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:

$$K_{t+1} = (1 - \delta) K_t + I_t$$

where $\delta$ is the rate at which capital depreciates (about 10% per year), or:

$$I_t = (K_{t+1} - K_t) + \delta K_t$$

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)
  
  $$Y=AK^\alpha N^{1-\alpha}$$
  
  

• (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:
  
  $$Y= F(K) = AK^\alpha$$
  
  

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):

$$max_{\{C1,C2\}} U(C1,C2)$$

subject to the constraints:

K2 + C1 = Y1

$$C2 = F(K2) + (1-\delta)K2$$

where

$$Y1 \equiv F(K1) + (1-\delta)K1$$

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:

$$max_{C1} U(C1,F(Y1-C1) + (1-\delta)(Y1-C1))$$

FOC's (first order conditions)

$$U_1(C1,C2) = (F_1 (K2) + (1-\delta)) U_2(C1,C2)$$

$$\frac{U_1(C1,C2)}{U_2(C1,C2)} = (F_1 (K2) + (1-\delta))$$

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)

$$max_{\{C1,C2\}} U(C1,C2)$$

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:

$$max_{\{K2,N\}} F(K2,N) +(1-\delta)K2 -(1+r)K2 -wN$$

or

$$max_{\{K2,N\}} F(K2,N) - (r+\delta)K2 -wN$$

The household's budget constraint can be rewritten in the familiar form:

$$C1 + \frac{C2}{1+r} = Y1+ \frac{w}{1+r}$$

so we know the FOC of the household's problem:

$$\frac{U_1(C1,C2) }{U_2(C1,C2)}=(1+r)$$

or

$$\frac{U_1(Y1-K2,(1+r)K2+w) }{U_2(Y1-K2,(1+r)K2+w)}=(1+r)$$

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:

$$F_1(K2,N)+(1-\delta) = (1+r)$$

F₂(K2,N) = w

The amount of resources that the household wants to carry on to the next period, K2^h=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:

$$S= F(K1)-C1=Y1-(1-\delta)K1-C1=K2^h-(1-\delta)K1$$

(remember the definition $Y1 \equiv F(K1) + (1-\delta)K1$).
The amount of capital the firms wants to have in place K2^f 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): $S=K2^h(r) - (1-\delta)K1$ and $I = K2^f(r) -(1-\delta) K1$ 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:

K2^h (r) = K2^f (r)

therefore at equilibrium the K2 must be such that: $\frac{U_1(C1,C2) }{U_2(C1,C2)}=(F_1(K2,N) + (1-\delta))$ Same equilibrium as in the planner's problem! (First Welfare Theorem, or, "prices do magic")