econ3102-005 chapter 9: a real intertemporal model … · chapter 9: a real intertemporal model of...
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ECON3102-005Chapter 9: A Real Intertemporal Model of
Investment
Neha Bairoliya
Spring 2014
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What do we study in this chapter?
• Construct a real intertemporal model that will serve as a basis forstudying money and business cycles in Chapters 11-13.
• Understand the investment decision of the firm.
• Show how macroeconomic shocks affect the economy.
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What’s new in this chapter?
• The key things added to chapter 8 are a work-leisure decision and arepresentative firm.
• We are moving from an endowment economy to a productioneconomy.
• The representative firm and consumer interact on the labor andconsumption markets.
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The Representative Consumer
• has h units of time each period, for work and leisure (`).
• today, earns real wage w(h − `) receives dividend income π, paystaxes T , and saves Sp.
• hence the current period BC:
C + Sp = w(h − `) + π − T
• tomorrow, earns real wage w ′(h − `′) receives dividend income π′ ,pays taxes T ′ , and receives Sp(1 + r).
• hence, the future period BC:
C ′ = w ′(h − `′) + π′ − T ′ + Sp(1 + r)
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The Representative Consumer
• This gives the consumer’s PVBC:
C +C ′
1 + r= w(h − `) +
w ′(h − `′)1 + r
+ π +π′
1 + r− T − T ′
1 + r
• The representative consumer chooses C ,C ′, `, `′ to be as well off aspossible, given this PVBC.
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The Representative Consumers Optimization Conditions
• The MRS of leisure for consumption is the price of leisure (wage):MRS`,C = w and MRS`′,C ′ = w ′.
• The MRS of current for future consumption is the price of currentconsumption in terms of future consumption (1 + r):MRSC ,C ′ = 1 + r
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The Representative Consumers Labor Supply
• Recall from Chapter 4 that as the real wage increases, thesubstitution effect ⇒ labor supply ↑.
• However, the income effect ⇒ leisure ↑, i.e. labor supply ↓.(assuming leisure is normal.)
• Here,we assume substitution>income effect, so that w ↑⇒ laborsupply ↑.
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Current Labor Supply and Real Interest Rate
• The price of current relative to future leisure is: w(1+r)w ′
• As r ↑, the consumer substitutes current for future consumption andcurrent for future leisure.
• Hence, assuming substitution>income effect, r ↑⇒ labor supply ↑.
• Labor supply ↓ as non-wage disposable income ↑ (income effect)(e.g: ↓ in lifetime taxes.)
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An Increase in the Real Interest Rate Shifts the CurrentLabor Supply Curve to the Right
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Effects of an Increase in Lifetime Wealth
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The Representative Firm
• Let z and z ′ denote present and future TFP. Current PeriodProduction function is:
Y = zF (K ,N)
• Future Period Production function:
Y ′ = z ′F (K ′,N ′)
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The Representative Firms Investment
• We assume it requires one unit of current consumption may betransformed into one unit of capital.
• The representative rm foregoes current profits, invests in currentcapital to increase capacity tomorrow:
K ′ = (1− d)K + I
• We assume that in the last period, the remaining capital (1− d)K isconverted back into the consumption good and sold to consumers.
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The Representative Firm’s Profit
• The objective of the firm is to max the PV of prots.
• Current Prots are:π = Y − wN − I
• Future Prots are:
π′ = Y ′ − w ′N ′ + (1− d)K
• The present value of prots is
π +π′
1 + r
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The Representative Firms Current Labor Demand
The optimization rule for labor demand from the firm is:
• The objective of the firm is to max the PV of prots. MPN = w .
• Recall from Ch 4 that because MPN is ↓ in N, the firms labordemand is also ↓.
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The Current Demand Curve for Labor Shifts Due toChanges in Current Total Factor Productivity z and in the
Current Capital Stock K
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The Representative Firms Investment Decision
• The marginal cost of investment is what is given up to make anadditional unit of investment. It equals one unit of the consumptiongood:
MC (I ) ≡ 1
.
• An additional unit of investment transforms into one unit of capitalthat tomorrow is used for production and sold after depreciation.
• Hence, the marginal benet of investment in terms of units of currentconsumption is:
MB(I ) ≡ MPk′ + 1− d
1 + r
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The Representative Firms Investment Decision
• The firm invests until: MC (I ) = MB(I ),
• ⇒ MB(I ) ≡ MPk′+1−d1+r = 1⇒ MPk′ − d︸ ︷︷ ︸
net marginal product of capital
= r .
• Hence, as r ↑⇒ MPk′ ↑⇒ K ′ ↓⇒ I = (K ′ − (1− d)K ) ↓.
• The net marginal product of capital is the marginal product ofcapital, net of its depreciation.
• Intuition: The bond and capital are the two assets in the economyand should yield identical returns (the rm often borrows to nanceinvestment).
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Optimal Investment Schedule for the Representative Firm
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The Optimal Investment Schedule Shifts to the Right ifCurrent Capital Decreases or Future Total Factor
Productivity Is Expected to Increase
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The Government
• Governments behavior is the same as in Chapter 8:
G +G ′
1 + r= T +
T ′
1 + r
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Competitive Equilibrium
• A CE in this economy is a situation in which consumers and rmsoptimize, taking prices and taxes as given, GBC holds and marketsclear.
