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Master Economics and Public PolicyECO 553. Economic Growth1

Lecture 5Public policies in the neoclassical growth model

Pierre Cahuc

Winter 2013-2014

1http://sites.google.com/site/eco553x/1 / 47

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1. Public debt

I Let us assume that there is a government that must …nancepublic expenditures g (t ) 0 at time t 0

I The economy is competitive (same model as in lecture 4)

I For the sake of simplicity, public expenditures are not in theutility function of the households

I The valuation of public expenditure is implicit (public good,e.g. justice, police, army...)

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1. Public debt

I The intertemporal budget constraint of the household is

Z ∞

0c (t )e

R t 0 r (x )dx dt = h0 + a0 G 0 (2)

I h0 + a0 G 0 is the wealth of the householdI h0 =

R ∞

0 w (t )e R t

0 r (x )dx dt stands for the human capitalcomponent of wealth

I a0 is the value of the assets at time t = 0

I

G 0 =R ∞

0 g (t )e R t

0

r (x )dx

dt is the present value of taxes.I Increases in taxes reduce the wealth of the household

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1. Public debt

I The program of the household is

maxfc (t )0,t 0g

Z +∞0

u (c (t ))e ρt dt

subject to

ȧ(t ) = w (t ) + r (t )a(t ) g (t ) c (t )

a(0) = a0

limt !∞ a(t )e R t

0 r (x )dx

= 0

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1. Public debt

I The solution of the maximization problem of the householdyields the Keynes-Ramsey rule

ċ (t )

c (t ) =

1

σ (c (t ))[r (t ) ρ]

and the transversality condition

limt !∞

a(t )u 0(c (t ))e ρt = 0

as in the model without tax

I The behavior of the …rm is the same as in the model without

tax:r (t ) = f 0(k (t )) δ (3)

w (t ) = f (k (t )) k (t )f 0(k (t )) (4)

where k (t ) = K (t )/L(t ).

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1. Public debt

I Equilibrium on the labor market and on the capital marketimplies:

L(t ) = 1, a(t ) = k (t )

I The equality a(t ) = k (t ), the demand for labor and forcapital (eq. (4) and (3)) allow us to write the instantaneousbudget constraint of the household as follows

k̇ (t ) = f (k (t )) δk (t ) c (t ) g (t )

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1. Public debt

I Finally, the competitive equilibrium path of fc (t ), k (t ), t 0g is de…ned by

k̇ (t ) = f (k (t )) δk (t ) c (t ) g (t )

limt !∞

k (t )u 0(c (t ))e ρt = 0

ċ (t )

c (t ) =

1

σ (c (t ))[f 0(k (t )) δ ρ]

k (0) = k 0 > 0

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1. Public debt

I Let us look at the steady state equilibrium

I Assume that g (t ) = g > 0

I Steady state equilibrium value of (k , c ) is de…ned by

f 0(k ) = ρ + δc = f (k ) δk g

I Public expenditures crowd out private consumption

I Public expenditures have no e¤ect on the equilibrium steadystate capital stock

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1. Public debt

I The Solow model yields a di¤erent prediction

I In the Solow model, the law of motion of k (t ) is

k̇ (t ) = sf (k (t )) δk (t ) g (t )

where s 2 (0, 1) denotes the saving rate of households

I In steady state:sf (k ) = δk + g

I The Solow model implies that k and then y decrease with g

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I

In the neoclassical model, the saving rate increases whenpublic expenditures are increased because the marginal utilityof consumption does not depend on public expenditures

I In the neoclassical model, the steady state equilibrium stockof capital does not depend on public expenditures

I The transitional dynamics of k (t ) induced by anunanticipated shock which increases public expenditures from0 to g > 0, is represented on the next …gure

I k (t ) = k if the economy is in the steady state initially

I If the economy is not in steady state initially, the transitionaldynamics of k (t ) depend on the properties of theinstantaneous utility function

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1. Public debt

k

c

c*’

dc/dt=0

dk/dt=0

k*

c*

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1. Public debt

I Let us now assume that there are public de…cits …nanced withpublic debt

I The lump sum tax paid by the representative household τ (t )is not any more equal to g (t )

I The public debt is denoted by b (t )I The instantaneous budget constraint of the government is

