Transcript Document

Chapter 9
A Real Intertemporal Model with
Investment
• In this chapter, we will complete a model
of real side of the economy.
• We will show how real aggregate output,
real consumption, real investment,
employment, the real wage, and the real
interest rate are determined in the
macroeconomy.
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Plan for Chap 9
•
•
•
•
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Basic set up and optimization conditions
Assumptions of labor supply
Consumption goods demand
Firm investment behavior
Optimal investment decision
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The Representative Consumer
• In this model we will bring together the
work-leisure choice form Chapter 4 with
the intertemporal consumption behavior
(consumption-saving decision) from
Chapter 8.
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• Utility function
U (C, C ', l , l ')
• Current BC
C  S  w(h  l )    T
p
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• Future BC
C '  w '(h  l ')   ' T ' (1  r)S
p
• Lifetime BC
C'
w '(h  l ')   ' T '
C
 w(h  l )    T 
1 r
1 r
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• Representative consumer’s problem
maxC ,C ',l ,l ' U (C, C ', l, l ')
Subject to lifetime BC
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• FOCs
U
U

 ,

C
C ' 1  r
U
U
w'
  w,

l
l ' 1  r
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The Optimization Conditions
U / l
MRSl .C 
w
U / C
U / l '
MRSl '.C ' 
 w'
U / C '
U / C
MRSC .C ' 
 1 r
U / C '
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The Determinants of Current
Labor Supply
• N=h-l increases when w increases.
Here we assume the substitution effect of a
change in w is always larger than the income
effect.
• N increases when the real interest rate r
increases.
w(1+r)/w’ is the current price of leisure relative to
the future price of leisure. And we assume again
the substitution effect is larger than the income
effect.
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• N decreases when the lifetime wealth
increases.
Because leisure is the normal good.
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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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The Current Demand for the
Consumption Goods
• Current consumption will decrease if real
interest rate r increases by assuming the
substitution effect dominates.
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• Holding constant Y and r, if lifetime wealth
increases, current consumption increases.
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Figure 9.5 An Increase in the Real
Interest Rate from r1 to r2 Shifts the
Demand for Consumption Goods
Down
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Figure 9.6 An Increase in
Lifetime Wealth for the
Consumer Shifts Up the Demand
for Consumption Goods.
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The Representative Firm
• Current Production Function
Y  zF ( K , N )
• Future Production Function
Y '  z ' F ( K ', N ')
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• Model investment goods as being
produced from output on a one-to-one
basis
K '  (1  d ) K  I
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• Current profits
  Y  wN  I
• Future profits
 '  Y ' w ' N ' (1  d ) K '
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• The Representative Firm’s problem
'
max N , N ', I V   
1 r
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The Optimization Conditions
• Employment decisions
F
MPN  z
w
N
F
MPN '  z '
 w'
N '
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Figure 9.7 The Demand Curve for
Current Labor Is the Representative
Firm's Marginal Product of Labor
Schedule
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Figure 9.8 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 investment decision
• The marginal cost of investment
MC ( I )  1
Because when I increases one unit, the
present value of profits V decreases one
unit accordingly.
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• The marginal benefit from investment
MP 'K  1  d
MB( I ) 
1 r
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• Firm will equate the marginal benefit and
marginal cost of invetsment
I.e.
MP 'K  1  d
1
1 r
MP 'K  d  r
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• Firm will invest until the net marginal
product of capital is equal to the real
interest rate. This is the optimal
investment rule.
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• Two types of shifts in the optimal
investment schedule
– When z’ increases, optimal investment
schedule shifts to the right.
– When K is higher, then optimal investment
schedule shifts to the left.
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Figure 9.9 Optimal
Investment Schedule for the
Representative Firm
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Figure 9.10 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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Optimal Investment: A
Numerical Example
• Paula, a small-scale farmer, has an apple
orchard. K=10.
• 10 trees can produce 100 bushels of apples.
Y=100.
• At the end of each period, 20% of trees die.
d=0.2.
• The liquidation price is one tree for one bushel
of apples.
• Real interest rate = 5%.
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Table 9.1 Data for Paula’s
Orchard
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Government
• Government has present-value BC
G'
T'
G
T 
1 r
1 r
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Competitive Equilibrium
• We will show how a CE, where supply
equals demand in the current-period labor
and goods markets, can be expressed in
terms of diagrams.
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The Current Labor Market and
the Output Supply Curve
• Given real interest rate r, current labor
market clears at the current wage w* and
the current equilibrium employment N*.
Hence the equilibrium current output Y* is
determined.
• When r changes, Y* will change
accordingly. We show the relationship in
the output supply curve Ys.
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on of
Equilibrium
in the
Labor
Market
Given the
Real
Interest
Rate r
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Figure 9.12 Construction of
the Output Supply Curve
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• Slope of Output Supply—Real Interest
Rate Effects
• Shifts in Output Supply
– Lifetime Wealth, e.g., an increase in G or G’
will shift labor supply curve to the right and
shifts the output supply curve to the right due
to the income effect on labor supply.
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– TFP. Increase in z will shift the output supply
curve to the right.
– Capital Stock K.
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Figure 9.13 An Increase in
Current or Future Government
Spending Shifts the Ys Curve
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Figure 9.14 An Increase in
Current Total Factor Productivity
Shifts the Ys Curve
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The Current Goods Market and
the Output Demand Curve
• Current good market clearing condition
Y  C (r)  I (r )  G
d
d
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• From this condition, we can trace out the
output demand curve Yd as a function of
real interest rate r.
• The slope of Yd is negative.
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Figure 9.15 The Demand for
Current Goods
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Figure 9.16 Construction of
the Output Demand Curve
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Shifts in Output Demand
•
•
•
•
•
Current Government Spending ↑
Present value of taxes ↓
Future Income ↑
Future Total Factor Productivity ↑
Current Capital Stock ↓
Will shift the output demand curve to the
right.
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Figure 9.17 The Output
Demand Curve Shifts to the
Right if Current Government
Spending Increases
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The Complete Real
Intertemporal Model
• Equilibrium in the
Labor Market
N  N (r*)  w*, N *
d
s
• Equilibrium in the
Goods Market

