c(in the above example, c = 2). Then the average transaction costs per month arekc = Y c/W . Instead of trying to figure out how much money is in the savings ac-count, simply compute the amount of interest which the cash in hand would haveearned, and minimize this opportunity cost. The average cash balance (money de-mand) is M = W/2, and the interest rate per time period is r (in the example, 1%per month) therefore the foregone interest is rW/2. Therefore one has to choose theW that minimizes

(23)rW

2+

cY

W

Compute the optimal W and the average number of withdrawals.


Answer. First-order condition r/2 − cY/W 2 = 0 or r/2 = cY/W 2 or 2/r = W 2/cY or

W =√

2cY/r. This is the optimal withdrawal amount; the money demand is half of this, i.e.,

M = W/2 =√

cY/2r. The average number of withdrawals is k = Y/W =√

Y r/2c.

• d. (1 points) If one would apply this formula to the above family, what wouldthe optimal withdrawal amount be?

Answer. The optimal withdrawals are W =√

2·2·3000.01

=√

1, 200, 000 = 200√

30 l 1095.44.

By contrast, the above discrete calculation resulted in W = 1000.

The supply of money is influenced by the Fed, the banking system, and thebehavior of the public.

Problem 4. (4 points) Here is a model of the money supply. The symbols usedare:


Cp Currency in the hands of the public (outside banks)Dp Bank depositsh Proportion of M which the public likes to hold as currencyM Money (not quite M-1 since M-1 also includes currency in banks!)RB Borrowed reserves (banks go to the “discount window”)RE Excess reserves (reserves held in excess of requirements)RR required reservesRU unborrowed reserves (created at initiative of Fed)z required reserve ratio.

Here are the equations:

M = Cp + Dp definition of M(24)

Cp = hM behavior of the public(25)

RR = zDp reserve requirement(26)

RU + RB = RR + RE + Cp sources and uses of reserves.(27)

Show that

(28) M =RU + RB −RE

h + z(1− h)


According to this formula, M is a function of RU and z (which are determined bythe Fed), h (determined by the public), and RE and RB (determined by the bankingsystem).

Answer. Write the required reserves RR in two ways:

RR = zDp = z(1− h)M(29)

RR = RU + RB −RE − Cp = RU + RB −RE − hM(30)

Eliminating RR gives

z(1− h)M = RU + RB −RE − hM,(31)

and solving for M gives

M =RU + RB −RE

h + z(1− h).(32)

Problem 5. In the usual Keynesian IS-LM models the money supply is assumedto be exogenous. If one models the money supply more carefully, one sees that themoney supply increases with the interest rate, since banks will want to keep theirexcess reserves low when interest rates are higher. Does this increase or decreasethe Keynesian government expenditure multiplier? You may either give an intuitive

2. WHO ARE THE NEO-CONSERVATIVES? 23

explanation, or use a simple ISLM model in which money supply depends on theinterest rate, and compute the multiplier.

Answer. It will increase their effect, because it will prevent the interest rate from rising somuch. For a little model, use perhaps

y = c(y) + i(r) + g c′ > 0, i′ < 0(33)

M(r)

P= l(r, y) lr < 0, ly > 0(34)

dy = c′dy + i′dr + dg(35)

M ′dr

P= ly dy + lr dr dr =

ly

M ′/P − lrdy(36)

dy(1− c′ − i′ly

M ′/P − lr) = dg.(37)

For M ′ = 0, the termly

M′/P−lris positive. For positive M ′, it becomes less positive or even

changes its sign. This means, a smaller amount is added to the denominator of the multiplier, thedenominator becomes smaller, hence the multiplier itself bigger.

2. Who are the Neo-Conservatives?

Here is something from the lbo-talk mailing list. This will not be in the exam,but I thought you might find it interesting.


Justin Schwartz wrote: the big “economic” theory that the neocons are into,insofar as they are into an economic theory, is actually not a theory about the econ-omy but about the polity. It is called public choice theory, and applies neoclassicalerconomics to government. The object is to show that govt is inefficicient becauseit is inevitably caputured by special interests, whether legislators seeking reelection,bureaucrats seeking power, labor unions seeking concessions, or employees who arelazy. (Never businesses seeking handouts and freebees!) For a quick overview of thissteaming load of ideological horseshit, see Dan Farber and Phil Frickey, Law andPublic Choice. Of the people listed below, Buchanan is a big public choice theorist.(Schumpeter, btw, is a genuinely great economist and indeed a great social thinker.)

Someone else wrote the following, don’t remember who:Many neocons have a soft spot for Schumpeter, Friedman, James Buchanan,

Thomas Sowell, George Gilder [not really and economist] and Mancur Olson. All ofthese economists, following James Madison and his experience of watching the Vir-ginia legislature, abhorred what came to be known as rent seeking behavior [Schum-peter may have been the most tolerant/understanding of the lot]; but it seems theRepublican Party has gone dialectical on a free market nono and turned it into a fulltime strategy for channeling taxpayer dollars to their districts and the corporationsthat send them the checks. It’s downright mercantilist in a way, which is why Ithink there’s something to be said for Frederic Lane’s and Charles Tilly’s approach

2. WHO ARE THE NEO-CONSERVATIVES? 25

to protection rents/rackets. Alexander Hamilton would probably feel enthralled andrepulsed.

I don’t see how this group would be neocons instead of just plain cons. Mirowski’snew book shows how some cold war social democrats came within the ambit of RANDand the Pentagon.

The neocons seem to revel in state power, not the shrivelled state that Friedmanand the conservatives want.

CHAPTER 3

Dynamics in Aggregate Supply and Demand

Chapter 2 in [HVdP02] discusses several models in which one or more of thevariables are given by accumulation over time (like: capital stock as accumulation ofinvestment, adaptive expectations, and also natural unemployment as accumulationof past experience).

1. Adaptive Expectations and Stability

They use a continuous-time model in which aggregate demand curve is given as afunction, called AD, and the aggregate supply curve is a straight line (they use squarebrackets to indicate that it is not a function bracket but simple multiplication). Since

27

28 3. DYNAMICS IN AGGREGATE SUPPLY AND DEMAND

time is continuous, the adaptive expectation becomes a differential equation:

Y = AD(G, M/P ) AD1 > 0, AD2 > 0(38)

Y = Y ∗ + φ[P − P e] φ > 0(39)

P e = λ[P − P e] λ > 0(40)

Endogenous variables are Y and P , exogenous variables G, M and Y ∗. The expectedprice level P e is a predetermined variable, they call it a state variable, because it isdetermined by the past time path of the system.

For formula (2.8) see (73) in Problem 9.Graphical solution, Figure (2.1). Ignore the upper panel, for us it is only neces-

sary to understand the lower panel of Figure 2.1. Start out with equilibrium as theintersection E0 of AS and AD curves. Then an increase in G shifts the AD curve upand to the right, while the AS curve is unchanged. I.e., the economy is now at A. A isan equilibrium in the sense that all markets clear, but now an equilibrium regardingexpectations: the actual price level at A is P ′, but the labor supply of households isbased on the expectation that price level is still P0. These expectations are graduallyrevised, and this causes the AS curve to gradually move up and to the left, and theintersection point converges towards E1.

1. ADAPTIVE EXPECTATIONS AND STABILITY 29

Then they do it with total differentiation: Differentiation of (2.1) yields

(41) dY = AD1 dG + AD2 d(M/P );

to get (2.9) use the fact that log(M/P ) = log M − log P and that for any functionZ, d log Z = dZ

Z ; therefore d(M/P )M/P = dM

M − dPP .

(2.9) and (2.10) are two equations in two unknowns, the endogenous dY anddP , dependent on the exogenous dG, dM , dY ∗, and the pre-determined dP e. Thissystem can be solved for dP and dY , these are equations (2.11) and (2.12). Theygive all the derivatives of the equilibrium dP and dY with respect to the exogenousvariables dG, dM , dY ∗.

Problem 6. Look at (2.11). Read off the following derivatives and determinetheir signs (either positive, negative, or undetermined). To simplify notation, youmay use the abbreviation α = (M/P 2)AD2.

∂P

∂G=(42)

∂P

∂M=(43)

∂P

∂Y ∗ =(44)


Likewise, from (2.12) read off the formulas and signs for the following derivatives:

∂Y

∂G=(45)

∂Y

∂M=(46)

∂Y

∂Y ∗ =(47)

(2.11) and (2.12) also give the derivatives of the equilibrium dP and dY with re-spect to the pre-determined variable dP e, but unlike the exogenous variables, whichcan be changed by policy decisions or supply shocks, the expected price level P e

moves on a timepath which depends on the time path of the actual P . This relation-ship is given by (2.3); and since the endogenous equilibrium P is a function of thetimepaths of the exogenous and predetermined variables, one might write (2.3) as

P e = λ[P (P e, G,M, Y ∗)− P e](48)

and this is what the book calls Ω:

P e = Ω(P e, G,M, Y ∗)(49)

1. ADAPTIVE EXPECTATIONS AND STABILITY 31

The derivatives of Ω can be determined most easily by plugging (2.11) into the totaldifferential of (2.3) which reads dP e = λ[dP − dP e]; the book does this in equation(2.13). All partials of Ω have determinate signs.

(49) is a differential equation for P e. Its analytical solution is given in (2.8)and a graphical solution in Figure 2.2: Since we know that Ω1 < 0 the graph of Ωlooks like the lower of the two downward sloping curves. Our starting point is anequilibrium situation at which expected prices have had time to adjust fully and tobe equal to actual prices. In this situation, P e = 0, i.e., our starting point is E0, theintersection of the downward sloping curve with the horizontal axis. Since Ω2 > 0,an increase in government expenditure shifts the curve up. This means, P e becomespositive, therefore P e gradually increases, until we get to point E1.

Note: if the graph of Ω were a upward sloping function, then the system wouldnot be stable.

Problem 7. (6 points) This is a simple question involving the math of contin-uous time: Assume prices rise at the constant rate π, i.e., p(t) = πp(t), and thebalance in your savings account at time t is A(t) monetary units. The bank adds in-terest to this account continuously at the rate r, and you withdraw from the bank (alsocontinuously) the withdrawal flow D(t), so that A(t) = rA(t) − D(t). You choosethis D(t) to be just the right amount so that in real terms your assets maintain their


value, i.e., that ddt

A(t)p(t) = 0. Using this D(t), the real rate of interest at time t can be

defined to be D(t)/p(t)A(t)/p(t) . Show that the real rate of interest is constant at r − π.

Answer. You have the following equations to work with:

p(t) = πp(t) A(t) = rA(t)−D(t)d

dt

A(t)

p(t)= 0(50)

Differentiate the third of these equations and plug in the other two:

(51) 0 =d

dt

A(t)

p(t)=

A(t)

p(t)−

A(t)

p(t)

p(t)

p(t)=

rA(t)−D(t)

p(t)−

A(t)

p(t)π = (r − π)

A(t)

p(t)−

D(t)

p(t).

Therefore

(52)D(t)

p(t)

/A(t)

p(t)= (r − π)

A(t)

p(t)

/A(t)

p(t)= r − π

2. Hysteresis

The evolution of expectations is a time-dependent cumulative effect on the de-mand side. Now we are also adding a time-dependent cumulative effect on the supplyside, by saying that the Unemployment get discouraged and therefore no longer seekfor work, therefore no longer put downward pressure on wages. In other words, thenatural rate of unemployment is a function of the past unemployment experience.

2. HYSTERESIS 33

The simplest way to model this is to use a discrete time model, and to say thatY ∗

t+1 = Yt because everyone who has unemployed at time t has already stopped seek-ing for work by time t+1. They assume that Y ∗ is also subject to exogenous shocks(for instance technology shocks, an increase in productivity which would allow moreto be produced with the same rate of unemployment, or an increase in “homelandsecurity” costs which would diminish the net output). adverse such shocks are calleddσ. Therefore one gets dY ∗

t+1 = dYt− dσ. If one plugs (2.12) into this one gets their(2.14). Notation: demand shocks are called dδ, and note that suddenly everythinghas time subscripts.

Now we are in discrete time, therefore instead of (2.3) we need (1.13) which canalso be written

(53) P et+1 = λPt + (1− λ)P e

t−1.

Note that this is no longer the same λ as in (2.3); the λ in (2.3) was constrained tobe positive, but this λ here is not only positive but it must be between 0 and 1. Thetotal differential of (53) is their (2.15).

The system (2.14) and (2.15) is a system of linear difference equations for thepredetermined variables dY ∗ and dP e in which the exogenous shocks dσ and dδconstitute the inhomogeneous part. In matrix notation, this system can be written


as [dY ∗

t+1

dP et+1

]=

1α + φ

[α −φα−λ (1− λ)α + φ

] [dY ∗

t

dP et

]+

1α + φ

[φdδt − (α + φ)dσt

λ dδt

](54)

This is my way of writing their (2.19). α = MP 2 AD2 > 0.

The stability properties of this system are determined by the coefficient matrix

(55) A =1

α + φ

[α −φα−λ (1− λ)α + φ

].

Problem 8. This is math only, but you need to know how to do this.

• a. (4 points) Show that the matrix

(56)1

α + φ

[α −φα−λ (1− λ)α + φ

]has eigenvalues 1 and α(1−λ)

α+φ and eigenvectors

(57)[−α1

]and

[φλ

].

Answer. Since eigenvalues and eigenvectors are given, it is not necessary to solve the char-acteristic equations. All you have to do is verify that the given vectors and values are indeed

2. HYSTERESIS 35

eigenvectors and eigenvalues, i.e., that the following matrix identities hold:

1

α + φ

[α −φα−λ (1− λ)α + φ

] [−α1

]=

[−α1

](58)

1

α + φ

[α −φα−λ (1− λ)α + φ

] [φλ

]=

α(1− λ)

α + φ

[φλ

].(59)

• b. (2 points) Show that the second eigenvalue α(1−λ)α+φ lies between 0 and 1

(assuming 0 < λ < 1, φ > 0, and α > 0).

Answer.

(60)α(1− λ)

α + φ=

1− λ

1 + φ/α

The numerator is between 0 and 1, and the denominator is greater than 1, therefore the fractionitself is between 0 and 1.

Let Λ be the matrix with the eigenvalues of A in the diagonal (in our case. . . ) and S be the matrix whose columns are the eigenvectors of A (in our case . . . ),which is nonsingular in our case. The eigenvalue property means AS = SΛ, thereforeS−1AS = Λ. Therefore the difference equation

xt+1 = Axt + at(61)


can be written as

S−1xt+1 = S−1ASS−1xt + S−1at(62)

or, if we define zt = S−1xt and bt = S−1at, this becomes

zt+1 = Λzt + bt.(63)

By recursive substitution one gets

zt+1 = bt + Λbt−1 + Λ2bt−2 + · · · =(64)

=∞∑

i=0

Λibt−i.(65)

The advantage here is that Λ is diagonal, therefore its powers are easily computed.bt are given in the book in (2.20). They are linear combinations of constant

vectors with dδt and dσt as coefficients. First let’s look at demand side shocks only,i.e., set all dσt = 0. Then we get (2.21) for the time path of zt; the first componentis always zero, and (2.22) shows how dY ∗

t and dP et depend on the second component

of zt. A permanent increase in government expenditure has a permanent effect onY ∗; the formula is given in (2.23), this is the sum of all the temporary multipliers.By contrast, a temporary increase in G which is reversed will not have a permanent

2. HYSTERESIS 37

effect, as one can see from (2.21), because the demand shocks dδt are multiplied byrising powers of α(1−λ)

α+φ (the second eigenvalue), which is smaller than 1.Now set all demand shocks zero and look at the supply shocks. The time path

of zt is given by (2.24): the first component is λ∑∞

i=0 dσt−1−i; i.e., this effect is notattenuated over time, therefore each one-time supply shock has an effect which willnever be reversed again.

Problem 9. This problem takes you through the solution of an inhomogeneouslinear differential equation by the method of the “variation of the constant.”

• a. Show that the general solution of the differential equation

(66) V (t) = rV (t)

can be written in the form V (t) = cer(t−t0), where c and t0 are arbitrary constants.This solution has the initial condition V (t0) = c.

Answer. Differentiate V (t) = cer(t−t0) and show that it satisfies (66).

• b. In order to get a solution of the inhomogeneous differential equation

(67) V (t) = rV (t) + H(t)

try a solution of the form V (t) = c(t)er(t−t0). (This is called “variation of theconstant” because the constant c is now turned into a variable.) (67) implies a


differential equation for c(t) which is easy to solve because c(t) depends only on t,not on c. Solve this to get

(68) V (t) =∫ t

τ=−∞H(τ)er(t−τ) dτ

Answer. If V (t) = c(t)er(t−t0) then

V (t) = c(t)er(t−t0) + c(t)rer(t−t0) = c(t)er(t−t0) + rV (t)(69)

In other words,

V (t)− rV (t) = c(t)er(t−t0)(70)

(67) means therefore

c(t) = H(t)e−r(t−t0).(71)

This can be solved for c(t) as follows:

c(t) =

∫ t

τ=−∞H(τ)e−r(τ−t0) dτ.(72)

To get the general solution for V (t), multiply (72) by er(t−t0):

V (t) =

∫ t

τ=−∞H(τ)er(t−τ) dτ(73)

2. HYSTERESIS 39

CHAPTER 4

Rational Expectations and Economic Policy

Of chapter 3 in [HVdP02], sections 3.1 and 3.2 are assigned. Section 3.1 dis-cusses [Mut61], and 3.2 [SW76].

Problem 10. This Problem derives a Lucas supply curve lnY = a+b(P −E[P ])as the aggregation of individual firms with rational expectations. Suppliers are locatedin scattered markets, with demand distributed unevenly over these markets. (In oneof his articles Lucas says the suppliers are located on different islands, and the buyersgo randomly from island to island.) Output of firm i at time t obeys the followingsimple supply curve:

(74) Yit = k(Pit

Pt

)h

.

41

42 4. RATIONAL EXPECTATIONS AND ECONOMIC POLICY

Here Yit is the output of firm i in time period t, k and h are positive constants, Pit

is the price of the firm’s product at time t, and Pt is the average price of all firms attime t. The higher the relative price the firm can fetch, the more the firm produces.

Now we will introduce uncertainty into this model. We will assume that the firmknows with certainty the price at which it sells its own product on the market, butthere is uncertainty regarding the general price level Pt. In this case, Pt in (74) mustbe replaced by P e

t , the firm’s expectation of what the general price level is. The firm’ssupply curve becomes therefore

Yit = k(Pit

P et

)h

,(75)

and after taking logarithms

lnYit = λ + h(lnPit − lnP et )(76)

where λ = ln k.The prices are modeled as random variables. Firm i’s price has the specification

(77) lnPit = ln Pt + εt + εit.