ḃ (t ) = g (t ) τ (t ) + r (t )b (t )

I ḃ (t ) is the current de…cit of the budget of the government

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1. Public debt

I One must also impose a No-Ponzi game condition to thegovernment

limt !∞

b (t )e R

t 0 r (x )dx = 0

I The intertemporal budget constraint of the government is

b 0 =Z ∞

0 [τ (t ) g (t )] e R

t

0 r (x )dx

dt (5)

I This equation shows that the discounted values of expenditures and taxes must reimburse the initial debt

I The instantaneous budget of the representative household is

the same as before (see eq. (1)) except that τ (t ) issubstituted for g (t )

ȧ(t ) = w (t ) + r (t )a(t ) τ (t ) c (t )

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1. Public debt

I But the capital market equilibrium is

a(t ) = k (t ) + b (t )

because the assets of the household are made of

I the private bonds whose counterpart is the stock of capital of the …rms

I the public bonds whose counterpart is the public debt

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1. Public debt

I

Therefore, the intertemporal budget constraint of thehousehold can be written asZ ∞

0c (t )e

R t 0 r (x )dx dt = h0 + k 0 + b 0 T 0 (6)

I h0 + b 0 + k 0 T 0 is the wealth of the householdI h0 =

R ∞

0 w (t )e R t

0 r (x )dx dt stands for the human capitalcomponent of the wealth

I b 0 is the value of the public bonds at time t = 0

I k 0 is the stock of capital of the …rms at time t = 0I T 0 =

R ∞

0 τ (t )e

R t 0 r (x )dx dt is the present value of taxes

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1 P bli d b

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1. Public debt

I Using the intertemporal budget constraint of the government

(5):b 0 =

Z ∞

0[τ (t ) g (t )] e

R t 0 r (x )dx dt

we can write the intertemporal budget constraint of thehousehold (6):

Z ∞

0c (t )e

R t 0 r (x )dx dt = h0 + k 0 + b 0 T 0

as follows:

Z ∞

0c (t )e

R t

0 r (x )dx dt = h0 + k 0 G 0 (7)

which is identical to equation (2) with a0 = k 0.

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1 P bli d b

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1. Public debt

I Therefore, the budget constraint of the household isindependent of the level of public debt

I The impact of public expenditures, fg (t ), t 0g on theconsumption of households and on the evolution of the stock

of capital is independent of the path of public debtfb (t ), t 0g

I The saving behavior of the household neutralizes the e¤ect of the public debt

I Neutrality of public debt, Ricardo equivalence, Barro

equivalence

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1 P bli d bt

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1. Public debt

I The neutrality of public debt relies on strong assumptions:

I perfectly rational householdsI optimization over an in…nite time horizonI perfect credit marketI non distortionary taxation

I Actually, public debt is not neutral, but empirical studies …ndthere is a negative correlation between public saving andprivate saving2

2see for instance Luiz de Mello, Per Mathis Kongsrud and Robert Price,2004, Saving Behaviour and the E¤ectiveness of Fiscal Policy , OCDE,

Economic Department paper, 397.21 / 47

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2. Distortionary taxation

I Until now, it has been assumed that taxes did not depend on:

1. capital income2. labor income

I Study of distortionary taxation on labor income:

I the distortion is empirically importantI can be studied with models that include labor supply

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2 Disto tio a ta atio

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I Measure of distortion: tax wedge

I Let W and P f respectively be the nominal wage received byan employee and the producer price index.

I t f the average rate of mandatory deductions from wages

borne by …rms

I The real labor cost for the employer is written:

w f = W (1 + t f )

P f

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2 Distortionary taxation

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I Denote by t c and t e respectively the average rate of indirecttaxes on consumption and the average rate at which earnedincome is taxed, net of bene…ts received

I Let P c represent the consumer price index exclusive of consumption taxes.

I The purchasing power of an employee takes the form:

w e = W (1 t e )

P c (1 + t c )

Eliminating the nominal wage W between the expressions of

w e and w f ,

we get:

w f = τ w e with τ = (1 + t c )(1 + t f )

(1 t e )

P c

P f

(8)

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I The term τ de…nes the wedge ; it measures the ratio betweenthe cost of labor borne by the employer and the purchasingpower of wages.