• We will focus on current period markets only.
• Eventually, in this model, we will show that this is WLOG, as theclearing of current markets implies that of future markets.
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The Output Supply Curve
• The next 2 slides explain the determination of the output supplycurve by the firm, as a function of r .
• We will find that because of the intertemporal substitution effecteffect on labor supply, the the output supply curve is ↑ in r .
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Determination of Equilibrium in the Labor Market Giventhe Real Interest Rate r
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Construction of the Output Supply Curve
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Shifts of the Output Supply Curve
• The output supply curve (function of r) shifts because of:
• A shift in the current labor supply curve (e.g. from a change inlifetime wealth) (by the income effect)
• A shift in the current labor demand curve (e.g from changes in z ,K).
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Shifts of the Output Supply Curve in response to a changein G ,G ′
• Hence an ↑ in G or G ′ ⇒ ↑ in lifetime taxes ⇒ ↓ in lifetime wealth⇒ leisure ↓⇒ labor supply ↑.
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Shifts of the Output Supply Curve in response to anincrease in G or G ′
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Shifts of the Output Supply Curve in Response to aChange in z (same for an increase in K )
• z ↑⇒ the production function shifts to the right (ch 4)
• Also, an ↑ in z ⇒ labor demand ↑ as MPN increases.
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Shifts of the Output Supply Curve in response to anincrease in z (same for an increase in K )
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The Output Demand Curve
• Let Y represents the representative consumers current income.
• Let C d(Y , r) denote the representative consumers current demandfor the consumption good ( a ↓ function of r assuming thesubstitution effect dominates).
Y d = C d(Y d , r) + I d(r) + G
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The Total Demand for current goods
• Since I d and G do not depend on current income Y , only C d
inuences C d(r) + I d(r) + G as a function of Y .
• Hence, the slope of Y d = MPC .
• We are not done yet. We need to get Y d as a function of r .
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The Total Demand for current goods
• Recall C d ↓ as r ↑ (assuming substitution > income effect).
• Recall I d ↓ as r ↑
• Hence, Y d = C d(Y d , r) + I d(r) + G ⇒ Y d ↓ as r ↑.
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An Increase in Current or Future Government SpendingShifts the Y d Curve
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But the magnitude of ∆Y∆G is different from what Keynesiantheory says!
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Here we have:
• MPC ≡ DBAB = FB−FD
Y d2 −Y d
1=
[Y d2 −Y
d1 ]−FD
Y d2 −Y d
1=
[Y d2 −Y
d1 ]−[(1−MPC)∆G
Y d2 −Y d
1
• ⇒ Y d2 − Y d
1 = G2 − G1 ⇒ ∆Y∆G = 1
• Not what you know from your Chapter on Keynes in Econ 1102!
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What went wrong in the Keynesian model?
• In Econ 1102, we had ∆Y = 11−MPC ∆G
• This is because the Keynesian theory forgets the impact of thechange in taxes on consumption.
• The new classical model is not perfect either, as it assumes perfectforesight.
• So far, we have only looked at the effect of ∆G on Y d . Whatabout the effect on Y s? Taxes ↑⇒ labor supply ↑ (income effect⇒ leisure ↓). This ⇒ Y s shifts to the right.
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General Equilibrium effects of an increase in G
• The horizontal shift of Y d is of length = ∆G .
• But because Y s does not shift to the right by the same magnitude,the change in Y is less than the original change in ∆G .
• The Keynesian model does not account for substitution effects andcannot capture the impact on y s .
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G crowds out private consumption and private investment
• Note from the previous graph, that r ↑ if the income effect( thatcauses leisure to ↓ and shifts Y s to the right) is not too high.Otherwise, r ↓.
• Lets assume conditions are met for an ↑ in r .
• Then, because of intertemporal substitution, current consumptionwill ↓.
• The ↑ in r also ⇒ ↓ in private investment. This crowding out effectis ignored in the Keynesian theory.
• Note further that the ↑ in r causes the consumer to substitutecurrent for future leisure, shifting Ns and Y s further to the right.
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General Equilibrium Effects of ↓ in K
• ⇒ I d ↑⇒ Y d shifts to the right.
• Also, the ↓ in K ⇒ a ↓ in MPN ⇒ Nd and Y s shifts to the left. Intotal, Y ∗ may ↑ or ↓.
• Hence, in equilibrium, r must necessarily ↑. So, current consumptionmust ↓.
• I d may ↑ (as K has ↓) or ↓ as r has ↑.
• As r ↑, the consumer substitutes current for future leisure, which⇒ Ns shifts to the right, depressing wages further. Overall, w may ↓or ↑.
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General Equilibrium Effects of ↓ in K
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General Equilibrium effects of an increase in TFP z
• As z ↑, MPN ↑, and so, Nd and Y s shifts to the right.
• Y s shifts to the right ⇒ r ↓⇒ Ns shifts to the left as the consumersubstitutes future for current leisure.
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General Equilibrium effects of an increase in TFP z