Here Pt is a nonrandom general trend level of prices, which is known to the firmswhen they make their supply decisions. In addition to this general trend, Pit is

4. RATIONAL EXPECTATIONS AND ECONOMIC POLICY 43

influenced by random factors (such as variations in the money supply) that affect theprices of all firms equally; they are called εt. Other shocks only affect the price offirm i; they are represented by εit. We assume that εt and εit have zero mean andare independent of each other, and var[εt] = σ2 and var[εit] = σ2

i . If there are manysmall firms, it follows that the average price level Pt satisfies

(78) lnPt = ln Pt + εt

since the many εit cancel each other out.The firms know Pt and the means and variances of εt and εit, and since they

know their own price, they know the actual value of εt+εit. But they do not know thevalues taken by εt and εit separately, and therefore cannot compute the actual valueof Pt. Their prediction of lnPt can be considered a signal extraction problem: theyknow the sum of signal plus noise, and they must judge how much of this is signal,and how much is noise. For their prediction they use a linear function of ln Pt andlnPit:

(79) lnP et = α ln Pt + β lnPit.

They use that linear predictor of lnPt which is unbiased and has lowest varianceamong all linear predictors. Arguably, this is the most rational thing they can do aslong as they don’t know the distribution of the error terms in full and can therefore


not use conditional expectations. (If the distributions are normal then the best linearpredictor is the conditional expectation.)

• a. (2 points) Show that for these expectations to be unbiased, i.e., for E[lnP et ] =

E[lnPt] to hold, the condition α+β = 1 must be satisfied, i.e., the expectation functionhas the form

(80) ln P et = α ln Pt + (1− α) ln Pit.

Answer. Follows from 0 = E[ln P et − ln Pt] = E[α ln Pt + β ln Pit − ln Pt] = α ln Pt + β ln Pt −

ln Pt. Divide by ln Pt to get α + β = 1.

• b. (4 points) Show that for the expectation errors to have lowest possible vari-ance among all unbiased expectations we need α = σ2

i /(σ2 + σ2i ).

Answer.

ln P et − ln Pt = α ln Pt + (1− α) ln Pit − ln Pt(81)

= α ln Pt + (1− α) ln Pt + (1− α)εt + (1− α)εit − ln Pt − εt(82)

= −αεt + (1− α)εit.(83)

var[ln P et − ln Pt] = α2σ2 + (1− α)2σ2

i .(84)

We have to find that α which makes this variance smallest: the first order condition is 0 = 2ασ2 −2(1− α)σ2

i , hence α(σ2 + σ2i ) = σ2

i .


• c. (2 points) Putting everything together show that the firm’s supply curvebecomes

lnYit = λ + hα(lnPit − ln Pt) with the same α =σ2

i

σ2 + σ2i

.(85)

Answer. Plug (80) into (76).

• d. Now I have to supply a step in the argument which I am not asking youto prove: If the number of firms is large (call it N) and all εi are small, then theaggregate supply curve of the whole economy is approximately

(86) Yt = Nk ·(

Pt

Pt

)hα

.

Answer. Here is a proof: Rewrite the individual supply curves as

ln Yit = λ + hα(εt + εit)(87)

Yit = k · ehαεt · ehαεit .(88)

The total output of the economy is therefore

Yt =∑

Yit = k · ehαεt ·∑

i

ehαεit .(89)


If N is large and the εi small, then the last term consists of many summands which are randomlydistributed around 1, and therefore approximately sum up to N . Finally use (78) to see thatεt = ln Pt − ln Pt. Taking logarithms one gets therefore ln Yt = λ + ln N + hα(ln Pt − ln Pt).

• e. (2 points) Assume that the general price level follows a random walk of theform lnPt = lnPt−1 + εt, where Pt−1 is known to the firms when they make theirsupply decisions for period t, but Pt is not. Write down what in this case Yt wouldbe. If one were to plot inflation against unemployment in this economy, would oneobtain a vertical Philips curve or would there be a tradeoff between inflation andunemployment?

Answer. This fits into the above framework if one writes ln Pt = ln Pt−1. There is a Phillipscurve.

Rational expectations therefore explains why a rational agent, who does not havemoney illusion, would still react to nominal quantities.

Problem 11. This question follows the original article [SW76] much moreclosely than [HVdP02] does. Sargent and Wallace first reproduce the usual ar-gument why “activist” policy rules, in which the Fed “looks at many things” and“leans against the wind,” are superior to policy rules without feedback as promotedby the monetarists.


They work with a very stylized model in which national income is represented bythe following time series:

(90) yt = α + λyt−1 + βmt + ut

Here yt is GNP, measured as its deviation from “potential” GNP or as unemploymentrate, and mt is the rate of growth of the money supply. The random disturbance ut

is assumed independent of yt−1, it has zero expected value, and its variance var[ut]is constant over time, we will call it var[u] (no time subscript).

• a. (4 points) First assume that the Fed tries to maintain a constant moneysupply, i.e., mt = g0 + εt where g0 is a constant, and εt is a random disturbancesince the Fed does not have full control over the money supply. The εt have zeroexpected value; they are serially uncorrelated, and they are independent of the ut.This constant money supply rule does not necessarily make yt a stationary timeseries (i.e., a time series where mean, variance, and covariances do not depend ont), but if |λ| < 1 then yt converges towards a stationary time series, i.e., any initialdeviations from the “steady state” die out over time. You are not required here toprove that the time series converges towards a stationary time series, but you areasked to compute E[yt] in this stationary time series.


Answer. Taking expected values in (90) we get

E[yt] = α + λ E[yt−1] + β E[mt](91)

= α + λ E[yt−1] + βg0(92)

In the steady state,

E[yt−1] = E[yt] = y∗,(93)

therefore (92) becomes

y∗ = α + λy∗ + βg0(94)

which can be solved

y∗ =α + βg0

1− λ.(95)

• b. (8 points) Now assume the policy makers want to steer the economy towardsa desired steady state, call it y∗, which they think makes the best tradeoff betweenunemployment and inflation, by setting mt according to a rule with feedback:

(96) mt = g0 + g1yt−1 + εt


Show that the following values of g0 and g1

g0 = (y∗ − α)/β g1 = −λ/β(97)

represent an optimal monetary policy, since they bring the expected value of the steadystate E[yt] to y∗ and minimize the steady state variance var[yt].

Answer. Plug (96) into (90):

yt = (α + βg0) + (λ + βg1)yt−1 + βεt + ut(98)

Take expected values to get

E[yt] = (α + βg0) + (λ + βg1) E[yt−1](99)

Now take limits assuming limt→∞ E[yt] = y∗:

y∗ = (α + βg0) + (λ + βg1)y∗.(100)

The coefficients g0 and g1 must therefore satisfy

y∗ =α + βg0

1− λ− βg1.(101)

One sees, there are many combinations g0 and g1 which bring about a given y∗. Which of

these achieve a steady state y with lowest variance around y∗? For this we have to apply thevariance operator to (98). Assume the economy has had enough time to adjust so that it is now


on its steady-state path. Since ut and εt are independent of each other and of yt−1, and since thevariance of the yt is constant over time by the definition of a steady state, one gets

var[y] = (λ + βg1)2 var[y] + β2 var[ε] + var[u](102)

var[y] =var[u] + β2 var[ε]

1− (λ + βg1)2.(103)

One minimizes this variance by choosing λ + βg1 = 0, or g1 = −λ/β. In this case, the mean of thesteady state y becomes y∗ = E[y] = α + βg0, or g0 = (y∗ − α)/β. The policy rule which gets thesteady state value y∗ with lowest possible variance is therefore

(104) mt =y∗ − α

β−

λ

βyt−1.

• c. (3 points) This is the conventional reasoning which comes to the resultthat a policy rule with feedback, i.e., a policy rule in which g1 6= 0, is better than apolicy rule without feedback. Sargent and Wallace argue that there is a flaw in thisreasoning. Which flaw?

Answer. It ignores the fact that equation (90) is a reduced form equation whose parametersdepend, among others, on the policy. The parameters α, λ, and β can therefore not be assumed toremain constant if the policy changes. To treat the problem correctly, one has to go down to thestructural equations.


• d. (5 points) A possible system of structural equations from which (90) can bederived are equations (105)–(107) below. Equation (105) indicates that unanticipatedincreases in the growth rate of the money supply increase output, while anticipatedones do not. This is a typical assumption of the rational expectations school (Lucassupply curve).

(105) yt = ξ0 + ξ1(mt − Et−1 mt) + ξ2yt−1 + ut

The Fed uses the policy rule

(106) mt = g0 + g1yt−1 + εt

and the agents know this policy rule, therefore

(107) Et−1 mt = g0 + g1yt−1.

Show that in this system, the parameters g0 and g1 have no influence on the timepath of y.

Answer. From (106) and (107) follows mt −Et−1 mt = εt, and plugging this into (105) gives

(108) yt = ξ0 + ξ1εt + ξ2yt−1 + ut.

The unanticipated, random part has an effect, in fact an undesirable effect, since it increases the

variance of yt. Whether or not a feedback rule is followed is therefore irrelevant from the point


of view of this model; however a simple rule might be better in terms of the cost of acquiring theinformation.

• e. (4 points) On the other hand, the econometric estimations which the policymakers are running seem to show that these coefficients have an impact. During acertain period during which a constant policy rule g0, g1 is followed, the econometri-cians regress yt on yt−1 and mt in order to estimate the coefficients in (90). Whichvalues of α, λ, and β will such a regression yield?

Answer. Plug (107) into (105):

(109) yt = (ξ0 − ξ1g0) + (ξ2 − ξ1g1)yt−1 + ξ1mt + ut.

This is the same form as (90) but now we can see how the reduced-form parameters α, λ, and βdepend on the structural parameters ξ0, ξ1, ξ2, and and g0, g1:

α = ξ0 − ξ1g0 β = ξ1 λ = ξ2 − ξ1g1, β = ξ1.(110)

The argument for the superiority of the feedback rested on the erroneous assumption that the policymakers can change g0 and g1 without this causing any changes in α and λ.

CHAPTER 5

Anticipation Effects and Economic Policy

In [HVdP02, chapter 4] π(t) is called “profit,” but it is really the dividendstream, that what the firm pays out to its owners after paying wages and afterinvesting. We assume that there is only one good, whose price is P = 1, and we callthe wage w (lower case letter because it is at the same time the real wage W/P ),and Φ

((I(t)

)is investment including the adjustment cost which may be a quadratic

function, for instance

Φ(I(t)

)= I(t) + b

[I(t)

]2(111)

is a possible adjustment function. With this notation, the dividend flow at time t is

π(t) = F(N(t),K(t)

)− w(t)N(t)− Φ

(I(t)

)(112)

53

54 5. ANTICIPATION EFFECTS AND ECONOMIC POLICY

and the goal is to make those hiring decisions N(t) and investment decisions I(t)which maximize the present discounted value of that flow at time 0 (which is thestock market valuation of the firm):

V (0) =∫ ∞

t=0

π(t)e−rt dt(113)

The capital stock K(0) at time 0 is exogenously given, and over time K(t) evolvesaccording to the differential equation

K(t) = I(t)− δK(t)(114)

where δ is the rate of depreciation.Which time paths of hiring N(t) and investment I(t) will give the highest possible

V (0)? For this we need “optimal control theory” or “calculus of variations.” [Dor69]and Errata [Dor70] give an economic interpretation of this procedure.

In order to solve this, we first have to define Tobin’s q(t), which is somethinglike the Lagrange multiplier. The book calls q(t) the shadow price of capital at timet; i.e, it indicates the increase of the present discounted value (as of time t) of thedividend stream after time t if one small unit of capital is added to the capital stock,assuming that from then on hiring and investment decisions are optimal for that(slightly increased) capital stock.

5. ANTICIPATION EFFECTS AND ECONOMIC POLICY 55

Using q(t), we must write down the Hamiltonian (or “auxiliary”) function:

H(t) = e−rt[π(t) + q(t)[I(t)− δK(t)]

]= e−rt

[F

(N(t),K(t)

)− w(t)N(t)− Φ

(I(t)

)+ q(t)[I(t)− δK(t)]

](115)

H(t) dt can be interpreted to be the direct and the indirect contribution of productionin the time interval between t and t + dt to the firm valuation V (0): the directcontribution is e−rtπ(t) dt, i.e., the present discounted value of the dividends paidout between t and t + dt, and the indirect contribution comes from the increasein the capital stock between t and t + dt. To derive this indirect contribution,note that [I(t) − δK(t)]dt is the net increase in the capital stock between t and dt,q(t)[I(t) − δK(t)] dt is, according to the definition of q(t), the present discountedvalue (as of time t) of future dividends generated by this additional capital, ande−rtq(t)[I(t) − δK(t)]dt is therefore the present discounted value of this additionaldividend stream as of time 0.

Three decisions have to be made, and these decisions are represented by threefirst-order conditions which can be formulated in terms of the Hamiltonian.

The first decision is: How many workers should be hired at any given point intime? This is the most unproblematic decision: of course you have to hire at the levelwhich maximizes your profit flow for that given time. More profit is always better,there is no tradeoff. This decision is represented by the first first-order condition, in


the book it is (4.6):

∂H(t)∂N(t)

= 0(116)

e−rt[F1

(N(t),K(t)

)− w(t)

]= 0(117)

F1

(N(t),K(t)

)= w(t)(118)

In other words, at every point in time, the firm must hire at the profit-maximizinglevel.

The critical decision will then be: how much of these profits should be invested.There are two kinds of tradeoffs involved here: one is the tradeoff between investingand paying out as dividends, and the other is the tradeoff between investing todayand investing tomorrow. These two decisions are intertwined, but this can be solvedas follows:

First assume we know how to spread out future investments over time optimally.Therefore we are confident that any increase in capital stock between t and t + dtwill be put to best use, which is represented by the “shadow price” q(t). The second


first-order condition says then

∂H(t)∂I(t)

= 0(119)

e−rt[Φ′

(I(t)

)− q(t)

]= 0(120)

Φ′(I(t)

)= q(t)(121)

This has a simple interpretation: Φ′(I(t)

)is the cost of the last dollar’s worth of

capital stock installed (which is usually more than a dollar, since there is a costpenalty for investing at high speed). Invest so much that the last dollar investedcosts q(t), i.e., invest until the increase in the present discounted value of the firmdue to this investment equal the cost of that investment.

But for this we need to know what q(t) is. The time path of q(t) is given by thethird first-order condition:

d[q(t)e−rt]dt

= −∂H(t)∂K(t)

(122)

e−rt[q(t)− rq(t)

]= −e−rt

[F2

(N(t),K(t)

)− δq(t)

](123)

F2

(N(t),K(t)

)+ q(t) = (r + δ)q(t)(124)


If we are on the optimal time path, small variations in the time profile of investmentshould not matter. I.e., it should not matter whether you add a small increment ofcapital dK at time t, or whether you put that money into the bank, draw intereston it between t and t + dt and then add the capital at time t + dt.

Let us exercise this through. In order to add an infinitesimal unit of capital dK attime t you need a money amount of Φ′

(I(t)

)dK which, by (121), is equal to q(t) dK.

Between t and t+dt, this extra unit of capital gives you an additional revenue flow ofF2

(N(t),K(t)

)dK dt, but this capital also depreciates and at the end you have only

(1− δ dt) dK units of this capital increment left. If you put the money into the bankbetween t and t+dt you get interest in the amount of rq(t) dK dt, and then you haveto buy not dK but (1 − δ dt) dK units of capital, at price q(t + dt) = q(t) + q(t) dt.These two scenarios must give the same outcome, i.e.,

−q(t) dK + F2

(N(t),K(t)

)dK dt = rq(t) dK dt− (q(t) + q(t) dt)(1− δ dt) dK

= rq(t) dK dt− q(t) dK − q(t) dK dt + δq(t) dK dt(125)

Here we left out the term with dt2 dK. The terms with dK only (no dt) cancel out,and the terms with dK dt give

F2

(N(t),K(t)

)= rq(t)− q(t) + δq(t)(126)

But this is exactly (124).


Problem 12. This problem works through a modification of the model for anoptimal investment path in which the adjustment cost Φ depends not only on invest-ment I but also on the capital stock K. It follows the grey box in [HVdP02, pp.83–85], the original article is [Hay82]. Φ(I, K) is assumed to be homogeneous ofdegree 1 in I and K, with Φ1 > 0, Φ2 < 0 (large firms experience less disruption fora given level of investment than small firms), Φ11 > 0, Φ12 < 0, and Φ22 > 0. Thedividend flow at time t is therefore

π(t) = F(N(t),K(t)

)− w(t)N(t)− Φ

(I(t),K(t)

)(127)

and the goal is to make those hiring decisions N(t) and investment decisions I(t)which maximize the present discounted value of that flow at time 0 (which is thestock market valuation of the firm):

V (0) =∫ ∞

t=0

π(t)e−rt dt(128)

The capital stock K(0) at time 0 is exogenously given, and over time K(t) evolvesaccording to the differential equation

K(t) = I(t)− δK(t)(129)


where δ is the rate of depreciation. The Hamiltonian function is in this case

H(t) = e−rt[F

(N(t),K(t)

)− w(t)N(t)− Φ

(I(t),K(t)

)+ q(t)[I(t)− δK(t)]

](130)

• a. (3 points) Evaluate the first first-order condition ∂H(t)∂N(t) = 0.

Answer. This is unchanged from (117) and (118):

∂H(t)

∂N(t)= 0(131)

e−rt[F1

(N(t), K(t)

)− w(t)

]= 0(132)

F1

(N(t), K(t)

)= w(t)(133)

• b. (3 points) Evaluate the second first-order condition ∂H(t)∂I(t) = 0.

Answer.

∂H(t)

∂I(t)= 0(134)

e−rt[−Φ1

(I(t), K(t)

)+ q(t)

]= 0(135)

Φ1

(I(t), K(t)

)= q(t)(136)


• c. (5 points) Evaluate the third first-order condition d[q(t)e−rt]dt = − ∂H(t)

∂K(t) .

Answer.

d[q(t)e−rt]

dt= −

∂H(t)

∂K(t)(137)

e−rt[q(t)− rq(t)

]= −e−rt

[F2

(N(t), K(t)

)− Φ2

(I(t), K(t)

)− δq(t)

](138)

F2

(N(t), K(t)

)+ q(t) = (r + δ)q(t) + Φ2

(I(t), K(t)

)(139)

• d. (3 points) Linear homogeneity of F means F (µN, µK) = µF (N,K). Derivefrom this the identity

F (N,K) = F1(N,K)N + F2(N,K)K(140)

Since Φ is homogeneous, the following equation holds as well (no separate proofrequired):

Φ(I,K) = Φ1(I, K)I + Φ2(I,K)K(141)

• e. (4 points) Show that

(142) π(t) = F2

(N(t),K(t)

)K(t)− Φ

(I(t),K(t)

)Answer. Plug (140) into (127) and use (133).


• f. (8 points) Show that

d[q(t)K(t)e−rt]dt

= −e−rtπ(t)(143)

from which follows (no separate proof required)∫ b

a

π(t)e−rt dt = q(a)K(a)e−ra − q(b)K(b)e−rb(144)

(because of the minus sign).

