Lecture Notes EC3102 06 Two Period Model Inv Handout

8/2/2019 Lecture Notes EC3102 06 Two Period Model Inv Handout

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EC3102: Lecture NotesTopic 06: Two-Period Model

(with Investment)

Aamir Raque Hashmi

National U of Singapore

February 13, 2012

Hashmi (National U of Singapore) Two-Period Model (Inv) February 13, 2012 1 / 63


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Outline

Investment decision by rms

Work-leisure decision by consumers

Derivation of output supply curveDerivation of output demand curve

Complete model of demand and supply (without money)

Eects of various shocks on the model economy



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Real Intertemporal Model

Current and future periods

Representative consumer

consumption/saving decision

work/leisure decision

Representative rm hires labor and invests in the current period andhires labor in the future period

Government spends and taxes in present and future and borrows on

the credit market



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Equation 10.1

Consumers current-period budget constraint:

C+ SP = w(h l) + T

Y w(h l) +



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Equation 10.2

Consumers future-period budget constraint:

C0 = w0 h l0+ 0 T0 + (1 + r) SPY0 w0

h l0

+ 0



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Equation 10.3

Consumers life-time budget constraint:

C+C0

1 + r= w(h l) + T+

w0 (h l0) + 0 T0

1 + ror

C+C0

1 + r= YT+

Y0 T0

1 + r

The choice variables for the consumer are: C, C0, I and I0



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Equation 10.4

Consumers current-period marginal condition:

MRSl,C = w



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Current-Period Marginal Condition

l

C

e

I

2423

w

Figure: Slope of the budget constraint is w & of the indierence curve is MRSl,C



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Equation 10.5

Consumers future-period marginal condition:

MRSl0,C0 = w0



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Equation 10.6

Consumers intertemporal marginal condition:

MRSC,C0 = 1 + r



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Consumers Current Labor Supply

Current labor supplied increases with the real wage (substitutioneects are assumed to dominate income eects)

S.E.: With the same sacrice in leisure, the consumer can buy moreconsumption. Consumpiton has become cheaper relative to leisure.Hence the consumer will buy more consumption and less leisureI.E.: The consumer feels richer and buys more of both consumptionand leisure



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Figure 10.1: Representative Consumers Current LaborSupply Curve



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Current labor supplied increases with an increase in the real interest

rate, through an intertemporal substitution eectS.E.: Tomorrows leisure becomes cheaper (because price of

tomorrows leisure is w0

1+r) so the consumer would like to have moreleisure tomorrow and less leisure today. Less leisure today means morework today.

I.E.: It depends on whether the consumer is a borrower or a lender. Aborrower (with C> w(h l)) will feel poorer and would like to reduceboth l and l0 and hence her labor supply will increase. A lender (withC< w(h l)) will feel richer and would like to increase both l and l0

and hence her labor supply will decrease.

We assume that the substitution eect is stronger than the incomeeect for the lender.

With this assumption, an increase in r will increase current laborsupply.



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Figure 10.2: Increase in the Real Interest Rate



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An increase in life-time wealth (e.g. a fall in taxes) reduces laborsupply



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Figure 10.3: Increase in Lifetime Wealth


C C D d f C i G d


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Consumers Current Demand for Consumption Goods

As current income (y) increases, current consumption (c) increases

The same relationship holds at the aggregate level


Fi 10 4 C i I i h I


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Figure 10.4: Consumption Increases with Income


I i h R l I R


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Increase in the Real Interest Rate

As the real interest rate increases, current consumption certainlydecreases for the borrowers

If we assume that the substitution eect is stronger than the incomeeect then the current consumption also decreases for the lenders


Fi 10 5 I i th R l I t t R t


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Figure 10.5: Increase in the Real Interest Rate


Fi 10 6 I i Lif ti W lth


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Figure 10.6: Increase in Lifetime Wealth


Equation 10 7


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Equation 10.7

Firms current-period production function:

Y = zF(K, N)


Equation 10 8


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Equation 10.8

Firms future-period production function:

Y0 = z0FK0, N0


Equation 10 9


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Equation 10.9

Evolution of the rms capital stock:

K

0

= (1

d) K+ I

where I (upper-case i) is investment and d is the depreciation rate ofcapital


Equation 10 10


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Equation 10.10

Firms current-period prots:

= YwN I


Equation 10 11


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Equation 10.11

Firms future-period prots:

0 = Y0 w0N0 + (1 d) K0

The capital stock is K at the beginning of the rst period,K0 = (1 d) K+ I at the end of the rst period and (1 d) K0 atthe end of the second period


Equation 10 12


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Equation 10.12

The rm maximizes the present value of prots:

V = +0

1 + r,

which can be written in an extended form as:

V = maxN, N0, I

(zF(K, N)wN I+ 11+r"

z0F

0@(1 d) K+ I

| { z } K0, N0

1Aw0N0+

(1 d)

0@(1 d) K+ I | { z }

K0

1A#).


