1 .(7 points) Consider the production function Q = 5 K.3 L.6 This firm faces a w
ID: 1141723 • Letter: 1
Question
1 .(7 points) Consider the production function Q = 5 K.3 L.6 This firm faces a wage of $10/unit and a rental price of capital of $30/unit.
a. Derive the total cost function, the marginal cost function, and the average cost function for this firm when capital is fixed at 2 units. At what quantity does MC=AC in the short-run? What is the equation of the supply curve of this firm in the short-run?
b) At what values are the MC and AC for the short-run function in part d) equal to the long-run function in part a)?
Explanation / Answer
Q = 5K0.3L0.6
K = 2
L0.6 = Q /5 x1.23
L = 0.162Q
Divide powers by 0.6
L = 2.227Q1.66
TC = wL + rK
C = 10 x 2.227Q1.66 + 30 x 2
a. C = 60 + 22.27Q1.66
MC = dC/dQ = 36.97Q0.66
AC = C/Q = 60/Q + 22.27Q0.66
MC = AC when AC is minimum
36.97Q0.66 = 60/Q + 22.27Q0.66
60 = 14.70Q1.66
Q = 4.081/1.66
Q = 2.32
Equation of Supply curve = MC = 36.97Q0.66
b. In long run all costs are variableand equilibrium is at output where long run marginal cost curve intersects long run average cost curve at its minimum. And long run cost is less than short run cost.
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