Economics 122.
Investment
Fall 2009
NOAA’s weather supercomputer
PET Scan of PIB molecule
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The Macroeconomics of Investment
Capital
• Produced, durable, used for further production
• Examples:
– tangibles (structures, equipment)
– intangibles (software, patents)
Basic role of investment in macro
• Short run: most volatile part of aggregate demand
– Recall decline of I in Great Depression
• Long run: key determinant of growth of potential output and major
way that governments affect economic growth.
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Investment decline in the Depression
1.4
Real GDP (1929 = 1)
Real I (1929 = 1)
1.2
1.0
0.8
0.6
0.4
0.2
0.0
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30
31
32
33
34
35
36
37
38
39
40
Note on data: Very convenient place is “FRED”: http://research.stlouisfed.org/fred2
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Investment-GDP ratio
Investment/GDP (nominal $)
.20
.18
.16
.14
.12
.10
60
65
70
75
80
85
90
95
00
05
10
Shaded areas are NBER recessions
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Today’s housing crunch
Housing Starts / Population
.014
.012
.010
.008
.006
.004
.002
.000
60
65
70
75
80
85
90
95
00
05
10
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The major theories of investment
1. Accelerator theory: states that investment is a function of
change in output
2. Neoclassical theory: Desired capital stock a function of output
and cost of capital
3. Q theory: Investment a function of Tobin’s Q (Q =ratio of
market value of K to replacement cost)
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1. Accelerator Theory
- Oldest and simplest theory is the accelerator model.
- Here, the idea is that there is a target capital-output ratio
K* = v Y
- Hence the desired change in investment is equal to the change
in output (I is gross investment):
I* = ΔK* + δK = v Δ Y + δK
-
The actual investment might differ from the desired, but this
is a simple and useful model. It shows why there is a close
relationship between investment and output change.
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Isoquants with fixed capital-output ratio
L
fixed K/L ratio in accelerator
K
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Graph shows relation between investment and change in output
(accelerator model)
I
δK+vΔY
δK
ΔY
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2. Neoclassical Theory
The mainstream theory is the neoclassical model.
- This is closely related to the accelerator model.
- The difference is that in the neoclassical model there is a
variable capital-output ratio
- Hence, K/Y depends upon relative factor prices, and in
particular upon the user cost of capital and taxes.
Using standard production theory
Start with aggregate production function:
Y = F(K,L)
Next figure shows the difference of the accelerator and
neoclassical models for an isoquant.
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Isoquants with fixed capital-output ratio v.
variable proportions
L
fixed K/L ratio in accelerator
Cobb-Douglas in neoclassical
K
- fixed K/L corresponds to accelerator model
- variable proportions (such as Cobb-Douglas)
corresponds to neoclassical model
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Investment Criteria
There are a number of return concepts.
–
–
–
–
present value (V0 today)
internal rate of return (i per year)
rental price of capital ($ per apartment per month)
cost of capital (usually, “user cost of capita”)
•
•
•
•
Central concept in macro theories of investment
Def. Cost of renting capital for one period
Appropriate for perfect capital market.
Estimate as imputed in most circumstances because firms own
capital (also for housing in NIPA)
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Formula for cost of capital
u ≈ (1+τ) pK [r + δ]
where
u = user cost of capital
pK = price of capital good
r = real interest rate
δ = depreciation rate
τ = effective rate of tax (or subsidy when negative) on capital
goods
Linkage to policy:
- through real interest rate
- through taxation of capital
In practice, u is complicated to measure; off to B School!
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Derivation and example:
• Buy a car, rent it for one period, and then sell at the end. No
inflation or taxes. r = .05.
• Pay $20,000 sell for 20,000(1-.1) = $18,000; collect rent u.
• What cost of capital (u) would just break even?
when pK = pK (1- δ)/(1+r) + u
20,000 = 18,000/(1.05) + u
Solve for break-even (competitive-market) rental:
u
= 20,000 – 18,000/1.05
= 20,000 – 17,143 = 2857
≈ pK (r+ δ) = 20,000(.15) = 3,000
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Cost of capital with no taxes and P = 1
This is a
slightly
more
realistic
version that
has both
debt and
equity
capital.
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Real cost of capital (% per year)
10
9
8
7
6
5
85
4
90
3
50
95
55
60
00
65
70
05
75
80
10
85
90
95
00
05
10
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Basic approach in the neoclassical model
We start with a production function; and then derive the demand for
capital.
