Economia V; Instructor: Marco Del Negro

Problem set 4
Solutions

The setup of the problem is as follows:

• The economy has two periods, 1 and 2.
• There is only one agent in the economy. Her income in periods 1 is equal to Y1=1. Her budget constraints in periods 1 and 2 are equal to: C1+K2=Y1, and C2=(1+r)K2 respectively. The supply of labor by the agent is inelastic and equal to 1 (N=1). The utility function of the agent is: $U(C1,C2)=\ln{C1}+\ln{C2}$
  
• There is only one firm in the economy. The firm produces using capital only and chooses the desired amount of capital in order to maximize profits:
  
  $$max_{K2} F(K2)+(1-\delta)K2-(1+r)K2$$
  
  
  where $F(K2)=2K2^{\frac12}$ and where capital depreciates completely ($\delta=1$).

a) Derive the intertemporal budget constraint of the agent from the budget constraints in periods 1 and 2:

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

substitute for C1 into the utility function and find the first order condition with respect to C2.
b) In the first order condition, express C1 and C2 as a function of K2, and r, using the budget constraints C1+K2=Y1 and C2=(1+r)K2
c) Using the first order condition, find K2 as a function of r. What you have found is the optimal amount of saving chosen by the agent, for any given interest rate. (In fact $K2=Y1-C1=\mbox{Income - Consumption = Savings}$). Plot the saving function.
d) Write the first order condition of the firm with respect to K2. Using the first order condition for the firm, find K2 as a function of r. What you have found is the optimal amount of investment chosen by the firm, for any given interest rate. (In fact, given that depreciation is complete I=K2). Plot the investment function in the same graph with the saving function.
e) Find the equilibrium real interest rate.
a) From the second budget constraint obtain:

$$K2=\frac{C2}{1+r}.$$

Substitute into the first budget constraint and obtain:

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

or $C1=Y1-\frac{C2}{1+r}$. Substitute into the utility function and obtain:

$$max_{C2} \ln(Y1-\frac{C2}{1+r})+\ln(C2)$$

The FOC is:

$$\frac1{C1}(-\frac1{1+r})+\frac1{C2}=0$$

or $\frac{C2}{C1}=(1+r)$.
b) Using the budget constraints C1+K2=Y1 and C2=(1+r)K2, the above expression becomes:

$$\frac{(1+r)K2}{Y1-K2}=(1+r)$$

c) Solving the FOC for K2 we find $K2=\frac{Y1}{2}$.
In this specific case, the saving function is not a function of the interest rate (i.e., saving is supplied inelastically by the household). d) The FOC for the firm is:

$$\frac1{\sqrt{K2}}-(1+r)=0$$

or $K2=\frac1{(1+r)^2}$.
e) In order to find the equilibrium real interest rate equate saving and investment:

$$S=\frac{Y1}{2}=I=\frac1{(1+r)^2}$$

and obtain

$$(1+r)=\sqrt(2)$$