1,suppose Q=100k½ w=40 and r=60
a, find the quantity of the labor and capital that the firm should demand in order tominimize the cost of producing 2000 units of output?
b, what is this minimum cost?
a. To minimize the cost of producing 2000 units of output, we can set up the cost function for the firm:
Cost = rK + wL
Where Cost is the total cost, r is the rental price of capital, K is the quantity of capital, w is the wage rate, and L is the quantity of labor.
Given Q = 100sqrt(K), we can solve for K in terms of Q:
Q = 100sqrt(K)
Q/100 = sqrt(K)
(K^2)/(100^2) = K
K^2 = (Q^2)/(100^2)
K = Q/100
Now, substituting this value of K into the cost function, we have:
Cost = r(Q/100) + wL
Cost = (60/100)(Q/100) + 40L
Cost = 0.6(Q/100) + 40L
Cost = 0.006Q + 40L
To minimize the cost, we can set the partial derivatives of the cost function with respect to Q and L equal to zero:
dCost/dQ = 0.006 = 0
dCost/dL = 40 = 0
Solving for Q and L:
0.006Q = 0
Q = 0
40L = 0
L = 0
Therefore, the firm should demand no labor or capital to minimize the cost of producing 2000 units of output.
b. Since there is no demand for labor or capital, the minimum cost of production would be zero.