Answer.

d[q(t)K(t)e−rt]

dt= e−rt

[q(t)K(t) + q(t)K(t)− rq(t)K(t)

](145)

Now use (139) to get rid of q(t) and (129) to get rid of K(t):

ert d[q(t)K(t)e−rt]

dt= K(t)

[−F2

(N(t), K(t)

)+ (r + δ)q(t) + Φ2

(I(t), K(t)

)]+

+ q(t)[I(t)− δK(t)

]+ rq(t)K(t)(146)

Eliminating those items which have been added and subtracted gives:

= −F2

(N(t), K(t)

)K(t) + Φ2

(I(t), K(t)

)K(t) + q(t)I(t)(147)


Now use homogeneity of Φ: Φ2K = Φ− Φ1I, and the second first-order condition (136):

= −F2

(N(t), K(t)

)K(t) + Φ

(I(t), K(t)

)= −π(t)(148)

where the last equal sign used (142).

• g. (2 points) An optimal investment path is usually assumed to satisfy the“transversality condition”

(149) limt→∞

q(t)K(t)e−rt = 0

If this would not be the case, too much capital would be accumulated in the long runwhich is not of any use for today’s generation. Use this to show that

(150) V (0) =∫ ∞

t=0

π(t)e−rt dt = q(0)K(0)

i.e., q(0) is not only Tobin’s marginal q but also Tobin’s average q.

Answer. Use∫∞0 = limb→∞

∫ b0 and the transversality condition.

CHAPTER 6

The Macroeconomics of Quantity Rationing

Instead of chapter 5 in [HVdP02] we will discuss Malinvaud’s model as explainedin [Mal77] or [Mal85]. (Equation numbers below refer to these two editions.)

One of Malinvaud’s main points is that Keynesian unemployment can only beunderstood if one looks at the relationship between goods markets and labor markets.I.e., instead of looking at one market only, we need to look at a general equilibriummodel. His “prototype model” is a nice little fully specified general equilibriummodel.

Briefly recall what a Walrasian equilibrium is: taking prices and wages as pa-rameters, workers determine how many hours they want to work and how much theywant to consume. Equilibrium prices are those at which everyone can work theseoptimal hours and make the optimal consumption purchases. According to the rules

65

66 6. THE MACROECONOMICS OF QUANTITY RATIONING

of this game, no trades are made before equilibrium is found. Therefore the questionnever arises how much to consume if one is unable to work as many hours as onewould like to work at the given wage rate, how much to work if there is a shortageof the desired consumption goods, or how much to produce if not all goods can besold that one would like to sell.

But empirically, these are exactly the kinds of questions real-life agents have tograpple with. When formulating their demands, workers keep in mind that theymay not be able to work as much at the going rate as they want to. Thereforethey determine their demand by maximizing their utility subject to this perceivedquantity constraint in the supply of their labor. This is their effective demand, asopposed to the notional demand formulated without the labor supply constraint.

This distinction was made in Keynes, and Patinkin was one of the first to for-malize it mathematically, see also [Clo65].

Firms are in a similar situation: they would like to hire more labor if they couldsell more, but they perceive a quantity limit of what they can sell, and therefore hirejust enough workers for this.

In equilibrium, the behaviors of consumers and firms validate each other. Firmscannot hire more labor and sell more because individuals consider their unemploy-ment in their purchasing decisions, and individuals cannot buy more because they

6. THE MACROECONOMICS OF QUANTITY RATIONING 67

cannot find jobs. They cannot find jobs because firms consider the demand restric-tions in their hiring decisions.

All this is complicated enough to warrant a mathematical formulation. Thismutual getting stuck can be called a short term equilibrium, in the sense that theactions of the different agents fit together. Every model has its own implied definitionof equilibrium. In Malinvaud’s model, this definition is:

• Prices are fixed• Amount of sales equal amount of purchases• Sales or purchases cannot exceed notional demand or supply• There cannot be rationed buyers and sellers in the same market.

Because of this last point, each market is either balanced, a sellers’ market, or abuyers’ market.

Let us look at the interrelation between goods market and and labor market. Dueto Walras’ law, either both markets are balanced, or both are unbalanced. Since eachmarket can be unbalanced in two ways, we get:

Buyers’ market for goods Sellers’ market for goodsBuyers’ market for labor Keynesian Unemployment Classical UnemploymentSellers’ market for labor impossible Repressed inflation


The fourth combination is not possible: Firms which cannot hire all the labor theywant would like to sell more than they are actually selling, i.e., the market for goodsis a seller’s market, not a buyer’s market.

1. Malinvaud’s Prototype Model

The model in [Mal77] or [Mal85] is fully specified. It gives a nice example of ageneral equilibrium model. Here is a list of the symbols Malinvaud uses:

1. MALINVAUD’S PROTOTYPE MODEL 69

a technological constant denoting decreasing returns to scale in productionC total consumption in nominal termsg government expenditures in real termsG government expenditures in nominal terms, G = pg.l maximum labor supply possible per consumerli labor sold by consumer i

m0 initial wealth plus nonlabor income of every consumermi money holding at end of period by consumer iN number of consumersp price of the goodu rate of unemploymentw wage rate

W sum of all wages, W = zwxi consumption of consumer iy total outputY total output in nominal terms, Y = pyz labor demand.

Government makes government expenditures which it finances by printing money.The consumers have a certain amount m0 of money at the beginning, they supply


labor, consume, and end up with a certain amount of money mi at the end. Firmsbuy labor and sell products; they accumulate their profits in the form of cash.

1.1. Consumers. All consumers have the same utility function

(151) Ui = x2i (l − li)

mi

p.

In Malinvaud’s booklets this is equation (21|29) (first|second edition). This utilityfunction has a very simple functional form but it is not strictly concave. HoweverUα with 0 < α < 0.25, which represents the same preferences, is strictly concave.Consumer i’s budget constraint is

m0 + wli − pxi = mi

which can also be written as

pxi + w(l − li) + mi = m0 + wl.(152)

Here the righthand side is the consumer’s exogenous “full income,” i.e., the moneythe consumer could spend if he or she were to work the maximum amount of hours.The lefthand side indicates the allocation of this full income to consumption, leisure,and money balances at the end of the period. In the unrationed case, consumers


pick those xi ≥ 0, 0 ≤ li ≤ l, and mi ≥ 0 satisfying their budget constraint whichmaximize this utility (“notional demand”).

The Lagrangian is

(153) L = x2i (l − li)

mi

p− λi(pxi + mi − wli −m0)

and the first-order conditions are

∂L∂xi

= 0 2xi(l − li)mi

p− λip = 0;(154)

∂L∂li

= 0 −x2i

mi

p+ λiw = 0;(155)

∂L∂mi

= 0 x2i (l − li)

1p− λi = 0.(156)

(156) gives λi; plugging this into (154) gives pxi = 2mi, i.e., consumers spend twiceas much on consumption as they keep in cash at the end of the period, and pluggingλi into (155) gives (l− li)w = mi, i.e., consumers “spend” as much money on leisurein terms of foregone earings as they keep at the end of the period.


If one plugs pxi = 2mi and (l − li)w = mi into the budget constraint (152) onegets 4mi = m0 + wl, which leads to the solutions given in (23|31):

xi =12p

(m0 + wl)(157)

li =1

4w(3wl −m0)(158)

mi =14(m0 + wl).(159)

In terms of full income, (158) reads

l − li =1

4w(m0 + wl)(160)

Consumers spend half of their full income on goods, a quarter on leisure, and aquarter they keep in form of cash.

1.2. Effective demand of consumers. These are the consumers’ notionaldemands. Now what are the effective demands of unemployed consumers, i.e., whatwould their demand be if they were unemployed and formulated their demand underthe assumption that they remained unemployed? Setting li = 0, their utility function


is x2i l

mi

p , which is maximized subject to pxi + mi = m0. This gives the Lagrangian

L = x2i l

mi

p− λi(pxi + mi −m0)(161)

and the demand equations

xi =23p

m0(162)

mi =13m0.(163)

Here, two thirds of full income m0 are spent on goods and one third is kept for thefuture. Since this is a maximization of the same utility function under additionalconstraints, the level of utility reached is lower than in the case of full employment.

Problem 13. (8 points) Derive the effective demand generated by unemployedworkers, i.e., maximize the utility function x2

i lmi/p s. t. pxi + mi = m0 with respectto xi and mi.

Answer. The Lagrangian is

(164) L = x2i l

mi

p− λi(pxi + mi −m0)


and the first-order conditions

∂L∂xi

= 0 2xi lm

p− λip = 0(165)

∂L∂mi

= 0 x2i l

1

p− λi = 0.(166)

Solving this gives

xi =2

3pm0(167)

mi =1

3m0(168)

Another kind of rationing occurs if the consumers cannot buy as much as theywant to buy, say the consumption of the consumer is rationed to be x. Then theymaximize their utility function with respect to li and mi for the given x. The solutionis

li =1

2w(px + wl −m0)(169)

mi =12(wl + m0 − px)(170)


Now assume the suppliers of labor are rationed because total labor demandis z. Malinvaud adopts the simplest plausible rule: people are either completelyunrationed, i.e., their labor supply is given by (158), or they are fully unemployed(i.e., work zero hours). The rate of unemployment u must be such that this laborsupply just adds up to z:

(171) z = N(1− u)1

4w(3wl −m0).

1.3. Notional Demand and Supply by Firms. Now look at the other side,the firms. Here Malinvaud assumes that output y requires a labor input

(172) z = y(1 +ay

2).

Problem 14. Show that (172) is equivalent to the production function

(173) y = f(z) =1a(√

1 + 2az − 1)

which expresses output y as a function of the labor input z.

Answer. (172) can be written as

y2 +2y

a−

2z

a= 0.(174)


For y2 + py + q = 0 the solution formula is y1,2 = − p2±

√p2

4− q; therefore

y1,2 = −1

a±

√1

a2+

2z

a(175)

Only the positive solution makes sense:

y = f(z) =1

a(√

1 + 2az − 1).(176)

Problem 15. Show that, when the technology is given by z = y(1 + ay2 ), the

notional supply of goods and demand for labor are

y =1a(p

w− 1)(177)

z =12a

(p2

w2− 1).(178)

Answer. Maximize profits py − wz with respect to y, using equation (172) which expresses

labor input z as a function of y. The objective function is

(179) py − wy(1 +ay

2) = (p− w)y −

awy2

2.

2. KEYNESIAN UNEMPLOYMENT 77

First order condition is p − w − awy = 0, or y = p−waw

= 1a( p

w− 1). To get z, plug this y into the

labor requirements equation (172) to get

(180) z =1

2a(

p2

w2− 1).

2. Keynesian Unemployment

In Keynesian unemployment, labor suppliers are rationed at z, and goods sup-pliers rationed at y. The assumption is that the employed workers can work as longas they want, only the unemployed workers are rationed in their labor supply.

We will show that under these conditions one obtains a Keynesian consumptionfunction. Aggregate nominal consumption depends on u. From (157) follows thattotal demand of all employed workers is N(1−u) 1

2p (m0 + wl), and due to (162), thetotal demand of all unemployed workers is Nu 2

3pm0. Summing up gives

C = N(1− u)12(m0 + wl) + Nu

23m0.(181)

The total wage income depends on u as well:

W =N

4(3wl −m0)(1− u).(182)


By eliminating u from these two equations we get a relation between C and W . Thiselimination is most easily done by multiplying the first equation by 6, the second by4, and then subtracting them from each other. Then the terms with u fall out.

6C = 3N(1− u)(m0 + wl) + 4Num0(183)

4W = N(3wl −m0)(1− u)(184)

6C − 4W = 3N(w0 + wl)−N(3wl −m0) = 4Nm0, or(185)

C =23(Nm0 + W ).(186)

This is a linear consumption function with MPS = 1/3. W affects consumptionbecause it is an indicator how many workers are unemployed. It looks as if theunemployed workers have a different utility function than the employed ones, butthis is only because in their consumption choice they take their unemployment intoconsideration.

The consumption function gives us C once we know W . From C we get nominalaggregate demand G + C. Firms are rationed in their output, i.e., they produceexactly as much as they can sell, therefore

Y = C + G(187)


In order to produce Y , firms need Yp (1+ aY

2p ) units of labor, therefore the wages theypay are

W = wY

p(1 +

aY

2p).(188)

For equilibrium, this W must be the same as the one entering the consumptionfunction above.

The three equations (186), (187), and (188) form a simple income expendituremodel. Solving it for Y one obtains

Y =23(Nm0 + w

Y

p(1 +

aY

2p))

+ G.(189)

Substitute py for Y and pg for G, and rearrange:

13(3p− 2w)y − 1

3awy2 = pg +

23Nm0.(190)

This is a quadratic equation which may have no real roots, i.e., Keynesian equilibriumexists only for certain combinations of the exogenous variables g, w, and p. In othercases, this equation has two solutions for y, but it can be shown that one of theseis always bigger than the notional supply. The upshot is: when it exists, Keynesianunemployment is uniquely determined.


Problem 16. Alternative derivation of (190): The following equations musthold between the endogenous variables z, u, and y: (here g is the exogenous govern-ment demand):

z = y(1 +ay

2). labor demand by firms constrained to output y, (172)

(191)

z =N

4w(3wl −m0)(1− u) Unemployment due to insufficient labor demand, (171)

(192)

y = g + N(1− u)12p

(m0 + wl) + Nu23p

m0 goods supply = demand

(193)

(194)

Furthermore, the following inequalities: 0 ≤ u ≤ 1, and y ≤ 1a ( p

w − 1) (i.e., supplyof goods is less than or equal to notional supply). Show that elimination of z and ufrom the above three equations gives (190).


With equation (190) one can do comparative statics. Taking total differentialsgives

(195)13(3p− 2w(1 + ay)

)dy = p dg − (y − g) dp +

23(y +

ay2

2) dw = 0.

Malinvaud evaluates the desirability of policy by two criteria: the effect of thesepolicies on the utilities of consumers, and the degree to which they decrease theexcess supplies in the labor and goods markets, since these excess supplies may leadto undesirable long run adjustments. In the Keynesian situation, higher y will lowerboth excess supplies.

First look at an increase in g. This gives

(196) dy =3p

3p− 2w(1 + ay)dg.

This is the government expenditure multiplier. The condition that output is belownotional supply, y ≤ 1

a ( pw − 1), can be written as

(197) 1 + ay ≤ p

w.

Therefore the denominator satisfies

(198) 3p > 3p− 2w(1 + ay) ≥ 3p− 2wp

w= p,


i.e., the multiplier lies between 1 and 3. If the marginal propensity to save is 1/3,the simple textbook multiplier is 3. The multiplier here is smaller, because in thecase of rationing workers receive less than their marginal product, i.e., part of theincreased output goes into profits, and in this model there is no consumption out ofprofits. g increases output and therefore lowers excess supplies. The utility of thoseemployed and those who remain unemployed is unchanged, but the utility of thosewho switch from unemployed to employed increases.

Now assume there are wage and price controls which change w and p proportion-ally by the same negative percentage amount dq: dp = p dq and dw = w dq. Then

(199)13(3p− 2w(1 + ay)) dy + (y − g)p dq − 2

3(y +

a

2y2)w dq = 0.

Since equation (190) can be written as

3py − 2w(y +a

2y2) = 3pg + 2Nm0, we can simplify(200)

(y − g)p− 23(y +

a

2y2)w = 2Nm0.(201)

Therefore one obtains a multiplier of the form

(202)dy

dq= − 3p

3p− 2w(1 + ay)23Nm0.

3. CLASSICAL UNEMPLOYMENT 83

3p3p−2w(1+ay) is the main multiplier, and 2

3Nm0 is the intercept in the consumptionfunction, i.e., this is a real balance effect. A negative dq is therefore beneficial in allrespects.

Raising w/p has two effects: on the one hand, output increases due to increaseddemand, but on the other hand, also the notional labor supply of those employedincreases. Since those who are employed are by assumption unconstrained in theirlabor supply and therfore work more, it is possible that the unemployment rate rises.But this is a perverse case; generally, in Keynesian unemployment, higher wages willreduce unemployment. In Keynesian unemployment, prices are so high and/or wagesso low that the autonomous demand (g and money balances) is insufficient to elicitfull employment.

3. Classical Unemployment

In classical unemployment, firms aren’t constrained in either market, but realwages are so high that firms do not find it profitable to employ all workers. Theunemployed workers are content to consume less than their notional goods demandbecause they are unemployed, and the employed workers are content to work lessthan their notional labor supply because their consumption is rationed.

Wages would have to be lowered to reduce unemployment. We will just setup the equations for classical unemployment, without solving them. Firms are not


constrained in either of the two markets; therefore output and labor demand are thenotional demands:

y =1a(p

w− 1)(203)

z =12a

(p2

w2− 1).(204)

Assumptions about the rationing on the demand side for goods: governmentand unemployed workers are not rationed, but the consumption of every employedworker is rationed at the amount x. The equality of sales and purchases on the goodsmarket requires therefore

(205) py = pg +2N

3m0u + N(1− u)px

If worker i is constrained to consume an amount x which is smaller than hisnotional demand, then he will also not want to work as much. He maximizes x2(l−li)mi s.t. m0 + wli − px −mi = 0. We already know how this has to be solved: he

3. CLASSICAL UNEMPLOYMENT 85

will spend as much on leisure as he will keep in mi, therefore

mi =m0 + wl − px

2, and(206)

l − li =m0 + wl − px

2w, i.e., li =

wl −m0 + px

2w.(207)

Hence, equilibrium of the labor market requires

(208) 2wz = N(1− u)(px + wl −m0).

(205) and (208) determine x and u simultaneously.Policy implications: Generally, an increase in prices or lowering of wages will

reduce classical unemployment and the excess demand of goods. This is the oppositeeffect than in Keynesian unemployment. But perverse cases are also possible: sincean increase in g cannot increase y, it will tighten the rationing of the employedworkers. This will actually reduce unemployment, since the employed workers willwant to work less. The newly employed workers gain utility, but the employedworkers lose utility, and there is also an increase in the excess demand for goods.Lowering prices and wages proportionally has a similar perverse effect via the realbalances.

Problem 17. (10 points) One of the main results of the quantity constrainedanalysis (such as Malinvaud’s) is that it allows to distinguish between Keynesian and


classical unemployment. How do these two kinds of unemployment differ, and whyis it important to distinguish between them?

Answer. In Keynesian unemployment, firms are constrained as suppliers in the goods market,in classical unemployment, firms are unconstrained.

In classical unemployment, there is a buyer’s market for labor and a seller’s market for goods.Firms, which buy labor and sell goods, are therefore unconstrained. But real wages are so highthat the firms do not employ all workers. Unemployed workers lower their purchases because they

cannot sell their labor, and employed workers lower the labor they sell because they cannot buyeverything they need.

