Transcript Chapter 9

Chapter 9
A Real
Intertemporal
Model with
Investment
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Chapter 9 Topics
• Construct a real intertemporal model that will serve as a
basis for studying money and business cycles in
Chapters 10-12.
• Understand the investment decision of the firm.
• Show how macroeconomic shocks affect the economy.
• Focus on the implications of future expectations for
current macroeconomic performance, and the
difference between temporary and permanent shocks.
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Real Intertemporal Model
• Current and future periods.
• Representative Consumer –
consumption/savings decision
• Representative Firm – hires labor and invests in
current period, hires labor in future
• Government – spends and taxes in present and
future, and borrows on the credit market.
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Model - Consumer
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Time endowment in each period , h.
Real wage in current and future period, (w, w’).
Lump-sum taxes (T, T’)
Dividend incomes (∏, ∏’).
Price takers: w, w’ and r are taken as given.
Two decisions:
– Saving in current period.
– (C, l) in current period and (C’, l’) in future period.
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Equation 9.1
Consumer’s current-period budget constraint:
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Equation 9.2
Consumer’s future-period budget constraint:
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Equation 9.3
Consumer’s lifetime budget constraint:
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Equation 9.4
Consumer’s current-period marginal condition:
Marginal condition for work-leisure decision from Ch. 4.
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Equation 9.5
Consumer’s future-period marginal condition:
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Equation 9.6
Consumer’s intertemporal marginal condition:
Marginal Condition for consumption-saving decision in Ch.8.
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Tradeoff of Intertemporal Leisure
Decision
• MRS l , l’ = w ( 1 + r ) / w’
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Consumer’s Current Labor
Supply Behavior
• Current labor supplied increases with the real
wage (substitution effects are assumed to
dominate income effects).
• Labor supply increases with an increase in the
real interest rate, through an intertemporal
substitution effect.
• An increase in lifetime wealth (e.g. taxes fall)
reduces labor supply.
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Figure 9.1 The Representative
Consumer’s Current Labor Supply
Curve
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Figure 9.2 An Increase in the Real
Interest Rate Shifts the Current Labor
Supply Curve to the Right
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Figure 9.3 Effects of an Increase
in Lifetime Wealth
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Equation 9.7
Firm’s current-period production function:
N is choice variable in current period.
That is, firms choose N to maximize current profits.
K is stock of capital in current period and firms take it as given.
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Investment
• Def: goods and service that are produced in current
period, yet not consumed.
• Investment in current period.
– One unit of consumption good can be transformed into one
unit of investment good.
• Since I uses some of c, c drops. Since I enhances future
period’s productive capacity (recall that MPK >0), c’
increases.
– I represents tradeoff between c and c’ (or between ∏, ∏’).
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Equation 9.9
Evolution of the firm’s capital stock:
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Equation 9.8
K’ = (1 – d) K + I.
I is investment made in current period, and d is depreciation rate.
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Equation 9.10
Firm’s current-period profits:
I: capture the foregone profits in current period since investing
in I units of capital requires I units of consumption goods in current
period. The profit drops by I.
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Equation 9.11
Firm’s future-period profits:
(1 – d) K’ : at the end of future period, capital can be liquidated in market.
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Equation 9.12
The firm maximizes the present value of profits,
Firms profit maximizing problem: given w, w’ and r, maximizing PV of profits
by choosing N, N’, and I.
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The Firm’s Labor Demand
As in Chapter 4, the firm’s labor demand
schedule is the marginal product of labor for the
firm, which is downward sloping.
MPN = w.
That is, the marginal product of N equals real wage.
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Figure 9.4 The Demand Curve for Current
Labor Is the Representative Firm’s Marginal
Product of Labor Schedule
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Figure 9.5 The Current Demand Curve for Labor
Shifts Due to Changes in Current Total Factor
Productivity z and in the Current Capital Stock K
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The Representative Firm’s
Investment Decision
• The firm invests to the point where the marginal
benefit from investment equals the marginal cost
(MB=MC).
• This decision process is similar to the optimal
demand for labor in current period.
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Equation 9.13
The marginal cost of investment is 1, as the firm
gives up one unit of current profits for each unit
it invests, so:
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Equation 9.14
The marginal benefit of investment is the
marginal product of future capital plus the
quantity of capital that will be left in the future
after depreciation, all discounted back to the
present:
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Equation 9.15
The firm’s optimal investment rule, obtained by
equating the marginal benefit and marginal cost
of investment:
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Equation 9.16
Simplified optimal investment rule:
Investing in K until the net marginal product of K equals real interest rate.
