PS7

Problem Set 7 — Solutions (Q1–Q2)

Part of: Macro-Economics Not for submission — related to material covered since the last problem set. Both questions worked here.

Optimal allocation of capital.

Assume that there are two firms in the economy with the following production functions:

where $0<\alpha\le\beta<1$, and $K_i$ is the capital allocated to firm $i$. Suppose that the economy has 100 units of capital that have to be allocated to the two firms: $K_1+K_2=100$. Finally, assume that the purpose is to maximize output in the economy, which we call the efficient allocation.

1. Write down the maximization problem of a social planner who would like to maximize output.
2. Derive the condition for the efficient allocation of capital between the two firms (you can either use a Lagrangian or substitute the constraint into the maximization problem).
3. Assume that $\alpha=\beta$ and that $A_1>A_2$. Is it efficient to allocate all the capital stock to firm 1? Explain clearly why or why not, or derive the efficient allocation.
4. Assume that $A_1=A_2=1$ and $\tfrac12=\beta>\alpha=\tfrac13$. The analytical solution is somewhat hard to derive — instead, compare aggregate GDP when (i) $K_1=K_2=50$, (ii) $K_1=49,\ K_2=51$, and (iii) $K_1=51,\ K_2=49$. Which allocation results in a higher GDP? What can you learn from this about the optimal allocation of capital between the two firms?

Efficient allocation with two factors and constant returns to scale.

Assume that there are two firms in the economy, and that they produce using the same production function:

where $i$ denotes a firm (either 1 or 2). For simplicity, assume that $A_1=A_2=1$.

1. Derive the marginal product of capital for each firm (since both firms have the same production function, you don't have to derive it twice).
2. Derive the marginal product of labor for each firm.
3. In class we discussed the idea that the efficient allocation requires that the marginal product of each factor should be equal across firms. Let's see how it works in this (special) case with two factors (NOTE: this part is not easy — if you get stuck, skip and move to the next part):
   • (a) Start with equating the marginal product of labor across the two firms. What does it imply about the ratio $\frac{K}{N}$ for the two firms (is it equal? is one greater than the other?)
   • (b) Now move to the marginal product of capital. Show that if the condition you found in the previous part holds, then it must be that the marginal product of capital is also equal across firms.
4. Consider the allocation $K_1=K_2=50$ and $N_1=N_2=50$. Is it efficient? (i.e. does it satisfy the conditions that the marginal product of labor is equal across firms and the marginal product of capital is equal across firms?)
5. Now consider the allocation $K_1=60,\ K_2=40,\ N_1=60,\ N_2=40$. Is it efficient?

One-page recap