Y ( r )  Y (r )  r , Y
d
s

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Figure 9.18 The Complete
Real Intertemporal Model
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Working with the Model
Two key messages in this chapter
• It does matter that whether the shock is
temporary or permanent
• The effects of a shock to the economy expected
in the future will have important macroeconomic
effects in the current period.
We will use several experiments to show them.
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A Temporary Increase in
Government Purchases
• Impact Effects (G↑, G’ unchanged)
– Labor supply: PV of taxes increases ↓lifetime
wealth decreases ↓ labor supply curve shifts
to the right
– Output supply: shifts to the right.
– Output demand: drops due to reduction in the
lifetime wealth, increases due to G ↑. But
since MPC<1, net effect is increase.
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• Equilibrium Effects
– Goods Market: Y ↑, r ↑
– Labor Market: w↓, N ↑
– C ↓, I ↓, increases in G crowds out both the
consumption and investment.
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Figure 9.19 A Temporary
Increase in Government
Purchases
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Test the Theory: WWII
• G↑
• Previously we knew during this period
Y ↑ and C↓
• We can also see I↓
• But real interest rate r was quite low
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Figure 9.21 Natural Log of
Real Investment, 1929–2002
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A Reduction in the Current
Capital Stock
• Impact Effects
– Labor demand: MPL decreases, so does the
labor demand
– Output supply: shifts to the left since Nd
decreases.
– Output demand: investment increases since
future MPK will increase. So output demand
shifts to the right.
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• Equilibrium Effects
– Goods Market: r↑, Y?
– Labor Market: N?, w↓
– C ↓ since r increases. But I? because MPK’
increases will induce more investment, but
increases in r will decrease the investment.
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Figure 9.22 The Equilibrium
Effects of a Decrease in the
Current Capital Stock
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An Increase in Current TFP
• Impact Effects
– Labor demand: MPL increases, shifts the
curve to the right
– Output supply: shifts to the right
– Labor supply: since r decreases, it will shift to
the left
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• General equilibrium effect
– Goods market: Y ↑, r ↓
– Labor market: w ↑, N? (likely increase)
– C ↑, I ↑
• Recall the business cycle facts, the model
predicts that consumption, employment
(likely), and the real wage are procyclical,
just as in the data.
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Figure 9.23 The Equilibrium
Effects of
an Increase in Current Total
Factor Productivity
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An Increase in Future TFP
• Impact Effects
– Output demand: z’ ↑ will increase MPK’, so
firm wants to invest more
– Labor supply: r ↑ will make it shifts to the right
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• Equilibrium Effects
– Goods market: Y ↑, r ↑
– Labor market: w↓, N ↑
– C?, I ↑. C is ambiguous since r ↑ will cause C
to fall, but Y ↑ will increase it.
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Figure 9.24 The Equilibrium
Effects of
an Increase in Future Total Factor
Productivity
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