I The wedge has two components.

1. (P c /P f ),

which is in‡uenced by the price of imports, becauseP c comprises import prices, whereas the producer price indexonly comprises prices of domestic goods (which can howeverbe indirectly in‡uenced by import prices).

2. The tax wedge, which hinges on the tax rates t c , t e and t f .

I Henceforth, we will focus only on the tax wedge by setting theratio (P c /P f ) equal to one.

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Tax wedge around 2005. The black bar represents the tax wedge without

consumption taxes. Source: OCDE

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Tax wedge around 2005. The black bar represents the tax wedge without

consumption taxes. Source: OCDE

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Direct taxes on earned income (income tax plus employees’and

employers’contributions), for a single person with no children paid at 100

percent of the average wage.

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I There are huge cross country di¤erences

I Edward Prescott, 2004, “Why Do Americans Work So Much

More than Europeans.", Federal Reserve Bank of Minneapolis,issue July, pp. 2-13.

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Taxes on labor income and weekly hours worked per individuals in the

age range 15-64 in 1970-1974 and 1993-1996. Source: Prescott (2004)

E¤ective marginal tax rate = (t c + t e + t f )/(1 + t c ) () fraction of

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Taxes on labor income and annual hours worked in 15 OECD countries

over the period 1970-2010. Each dot corresponds to a year-country

observation. Source McDaniel (2011 and www.caramcdaniel.com) and

OECD.

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I In order to show that di¤erences in hours worked acrosscountries may re‡ect di¤erences in taxation, Prescott uses amodel of intertemporal labor supply

I We present here a static version that highlights the main

thrust of the argument.I The static version corresponds to the steady state of the

neoclassical growth model

I We shall …rst remind the issue of tax incidence and then

analyze the impact of taxes on hours worked

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y

I In steady state, we have shown that F K (K , L) = r + δ, withr = ρ, which implies f 0(k ) = ρ + δ where k = K /L

I Therefore we get

w e = F L(k , 1)T f T e

where k is determined by f 0(k ) = ρ + δ.

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I Equationw e = F L(k , 1)T f T e

shows two important properties

1. First, the impact of taxes on the net wage is identical, whetherit is the employer or the employee who is paying taxes to the…sc.

2. Second, a $1 tax on labor induces a $1 decrease in the netwage.

I This result holds good in a framework where the marginalproductivity of labor is constant, so that labor demand isin…nitely elastic with respect to the wage.

I

In steady state with constant returns to scale, the marginalproductivity of labor is independent of the quantity of labor Lbecause …rms adjust the stock of capital to keep thecapital/labor ratio k constant when the quantity of laborchanges.

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I It should be noted that empirical evaluations of the incidence

of taxes generally …nd that changes in taxes have a strongimpact on net wages, which move in the direction opposite tothat of the taxes.

I These …ndings suggest that using a model where a $1 tax onlabor induces a $1 decrease in net wage can be an acceptableapproximation in the long run

I This is certainly one of the reasons why the theory of optimaltaxation, stemming from the seminal paper of Mirrlees (1971)makes, in most cases, such an assumption (see Piketty and

Saez, 2013)I Henceforth, we will assume, without loss of generality, that

F L(k , 1) = 1 for the sake of simplicity.

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I The preferences of the representative individual are describedby the utility function:

U = log c + α log(1 h)

where c and h designate consumption and hours worked, ormore precisely the proportion of disposable time dedicated to

working.I The parameter α > 0 speci…es the value of leisure time for the

household

I Each unit of labor produces one unit of good, which implies,

together with the zero pro…t condition, that the wage is equalto 1, assuming that …rms pay no tax.

I Taxes on consumption and on labor earnings are assumed tobe proportional for the sake of simplicity.

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I The budget constraint of the consumer is

(1 + τ c )c = h(1 τ h ) + T (9)

where τ c and τ h stand respectively for consumption tax rateand the rate of tax on labor earnings, and T denotes lumpsum transfers from the government.

I The budget constraint (9) gives the value of c as a function of h.

