Suppose that the economy consists of two types of firms: type A firms which produce output using a technology Qa=Min(1/3K,L) and type B firms which produce output using a technology Qb=Min(1/2K,L). Type A output sells for $1.00, and type B sells for $.50. Assuming that both firms earn zero profits, what is the wage rate and the rate of return to capital?

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economyst · Answer

Im stuck. I get a result where either the wage rate or the price of capital is negative. (I presume Min(1/3K,L) means the minimium of (1/3)*K or L).
So, to produce 1 more unit of Qa, the firm must use 3 units of K and 1 unit of L. That is MPa=3K+L. Similarly, MPb=2K+L.
Let z be the price of K and w be the wage rate. So the marginal cost of for A is 3Kz + Lw. The MC for B is 2Kz + Lw
Now then, each firm will optimize when MC=MR. So, for Firm A: 3Kz+Lw=1.00 and for B: 2Kz+Lw=0.50 Using algebra, I get Kz=0.50. Which implies one of three possibilities (for firm A) Either w must be negative, or 3Kz+Lw is greater than 1.00, which means the firm produces an infinite amount and earns boatloads of profits, or 3Kz+Lw is less than 1.00, which means the firm doesnt produce at all.

I am curious what your instructor says is the final answer.