Answer:
Detailed step-wise answer is given below:
Explanation:
q = min{L, 3K}
(a)
This is a fixed proportion production function which signifies that L and K are perfect complements. In this case, cost is minimized when
L = 3K, or
L/K = 3
(b)
When q = 90, we have
90 = min{L, 3K}
90 = min{L, L}
L = 90
K = L/3 = 90/3 = 30
The (L/K) ratio does not change with a change in (w/r), since the inputs are used in fixed proportion.
(c)
Generalized isocost line: C = wL + rK
C = wL + r x (L/3)
3C = 3wL + rL
3C = L x (3w + r)
L = 3C / (3w + r) [Generalized factor demand function for L]
K = [3C / (3w + r)] / 3 = C / (3w + r) [Generalized factor demand function for K]
(d)
Since q = min{L, 3K} and L = 3K,
q = min{L, L} = L
K = L/3 = q/3
Total cost (C) = wL + rK = wq + r x (q/3)
(e)
Marginal cost (MC) = dC/dq = w + (r/3)
Average cost (MC) = C/q = w + (r/3)
Therefore, MC and AC are independent of q.