In Keynesian unemployment, the employed workers are on their notional demand curve for

goods, in classical unemployment, they are not.Keynesian unemployment is mitigated by higher wages, classical unemployment by lower wages.

Problem 18. (10 points) Explain in words, as precisely as you can, exactlydescribing which agents are rationed and which are not, how a Keynesian and howa classical unemployment “equilibrium” can come about in Malinvaud’s model. Howwould a rise in wages affect output in both situations?

4. Repressed Inflation

Mathematically this is the simplest case because everyone is employed. Butsince consumers cannot buy as much as they want, they supply less labor than their

4. REPRESSED INFLATION 87

notional labor supply. Firms, on the other hand, cannot produce more since theycannot find enough workers. This was the situation in the Soviet-Union type socialistcountries, where people had comfortable amounts of money, but there were so fewgoods on the market that they could not really use their money, therefore they werenot motivated to work hard.

Problem 19. The following model is a simplification of Malinvaud’s prototypemodel insofar as employed workers do not choose how many hours they work, but thelength of their working day is fixed at l hours. Here is a list of the symbols you mayneed:


a technological constant indicating decreasing returns to scale in productiong government expenditures in real termsl length of workday (the same for all employed workers)m0 initial wealth plus nonlabor income of every consumermi money holding at end of period by consumer iN number of workers/consumersp price of the goodu rate of unemploymentw wage ratexi consumption of consumer iy total outputz labor demand in labor hours. Note that z = N(1− u)l.

The workers/consumers have utility function Ui = x2i

mi

p , government has govern-ment expenditures g, and the technology is such that for producing output y oneneeds the amount z of labor, where

(209) z = y(1 +ay

2).


• a. (4 points) Show that the notional demand for labor z and the notional supplyof goods y are

z =12a

(p2

w2− 1)(210)

y =1a(p

w− 1).(211)

Answer. The answer is exactly the same as the answer to Question 15. Maximize profitspy − wz with respect to y, using equation (209) which expresses labor input z as a function of y.The objective function is

(212) py − wy(1 +ay

2) = (p− w)y −

awy2

2.

First order condition is p − w − awy = 0, or y = p−waw

= 1a( p

w− 1). To get z, plug this y into the

labor requirements equation (209) to get, after some tedious calculations,

(213) z =1

2a(

p2

w2− 1).


• b. (4 points) Show that the goods and money demand of an employed workeri is

xi =23p

(m0 + wl)(214)

mi =13(m0 + wl)(215)

and after you have dreived the above, it should be very easy to show that the goodsand money demand of an unemployed worker j is

xj =23p

m0(216)

mj =13m0.(217)

These workers take their labor hours as given and choose consumption and moneybalances which maximize their utility.

Answer. The employed worker maximizes x2i

mip

s.t. m0 + wl − pxi −mi = 0. Lagrangian is

L = x2i

mip− λ(m0 + wl− pxi −mi). First order conditions are 2xi

mip

+ λip = 0 andx2

ip− λi = 0.

Eliminate λi to get 2mi/p = xi (there is another solution: xi = 0, mi = m0, which does notmaximize utility but gives the lowest possible utility U = 0). Together with the budget constraintthis gives the above. For unemployed workers the same formulas are valid, simply set l = 0:


• c. (2 points) Show that aggregate demand (total consumption plus governmentexpenditures) when the rate of unemployment is u, and neither employed nor unem-ployed workers are constrained in consumption, is, in real terms,

(218) y = g +2N

3p

(m0 + w(1− u)l

).

Answer. Add g and N(1−u) times the notional demand of employed workers, and Nu timesthe notional demand of unemployed workers:

(219) y = g + N(1− u)2

3p(m0 + wl) + Nu

2

3pm0 = g +

2

3p(Nm0 + wN(1− u)l).

Note that in this second formulation, N(1− u)l = z.

• d. (3 points) Assume the firm hires the number of workers needed to producea given output y, which is below full employment output. Show that the rate ofunemployment u satisfies

(220) 1− u =y

Nl(1 +

ay

2).

Answer. Demand in labor units: z = y(1 + ay2

). The relation between labor units and

unemployment is z = N(1− u)l; therefore 1− u = yNl

(1 + ay2

).

• e. (2 points) This is only tedious math. Equations (218) and (220) are tworelations between y and u valid in Keynesian unemployment; one is from the demand


side, the other from the supply side. Show that eliminating u from these equationsgives

(221) (3p− 2w)y − awy2 = 3pg + 2Nm0

Answer. Start with (218):

y = g +2N

3p

(m0 + (1− u)wl

)(222)

Plug the 1− u from (220) into this:

= g +2N

3p

(m0 +

y

Nl(1 +

ay

2)wl

)(223)

= g +1

3p

(2Nm0 + wy(2 + ay)

)(224)

(3p− 2w)y − awy2 = 3pg + 2Nm0.(225)

• f. From (221) derive the following expression for the government expendituremultiplier in Keynesian unemployment:

dy =3p

3p− 2w(1 + ay)dg(226)


Show that the government expenditure multiplier

k =3p

3p− 2w(1 + ay)(227)

is greater than 1 but smaller than 3 as long as y is smaller than the notional demandgiven by (211).

Answer. Taking total differential of (221) with respect to y and g gives

(3p− 2w) dy − 2awy dy = 3p dg(228) (3p− 2w(1 + ay)

)dy = 3p dg(229)

from which the formula for k follows. To show that k > 0 start with

y <1

a(

p

w− 1)(230)

ay <p

w− 1(231)

1 + ay <p

w(232)

0 < w(1 + ay) < p(233)

3p > 3p− 2w(1 + ay) > 3p− 2p = p > 0(234)

1 <3p

3p− 2w(1 + ay)<

3p

p= 3(235)


• g. (1 points) For comparison, here is a simple income-expenditure model whichhas the same marginal propensity to consume as the above model in (214), namely2/3:

y = g + c c = c0 +23y(236)

What is the government expenditure multiplier in this model? Is it smaller or biggerthan the multiplier in the Keynesian unemployment model?

Answer.

(237) y = 3(g + c0)

therefore the multiplier is 3, which is bigger than the above multiplier.

Problem 20. New version of (??) with explicit numbers, but N is still as pa-rameter: The following data describe an unemployment equilibrium situation in this


model.

a =1

50N(238)

g = N(239)

l = 15(240)

p =m0

15(241)

w =m0

16(242)

y =10N

3.(243)

CHAPTER 7

Government Budget Deficit

This is [HVdP02], chapter 6.

1. Ricardian Equivalence

Ricardian equivalence says: only the government expenditures matter for house-hold behavior; the time path of taxes does not matter.

Here is Barro’s proof. We should know it in order to intelligently discuss thesepositions.

• Two periods, present and future. C1 consumption in present, C2 in thefuture.

97

98 7. GOVERNMENT BUDGET DEFICIT

• Consumers’ life time utility depends on life time consumption: V (C1, C2).They choose that consumption path which maximizes their lifetime utilitysubject to their budget constraint. Therefore let us look at this budgetconstraint.

• Assets at beginning of first period A0, end of first period A1, end of secondperiod A2. These assets pay interest: at beginning of first period consumergets rA0, and at beginning of second period rA1.

• Besides these interest payments, consumers have incomes Y1 and Y2.• Proportional tax rate is t1 and t2. Interest income is exempt from taxes!

([HVdP02, pp. 139/40] shows that Ricardian equivalence does not hold ifinterest income is taxed as well.)

From this we get the accounting identities

A1 = (1 + r)A0 + (1− t1)Y1 − C1(244)

A2 = (1 + r)A1 + (1− t2)Y2 − C2(245)

In order to turn these accounting identities into budget restrictions we make twoassumptions:

• the consumer can freely lend or borrow at the prevailing interest rate, i.e.,A1 can be any number, negative or positive.

• the consumer leaves zero assets at his or her death.

1. RICARDIAN EQUIVALENCE 99

Mathematically, this means we set A2 = 0 and eliminate A1 from the equationsystem. If A2 = 0 then (245) can be solved for A1. Setting this equal to (244) gives

C2 − (1− t2)Y2

1 + r= (1 + r)A0 + (1− t1)Y1 − C1(246)

By reordering the terms one can get this in the form: present discounted value ofconsumption = full income:

C1 +C2

1 + r= (1 + r)A0 + (1− t1)Y1 +

(1− t2)Y2

1 + r(247)

Now look at the government budget constraint. B0 is the government debt atthe beginning of the first period, B1 at the end of the first, and B2 at the end of thesecond period. Government expenditures are G1 and G2. These must be financed bytaxes or issuing debt. The interest rate at which the government lends and borrowsis the same r that is valid for households. Again the accounting identities are

(1 + r)B0 + G1 − t1Y1 = B1(248)

(1 + r)B1 + G2 − t2Y2 = B2(249)


Again we assume B2 = 0 and B1 unrestricted, therefore the same procedure as aboveyields

(1 + r)B0 + G1 − t1Y1 =t2Y2 −G2

1 + r(250)

(1 + r)B0 + G1 +G2

1 + r= t1Y1 +

t2Y2

1 + r(251)

I.e., the present value of the net liabilities of the government is equal to the presentvalue of its tax revenues.

Now let’s look at the link between government and households: since governmentdebt is the only asset that the households can hold, equilibrium requires A0 = B0,A1 = B1, A2 = B2. Walras’s law says: if two of these identities hold, then the thirdidentity holds as well. A2 = B2 holds by assumption, because both are assumed tobe zero. Therefore only one other identity must hold, say A0 = B0.

If these are the assumptions of the model, then Ricardian equivalence holds.Only the government expenditures matter for household behavior, the distributionof the tax burden over the first and second periods does not matter. To prove this,substitute B0 for A0 in the household budget constraint (247):

C1 +C2

1 + r= (1 + r)B0 + (1− t1)Y1 +

(1− t2)Y2

1 + r(252)


Now use the government budget constraint (251) in order to express B0 in terms ofgovernment expenditures and tax revenues:

= t1Y1 +t2Y2

1 + r−G1 −

G2

1 + r+ (1− t1)Y1 +

(1− t2)Y2

1 + r(253)

and simplify:

= Y1 −G1 +Y2 −G2

1 + r(254)

After simplification, the righthand side of the household budget constraint only de-pends on G1 and G2, its dependence on taxes has cancelled out. If you look at thesystem as a whole, therefore, the household’s budget constraint does not dependon taxes. Since their utility function does not depend on taxes either, this means,household consumption does not depend on taxes, it only depends on govermentexpenditures.

In other words, if taxes are reduced in the first period (but government expen-ditures in both periods remain the same), then households will not increase con-sumption, but they will save their tax windfall so that they can pay the necessarilyincreased taxes in the second period.

Now let’s do a more specific model.• ρ is rate of time preference, higher ρ means people are more impatient.


• Lifetime utility V is additive with time preference ρ:

(255) V (C1, C2) = U(C1) +1

1 + ρU(C2)

• In order to get simple formulas, the one-period utility function is assumedto be

(256) U(Ct) = log Ct

For notatinal convenience define

(257) Ω = Y1 −G1 +Y2 −G2

1 + r

Then the following holds:

Problem 21. Show the following: the consumption path C1, C2 which maximizes

log C1 +1

1 + ρlog C2 subject to C1 +

C2

1 + r= Ω(258)

is given by

C1 =1 + ρ

2 + ρΩ C2 =

1 + r

2 + ρΩ(259)


Answer. The Lagrange function is

(260) L = log C1 +1

1 + ρlog C2 + λ

(Ω− C1 −

C2

1 + r

)This gives the first-order conditions

∂L∂C1

= 01

C1= λ(261)

∂L∂C2

= 01

(1 + ρ)C2=

λ

1 + r(262)

∂L∂λ

= 0 C1 +C2

1 + r= Ω(263)

Divide (261) by (262) to get

(264)C2

C1=

1 + r

1 + ρ

and use this together with the household budget constraint to get (259).

Problem 22. (8 points) Which modifications of Barro’s basic model will causeRicardian Equivalence to fail?

Answer. Y depends on taxes.Interest income is taxable.Households cannot borrow at same interest rates as government.

The fact that government lives longer than households does not necessarily cause failure, aslong as households care about their offspring and make bequests.


Population growth.Irrationality, myopic behavior, lack of information.“Bird in Hand” issue.The fact that the debt is sold to foreigners does not matter.

2. Tax Smoothing

If Ricardian equivalence holds, there is an extra degree of freedom, because thetime distribution of taxes does not affect final demand. This allows us to choose thetime distribution of taxes in such a way that the welfare loss of taxation is minimized.An example of such a welfare loss function is

(265) LG =12t21Y1 +

12

t22Y2

1 + ρG

If ρG = r then tax rates must be equal in the two periods in order to minimize thiswelfare loss. Government lending is used to keep the tax rates constant. This is whythis theory is called “tax smoothing.”

CHAPTER 8

Labor Market

1. Unemployment Stylized Facts

(1) Unemployment rate fluctuates over time.(2) Unemployment fluctuates more between business cycles than within busi-

ness cycles.(3) Rise in European unemployment coincieds with an enormous increase in

long term unemployment.(4) In the very long run unemployment shows no trend.(5) Unemployment differs between countries.(6) Few unemployed have themselves chosen to be unemployed.

105

106 8. LABOR MARKET

(7) Unemployment differs a lot between age groups, occupations, regions, races,and sexes.

2. Minimum Wage too High?

The fact that the unemployment rate of skilled laborers is much lower than thatof unskilled laborers has been used to “prove” that unemployment is caused by a toohigh minimum wage.

Here is this proof. Short run production function has two arguments, unskilledand skilled labor:

Y = F (NU , NS); F1 > 0, F2 > 0, F11 < 0, F22 < 0.(266)

What about F12? Assumption that NU and NS are substitutes:

F12 < 0(267)

Firm takes output price P and wages WU and WS as given and choses those NU

and NS which maximize

(268) Π = PF (NU , NS)−WUNU −WSNS

The first order conditions define the labor demands NDU and ND

S :

PF1(NDU , ND

S ) = WU ; PF2(NDU , ND

S ) = WS(269)

2. MINIMUM WAGE TOO HIGH? 107

or, writing wU = WU/P , wS = WS/P ,

F1(NDU , ND

S ) = wU , F2(NDU , ND

S ) = wS .(270)

If the production function is well behaved, this can be solved for NDU and ND

S to givethe demand functions, which depend on both prices:

NDU = ND

U (wU , wS), NDS = ND

S (wU , wS).(271)

In order to get the partial derivatives of these demand functions, we take totaldifferentials of the First Order Conditions:

F11(NDU , ND

S ) dNDU + F12(ND

U , NDS ) dND

S = dwU ;

(272)

F21(NDU , ND

S ) dNDU + F22(ND

U , NDS ) dND

S = dwS ;(273)

This can be written in matrix notation (note that F21 = F12)[F11 F12

F12 F22

] [dND

U

dNDS

]=

[dwU

dwS

](274)

108 8. LABOR MARKET

Therefore [dND

U

dNDS

]=

[F11 F12

F12 F22

]−1 [dwU

dwS

]=

1F11F22 − F 2

12

[F22 −F12

−F12 F11

] [dwU

dwS

](275)

This information can also be written in the form of partial derivatives:

∂NDU

∂wU=

1F11F22 − F 2

12

F22 < 0∂ND

U

∂wS= − 1

F11F22 − F 212

F12 > 0(276)

∂NDS

∂wU= − 1

F11F22 − F 212

F12 > 0∂ND

S

∂wS=

1F11F22 − F 2

12

F11 < 0(277)

For the signs we used the assumption F11F22−F 212 > 0 which, according to the book,

holds for every well-behaved production function—why?These are the demand functions for the two kinds of labor. How about the supply

functions? The simplest assumption is that labor supply does not depend on wagesat all, it is perfectly inelastic:

NSU = NU NS

S = NS(278)

2. MINIMUM WAGE TOO HIGH? 109

Therefore the equilibrium (real) wages wU and wS which give full employmentin both labor markets are gained by setting demand equal to supply:

NU = NDU (wU , wS) NS = ND

S (wU , wS)(279)

The usual graphical representation draws the demand curve NDU (wU , wS) against

wU with wS as a shift parameter, and NDS (wU , wS) against wS with wU as a shift

parameter. This gives Figure 7.6 in the book on p. 169, for the time being look onlyat the lower of the two parallel demand curves in each diagram.

Now make the assumption that the minimum wage is above the equilibrium wagefor unskilled labor. This means, on the right hand panel you move from EU

0 to A,with lots of unemployment. But higher unskilled wages will cause employers to shiftto more skilled labor, therefore you move from ES

0 to ES1 and at the same time from

A to EU1 . (All 4 signs from (276) and (277) were used here.)

If this is the reason for unemployment, what are the policy options?

• Abolish minimum wage. The authors say this may be difficult politically.But some argue it is also difficult economically: if wages are below sub-sistence levels, then these costs have to paid in other ways: people needfood stamps, subsidized housing, people cannot afford health care, or theyget evicted from their homes because they fall behind in rent, and there

110 8. LABOR MARKET

is crime etc. (This is a little bit covered in the first point of the efficiencywage theory.)

• Subsidize unskilled labor• Government should directly employ unskilled labor• Government should invest in retraining projects to turn unskilled workers

into skilled workers.

3. Efficiency Wages

Efficiency wage theory says that net productivity of workers is a function of thewage they receive. There are several possible transmission mechanisms:

• Higher wages means better nutrition means higher productivity.• Lower wages mean higher labor turnover, i.e., firm-specific human capital

is lost.• Firm has imperfect information about the characteristics of the worker and

is willing to pay higher wages in order to get higher quality workers.• Firm pays high wages as disciplining device: worker is fired if caught shirk-

ing.• Workers’ performance depends on whether they believe they are treated

fairly.

3. EFFICIENCY WAGES 111

How does efficiency wage theory explain unemployment? Efficiency wages are rigid,they cannot fall to the market-clearing level because this would lower productivitytoo much.

Here is a model: The input into the production function is L = E ·N where Nare the hours worked, and E is the “effort” which depends positively on the wage Wpaid by the firm and negatively on the wage WR paid elsewhere:

E = e(W,WR) e1 > 0 e2 < 0(280)

The firm chooses that level of employment N and wage W which maximizes profits:

(281) Π = PF(e(W,WR) ·N

)−WN

First order conditions:∂Π∂N

= 0 PF ′(e(W,WR) ·N

)· e(W,WR)−W = 0(282)

∂Π∂W

= 0 PNF ′(e(W,WR) ·N

)· e1(W,WR)−N = 0

which can be simplified to

PF ′(e(W,WR) ·N

)· e1(W,WR) = 1(283)

112 8. LABOR MARKET

Solve (282) for PF ′(e(W,WR) ·N

)and plug this into (283):

(284)W · e1(W,WR)

e(W,WR)= 1

This means that the tangent of the effort function goes through the origin, see Figure7.10.