Demand for capital:
Assume a Cobb-Douglas production function
Y = A Kβ L1-β
Then derive the value of MPK:
p MPK = p ∂Y/∂K=p βA Kβ-1 L1-β = pβY/K
Finally, the firm sets its desired capital stock (K*) by equating
the cost of capital to the MPK:
u =p∂Y/∂K=pβY/K
Now normalize prices = 1 for simplicity and ignore taxes:
u = (r+δ) = βY/K
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This yields a demand for capital (K*):
K* = βY/(r+δ)
Note the various components of the demand for capital
–
–
–
–
output (like accelerator)
interest rate (r)
depreciation rate (δ)
With taxes: K* = βY/[(r+δ) (1+τ)]
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- Note that the impact of interest rates on investment is
powerful but depends importantly on the lifetime of the
capital:
User cost of capital Change in user
Depreci- (per $ capital) cost with increase
Lifetime ation rate at real interest rate:
in r from
2% to 4%
Investment item (years)
2%
4%
δ
Structures
50
0.02
4%
6%
50.0%
Major
equipment
10
0.10
12%
14%
16.7%
Computers
3
0.40
42%
44%
4.8%
Inventories
1
1.00
102%
104%
2.0%
- Where biggest impacts? Housing
- Where smallest? Computers and inventories
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From demand for capital to demand for investment
• How do we get investment demand (or equilibrium
investment)?
• Generally, go from demand for capital to demand for
investment
• Several approaches:
- Costs of adjustment of investment
- Capacity in the capital goods industry (Boeing aircraft)
- Construction lags (power plants)
- Internal funds constraint
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3. Q theory of investment
Idea here is that investment is determined by relationship between the
value of firms or houses and the cost of new or replacement capital.
Keynes:
“The daily revaluations of the Stock Exchange, though they are
primarily made to facilitate transfers of old investments between
one individual and another, inevitably exert a decisive influence on
the rate of current investment. For there is no sense in building up a
new enterprise at a cost greater than that at which a similar existing
enterprise can be purchased; whilst there is an inducement to spend
on a new project what may seem an extravagant sum, if it can be
floated off on the Stock Exchange at an immediate profit.”
Tobin:
"It is common sense that the incentive to make new capital investments
is high when the securities giving title to their future earnings can
be sold for more than the investments cost, i.e., when q exceeds
one."
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The collapse in shipping prices on dry cargo
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A glut of cargo ships
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More formally:
Q = (market value of K)/(replacement cost of K)
Example:
- Cargo ships are selling for a Q of 0.25
- E.g., cost of production is $20 million, but ships sell for $5 million
- How is this possible?
• Inelastic supply and high demand for cargo → low rentals
• PV of ships is low.
• Therefore, little to no shipbuilding, and the stock gradually
depreciates or is scrapped
How does Q affect investment?
- Because Q < 1, shipping firms buy old ships rather than build new
ones
- This depresses investment.
- Therefore I/K = f(Q), f’(Q) > 0.
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Rental of
ships
S2009
Current
ship
rental
Demand for shipping
Number of ships
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Rental of
ships
Sfuture
S2009
“cost of capital” for new ships
Current
ship
rental
Demand for shipping
Number of ships
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Q
I/K = f (Q)
1
I/K
δ
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Investment ratio and Q for housing
1.7
900
I housing (right scale)
1.6
800
1.5
Q of housing
Residential construction (2005 $)
1.4
700
1.3
600
Q (left scale)
1.2
1.1
500
1.0
400
0.9
0.8
Housing bubble:
1.Note that Q rose
about 50 percent
from mid-1990s.
2. Note huge decline
in residential
construction (I)
88
90
92
94
96
98
00
02
04
06
08
10
300
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Summary of investment theory
1.. The major components of investment are residential, business plant
and equipment, software, and inventories.
2. These are among the most volatile components of output in the short
run.
3. In equilibrium, demand for capital determined where the cost of
capital equals the marginal productivity of capital.
4. The major theories are the accelerator theory, the neoclassical theory,
and the Q theory. These apply differently in different sectors.
5. Economic policy affects investment through both monetary and
fiscal policy:
• monetary policy through real interest rate and stock prices
• fiscal policy through things like depreciation policy and
investment tax credits.
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