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I Carrying this value into the utility function U and derivingwith respect to h, we …nd that the optimal value of h veri…es:

(1 τ h)

h(1 τ h) + T α

1

1 h = 0 ()

1τ h1+τ c

c = α

1

1 h

I This equation may also be written:

(1 τ ) = αc

1 h (10)

where

τ = τ c + τ h

1 + τ c (11)

is the e¤ective marginal tax rate on labor income, which is thefraction of labor income that is extracted in the form of taxes.

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I The budget constraint of the government implies that lump

sum transfers T are equal to tax receipts, which implies thatT = τ c c + hτ h.

I Using the budget constraint (9) of the household, we …nd thatc = h, i.e. that consumption equals labor income.

I Substituting this equality into (10), we get the equilibriumnumber of hours worked:

h = 1α

1τ + 1. (12)

I This equation shows that the duration of hours worked doesdecrease with the e¤ective marginal tax rate.

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I Prescott computes the average across-country value of τ andsets the value of α equal to 1.54, in order to match theaverage number of weekly hours worked in the model with theactual average value for the G7 countries.

I This allows him to argue that across-country variation in taxesexplains most of the di¤erences in hours worked.

I A question that arises is whether the elasticity of labor supplyimplied by the calibrated model is compatible with the usual

microestimates of the elasticity of labor supply.

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I The literature distinguishes 3 types of labor supply elasticity1. Marshall, or uncompensated

I max U subject to the budget constraintI income and substitution e¤ects

2. Hicks, or compensatedI min expenditure subject to U Ū I substitution e¤ects only

3. Frisch

I constant marginal utility of wealthI temporary changes in labor income

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I In the model of Prescott,I permanent changes in taxesI compensated by changes in transfers, implying that changes in

taxes have no income e¤ects.

I

Hicksian elasticity of labor supplyI Equation (12) allows us to compute the Hicksian elasticity of

labor supply with respect to net labor income (1 τ ), equalto:

dlog h

dlog(1 τ ) =

α

α + 1 τ .

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I In the data of Prescott,I the average value of 1 τ amounts to 0.53.I α equal to 1.54.I Accordingly, the Hicksian elasticity of labor supply predicted by

the model is about 0.74

I This is only a little greater than usual estimated values of theHicksian elasticity which is about 0.5

I Hence the model …ts the data surprisingly well in this dimension

I Results on Frischian elasticity are less convincing (see tutorial)

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I What are the lessons of this exercise?

I Prescott instructs us that di¤erences between countries in thetaxation of labor income can explain large di¤erences in hoursworked, for verisimilar values of the Hicksian elasticity of labor

supply.I However, the macro elasticity of hours worked depends on the

composition of the population, which can comprisedemographic groups with very heterogeneous behaviors, asstressed by Blundell et al. (2013).

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I In practice , the impact of taxes on hours worked depends onthe bundle of features of each separate …scal system, for thereis a wide range of average and marginal rates varying withamount and kind of income, and with types of households andtheir share in the population.

I In light of this, it would be erroneous to conclude that acountry with a higher e¤ective marginal tax rate on laborincome, measured as it is in Prescott’s type of study – that is,as an average value – has a tax system that is necessarily

more detrimental to the supply of hours worked.

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References( )

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Blundell, R., Bozio, A. and Laroque, G., (2013), “Extensive and intensive

margins of labour supply”, Fiscal Studies, 34(1), pp. 1-29.

Chetty R., Guren, A., Manoli, D. and Weber, A. (2011), “Are Micro and

Macro Labor Supply Elasticities Consistent? A Review of Evidence on the

Intensive and Extensive Margins”, American Economic Review, Papers

and Proceedings, 101, pp. 471-475.

McDaniel, C. (2011), "Forces Shaping Hours Worked in the OECD,

1960-2004", American Economic Journal: Macroeconomics, 3(4), pp.27-52.

Mirrlees, J. (1971), “An Exploration in the Theory of Optimum Income

Taxation”, Review of Economic Studies, 38, pp. 175-208.

Prescott, E. (2004), "Why do Americans work so much more than

Europeans?", Federal Reserve Bank of Minneapolis Quarterly Review,28(1), pp. 2–13.

Piketty, T. and Saez, E., (2013), "Optimal Labor Income Taxation", in

Handbook of Public Economics, edited by Auerbach, A., Chetty, R.,

Feldstein, M. and Saez, E., Volume 5, pp. 391-474

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