This model is not yet fully specified. Here is a full model. The effort function is

E = (W −WR)ε 0 < ε < 1(285)

and WR is an average of the wage W paid by the other firms and the unemploymentbenefit B:

(286) WR = (1− U)W + UB = W (1− U + βU)

where β = B/W . Unemployment benefits are assumed to be a constant portion ofW , i.e., β is by assumption a constant.

3. EFFICIENCY WAGES 113

(284) reads hereW

e(W,WR)e1(W,WR) ≡ W

(W −WR)εε(W −WR)ε−1 =

W

W −WRε = 1

ε =W −WR

W= 1− WR

W(287)

W =WR

1− ε(288)

In other words, the firm gives a markup of 1/(1 − ε) over the wages that can beobtained elsewhere. If all firms do this, i.e., if W = W , then (286) becomes WR =W (1− U + βU). Plug this into (288) to get

W =W (1− U + βU)

1− ε(289)

1 =1− U + βU

1− ε(290)

1− ε = 1− U + βU(291)

ε = U(1− β)(292)

U =ε

1− β.(293)

CHAPTER 9

Trade Unions

Modern neoclassical economics says that markets are efficient and desirable al-most everywhere. Trade unions are considered a cartel, i.e., a distortion of freemarkets. Marxists and classical economists have a different view of trade unions. Inclassical theory, the primary institutional element in capitalism is not the marketiself, but the allocation of resources which the market brings about. Markets are thesurface institution, the social interaction, by which an efficient allocation of laborand other resources is organized. But the primary thing is this allocation itself, notthe market.

For instance, the wage bargain between employer and employee is the institutionthrough which the employee receives the means by which they can live. If a freemarket, without monopolies or other interferences, does not have this outcome then

115

116 9. TRADE UNIONS

this must be considered a market failure. Not a failure of the market to function asmarket, but a failure of the market to perform its role. This kind of market failurehas its real costs to the economy. If people earn less than they need to live, thenthey need food stamps, get evicted, cannot pay their medical bills, cannot get childcare, turn to crime, etc. This affects their performance at work and shifts the coststo others. An employer who pays less than a living wage takes more than he or shegives.

Marx and other classical economists, [Dun60] for instance, argued that thepower differential between individual workers and capitalists is so great that theworker will not be able to get the value of labor-power in a free market interaction.This is where unions come in: they re-dress the balance and allow workers to get thevalue of their labor-power.

You will find such theories in classical writings, but not in neoclassical economics—because neoclassical economics does not treat the market as a surface relation throughwhich some underlying deeper relations are mediated, but they look at the marketas such.

Mainstream economics does not see the market as a surface mechanism thatenforces something underlying (socially necessary labor-time). Markets are by def-inition “right.” Therefore mainstream economics looks at the bargaining processitself. This is a special case of the “Edgeworth box.”

9. TRADE UNIONS 117

Union as a monopoly: union is able to set wage, and employers determine em-ployment accordingly. Unions choose the best point on the labor demand curve.

“Right to manage” model: Wage is set by a bargaining process between unionand employers, and then employers determine employment accordingly. Accordingto the bargaining strength this can lead to any point on labor demand curve betweenthe monopoly position and the competitive equilibrium position.

The problem with those models is: the outcomes are pareto inefficient. It wouldbe more efficient if unions and employers would bargain not only about wages butalso about employment. This gives any point on the “contract curve.”

CHAPTER 10

Search Theory

This chapter closely follows [Pis00]. The basic analytical instrument in searchtheory is the matching function G (comparable to production function or moneydemand function). Its arguments are two stock variables: the number of unemployedworkers (call it for now U , the book does not have a notation for this), and thenumber of vacancies (call it V). Its value is a flow, namely, the rate of matching, inthe following sense: G(U ,V)dt is the expected value of the number of matches in theinfinitesimal time interval dt.

Properties of the matching functions: G1 > 0, G11 < 0, G2 > 0, G22 < 0,G11G22 − G2

12 > 0, G(U , 0) = G(0,V) = 0. As is often the case, the book assumes

119

120 10. SEARCH THEORY

constant returns to scale as well:

(294) G(λU , λV) = λG(U ,V)

Matches occur randomly, and the function G gives the expected value of the totalnumber of matches in the economy per unit of time. This expected value does not yetgive the full probability distribution; what else do we know about this probabilitydistribution? In order to specify this distribution fully we make the assumptionthat all individual vacancies follow independent and identically distributed Poissonprocesses, and the same is true for all individual unemployed workers.

In order to get the right expected value for the sums, the parameter of the Poissonprocess which each individual vacancy goes through must be G(U,V)

V . In other words,for each of V vacancies that exist at time t, the probability that they will be filledin the time interval between t and t + dt is G(U,V)

V dt, therefore the expected value ofthe total number of matches is V G(U,V)

V dt = G(U ,V)dt, as postulated.

Problem 23. For each vacancy i (i = 1, . . . ,V) define the random variable ni

as follows: ni = 1 if the ith vacancy has been filled between t and t + dt, and ni = 0otherwise. ni is called the “indicator function” for the event “the ith vacancy is filledbetween t and t + dt.” The total number of vacancies filled in the time interval dt isthen

∑Vi=1 ni. Show that E[

∑ni] = G(U ,V) dt.

1. RELATIONSHIP BETWEEN UNEMPLOYMENT RATE AND LABOR MARKET TIGHTNESS121

Answer. From

(295) E[ni] = 0 · Pr[ni = 0] + 1 · Pr[ni = 1] = Pr[ni = 1] =G(U ,V)

Vdt.

follows

(296) E[∑

ni] =∑

E[ni] = VG(U ,V)

Vdt = G(U ,V) dt QED.

1. Relationship between Unemployment Rate and Labor MarketTightness

Assuming constant returns to scale, and setting λ = 1/V in (294), one getsthe result that the parameter of the matching process of an individual vacancy is afunction of V/U :

G(U ,V)V

= G(U/V, 1) ≡ q(θ)(297)


where θ = V/U is the vacancy-unemployment rate, which can be considered a mea-sure of labor market tightness. Due to the properties of the Poisson distribution,q(θ) dt is the probability that a given vacancy will be filled in the time interval dt.Likewise, the parameter of the job search process of an individual unemployed workeris

G(U ,V)U

=VU

G(U ,V)V

= θq(θ)(298)

Problem 24. The function

(299) U ,V 7→ G(U ,V)

is a matching function with the usual properties: G1 > 0, G11 < 0, G2 > 0, G22 < 0,G11G22 −G2

12 > 0, G(U , 0) = G(0,V) = 0, and constant returns to scale:

(300) G(λU , λV) = λG(U ,V)

• a. (2 points) Show that

(301) G(U ,V) = UG1(U ,V) + VG2(U ,V)

• b. (2 points) Show that the derivatives are homogeneous of degree zero, e.g.,

(302) G1(λU , λV) = G1(U ,V)

and the same is valid for G2.


• c. (2 points) Show that for all U 6= 0 and V,

(303) G1(U ,V) <G(U ,V)U

Answer. It is easiest to start with (301):

(304) G(U ,V) = UG1(U ,V) + VG2(U ,V) > UG1(U ,V)

and then divide by U .

• d. (5 points) Define θ = VU . By the definition of search functions,

(305) q(θ) =G(U ,V)V

is the Poisson parameter of the search process of an individual firm, and

(306) f(θ) =G(U ,V)U

is the Poisson parameter of the search process of an individual worker. Let

(307) η(θ) = − θ

q(θ)q′(θ)


be the elasticity of q. Show that

(308) η(VU

)=

UG(U ,V)

G1

(U ,V

).

Answer.

(309) q(θ) =G(U ,V)

V= G(U/V, 1) = G(1/θ, 1).

Therefore

q′(θ) = −1

θ2G1(

1

θ, 1) ≤ 0(310)

η(θ) = −θ

q(θ)q′(θ) =

1

θq(θ)G1(

1

θ, 1)(311)

=UV

VG(U ,V)

G1

(UV

, 1)

(312)

=U

G(U ,V)G1

(U ,V

)(313)

• e. (1 points) Show that 0 < η < 1.

Answer. 0 < η clear, and in order to see η < 1 multiply both sides of (303) by UG(U,V)

and

use (308).


• f. (2 points) Show that the elasticity of f is 1− η.Answer.

(314)θ

f(θ)f ′(θ) =

θ

θq(θ)

(q(θ) + θq′(θ)

)= 1 +

θ

q(θ)q′(θ) = 1− η(θ)

We also assume that jobs do not last for ever, but that each filled job is eventuallydestroyed, again following a Poisson process, whose parameter s is exogenous. In thetime interval dt therefore the proportion s dt of all filled jobs disappears. If N is thesize of the workforce, then this means that in every time interval dt, on the averages (N − U) dt jobs disappear (where s is a constant number) and G(U ,V) dt jobs arecreated. In equilibrium these two processes must balance each other, therefore

s(N − U) = G(U ,V)(315)

Divide by N and introduce the unemployment rate U = U/N :

s(1− U) =G(U ,V)

N(316)

=G(U ,V)V

VUUN

(317)

= q(θ)θU(318)


This allows us to determine the equilibrium unemployment rate associated with thevacancy-unemployment ratio θ. For this, collect all terms with U on the rhs:

s = (s + θq(θ))U(319)

U =s

s + θq(θ)(320)

This so-called Beveridge curve is the first key equation in the model. It expressesthe endogenous U as a function of the exogenous parameter s and the endogenousθ. (Instead of considering U and V as two different endogenous variables, it is moreconvenient to consider U and θ as two different endogenous variables.)

2. Firms and Workers

Firm owners value their firms by the present discounted value of the expectedvalue of their income flow. Each firm has one job only. If the job is filled, thefirm hires K units of capital and produces. If the job is empty, the firm incurs aconstant flow of search costs of γ0 per time unit. The arbitrage equations are easiestto understand if we multiply them by dt:

rJV dt = −γ0 dt + q(θ) dt [JO − JV ](321)

rJO dt =(F (K, 1)− (r + δ)K − w

)dt− s dt [JO − JV ](322)

2. FIRMS AND WORKERS 127

Present discounted value of expected returns means: the owner of the firm is riskneutral; only the expected values of the returns matter to him. He is indifferentbetween putting an amount representing the value of his firm into the bank, orreceiving the expected value of the return accruing to the firm.

First look at the firm with a vacancy: During the time interval between t andt + dt, putting the money into the bank would earn him interest rJV dt. The searchcost would be γ0 dt, and the probability of filling the vacancy would be q(θ) dt, inwhich case the value of the firm would jump from JV to JO.

Now look at the firm whose job is filled. During the time interval between t andt + dt, putting the money into the bank would earn the owner interest rJO dt. Theoperating profit would be

(F (K, 1)− (r + δ)K − w

)dt, and with probability s dt he

would lose the worker, in which case the value of his firm would revert to JV .Next steps after establishing these arbitrage conditions: Free entry means JV =

0. (321) gives therefore

(323) JO =γ0

q(θ)

In order to evaluate (322) we first have to determine the value of K: its choicemaximizes the flow of profits:

(324) maxK

F (K, 1)− (r + δ)K − w


The first-order condition is

(325) F1(K, 1) = r + δ

From this follows F (K, 1)− (r+δ)K = F (K, 1)−KF1(K, 1) = 1 ·F2(K, 1), therefore(322) becomes

(326)F2(K, 1)− w

r + s=

γ0

q(θ)

Workers value their income in a similar fashion by the present value of theirexpected incomes. The present value of the expected incomes of employed workers(YE) and of unemployed workers (YU ) can be derived from the following arbitrageconditions:

rYU dt = z dt + θq(θ) dt [YE − YU ](327)

rYE dt = w dt− s dt [YE − YU ](328)


rYU can be considered the “reservation wage,” the income for which the worker wouldbe willing to stop searching for a job. Solving this for rYU and rYE gives:

rYU =(r + s)z + θq(θ)w

r + s + θq(θ)(329)

rYE =sz +

(r + θq(θ)

)w

r + s + θq(θ)=

r(w − z)r + s + θq(θ)

+ rYU(330)

Both are weighted averages between the unemployment benefit z and the wage wwhich the workers would get if employed. If r = 0 then YU = YE , since all workerswill go back and forth between employed and unemployed, and if r = 0 it does notmatter which comes first.

When workers and employers meet, then an “economic rent” arises, i.e., a benefitwhich cannot be competed away, because now they no longer have to incur searchcosts, but they can start producing. The wage agreed between worker and employerreflects the distribution of this benefit between worker and employer. If the wage isw = rYU , the worker’s reservation wage, then the capitalist captures all the surplus;and if the wage is w = F2(K, 1), the marginal product of labor, then the workercaptures all the surplus. (Note: (1) the reservation wage depends on the wage theworker can expect if working elsewhere, if he or she declines the job in firm i wherethe bargaining takes place. (2) Regardless of the wage, the capital stock K will


always be at the profit maximizing level, i.e., the one that satisfies (325). The bookintroduces this assumption only later, but it is simpler to introduce it already here.)

There is no economic law which determines the outcome of this bargaining,it depends on the circumstances. Neoclassical economists don’t like situations inwhich there is economic indeterminacy. Therefore they invoke the (generalized)Nash equilibrium, which introduces the parameter β, the bargaining strength, andfinds pareto-efficient outcomes between each party’s “fallback position” based on thevalue of β. If one goes through the formalism of the generalized Nash equilibrium,the outcome is very simple: the wage in firm i is the weighted average between thetwo extreme positions, with β being the weight attributed to the marginal productof labor:

(331) wi = (1− β)rYU + βF2(K, 1)

For our purposes here, let’s forget about Nash equilibria and just assume that (331)holds. On the one hand, this assumption states the obvious, namely, that the wagewi is somewhere between the two extreme positions. The strongest part of thisassumption is that the exogenous parameter β is the same for each firm. In otherwords, we are assuming here that all workers have the same bargaining strength.

Now the model is closed by the assumption that the wage which is the result ofthis bargaining is equal to the wage the worker can expect elsewhere, since all firms


do the same. I.e., the wi from (331) is the same as the w in the other equations. Ifone uses this, one gets, after some math:

(332) w = (1− β)z + β[F2(K, 1) + θγ0]

The model consists therefore of the four equations (325), (326), (332), and (320).

Problem 25. This question discusses a version of the search theory model whichis mathematically simpler than the one in the textbook since it does not have capitalstock, and it assumes that the coefficient β indicating bargaining strength is 1/2. Itis similar to, but simpler than, the version of the model given in [Pis00, chapter 1].But I tried to make the narrative explaining the model more cogent.

U is the number of unemployed workers, and V the number of vacant positionsin the economy. The function

(333) U ,V 7→ G(U ,V)

is a job-search matching function with the usual properties: G1 > 0, G11 < 0, G2 > 0,G22 < 0, G11G22 −G2

12 > 0, G(U , 0) = G(0,V) = 0, and constant returns to scale:

(334) G(λU , λV) = λG(U ,V)

Define θ = VU and define

(335)G(U ,V)V

= G(1θ, 1) =: q(θ)


• a. (2 points) Based on this matching function, the model assumes that on theaverage G(U ,V) dt jobs are created in every time interval dt. Another assumptionof the model is that on an average the proportion s dt of all filled jobs disappearsagain in each time interval dt, where s is an exogenous parameter. In other words,if N is the size of the workforce, therefore N − U is the number of filled jobs, onthe average s (N − U) dt jobs disappear in time interval dt. Derive from this the“Beveridge curve” (336) which gives the equilibrium unemployment rate associatedwith the vacancy-unemployment ratio θ:

U =s

s + θq(θ)(336)

where U = UN is the rate of unemployment,

Answer. On the average s (N − U) dt jobs disappear and G(U ,V) dt jobs are created in timeinterval dt. In equilibrium these two processes must balance each other, therefore

s(N − U) = G(U ,V)(337)


Divide by N

s(1− U) =G(U ,V)

N(338)

=G(U ,V)

VVUUN

(339)

= q(θ)θU(340)

Now collect all terms with U on the rhs:

s = (s + θq(θ))U(341)

This gives (336).

• b. (3 points) Firms are small; each firm has one job only which is vacant whenthe firm enters the market. Vacant firms engage in hiring, which costs γ0 per timeunit. If the job is filled, the firm produces. Its output is 1 product per time unit. Thefirm is able to sell this product instantly at price p. If r is the interest rate and wthe expected value of the wage paid by the firm, this gives the two arbitrage equations

rJV dt = −γ0 dt + q(θ) dt [JO − JV ](342)

rJO dt = (p− w) dt− s dt [JO − JV ](343)

Give a verbal justification of these two equations (and do not forget to give the defi-nitions of JV and JO).


Answer. First look at (342) which focuses on the firm with a vacancy: During the time intervalbetween t and t + dt, putting the money into the bank would earn the owner interest rJV dt. Thesearch cost would be γ0 dt, and the probability of filling the vacancy would be q(θ) dt, in which casethe value of the firm would jump from JV to JO.

Now look at (343) which focuses on the firm whose job is filled. During the time intervalbetween t and t + dt, putting the money into the bank would earn the owner interest rJO dt. Theoperating profit would be (p − w) dt, and with probability s dt the firm would lose the worker, in

which case the value of the firm would revert to JV .

• c. (1 points) You should skip this and simply use the result given below, unlessyou really have extra time at the end of the exam. It is tedious math and will get youonly one point. (342) and (343) can be considered two equations in the two unknownsJO and JV . Show that solving these equations gives

JV =(p− w)q(θ)− γ0(r + s)(r + q(θ)

)(r + s)− q(θ)s

(344)

JO =(p− w)

(r + q(θ)

)− γ0s(

r + q(θ))(r + s)− q(θ)s

(345)

Answer. Collect terms with JV , JO, and constants:(r + q(θ)

)JV − q(θ)JO = −γ0(346)

−sJV + (r + s)JO = p− w(347)


This can be written in matrix form:

(348)

[r + q(θ) −q(θ)−s r + s

] [JV

JO

]=

[−γ0

p− w

]Inversion of the matrix gives

(349)

[JV

JO

]=

1(r + q(θ)

)(r + s)− q(θ)s

[r + s q(θ)

s r + q(θ)

] [−γ0

p− w

]

• d. (5 points) Free entry means that JV = 0. Show that this implies

(350) w = p− (r + s)γ0

q(θ)

and

(351) JO =γ0

q(θ)

Answer. For JV in (344) to be zero the parameters must be such that (p−w)q(θ)−γ0(r+s) =0. From this follows

(352)p− w

r + s=

γ0

q(θ)


and (350). The shortest way to derive (351) is to set JV = 0 in (342). One can als derive (351)from (345) using (352), but this is tricky. Start with (345):

JO =(p− w)

(r + q(θ)

)− γ0s(

r + q(θ))(r + s)− q(θ)s

(353)

Now multiply the whole fraction by γ0q(θ)

, and in order to balance this out, multiply the first term

in the numerator by r+sp−w

and the second term byq(θ)γ0

=γ0

q(θ)

(p− w)(r + q(θ)

)r+sp−w

− γ0sq(θ)γ0(

r + q(θ))(r + s)− q(θ)s

=γ0

q(θ)

(r + q(θ)

)(r + s)− q(θ)s(

r + q(θ))(r + s)− q(θ)s

.(354)

• e. (3 points) The present value of the expected incomes of employed workers(YE) and of unemployed workers (YU ) can be derived from the following arbitrageconditions:

rYU dt = z dt + θq(θ) dt [YE − YU ](355)

rYE dt = w dt− s dt [YE − YU ](356)

where z is the unemployment benefits. Explain these conditions verbally.