The Firms Labor Demand


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The Firm s Labor Demand

The rms labor demand schedule is the marginal product of labor forthe rm, which is diminishing

To see this, nd the rst-order condition for N :

MPN =zF (K, N)

N= w


Figure 10 7: The Firms Labor Demand


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Figure 10.7: The Firm s Labor Demand


The Firms Labor Demand


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The Firm s Labor Demand

As z or K increase, MPN increases


Figure 10.8: The Eect of Increase in z or K on Labor


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Figure 10.8: The Eect of Increase in z or K on LaborDemand


Equation 10.16: Firms Investment Decision


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q

The rm invests to the point where the marginal benet frominvestment equals the marginal cost

To see this, nd the rst-order condition for I :

1 + 11 + r

2664z0F (K0, N0)K0 | { z } MPK0

K0

I|{z}1

+ (1 d)3775 = 0

=)

MPK0 d = r


Figure 10.9: Optimal Investment Schedule


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g p


Figure 10.10: The Eect of Increase in z or Decrease in K


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gon Investment Schedule


Shifts in Investment Schedule


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It is obvious that as z0 increases, MPK0 increases

To see the intuition for the inverse relationship between K and MPK0

suppose that initially the rm demands I1 units of investment at rate ofinterest r

Next assume that K1 decreases to K2This implies that at I1, there will be too little future capital stock:[(1 d)K2 + I1] < [(1 d)K1 + I1]Given that MPK0 is diminishing, a lower K

0 will increase MPK0 relativeto the given rate of interest r

Hence, to satisfy the equilibrium condition, the rm must increase itsinvestment


Shifts in Investment Schedule: Increase in Default


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Premium

Let x denote the default premium

The optimality condition for investment becomes:

MPK0

d = r+ x.

Which can be rewritten as:

MPK0 d x = r.

This implies that an increase in default premium will reduce thedemand for investment at a given rate of interest


Figure 10.11: The Eect of Increase in Default Premium


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Figure 10.12: Investment and the Interest Rate Spread


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Figure 10.13: Investment and the Interest Rate Spread


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Equation 10.18


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The governments present-value budget constraint:

G+ G0

1 + r = T+ T0

1 + r.


Figure 10.14: Equilibrium in the Labor Market for a given


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Real Interest Rate


Figure 10.15: The Output Supply Curve


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Figure 10.16: Increase in Current or Future GovernmentS di


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Spending

As G or G0 increases

T and/or T0 increases (to balance the budget)

Lifetime wealth decreases

The consumer works more and hence labor supply increases

This is a supply-side eect of a change in G or G0


Figure 10.16: Increase in G or G


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Figure 10.16: Increase in z


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Equation 10.19


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So far it was the supply side

Now we move on to the demand side

Total demand for output is equal to the sum of demand for

consumption goods, demand for investment goods and demand forgoods from the government:

Yd = Cd

Yd, r+ Id (r) + G.


Figure 10.18: Demand for Current Goods


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Eect of Increase in Expenditure on Demand: TheMultiplier


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Multiplier

An increase in expenditure leads to a more than one-for-one increasein output:

Yd = MPC Yd + E.

=)

Yd =1

1MPC

| {z }The MultiplierE


Figure 10.19: Output Demand Curve


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Figure 10.19: Output Demand Curve


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This is the same as the IS curve

As r increases, the consumers consume less (credit card, durables oninstalments, house, car etc.) and they invest less

The story is: demand for credit is low hence the supply of creditshould also be low to clear the market. The supply of credit will below if the level of income is low


Figure 10.20: Shift in Output Demand Curve (Increase inG )


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G)


Shifts in Output Demand Curve


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An increase in G shifts the curve to the right

A decrease in taxes (working through consumption), an increase in

future productivity (working through investment) and a decrease incurrent capital (working through investment) will also shift the curveto the right


Figure 10.21: The Complete Real Intertemporal Model


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Experiments using the Real Intertemporal Model


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An increase in G [Figures (10.22) and (10.23)]

A decrease in K [Figure (10.24)]

An increase in z [Figure (10.25)]

An increase in z0 [Figure (10.26)]


Figure 10.22: The Eect of an Increase in G on Demand


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Figure 10.23: An Increase in G


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Figure 10.23: An Increase in G


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C+ I+ G curve shifts up =) YD shifts to the right

T increases (life-time wealth decreases) and NS shifts to the right

Because of the shift in YD, r goes up =) further increase in NS

Increase in NS will shift the YS curve to the right

Final outcome: employment increases; output increases, real wagedecreases; rate of interest increases, and investment declines(crowding out). The eect on consumption is unclear


Figure 10.24: The Equilibrium Eect of a Decrease in K


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A Decrease in K


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Marginal product of labor declines (the same labor have to work withless capital) and ND curve shifts to the left

For a given level of employment, output [Y = zF(K, N)] falls andthe production function shifts down

The output supply curve shifts to the left

The marginal product of K0 increases and output demand curve shiftsto the right

The equilibrium real interest rate increases but the net eect on

equilibrium output is uncertain


Figure 10.25: The Equilibrium Eect of an Increase in z


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An Increase in z


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The marginal product of labor increases and the labor demand curveshifts to the right

The same quantities of labor and capital produce more output andhence the production function shifts up

The output supply curve shifts to the right

The equilibrium real interest rate will drop. This will shift the laborsupply curve to the left. However, we assume this eect to be small

The equilibrium output increases and real rate of interest decreases


Figure 10.26: The Equilibrium Eect of an Increase in z


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An Increase in z


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The marginal product of K0 increases and the output demand curveshifts to the right

The equilibrium output and real interest rate increaseThe increase in real interest rate will shift the labor supply curve tothe right. Once again, we assume this eect to be small


Lecture Notes EC3102 06 Two Period Model Inv Handout

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