CLASS #10: THE INTEREST RATE IN A CLOSED ECONOMY WITH PRODUCTION

From last class, we learned that there exists an interest rate such that

Savings = Investment

• what does the equilibrium interest rate depend upon?
• what does investment in equilibrium depend upon?

In order to address these questions, we will focus on a model with complete depreciation (simple to solve): capital disappears from one period to the other ($\delta =1$). Therefore

Investment = Capital Stock.

Since the "social planner"'s equilibrium and the "market"'s equilibrium are the same, we are going to solve the first one (because it is easier).

Assuming logarithmic utility, and Cobb-Douglas production function, the planner's problem is:

$$max_{\{C1,C2\}} ln(C1) +\beta ln(C2)$$

subject to the constraints:

K2+C1=Y1

and

$$C2 = A K2^\alpha$$

We know from last time that the FOC deliver the relationship MRS = MRT:

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

which implies

$$\frac{C2}{C1} = \beta A\alpha K2^{\alpha-1}$$

Using $C2 = A K2^\alpha$ and K2=Y1-C1 we obtain:

$$K2 =\frac{\alpha\beta}{1+\alpha\beta} Y1$$

and

$$C1 =\frac{1}{1+\alpha\beta} Y1$$

Implications of the above formula: the household invests a fraction $\frac{\alpha\beta}{1+\alpha\beta}$ of the available resources, and consumes the remainder.
The formula says that:

• Investment is pro-cyclical
• The fraction of resources invested depends positively on:
  • the share of capital in the production function $\alpha$. What is the intuition?
  • the time discount $\beta$. What is the intuition?

The equilibrium (real) interest rate

As we learned in the previous class, in equilibrium the interest rate is determined by the relationship:

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

(we are using the fact that the planner's equilibrium and the market equilibrium are the same)

In this specific model:

$$(1+r) = A\alpha K2^{\alpha-1}$$

Since $K2 =\frac{\alpha\beta}{1+\alpha\beta} Y1$, then the interest rate is given by the formula:

$$(1+r)=A\alpha (\frac{\alpha\beta}{1+\alpha\beta} Y1)^{\alpha-1}$$

How does the equilibrium real interest rate changes with:

• technology shocks (A)?
• the available amount of resources (Y1)?
• the share of capital in the production function $\alpha$?

take logs:

$\begin{array}{lll} r & \simeq & \ln(1+r) \\ & = & \ln{A}+(\alpha-1)\ln{Y1}+\ln{\alpha}+(\alpha-1)\ln{\frac{\alpha\beta}{1+\alpha\beta}} \end{array}$

The real interest rate:

• increases with a positive technology shock
• low during booms, high during recessions
• increases with $\alpha$

Investment and the Government

Introduce the government in the model, assuming for simplicity that the government spends an amount of resources G1 in period 1 only. The planner's problem is now:

$$max_{\{C1,C2\}} ln(C1) +\beta ln(C2)$$

subject to the constraints:

K2+C1=Y1-G1

and

$$C2 = A K2^\alpha$$

How do we solve this model? Call $\bar{Y1}\equiv Y1-G1$. The problem is exactly the same one studied above. The solution is:

$$K2 =\frac{\alpha\beta}{1+\alpha\beta} \bar{Y1}$$

$$r \simeq \ln{A}+(\alpha-1)\ln(\bar{Y1})+\ln{\alpha}+(\alpha-1)\ln{\frac{\alpha\beta}{1+\alpha\beta}}$$

What happens to Investment? and to the interest rate?

$$K2 =\frac{\alpha\beta}{1+\alpha\beta} (Y1-G1)$$

investment decreases: Crowding Out.
Intuition? Permanent income: if your resources today decline because of government spending, you do not want to reduce your consumption today only, but also your consumption tomorrow. So you reduce your investment.

$$r \simeq \ln{A}+(\alpha-1)\ln(Y1-G1)+\ln{\alpha}+(\alpha-1)\ln{\frac{\alpha\beta}{1+\alpha\beta}}$$

The interest rate increases. Why? In the graph seen in the last class, the supply of savings is shifting to the left. The demand for investment by the firm is unchanged. As a consequence, the interest rate increases.