Download 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.
2
Plan for Chap 9
•
•
•
•
•
Basic set up and optimization conditions
Assumptions of labor supply
Consumption goods demand
Firm investment behavior
Optimal investment decision
3
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.
4
• Utility function
U (C , C ', l , l ')
• Current BC
C  S  w(h  l )    T
p
5
• 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
6
• Representative consumer’s problem
max C ,C ',l ,l ' U (C , C ', l , l ')
Subject to lifetime BC
7
• FOCs
U
U

 ,

C
C ' 1  r
U
U
w '
  w,

l
l ' 1  r
8
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 '
9
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.
10
• N decreases when the lifetime wealth
increases.
Because leisure is the normal good.
11
Figure 9.1 The Representative
Consumer's Current Labor
Supply Curve
12
Figure 9.2 An Increase in the
Real Interest Rate Shifts the
Current Labor Supply Curve to
the Right
13
Figure 9.3 Effects of an
Increase in Lifetime Wealth
14
The Current Demand for the
Consumption Goods
• Current consumption will decrease if real
interest rate r increases by assuming the
substitution effect dominates.
15
• Holding constant Y and r, if lifetime wealth
increases, current consumption increases.
16
Figure 9.5 An Increase in the Real
Interest Rate from r1 to r2 Shifts the
Demand for Consumption Goods
Down
17
Figure 9.6 An Increase in
Lifetime Wealth for the
Consumer Shifts Up the Demand
for Consumption Goods.
18
The Representative Firm
• Current Production Function
Y  zF ( K , N )
• Future Production Function
Y '  z ' F ( K ', N ')
19
• Model investment goods as being
produced from output on a one-to-one
basis
K '  (1  d ) K  I
20
• Current profits
  Y  wN  I
• Future profits
 '  Y ' w ' N ' (1  d ) K '
21
• The Representative Firm’s problem
'
max N , N ', I V   
1 r
22
The Optimization Conditions
• Employment decisions
F
MPN  z
w
N
F
MPN '  z '
 w'
N '
23
Figure 9.7 The Demand Curve for
Current Labor Is the Representative
Firm's Marginal Product of Labor
Schedule
24
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
25
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.
26
• The marginal benefit from investment
MP 'K  1  d
MB( I ) 
1 r
27
• Firm will equate the marginal benefit and
marginal cost of invetsment
I.e.
MP 'K  1  d
1
1 r
MP 'K  d  r
28
• Firm will invest until the net marginal
product of capital is equal to the real
interest rate. This is the optimal
investment rule.
29
• 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.
30
Figure 9.9 Optimal
Investment Schedule for the
Representative Firm
31
Figure 9.10 The Optimal
Investment Schedule Shifts to the
Right if Current Capital Decreases
or Future Total Factor Productivity
Is Expected to Increase
32
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%.
33
Table 9.1 Data for Paula’s
Orchard
34
Government
• Government has present-value BC
G'
T'
G
T 
1 r
1 r
35
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.
36
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.
37
on of
Equilibrium
in the
Labor
Market
Given the
Real
Interest
Rate r
38
Figure 9.12 Construction of
the Output Supply Curve
39
• 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.
40
– TFP. Increase in z will shift the output supply
curve to the right.
– Capital Stock K.
41
Figure 9.13 An Increase in
Current or Future Government
Spending Shifts the Ys Curve
42
Figure 9.14 An Increase in
Current Total Factor Productivity
Shifts the Ys Curve
43
The Current Goods Market and
the Output Demand Curve
• Current good market clearing condition
Y  C (r )  I (r )  G
d
d
44
• 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.
45
Figure 9.15 The Demand for
Current Goods
46
Figure 9.16 Construction of
the Output Demand Curve
47
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.
48
Figure 9.17 The Output
Demand Curve Shifts to the
Right if Current Government
Spending Increases
49
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

50
Figure 9.18 The Complete
Real Intertemporal Model
51
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.
52
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.
53
• Equilibrium Effects
– Goods Market: Y ↑, r ↑
– Labor Market: w↓, N ↑
– C ↓, I ↓, increases in G crowds out both the
consumption and investment.
54
Figure 9.19 A Temporary
Increase in Government
Purchases
55
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
56
Figure 9.21 Natural Log of
Real Investment, 1929–2002
57
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.
58
• 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.
59
Figure 9.22 The Equilibrium
Effects of a Decrease in the
Current Capital Stock
60
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
61
• 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.
62
Figure 9.23 The Equilibrium
Effects of
an Increase in Current Total
Factor Productivity
63
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
64
• 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.
65
Figure 9.24 The Equilibrium
Effects of
an Increase in Future Total Factor
Productivity
66