• f. (1 points) Here is another part of the question which only involves tediousmath. Don’t derive the following two equations unless you have lots of extra time.


Just use the results results given here for the other parts of the question. Solving(355) and (356) for rYU and rYE gives:

rYU =(r + s)z + θq(θ)w

r + s + θq(θ)(357)

rYE =sz +

(r + θq(θ)

)w

r + s + θq(θ)=

r(w − z)r + s + θq(θ)

+ rYU(358)

• g. (2 points) Firm i takes the product price p and the interest rate r as given,but it does not take the wage wi as given from the market. Instead, the wage isdetermined by individual bargaining between firm and worker. After firm i has hireda worker with wage wi, the present discounted value of its income stream J i

O satisfiesan arbitration equation analogous to (343):

rJ iO dt = (p− w) dt− s dt [J i

O − JV ](359)

JV , on the other hand, does not depend on wi but on the average wage w that canbe expected, therefore JV = 0 as before. Show that

J iO − JV =

p− wi

r + s(360)

What is the highest wage which the firm could pay without suffering a decline in itsvacancy value? Both parts of this question need very simple, almost trivial math.


Answer. From (359) follows

rJiO = (p− wi)− sJi

O(361)

JiO =

p− wi

r + s(362)

Since JV = 0, (360) follows. From (360) one can see immediately that any wage wi < p will increasethe firm’s value from its zero vacancy value JV = 0.

• h. (4 points) The present discounted value of an unemployed worker’s incomestream depends on w, not on wi, i.e., the equation (357) is still valid. But once aworker finds a job with wage wi, the present discounted value of his or her incomestream is governed by an equation analogous to (356):

rY iE dt = wi dt− s dt [Y i

E − YU ](363)

where YU , given by (357), does not depend on wi. Derive from this

Y iE − YU =

wi − rYU

r + s.(364)

Derive from this the “reservation wage” of a worker, i.e., the smallest wage a workerwould accept without diminishing the present discounted value of his or her incomestream as an unemployed worker.


Answer. One first gets

Y iE =

wi

r + s+

s

r + sYU(365)

now add and subtract rr+s

YU from the rhs:

Y iE =

wi

r + s−

r

r + sYU +

r

r + sYU +

s

r + sYU

=wi − rYU

r + s+

r + s

r + sYU(366)

Clearly the reservation wage is rYU .

• i. (2 points) The actual wi must be somewhere between the worker’s reservationwage and the maximum wage a firm is willing to pay. In order to have a mathematicalmodel which wage will be picked, make the assumption that the bargaining power offirms and workers are equal. so that the actual wage is the arithmetic mean of thetwo limit values:

(367) wi =rYU + p

2Show that this implies

Y iE − YU = J i

O − JV = J iO (because JV = 0)(368)


i.e., if measured by present discounted value, the benefits from finding a job are alsodivided equally between worker and employer.

Answer. One verifies that with this wi,

wi − rYU = p− wi,(369)

from which (368) follows.

• j. (4 points) Finally assume that all firms and workers come to the samebargaining outcome, i.e., from Y i

E − YU = J iO for all i follows YE − YU = JO. Plug

this into (355) and use (351) to get

(370) w = wi =z + p + θγ0

2With this, all the model equations are derived, now comparative statics experimentscould be done with this model (not done here).

Answer. (355) becomes

rYU = z + θq(θ)[YE − YU ]

= z + θq(θ)JO(371)

Now use (351)

= z + θq(θ)γ0

q(θ)= z + θγ0(372)

3. POLICY IMPLICATIONS 141

from which (370) follows.

3. Policy Implications

The book shows how to solve this system graphically, but we will here simplyreproduce the comparative-statics and policy experiments given in the book, withouttheir derivations:

(1) An increase in the unemployment benefit z increases the wage rate and theunemployment rate and lowers the vacancy rate.

(2) An increase in the exogenous rate of job destruction s lowers the wages andincreases unemployment; the effect on the vacancy rate is indeterminate.

(3) An increase in the payroll tax paid by firms lowers the wage and the vacancyrate and raises the unemployment rate.

(4) An increase in the labor income tax increases the wage rate and the unem-ployment rate and lowers the vacancy rate.

(5) An interesting and novel experiment is: if firms get a government grant bwhen they hire a worker, but must repay the grant when they destroy this job again,this raises wages and the vacancy rate and lowers unemployment. The book comparesthis payment b to deposits on empty bottles.

(6) An extension of this model can also be used to explain persistence of unem-ployment: if unemployed workers lose their skills, and therefore become less attractive


to firms and have longer search times, then temporary shocks in unemployment havepermanent effects.

CHAPTER 11

Dynamic Inconsistency and Credibility

In the discussion of the effectiveness of policies under rational expectations, theissue of time inconsistency has been discovered. Here the two examples from Shef-frin’s booklet [She83] are instructive.

First example: In order to encourage research, government offers patent protec-tion. After the research has been done (a remedy to cancer, AIDS), then it wouldbe optimal for government to abolish patent rights.

Second example: assume there is a grain shortage, and people who have storedgrain make tremendous profits. Then it seems a good policy if government imposesan excess profits tax on stored grain. However such a tax might discourage thestoring of grain for the next shortage.

143

144 11. DYNAMIC INCONSISTENCY AND CREDIBILITY

Time inconsistency means: government has the temptation to cheat, renegingon earlier commitments. This is part of the rules-discretion debate: the breaking ofcommittments has been associated with discretion, which is undesirable.

We will only do Section 1 of chapter 10.

1. A Simple Model with Dynamic Inconsistency

Linear Lucas supply curve

y = y + α[π − πe] + ε α > 0(373)

Here y is the full employment level of output. If inflation is higher than expected,workers overestimate their wages and supply more labor than is in their interest.

The Fed controls the inflation rate. It knows πe and observes ε and then sets π,and y is then determined by (373). It considers y as too low, and uses policy to shiftoutput to y∗ with inflation as low as possible. Its policy objective is to minimize thecost function

(374) Ω =12[y − y∗]2 +

β

2π2

The first-order conditions tell us that the Fed will choose the inflation rate

(375) π = −α

β[y − y∗]

1. A SIMPLE MODEL WITH DYNAMIC INCONSISTENCY 145

which lies on the “social expansion path” shown as a dashed line in Figure 10.1.

Problem 26. The economy is somewhere on the Lucas supply curve

y = y + α[π − πe] + ε α > 0(376)

Here y is output, y is the full employment level of output, π is the inflation rate, andπe is the expected inflation rate.

• a. (3 points) Give a brief verbal explanation of the theory behind the Lucassupply curve.

Answer. If inflation is higher than expected, workers overestimate their wages and supplymore labor than is in their interest.

• b. (5 points) The Fed knows πe and ε and sets π. y is then determined by(376). The Fed’s goal is to shift output close to y∗ with inflation as low as possible.It minimizes the cost function

(377) Ω =12[y − y∗]2 +

β

2π2

Write down the Lagrangian and the first-order conditions for this minimization, andshow that

(378) π = −α

β[y − y∗]


Answer. Fed chooses π and y to minimize (377) subject to (376). Lagrangian is

(379) L =1

2[y − y∗]2 +

β

2π2 + λ(y − y − α[π − πe]− ε

and first-order conditions are

∂L∂π

= 0 βπ − λα = 0 λ =βπ

α(380)

∂L∂y

= 0 y − y∗ + λ = 0 λ = y∗ − y(381)

(382)

Eliminate λ from these and solve for π:

(383) π = −α

β[y − y∗]

The scenario is therefore: in every time period there is a different supply curve,determined by a different ε and πe. In y, π space all these supply curves are parallelstraight lines. The yD and πD attained by discretionary policy are determined bythe intersection of this supply curve with the expansion path. As a function of πe

and ε one gets

(384) πD =α2πe + α[y∗ − y − ε]

α2 + β

1. A SIMPLE MODEL WITH DYNAMIC INCONSISTENCY 147

So far we haven’t specified how expectations are formed. Assume rational ex-pectations. The public knows the policy-maker’s goals and tactics, but they do notknow the value of ε and therefore predict ε by 0. (In other words, the policy makershave more information than the public. This is why policy can be effective.) Thepublic can compute (384) and therefore the expected rate of inflation satisfies

πe =α2πe + α[y∗ − y]

α2 + β.

Solving for πe gives

πe =α

β[y∗ − y](385)

If ε = 0 then this rational expectations equilibrium point is point ED in Figure 10.1in the book.

Problem 27. (7 points) Compute the rational expectations equilibrium if thepublic has the same information as the policy makers, i.e., they observe ε beforeforming their expectations.

Answer. In this case,

πe =α2πe + α[y∗ − y − ε]

α2 + β.


Solving for πe gives

πe =α

β[y∗ − y − ε](386)

Now plug this and (375) into the Lucas supply curve (373):

y = y −α2

β[y − y∗]−

α2

β[y∗ − y − ε] + ε(387)

= y −α2

β[y − y − ε] + ε(388)

Here you can already see that y∗ falls out. But solve for y:

β2y = β2y − α[y − y − ε] + β2ε(389)

(β2 + α)y = (β2 + α)y + (β2 + α)ε(390)

y = y + ε(391)

This y does not depend on the target y∗ which the policy maker is aiming at, i.e., the policy isineffective.

Let’s go back to the situation where the government has more information thanthe economic agents. Then its policy has an effect, but this policy is suboptimal.The following policy would be better: the government simply announces that it willchoose πR = 0. The subscript R for “rule.” If the public believes this, then πe = 0.

2. MODELLING REPUTATION 149

With πe = 0 and still, as in our example before, ε = 0, the Lucas supply curve (373)becomes

y = y + απ.(392)

If the government follows through on its announcement, it sets π = 0, which gets us topoint ER in Figure 10.1. Note that ER is not on the expansion path! In other words,this policy rule is not believable because it is inconsistent : once everyone expectesπ = 0, the government is tempted not to follow through on its announcement π = 0but to generate an inflation surprise to move the economy to the “cheating point”EC . And if the public expects the government to cheat, i.e., to always choose theintersection of the Lucas supply curve with the expansion path, its expectations areformed according to formula (385), which throws us back to ED. Therefore: unlessthe government can obligate itself believably to stick to πR = 0, the outcome will bethe worst of the three outcomes shown in Figure 10.1.

2. Modelling Reputation

Section 10.1.3 gives a model by Barro and Gordon in which the government hasthe option to destroy or maintain its reputation to follow through on its announcedpolicies. In the first period t = 0, the public believes the government. And if thegovernment followed through on its announcement in period t − 1, then the public


believes that it will follow through again in period t. But if the government did notfollow through on its announcement in t − 1, then the public will believe that thegovernment is going to pick the intersection of the supply curve with the expansionpath at time t and therefore the public will use formula (385) to form its expectationsfor time t,

To evaluate the implications of this model, look at two alternative scenarios:(1) the government announces a constant inflation rate and follows through with

it indefinitely. This gives a cumulative cost which is the present discounted value ofall the one-period cost functions.

(2) there is a single act of cheating by the government, in the following way: Thegovernment announces a constant inflation rate; the public believes, but in period0 the government cheats. This gives a higher level of the social welfare function forthis one period. Since the government knows that in response to its cheating, thepublic will uses the rational expectations formula next time around, it announcesfor period 1 that it will use the discretionary policies which the public expects. Thepublic believes it, therefore the outcome in 1 is YD. Now the public believes thegovernment again, and assume that from now on the government will no longercheat.

If the parameters are such that this one-time cheating gives a lower cumulativecost for the government than staying honest, then cheating more than once will be

2. MODELLING REPUTATION 151

even better, and the policy objective is not enforceable. If the one-time cheatinggives higher cumulative cost, then the policy objective is enforceable.

Problem 28. (5 points) Based on the rational expectations hypothesis, there hasbeen a debate whether the following innocuous seeming rule should always be followedby policymakers: “Whatever situation the policymakers inherit, they should alwaysuse their policy tools to get the best possible outcome given that situation.” Describethe issue, if possible, with illustrative examples.

Answer. Examples: taxation of capital goods, independence of central banking.

Problem 29. (5 points) Which implications does the Rational Expectations Mar-ket Clearing assumption have for policies? Discuss also the issues of time inconsis-tency.

CHAPTER 12

Open Economy Macro

1. Which Measure of National Product?

Difference between NNP and NDP: net national product NNP is the productof US nationals, whether or not it is produced in the US. Profits from overseasinvestments owned by US corporations are part of NNP. This is not a very usefulconcept. A Marxist would say: this concept makes the fallacious assumption thatwhenever someone gets an income, that person has produced an equivalent of thisincome.

A more useful concept is net domestic product: NDP is product produced in theUSA, whether or not the producers were US citizens or the production equipmentUS owned.

153

154 12. OPEN ECONOMY MACRO

Difference between NDP and GDP: GDP is all final products (where machineryand factory buildings are considered final products). NDP is GDP minus capitalconsumption allowance, because depreciation is part of the price of the final products,and if we don’t subtract it out, it is double counted.

In theory, NDP would be the best measure of the national product. The problemwith it is that the capital consumption allowance is very difficult to measure, andthe life length of machinery is not entirely a technological given but also depends oneconomic decisions. This latter point made in [Kal72, pp. 1/2]. Therefore, GDP ispreferable for most practical purposes.

After all this is said, the book does not distinguish between NDP and NNP, butcalls it Y .

2. Accounting Identities

National income identity: All consumption plus investment plus governmentexpenditure plus exports must either be produced domestically (Y ) or importedIM).

Y + IM = C + I + G + EX(393)

Y = C + I + G + (EX − IM)(394)

Y = A + (EX − IM)(395)

2. ACCOUNTING IDENTITIES 155

This last equation is relevant because in the open-economy IS-LM model, that partof A which is not G is modeled as a function of Y and r, and EX − IM is modeleda function of Y and Q, the relative price of foreign goods (which increases when thedomestic currency is devalued).

But let’s continue with the underlying accounting identities. TR is internationaltransfer receipts (development aid received and money sent home from migrant work-ers working abroad), T is domestic taxes net of domestic transfers. Add TR − T toboth sides of (394) to get “disposable income” Y + TR− T :

Y + TR− T = C + I + (G− T ) + (EX + TR− IM)(396)

Now rearrange that the current account surplus CA = EX +TR− IM is on the rhs:

Y + TR− T − C − I + (T −G) = (EX + TR− IM)(397)

and use the definition of savings S = Y + TR− T − C:

(S − I) + (T −G) = (EX + TR− IM)(398)

i.e., the current account surplus CA is identically equal to the private sector savingssurplus S − I plus the government budget surplus T −G.

Equation (11.5) in the book must be extended:

(399) CA = ∆NFA = ∆NFAcb −KI


The current account surplus CA gives rise to increases in NFA (net foreign assets),and these can be split up into increases of net foreign assets held by the central bank∆NFAcb minus net private capital inflows KI.

To understand the effects of ∆NFAcb look at the balance sheet of the centralbank:

Assets LiabilitiesNet foreign assets NFAcb

Domestic credit DC High powered money H

H is connected to the money stock by the money multiplier. The important pointhere is that open-market purchases (increasing DC held by the central bank and atthe same time H) are not the only way to increase H; purchases of foreign assets(holding DC constant but increasing NFAcb) also increase H. But purchases of for-eign assets which are “sterilized” by simultaneous open-market sales will not increaseH.

3. THE MUNDELL-FLEMING MODEL 157

3. The Mundell-Fleming Model

Problem 30. [Wil83, pp. 142–44] Look at the following income expendituremodel with imports and exports:

Y = C + I + G + X −M(400)

C = c0 + cY(401)

M = m0 + mY(402)

Solve this system for Y in terms of the exogenous variables C, G, I, and X, andshow that in relation to the one-country income-expenditure model, the multiplyer issmaller, but the multiplicand is bigger.

Answer. The multiplier formula is

(403) Y =1

s + m(c0 + I + G + X −m0)

The open-economy IS/LM model is named after various papers of Mundell andindependently of Fleming, see [Mun68, chapters 15–18] and [Fle62]. It is a mis-nomer because this theory was already present in Meade [Mea51].

It introduces a link to the rest of the world into the IS/LM model in two places:(1) part of domestic absorption is imports and exports, which depend on Y and Q.


(2) Financial markets affect exchange rates and interest rates. (This is the mon-etary approach to the Balance of Payments.)

Here are the equations of the model:

Y = A(r, Y ) + G + X(Y, Q)(404)

MD/P = L(r, Y )(405)

MS = µ[NFAcb + DC](406)

MD = MS(407)

X(Y, Q) + KI(r − r∗) = ∆NFAcb(408)

Here X are net exports EX − IM (international transfer payments are ignored).P is the domestic price level, P ∗ the foreign price level, E the nominal exchangerate (domestic currency price of one unit of foreign currency), and Q = EP ∗/P therelative price of foreign goods. µ is the money multiplier. NFAcb is net foreignassets held by the central bank, in equilibrium ∆NFAcb = 0, and DC is domesticcredit, and KI are capital inflows. DC is policy-determined, it can be increased ordecreased by open-market purchases or sales, but NFAcb is endogenous, it dependson the trade balance or capital movements, etc.

Graphical representation of the model: One can still draw the IS and LM curvesin the r, Y plane; but the IS curve also depends on Q. In addition, one has a third


curve in the r, Y plane, the balance of payment curve

(409) X(Y, Q) + KI(r − r∗) = 0

The BP curve can be vertical (capital immobility), positively sloped, or horizontal(perfect capital mobility, in which the domestic interest rate is equal to the exoge-nously given foreign interest rate r∗). Points to the left and/or below the BP curvehave a balance-of-payments deficit, and to the right and/or above a surplus.

The economic substance of these models can be found in the assumptions whathappens when the different curves shift.

3.1. Exchange rates fixed, capital immobile. This is an extreme case inwhich neither monetary nor fiscal policy can be effective. BP curve is vertical. Fig11.2 on p. 266.

Assume equilibrium output is below full employment output YF and the policymaker wishes to use economic policy to change this.