Intuition: opportunity cost of investing in K is real interest rate, the rate
of return on other forms of assets, such as bond, in economy.
If firms do not invest, but trade output for bond. Then 1 unit of output in
current period gives firms (1 + r) units of future period goods.
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Figure 9.6 Optimal Investment
Schedule for the Representative Firm
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Figure 9.7 The Optimal Investment Schedule Shifts to
the Right if Current Capital Decreases or Future Total
Factor Productivity Is Expected to Increase
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Equation 9.18
The government’s present-value budget constraint:
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Derive Competitive Equilibrium
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Labor market: supply = demand.
Derive the relation between Ys and r.
Derive the relation between Yd and r.
Good market: supply = demand.
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Figure 9.10 Determination of Equilibrium
in the Labor Market Given the Real
Interest Rate r
Ns is upward-sloping under the
assumption that sub. effect dominates
income effect.
Position of Ns is affected by r and we.
Given r, equilibrium N is determined by
Nd and Ns.
Given z and K, the equilibrium N
determines equilibrium value of Y.
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Figure 9.11 Construction of
the Output Supply Curve
Ys: output net of current period I.
IS curve: relation between Ys and r.
IS curve is upward-sloping.
At each point on Ys, the labor market is
in equilibrium. That is, for a given r, LM is in
equilibrium when firms produce Y.
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Figure 9.12 An Increase in Current or
Future Government Spending Shifts
the Ys Curve
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Figure 9.13 An Increase in Current
Total Factor Productivity Shifts the Ys
Curve
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Equation 9.19
Total demand for output equals demand for
consumption goods plus demand for investment
goods, plus demand for goods from government:
So Id and G are both independent of Yd.
Cd and Id are both negatively related to r.
As a result, when r increases, Yd decreases.
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Figure 9.14 The Output
Demand Curve
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Figure 9.15 A Shift in the
Output Demand Curve
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Figure 9.16 The Complete
Real Intertemporal Model
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Define of Competitive Equilibrium
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CE is a set of endogenous quantities (c*, I*, Nd*, Ns*, Y*) and
endogenous prices, such as w and r, such that
Given (w*, r*, y*), (c*, l* or Ns*) solves consumers’ utility max
problem in current period.
Given (w*, r*), (Nd*, I*) solves firms’ profit max problem in
current period.
Labor market clears where Ns = Nd=N*, and the real wage w* in
current period is determined by intersection of Nd and Ns (r).
Good market clears where Ys = Yd=Y*, and the real interest rate
r* is determined by intersection of Yd and Ys .
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Experiments Using the Real
Intertemporal Model
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G increases temporarily.
G increases permanently.
K decreases.
z increases.
z’ increases.
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Temporary Increase in
Government Purchases
• Output increases, real interest rate increases, real
wage falls, consumption and investment
decrease, employment rises.
• Government spending crowds out both
consumption and investment.
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Figure 9.17 A Temporary Increase in
Government Purchases
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Permanent Increase in
Government Purchases
• Output increases by the amount of the increase
in government spending.
• No change in the real interest rate, investment
and consumption unchanged, real wage falls,
employment rises.
• No crowding out of C and I, but leisure falls for
the consumer.
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Figure 9.18 A Permanent Increase in
Government Expenditures
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A Decrease in the Current
Capital Stock
• This could arise due to a war or natural disaster.
• Output may rise or fall, depending on how large
the output demand effect is relative to the output
supply effect.
• The real interest rate rises, the real wage falls,
employment may rise or fall.
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Figure 9.20 The Equilibrium Effects of a
Decrease in the Current Capital Stock
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Current Total Factor
Productivity Increases
• Real interest rate falls, consumption and
investment rise, employment rises, real wage
rises.
• Productivity shocks are a potential explanation
for business cycles – see Chapter 11.
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Figure 9.21 The Equilibrium Effects of an
Increase in Current Total Factor
Productivity
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Total Factor Productivity
Expected to Increase in Future
• Output demand curve shifts right.
• Real interest rate rises, investment increases,
consumption may rise or fall, employment rises,
real wage falls, output rises.
• Important in explaining investment boom in the
1990s.
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Figure 9.22 The Equilibrium Effects of
an Increase in Future Total Factor
Productivity
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Figure 9.23 Percentage Deviations
From Trend in GDP and Investment,
1990–2006
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Figure 9.24 Investment as a
Percentage of GDP, 1990–2006
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Figure 9.25 S and P 500
Stock Price Index, 1990–2005
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