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CLASS #9: INVESTMENT IN A CLOSED ECONOMY
- Understanding investment (capital accumulation) is crucial to
understand growth
- Investment is the most volatile component if GDP:
to understand business cycle one has to understand the investment
What is investment?
- "Investment" is adding capital stock to the economy (including human
capital).
- The component I of the GDP is more narrowly defined as spending
for new capital goods: 1) machinery and equipment, 2) residential and
non residential buildings (1+2="fixed investment"), 3) additions to the
inventory stock
- Buying existing capital (i.e., purchasing stocks) is not investment
- A crucial characteristic of investment is that you need time
to build: only the capital that was accumulated in previous periods can
be used in production today
the relationship between capital and investment is given by:
where
is the rate at which capital depreciates (about 10% per year), or:
Investment = increase in capital + replacement of existing capital that
depreciated
- How do interest rates affect investment?
- What is the relationship between the MPK (marginal productivity
of capital) and investment?
- Can government policies affect the investment rate?
The Intertemporal Model with Investment and Production
same model we are used to, except that:
- people do not obtain output from trees (endowment economy), but
from production
- the production function has as inputs capital and labor (Cobb Douglas)
- (for the time being) people do not choose how much labor to
supply: set N=1. the production function is then a function of capital only:
The Centralized Economy ("Social Planner")
In the real world, people put their savings in the bank, and the allocation
of savings among firms is governed by a market mechanism: there is
an interest rate that equilibrates supply and demand for loans.
Also, there is a wage that equilibrates supply and demand for labor.
For the time being, no market: a "social planner" is telling people how
much to work and to save, so to maximize their utility.
Alternatively, think that the representative household owns directly the
firm.
Social planner's problem (two periods):
subject to the constraints:
K2 + C1 = Y1
where
is the amount of resources inherited from the previous period.
How much is the household going to consume this period? and how much is it
going to save as capital?
substitute for K2: K2=Y1-C1, and for C2:
FOC's (first order conditions)
MRS = MRT
(Marginal rate of substitution = Marginal rate of transformation)
The Market Economy (decentralized equilibrium)
Now there are households and firms. The household decides how much to
consume and to save, taking as given the interest rate r, and works for
a given wage w (N=1)
subject to
K2+C1=Y1
and
C2=(1+r)K2+w
The firm decides how much capital to rent, and how much labor to hire
(the capital it rents depreciates over the period) given r and w:
or
The household's budget constraint can be rewritten in the
familiar form:
so we know the FOC of the household's problem:
or
These first order conditions determine the amount K2 that the household wants to save as a
function of the interest rate.
The FOC for the firm are:
F2(K2,N) = w
The amount of resources that the household wants to carry on to the
next period,
K2h=Y1-C1
is a positive function of the interest rate
(if the substitution effect prevails over the income effect).
Savings is defined as income-consumption, or:
(remember the definition
).
The amount of capital the firms wants to have in place K2f is a
negative function of the interest rate.
Remember that the economy had inherited some capital (K1) from the previous
period. From the definitions of saving (S) and investment (I):
and
We obtain that saving depends positively on the interest rate, while
investment depends negatively on the interest rate
Equilibrium in the market economy
Closed economy: Savings = Investment
Or, the equilibrium interest rate must be such that:
K2h (r) = K2f (r)
therefore at equilibrium the K2 must be such that:
Same equilibrium as in the planner's problem!
(First Welfare Theorem, or, "prices do magic")
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Marco Del Negro
2000-02-08