Monetary policy: the policy maker conducts an open market purchase, whichshifts the LM curve to the right. The IS curve stays put, therefore we go from e0 toe′. The book sounds as if this was an instantaneous shift, but it is not. But even asthis shift just begins, interest rates fall, investment rises, and Y rises. This meansthe country moves into an area where there is a Balance of Payments deficit. Sincethe monetary authority is committed to maintaining the exchange-rate, it has use its


foreign reserves to service this deficit. In this way, the foreign reserves of the centralbank are gradually drawn down. This shifts the LM curve back to the left, until it isback at its starting position. I.e., monetary policy is effective in the short run, butnot the long run.

Fiscal policy: policy maker issues bonds and uses the proceeds for governmentspending (i.e., the money supply is unchanged). This moves the economy to e′′.Again, there is a BoP deficit, drawing down the foreign reserves of the central bank,and therefore decreasing the money stock. This gradually raises interest rates, untilthe economy is at e1.

3.2. Exchange rate fixed, capital perfectly mobile. Here monetary policyis ineffective, but fiscal policy is highly effective. BP curve is horizontal, since r = r∗.Figure 11.3.

The book says on p. 268: Monetary policy shifts LM curve to the right, toe′. But at e′ the domestic interest rate is below the world interest rate. Massiveand instantaneous capital outflows (to buy the more profitable foreign assets) putsupward pressure on the exchange rate, i.e., pressure to devalue the home currency.The central bank has to sell foreign currency in exchange for home currency. Thisdecreases the money supply, therefore LM curve immediately shifts back to e0.

This sounds as if the economy would instantly move to e′, with a lower r anda higher Y , and then instantly move back. I don’t believe this; shifts along the IS


curve are not instantaneous. I think figure 11.3 in the book is misleading. [dG97,Figure 2.3 on p. 20] has a better picture with a U-turn arrow at the LM curve.

I would try the following explanation: the Fed makes an open-market purchase,i.e., buys T-bills for cash. In a closed economy, this additional cash and loss of T-billsin the portfolios of banks and private investors will cause the interest rate to fall, andbanks make more loans to businesses, which increases Y . But in the open economy,there is no need for domestic investors to accept a lower interest rate: they simplybuy foreign assets at the old interest rate r∗. This puts an upward pressure on theexchange rate, and the central bank is forced to sell foreign currency in exchangefor cash. At the end of the day, the central bank holds more T-bills and less foreigncurrency, and the public holds fewer T-bills and more foreign bonds, but there is noeffect on the money stock or the interest rate. I.e., monetary policy is completelyineffectual. It is offset by asset shifts by the private investors. In this scenario, theinterest rate never falls more than the spread necessary for the investors to maketheir arbitrages buying foreign assets in exchange for domestic assets. All this ispredicated on there being 100 percent confidence that the exchange rate will notchange.

Fiscal policy: government bonds are sold, and the receipts used for governmentexpenditure. This shifts the IS curve to the right. Businesses face more demand, andthey have to borrow in order to invest. This puts upward pressure on the interest


rate, which causes capital inflows from abroad. This puts downward pressure onthe exchange rate, and the central bank has to buy foreign currency in exchangefor domestic currency. This causes the money supply to increase, and the LM curveshifts outward until it intersects the IS curve at point e1. The fiscal policy is thereforevery effective.

Problem 31. (5 points) What is wrong with the following description of theeffects of fiscal policy under perfect capital mobility with fixed exchange rates: Gov-ernment bonds are sold and the receipts are used for government expenditure. Thisshifts the IS curve to the right. Higher Y means higher imports, which puts pressureon the exchange rate. The Central Bank has to sell its foreign exchange reserves.This raises the interest rate, so that the LM curve shifts to the left, until we end upat the same Y as before but a higher interest rate.

Answer. This is exactly the answer for immobile capital, i.e., it ignores capital flows due

to the higher domestic interest rate. The downward pressure on the exchange rate due to theinstantaneous capital inflows is stronger than the upward pressure due to increased import flows.In the overshooting result of the Dornbusch model one can see these two contrary effects acting in

opposite directions.


Note that an increase in Y also increases imports, which tends to increase theexchange rate. But the downward pressure on the exchange rate due to the instan-taneous capital inflows is stronger. One can see this interaction in the overshootingresult of the Dornbusch-model.

3.3. Exchange rate flexible with perfect capital mobility. Here the ef-fectivity of policies is reversed: Monetary policy is shown in Figure 11.4. The bookacts as if the economy first moves from e0 to e′ and then to e1. I like the explanationin [dG97, p. 17] better. De Grauwe says, using the notation of Figure 11.4 in ourbook:

We analyze . . . a domestic increase in the money stock. . . . Thisshifts the LM curve to the right. As the domestic interest rateis constrained to remain unchanged (in the absence of a changein the foreign interest rate), we cannot move to point e′, whichwould be the new equilibrium point in a closed economy. Someother shift will have to occur. This shift occurs in the IS curve.

How does this shift come about? The open-market purchase exerts a downwardpressure on the domestic interest rate, which causes an outflow of capital. Thiscauses the exchange rate to rise, i.e., the currency devalues. (The central bank is notforced to sell its foreign currency reserves because it is not obligated to maintain the


exchange rate.) This makes domestic goods more competitive abroad, and outputrises. Monetary policy is effective, but it is a “beggar-thy-neighbor” effect. If othercountries devalue too, this will make the policy ineffective.

Now what about fiscal policy? This is shown in Figure 11.5. An outward shift ofthe IS curve due to more government expenditure puts upward pressure on the in-terest rate, which leads to capital inflows, the exchange rate falls (the home currencyappreciates), this decreases exports. I.e., due to currency appreciation, the increasedgovernment expenditures completely “crowd out” exports, so that there is no effecton Y .

3.4. Shortcomings of the Mundell-Fleming Model. This follows [dG97,p. 27–29]. First de Grauwe argues that the predictions of the model do not fittogether with the data.

Mundell-Fleming predicts that monetary expansion in a flexible exchange ratesystem leads to a permanent increase in output. Empirical evidence has massivelycontradicted this prediction.

Mundell-Fleming predicts that countries that keep a fixed exchange-rate canpermanently increase output by expansionary fiscal policies. Again, no systematicevidence has been found to sustain this prediction. If anything, a reverse correla-tion: countries experiencing increasing budget deficits have also experienced currencydepreciations in last 25 years.


Theoretical flaws:Assumes fixed wages and prices.As in the closed-economy IS/LM model, effects of continuing flows on the out-

standing stocks are not modeled. Therefore no good guidance for long-term prospects.(For instance: higher government expenditures mean an increasing government debtwhich has to be serviced. But the model does not track government debt.)

Also does not allow for rational expectations.(The following point does not come from de Grauwe:) In the version discussed

above, the model allows exchange rates to vary, but the agents always expect ex-change rates to stay constant.


4. Dornbusch Model

Extension of Mundell-Fleming which incorporates perfect foresight regardingexchange rates. Everything in logs, therefore the coefficients are elasticities:

y = −εY Rr + εY Q[p∗ + e− p] + εY Gg(410)

m− p = −εMRr + εMY y(411)

r = r∗ + ee(412)

p = φ[y − y](413)

ee = e(414)

y is output, y full employment output (exogenous), p domestic price level, p∗ foreignprice level (exogenous), m money supply (exogenous), e nominal exchange rate, ggovernment expenditures (exogenous), ee expected exchange rate. All these are inlogs. r and r∗ are domestic and foreign interest rates (no logs taken here, r∗ isexogenous).

(410) is a log-linearized form of the IS curve (404), and (411) of the LM curve(405). The model assumes perfect capital mobility and flexible exchange rates, butinstead of r = r∗ above it takes expectations about the exchange rate in considera-tion, which gives (412). (413) is the Phillips curve: prices are not allowed to makediscrete jumps, in this sense they are sticky, but they change continuously, with the

4. DORNBUSCH MODEL 167

adjustment speed determined by the output gap. (414) is perfect foresight regardingthe exchange rate.

How to solve this model? The book first solves (410) and (411) for y and r, i.e.,computes the intersection of IS and LM curves, and then plugs this into the otherequations of the model to get a system of linear differential equations for e and p. 3of the 4 coefficients have determinate signs, and an additional assumption determinesthe sign of the last coefficient as well.

The phase diagram is then given in Figure 11.16. The system has saddle pointinstability, i.e., it is unstable everywhere except on the saddle path. If there is a shock,then e jumps discontinuously to the saddle path leading to the new equilibrium.

Exercise this through: Figure 11.7 illustrates unanticipated fiscal policy. Aswe know from before, increased government expenditures leads to an appreciatedexchange rate, and therefore crowds out exports on a one-for-one basis. Old equi-librium at a0, new equilibrium a1 at same price level and income level, but lowerexchange rate. If the fiscal policy is announced, then the economy jumps from a0

to a′ at the time of announcement, then between announcement and implementa-tion gradually moves from a′ to a′′ (because now the currency is overvalued, thereare too few exports, therefore unemployment), then gradually moves from a′′ to a1

after implementation (because now the government expenditures have kicked in andunempoyment is eliminated again).


Unanticipated and permanent expansionary monetary policy gives overshooting:in Figure 11.18, old equilibrium is a0, new equilibrium a1. a1 is strictly northeastof a0 because the real exchange rate e − p remains the same. Adjustment path:economy jumps from a0 to a′ and then gradually moves to a1. Note that a′ is higherthan a1. i.e., the exchange rate overshoots.

At the end of the 70s this elegant model gave comfort to the economists: thesurprising volatility of exchange rates after the introduction of flexible exchange ratescould be the rational response of agents with perfect foresight.

CHAPTER 13

Money

1. Errors of Neoclassical Theory

I’d like to briefly discuss what I consider wrong with the neoclassical theory ofmoney. The mistake which this theory makes with money is similar to the mistake itmade with wages. Therefore, please review what I wrote at the beginning of chapter9 about the “shallow” view of markets in neoclassical economics.

This shallow view of the market makes it impossible to understand money. Neo-classical economists consider money as something that removes frictions in the mar-ket, so that transactions can go more smoothly. In Marx’s theory, money is thedevice which allows the market to perform its underlying role. What is this role?According to the labor theory of value, prices are proportional to labor content, and

169

170 13. MONEY

the market is the mediator which allocates the labor to the different branches ofproduction on an equal basis. For this allocation, money is needed as the surfacemanifestation of labor, the inner measure of value.

In Marx’s theory, capitalists make profits because workers produce more valuethan they receive. Marx stresses that the division of “value added” into profits andwages does not tell us how this value was generated, but only tells us how this valueis distributed in a class society. The source of this value is labor alone. Neoclassicaleconomics does not even have the tools to formulate this proposition. It cannot dis-tinguish between what the laborer receives and what the laborer produces, becauseit does not look beyond the market, it does not see the market at the surface ap-pearance of some underlying sphere of production. According to neoclassical theory,the wage is the contribution which the laborer makes to the value of the product.

This is why neoclassical theory cannot explain why there is money. If value istautologically defined by the distribution of the product, i.e., if one can arrive at thevalue of a product by simply adding up materials, labor, depreciation, profit, interest,and taxes, then money is not necessary. Money is needed as the institution whichensures that the valuation of the goods will indeed come from labor. A good monetarysystem is one which channels the profit motive of the capitalists into “honestly”exploiting labor, rather than trying to enrich themselves by speculation or othertricks in the circulation sphere. If the economy is going smoothly because everyone

2. FUNCTIONS OF MONEY 171

can sell what they produce and get the income they expect, then it is time for themonetary authority to make a reality check, make the distinction between valueproduced and value distributed: do prices and profits really represent surplus-value,or is the economy engaged in an inflationary pyramid scheme? And they will usemonetary policy accordingly.

2. Functions of Money

If you ask someone what money is, they will pull a dollar bill out of their walletand say “this is money.” Why is this an unsatisfactory answer?

If you ask someone what a radio is, they will point to a radio in the living roomand say: “this is a radio.” If you are not satisfied with this answer, you can openthe radio and look inside, and you see the elements which make the radio work.With dollar bills it is different. The powers of money do not come from the physicalproperties of the dollar bill. This dollar bill represents a social relation.

This is why the textbook shifts the question “what is money?” to the question“what does money do?”, i.e., what are the functions of money? Three functions arecommonly cited: (1) medium of exchange (2) medium of account (3) store of value.

The idea here is: since money is a social relation, it must come from an agreementbetween the economic agents. They agree to accept a certain object in exchange forother, valuable goods, because this agreement will benefit them.

172 13. MONEY

Here neoclassical economics cannot ask the right questions about money becauseit starts with a foreshortened view of social relations. They think social relations arereducible to individual agency, therefore the universal acceptance of money can onlybe explained if everyone benefits from it. Marx’s answer is different: people needmoney because their labors are interlinked in a very specific way. This is not theirchoice, but this is a fact of life which they are born into.

But let us continue with those functions of money which might induce people toagree on accepting a worthless piece of paper in exchange for their valuable goods.Here the role of money as medium of exchange takes first place, because the othertwo roles can also be taken by non-money commodities. Note that for Marx, thefirst function of money is measure of value, not means of circulation.

Why does barter need a general means of exchange? Neoclassical answer: be-cause this makes exchanges easier or even possible:

(1) if not every trader comes face to face with each other trader at the sametime, but if trades meet each other in pairs randomly, then relying on the doublecoincidence of wants will make trade much more cumbersome.

(2) Even under certainty and full information there are situation in which barteris not only made difficult but even made impossible without a general means ofexchange.

Medium of account:

3. OVERLAPPING GENERATIONS MODEL OF MONEY 173

It makes sense to make one commodity the numeraire. In direct barter, withoutthe intermediary of money, you have n(n− 1) different exchange ratios; with moneyyou have only n differnt prices.

Marx would say: we cannot know how people would trade unless we know howthey got possession of the things they trade. If every trader is the producer of thecommodities he or she trades, then the trades amount to a pooling of labor amongthe traders on an equal basis. Since there is really only one thing that is traded,namely, labor, it makes sense to measure everything that is traded in one commodity,which in this way becomes the representative of abstract human labor. Thereforethe “measure of value” function of money is primary.

Store of value.

3. Overlapping Generations Model of Money

This is [HVdP02, section 12.3.1]. Interesting comments about this model canalso be found in Lucas’s Nobel lecture [?, p. 672–75].

Discrete time t goes from −∞ to +∞. There is no growth, in each period thereis 1/2 young agent and 1/2 old agent, i.e., together 1 agent. Young agents get theendowment Y (they work) and old agents are retired and get a monetary transferpayment, which in real terms is Tt, from the government.

174 13. MONEY

Note that Tt is the government transfer payment in real terms, i.e., the dollaramount of transfer payments is PtTt. I find this misleading, since the governmentjust prints currency in the amount PtTt, there is no link to the purchasing powerwhen it does that.

Agents who are young in period t consume CYt and store the amount Kt of the

product. They sell the rest of the product at price Pt and keep the proceeds fromthis sale, Mt, until they are old. Their budget constraint in period t, denominatedin units of output Y , not monetary units, is

(415) Y = CYt + Kt + Mt/Pt

The things stored change their value to Kt/(1 + δ) where δ > −1. (In intergenera-tional transfers it makes sense that some goods cannot be kept in kind from youngto old.) Therefore the consumption of the old people in period t + 1 is

(416) COt+1 = Kt/(1 + δ) + Mt/Pt+1 + Tt+1.

They choose CYt , Kt and Mt to maximize their lifetime utility function

(417) V Yt = U(CY

t ) +1

1 + ρU(CO

t+1)

For the solution of this maximization it is more convenient to use mt = Mt/Pt insteadof Mt as choice variable, this is what the book is doing. One of the results of this

3. OVERLAPPING GENERATIONS MODEL OF MONEY 175

maximization is that the transfer from young to old takes place either by storing thegood or by saving money, but not both (except in the special case that both havethe same cost or return).

Assuming the parameters are such that money is used to transfer the incomeinstead of hoarding the goods themselves, this utility maximization gives rise to ademand function for real money balances

(418) Mt = Ptm(Tt+1, πt)

where πt = Pt+1−Pt

Ptis the (forward-looking) inflation rate at time t. The function

m depends on all the exogenous variables and parameters of the system, but Tt+1

and πt have been singled out here because they are endogenous, they are set to suchvalues that demand and supply of money are equalized.

Let’s therefore talk about money supply: each period, the government printsmoney and gives this newly printed money to the old generation as transfer payment.The assumption is that this newly printed money generates a uniform growth rate µfor the whole money stock: Mt+1 = (1 + µ)Mt.

The story of the model is therefore the following: every year, the governmentprints new money and gives it to the retired people. Will the retired generationbe able to use this money to buy things from the working generation? Only if theworking generation has use for this money. How can the working generation use

176 13. MONEY

this money? They can save it in order to supplement their own retirement transferreceipts in when they are old.

Solution of the model: if the utility function is logarithmic, U(x) = log x, and ifmoney qualifies for transferring wealth, i.e., π < δ, then the solution is

(419) Mt =PtY

2 + ρ + µ(1 + ρ)

i.e., real money balances are constant, the growth rate of the money stock is equalto the inflation rate.

The particulars of the solution are not as important as the fact that there is acompetitive solution. This model shows that money can be explained by its role forintergenerational wealth transfers. Instead of a co-operative solution in which thegenerations share, this is a competitive market solution. The government merelytransfers the monetary assets between the periods. It does not make any guaranteesregarding purchasing power, on the contrary, it dilutes purchasing power by printingmore money. Nevertheless the young find it in their interest to accept this money inthe sale of their product to the old.

This is not the raison d’etre of money, but it shows that the various functions ofmoney can sustain it, even if the link to the fundamentals is cut. Lipietz’s cartooncharacter running off the cliff.

4. OPTIMAL QUANTITY OF MONEY 177

4. Optimal Quantity of Money

This is no longer a model trying to explain why money exists, but it asks howmuch money should be printed. If the reason for money is to diminish transactioncosts, then lots of money should be printed, since money itself is costless. Obviously,this is not what is happening. In my mind, this “full liquidity” result is indirectproof that transaction costs cannot be the primary purpose of money (although it isusually not seen that way).

Two-period model, utility depends on consumption and real money balances:

(420) V = U(C1,M1

P1) +

11 + ρ

U(C2,M2

P2)

There is a slight inconsistency with earlier assumptions: in this model, the moneybalances at the end of the period enter the utility function, while in earlier modelsin the book, it was those at the beginning. This is how this model is set up inthe literature. Both assumptions are simplifications, one as good as the other. Ifone wanted to be more precise, one might want to use perhaps the average betweenbeginning and ending balances. As a consequence of this assumption, money held atthe end of the second period is not zero, since this money balance enters the utilityfunction.

178 13. MONEY

Production is exogenous, the same amount Y in both periods, and in each periodthe government prints money and gives it to the economic agents as a lump-sumpayment. As in the overlapping generations model, nominal money supply grows atrate µ, i.e.,

Mt −Mt−1 = µMt−1 t = 1, 2(421)

and this newly printed money is paid out as transfer payments, i.e., the governmentbudget constraint is

Mt −Mt−1 = PtTt t = 1, 2(422)

Note that Tt is again real transfer payments.The household budget identities for the two periods are therefore

PtY + Mt−1 + PtTt = PtCt + Mt t = 1, 2(423)

Households choose Ct and Mt to maximize utility subject to this budget constraint.

