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?
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.
To find the wage rate and the rate of return to capital, we need to first determine the optimal inputs of labor and capital for each type of firm and then use these inputs to calculate the wage rate and the rate of return to capital.
For Type A firms, the technology is given as Qa = Min(1/3K, L). This means that the firm takes the minimum value of 1/3 of capital (K) and labor (L) inputs to produce output Qa. Since Type A firms earn zero profits, we can assume that the revenue from selling output equals the cost of inputs. Therefore, the revenue can be written as: $1.00 * Qa = $1.00 * Min(1/3K, L).
Similarly, for Type B firms, the technology is given as Qb = Min(1/2K, L). The revenue from selling output can be written as: $0.50 * Qb = $0.50 * Min(1/2K, L).
To find the wage rate and the rate of return to capital, we need to set up the equations for cost minimization and solve them simultaneously.
Cost minimization for Type A firm:
1/3K * R + L * W = $1.00 * Qa
Cost minimization for Type B firm:
1/2K * R + L * W = $0.50 * Qb
Here, R represents the rate of return to capital and W represents the wage rate.
Since both Type A and Type B firms earn zero profits, we can set the revenue equal to the cost of inputs in each equation.
By re-arranging the equations, we get:
R = [3 * Qa - 3 * L * W] / K
R = [2 * Qb - 2 * L * W] / K
We can then equate the two expressions for R to eliminate the variable K:
[3 * Qa - 3 * L * W] / K = [2 * Qb - 2 * L * W] / K
By canceling out the K, we get:
3 * Qa - 3 * L * W = 2 * Qb - 2 * L * W
Rearranging the equation, we find:
3 * Qa - 2 * Qb = L * W
Now, we can solve for the wage rate (W) in terms of the output quantities (Qa and Qb) and the inputs (L):
W = (3 * Qa - 2 * Qb) / L
Similarly, we can solve for the rate of return to capital (R) in terms of the output quantities (Qa and Qb) and the inputs (L):
R = [3 * Qa - 3 * L * W] / K
Now we can substitute the expression for W in terms of Qa, Qb, and L into the equation for R to determine the rate of return to capital.