Problem 32. Notation: mt = Mt/Pt are the real money balances. Householdschoose those C1, m1, C2, and m2 which maximize the lifetime utility function

V = U(C1,m1) +1

1 + ρU(C2,m2)(424)


subject to the budget constraints

P1Y + M0 + P1T1 = P1C1 + P1m1(425)

P2Y + P1m1 + P2T2 = P2C2 + P2m2(426)

• a. Write down the Lagrange function and the first-order conditions and derive(12.83) and (12.84) in the book.

Answer. Lagrangian:

L = U(C1, m1) +1

1 + ρU(C2, m2) +(427)

+ λ1(P1Y + M0 + P1T1 − P1C1 − P1m1) +

+ λ2(P2Y + P1m1 + P2T2 − P2C2 − P2m2)

First-Order Conditions:

∂L∂C1

= 0 U1(C1, m1) = λ1P1(428)

∂L∂C2

= 01

1 + ρU1(C2, m2) = λ2P2(429)

∂L∂m1

= 0 U2(C1, m1) + λ2P1 = λ1P1(430)

∂L∂m2

= 01

1 + ρU2(C2, m2) = λ2P2(431)

180 13. MONEY

Equating (429) and (431) gives

U1(C2, m2) = U2(C2, m2).(432)

Solve (431) for λ2, plug this into (430), and equate with (428) to get

U1(C1, m1) = U2(C1, m1) +P1

P2

1

1 + ρU1(C2, m2)(433)

There are two markets in this economy: the goods market and the money market.Due to Walras’ law, the goods market clears if and only if the money market clears.Therefore one usually concentrates on one market, say the goods market. One has toderive the demand for goods and then determine those prices at which the demandfor goods is equal to the supply. At these prices, the money market will clear as well.

In the present model things are a little different: it is not necessary to computethe prices in order to know the quantities that clear the goods market. If one plugsthe government budget constraint (422) into the personal budget constraints (423)one gets immediately

C1 = Y C2 = Y(434)

Prices are therefore only determined by the market-clearing conditions for the moneymarket. Due to Walras’s law, if the money market clears, the goods market clears


as well, therefore we can use (434). If we plug (434) into the first-order conditionsand furthermore, since (421) gives Mt = Mt+1

1+µ , multiply each term of (433) by eitherm1 = M1

P1or by M2

(1+µ)P1, we get

U1(Y, m2) = U2(Y, m2).(435) (U1(Y,m1)− U2(Y, m1)

)m1 =

m2

(1 + ρ)(1 + µ)U1(Y, m2)(436)

These equations allow us to solve for the equilibrium real money balances. This isa recursive system: first determine m2 from (435), then plug this m2 into (436) inorder to get m1. You are going backward in time.

Now let’s make the model a little more specialized: utility is “additively separa-ble”

(437) U(Ct,mt) = u(Ct) + v(mt).

Marginal utility of consumption is declining but positive:

u′(Ct) > 0 u′′(Ct) < 0(438)

182 13. MONEY

But with money balances, satiation is possible:

v′(mt) > 0 for 0 < mt < m∗(439)

v′(mt) = 0 for mt = m∗(440)

v′(mt) < 0 for mt > m∗(441)

Plug (437) into (435) and (436):

u′(Y ) = v′(m2)(442) (u′(Y )− v′(m1)

)m1 =

m2

(1 + ρ)(1 + µ)u′(Y )(443)

This has to be solved for m1 and m2. Graphical solution in Figure 12.8 on p. 343.The “full liquidity” result: The optimal real money balances, which the economic

agents wish to hold in the two periods, depend on taste and endowment parameters.Write them as

m∗1 = m∗

1(ρ, Y, µ) m∗2 = m∗

2(ρ, Y, µ)(444)

In the separable case, m∗2 does not depend on µ, write it as m∗

2(ρ, Y ). (It also doesnot depend on ρ, but the book still carries ρ along as argument.) The utility which


the economic agents achieve with these money balances are

(445) V = u(Y ) + v(m∗

1(ρ, Y, µ))

+1

1 + ρ

[u(Y ) + v

(m∗

2(ρ, Y ))]

A utilitarian utility maker chooses that growth rate µ which maximizes this utility.First-order condition

∂V

∂µ= 0 v′

(m∗

1(ρ, Y, µ))∂m∗

1

∂µ= 0(446)

This simplifies to

(447) v′(m∗

1(ρ, Y, µ))

= 0

i.e., the desired money stock is that at which marginal utility of money is zero, asspecified in (440).

CHAPTER 14

New Keynesian Economics

Look at the encyclopedia entry by N. Gregory Mankiw at

www.econlib.org/library/Enc/NewKeynesianEconomics.html.

Here is a printout of this essay:New Keynesian economics is the school of thought in modern macroeconomics

that evolved from the ideas of John Maynard Keynes. Keynes wrote The GeneralTheory of Employment, Interest, and Money in the thirties, and his influence amongacademics and policymakers increased through the sixties. In the seventies, however,new classical economists such as Robert Lucas, Thomas J. Sargent, and Robert Barrocalled into question many of the precepts of the Keynesian revolution. The label “new

185

http://www.econlib.org/library/Enc/NewKeynesianEconomics.html

186 14. NEW KEYNESIAN ECONOMICS

Keynesian” describes those economists who, in the eighties, responded to this newclassical critique with adjustments to the original Keynesian tenets.

The primary disagreement between new classical and new Keynesian economistsis over how quickly wages and prices adjust. New classical economists build theirmacroeconomic theories on the assumption that wages and prices are flexible. Theybelieve that prices “clear” markets—balance supply and demand—by adjusting quickly.New Keynesian economists, however, believe that market-clearing models cannot ex-plain short-run economic fluctuations, and so they advocate models with “sticky”wages and prices. New Keynesian theories rely on this stickiness of wages and pricesto explain why involuntary unemployment exists and why monetary policy has sucha strong influence on economic activity.

A long tradition in macroeconomics (including both Keynesian and monetaristperspectives) emphasizes that monetary policy affects employment and productionin the short run because prices respond sluggishly to changes in the money supply.According to this view, if the money supply falls, people spend less money, and thedemand for goods falls. Because prices and wages are inflexible and don’t fall imme-diately, the decreased spending causes a drop in production and layoffs of workers.New classical economists criticized this tradition because it lacked a coherent theo-retical explanation for the sluggish behavior of prices. Much new Keynesian researchattempts to remedy this omission.

14. NEW KEYNESIAN ECONOMICS 187

Menu Costs and Aggregate-Demand ExternalitiesOne reason that prices do not adjust immediately to clear markets is that ad-

justing prices is costly. To change its prices, a firm may need to send out a newcatalog to customers, distribute new price lists to its sales staff, or in the case of arestaurant, print new menus. These costs of price adjustment, called “menu costs,”cause firms to adjust prices intermittently rather than continuously.

Economists disagree about whether menu costs can help explain short-run eco-nomic fluctuations. Skeptics point out that menu costs usually are very small. Theyargue that these small costs are unlikely to help explain recessions, which are verycostly for society. Proponents reply that small does not mean inconsequential. Eventhough menu costs are small for the individual firm, they could have large effects onthe economy as a whole.

Proponents of the menu-cost hypothesis describe the situation as follows. Tounderstand why prices adjust slowly, one must acknowledge that changes in priceshave externalities—that is, effects that go beyond the firm and its customers. Forinstance, a price reduction by one firm benefits other firms in the economy. When afirm lowers the price it charges, it lowers the average price level slightly and therebyraises real income. (Nominal income is determined by the money supply.) Thestimulus from higher income, in turn, raises the demand for the products of all firms.


This macroeconomic impact of one firm’s price adjustment on the demand for allother firms’ products is called an “aggregate-demand externality.”

In the presence of this aggregate-demand externality, small menu costs can makeprices sticky, and this stickiness can have a large cost to society. Suppose that GeneralMotors announces its prices and then, after a fall in the money supply, must decidewhether to cut prices. If it did so, car buyers would have a higher real income andwould, therefore, buy more products from other companies as well. But the benefitsto other companies are not what General Motors cares about. Therefore, GeneralMotors would sometimes fail to pay the menu cost and cut its price, even thoughthe price cut is socially desirable. This is an example in which sticky prices areundesirable for the economy as a whole, even though they may be optimal for thosesetting prices.

The Staggering of PricesNew Keynesian explanations of sticky prices often emphasize that not every-

one in the economy sets prices at the same time. Instead, the adjustment of pricesthroughout the economy is staggered. Staggering complicates the setting of pricesbecause firms care about their prices relative to those charged by other firms. Stag-gering can make the overall level of prices adjust slowly, even when individual priceschange frequently.


Consider the following example. Suppose, first, that price setting is synchronized:every firm adjusts its price on the first of every month. If the money supply andaggregate demand rise on May 10, output will be higher from May 10 to June 1because prices are fixed during this interval. But on June 1 all firms will raise theirprices in response to the higher demand, ending the three-week boom.

Now suppose that price setting is staggered: Half the firms set prices on the firstof each month and half on the fifteenth. If the money supply rises on May 10, thenhalf the firms can raise their prices on May 15. Yet because half of the firms will notbe changing their prices on the fifteenth, a price increase by any firm will raise thatfirm’s relative price, which will cause it to lose customers. Therefore, these firms willprobably not raise their prices very much. (In contrast, if all firms are synchronized,all firms can raise prices together, leaving relative prices unaffected.) If the May 15price setters make little adjustment in their prices, then the other firms will makelittle adjustment when their turn comes on June 1, because they also want to avoidrelative price changes. And so on. The price level rises slowly as the result of smallprice increases on the first and the fifteenth of each month. Hence, staggering makesthe price level sluggish, because no firm wishes to be the first to post a substantialprice increase.

Coordination Failure


Some new Keynesian economists suggest that recessions result from a failure ofcoordination. Coordination problems can arise in the setting of wages and pricesbecause those who set them must anticipate the actions of other wage and pricesetters. Union leaders negotiating wages are concerned about the concessions otherunions will win. Firms setting prices are mindful of the prices other firms will charge.

To see how a recession could arise as a failure of coordination, consider thefollowing parable. The economy is made up of two firms. After a fall in the moneysupply, each firm must decide whether to cut its price. Each firm wants to maximizeits profit, but its profit depends not only on its pricing decision but also on thedecision made by the other firm.

If neither firm cuts its price, the amount of real money (the amount of moneydivided by the price level) is low, a recession ensues, and each firm makes a profit ofonly fifteen dollars.

If both firms cut their price, real money balances are high, a recession is avoided,and each firm makes a profit of thirty dollars. Although both firms prefer to avoida recession, neither can do so by its own actions. If one firm cuts its price while theother does not, a recession follows. The firm making the price cut makes only fivedollars, while the other firm makes fifteen dollars.

The essence of this parable is that each firm’s decision influences the set ofoutcomes available to the other firm. When one firm cuts its price, it improves the


opportunities available to the other firm, because the other firm can then avoid therecession by cutting its price. This positive impact of one firm’s price cut on the otherfirm’s profit opportunities might arise because of an aggregate-demand externality.

What outcome should one expect in this economy? On the one hand, if eachfirm expects the other to cut its price, both will cut prices, resulting in the preferredoutcome in which each makes thirty dollars. On the other hand, if each firm expectsthe other to maintain its price, both will maintain their prices, resulting in the inferiorsolution, in which each makes fifteen dollars. Hence, either of these outcomes ispossible: there are multiple equilibria.

The inferior outcome, in which each firm makes fifteen dollars, is an exampleof a coordination failure. If the two firms could coordinate, they would both cuttheir price and reach the preferred outcome. In the real world, unlike in this parable,coordination is often difficult because the number of firms setting prices is large. Themoral of the story is that even though sticky prices are in no one’s interest, pricescan be sticky simply because people expect them to be.

Efficiency WagesAnother important part of new Keynesian economics has been the development

of new theories of unemployment. Persistent unemployment is a puzzle for economictheory. Normally, economists presume that an excess supply of labor would exert a


downward pressure on wages. A reduction in wages would, in turn, reduce unem-ployment by raising the quantity of labor demanded. Hence, according to standardeconomic theory unemployment is a self-correcting problem.

New Keynesian economists often turn to theories of what they call efficiencywages to explain why this market-clearing mechanism may fail. These theories holdthat high wages make workers more productive. The influence of wages on workerefficiency may explain the failure of firms to cut wages despite an excess supply oflabor. Even though a wage reduction would lower a firm’s wage bill, it would also–ifthe theories are correct—cause worker productivity and the firm’s profits to decline.

There are various theories about how wages affect worker productivity. Oneefficiency-wage theory holds that high wages reduce labor turnover. Workers quitjobs for many reasons—to accept better positions at other firms, to change careers,or to move to other parts of the country. The more a firm pays its workers, thegreater their incentive to stay with the firm. By paying a high wage, a firm reducesthe frequency of quits, thereby decreasing the time spent hiring and training newworkers.

A second efficiency-wage theory holds that the average quality of a firm’s workforce depends on the wage it pays its employees. If a firm reduces wages, the bestemployees may take jobs elsewhere, leaving the firm with less productive employeeswho have fewer alternative opportunities. By paying a wage above the equilibrium


level, the firm may avoid this adverse selection, improve the average quality of itswork force, and thereby increase productivity.

A third efficiency-wage theory holds that a high wage improves worker effort.This theory posits that firms cannot perfectly monitor the work effort of their em-ployees and that employees must themselves decide how hard to work. Workers canchoose to work hard, or they can choose to shirk and risk getting caught and fired.The firm can raise worker effort by paying a high wage. The higher the wage, thegreater is the cost to the worker of getting fired. By paying a higher wage, a firminduces more of its employees not to shirk and, thus, increases their productivity.

Policy ImplicationsBecause new Keynesian economics is a school of thought regarding macroeco-

nomic theory, its adherents do not necessarily share a single view about economicpolicy. At the broadest level new Keynesian economics suggests—in contrast to somenew classical theories—that recessions do not represent the efficient functioning ofmarkets. The elements of new Keynesian economics, such as menu costs, staggeredprices, coordination failures, and efficiency wages, represent substantial departuresfrom the assumptions of classical economics, which provides the intellectual basis foreconomists’ usual justification of laissezfaire. In new Keynesian theories recessionsare caused by some economy-wide market failure. Thus, new Keynesian economics


provides a rationale for government intervention in the economy, such as counter-cyclical monetary or fiscal policy. Whether policymakers should intervene in practice,however, is a more difficult question that entails various political as well as economicjudgments.

About the AuthorN. Gregory Mankiw is a professor of economics at Harvard University.Further Reading: [MR91], [Rot87].Copyright of the encyclopedia 1993, 2002 David R. Henderson. All rights re-

served.

Problem 33. What is New Keynesian Economics? Explain menu costs, stag-gering of prices, coordination failure, and efficiency wages. What are the policyimplications of New Keynesian Economics?

Bibliography

[Clo65] R. Clower. The Keynesian counter-revolution: A theoretical appraisal. In F. H. Hahnand F. P. R. Brechling, editors, The Theory of Interest Rates. Macmillan, London, 1965.66

[dG97] Paul de Grauwe. Paradigms of macroeconomic policy for the open economy. In Macroe-conomic Policy in Open Systems, number 5 in Handbook of Comparative Economic

Policies, chapter 2, pages 14–54. Greenwood Press, Westport and London, 1997. 161,163, 164

[Dor69] Robert Dorfman. An economic interpretation of optimal control theory. American Eco-

nomic Review, 59(5):817–831, December 1969. Jstor. 54[Dor70] Robert Dorfman. An economic interpretation of optimal control theory (errata). Amer-

ican Economic Review, 60(3):524, June 1970. Jstor. 54

[Dun60] T. J. Dunning. Trades’ Unions and Strikes: Their Philosophy and Intention. London,1860. 116

195

http://links.jstor.org/sici?sici=0002-8282%28196912%2959%3A5%3C817%3AAEIOOC%3E2.0.CO%3B2-O

http://links.jstor.org/sici?sici=0002-8282%28197006%2960%3A3%3C524%3AAEIOOC%3E2.0.CO%3B2-D

196 BIBLIOGRAPHY

[Fle62] J. M. Fleming. Domestic financial policies under fixed and under flexible exchange rates.International Monetary Fund Staff Papers, pages 369–79, 1962. 157

[Hay82] F. Hayashi. Tobin’s marginal q and average q: A neoclassical interpretation. Economet-rica, 50:213–224, 1982. 59

[HVdP02] Ben J. Hejdra and Frederick Van der Ploeg. Foundations of Modern Macroeconomics.Oxford University Press, 2002. 7, 11, 17, 27, 41, 46, 53, 59, 65, 97, 98, 173

[Kal72] Michal Kalecki. Selected Essays on the Economic Growth of the Socialist and the MixedEconomy. Cambridge University Press, 1972. 154

[Mal77] E. Malinvaud. The Theory of Unemployment Reconsidered. Basil Blackwell, 1977. 65,68

[Mal85] E. Malinvaud. The Theory of Unemployment Reconsidered. Basil Blackwell, second edi-tion, 1985. 65, 68

[Mea51] J. Meade. The Theory of International Economic Policy, Vol. I: The Balance of Pay-ments. Oxford University Press, Oxford, 1951. 157

[MR91] N. Gregory Mankiw and David Romer, editors. New Keynesian Economics. 1991. 2

volumes. 194[Mun68] Robert A. Mundell. International Economics. Macmillan, New York, 1968. 157[Mut61] John F. Muth. Rational expectations and the theory of price movements. Econometrica,

29:315–335, 1961. 41[Pis00] Christopher A. Pissarides. Equilibrium Unemployment Theory. MIT Press, Cambridge,

MA and London, second edition, 2000. 119, 131[Rot87] Julio Rotemberg. The New Keynesian Microfoundations. NBER Macroeconomics An-

nual 1987. 1987. edited by Stanley Fischer. 194

[Sar87] Thomas J. Sargent. Macroeconomic Theory. Academic Press, second edition, 1987. 15

BIBLIOGRAPHY 197

[She83] Steven Sheffrin. Rational Expectations. Cambridge University Press, 1983. 143[SW76] Thomes J. Sargent and Neil Wallace. Rational expectations and the theory of economic

policy. Journal of Monetary Economics, 2:169–183, 1976. 41, 46[Wil83] John Williamson. The Open Economy and the World Economy. Basic Books, New York,

1983. 157

Class Notes 7008 Macroeconomics Spring 2003 Hans G. Ehrbarcontent.csbs.utah.edu/~ehrbar/macrec.pdf · Class Notes 7008 Macroeconomics Spring 2003 Hans G. Ehrbar Economics Department,

Documents