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q = F(K, L) = K1/3L1/2
In long run, cost is minimized when MPL/MPK = wL/wK = 3/2
MPL = q/L = (1/2) x K1/3 / L1/2
MPK = q/K = (1/3) x L1/2 / K2/3
MPL/MPK = [(1/2) x K1/3 / L1/2] / [(1/3) x L1/2 / K2/3] = (3/2) x (K/L) = 3.2
K/L = 1
L = K
Substituting in production function with q = 32,
K1/3L1/2 = 32
L1/3L1/2 = 32
L5/6 = 32
Taking (6/5)th root,
L = 64
K = 64
MPL/MPK = K/L = 3/2
K = 3L/2
Substituting in generalized production function,
(3L/2)1/3L1/2 = q
(3/2)1/3L1/3L1/2 = q
1.14 x L5/6 = q
L5/6 = q / 1.14 = 0.88q
Taking (6/5)th root,
L = (0.88)6/5q6/5 = 0.85 x q6/5
K = (3/2) x L = 1.5 x 0.85 x q6/5 = 1.28 x q6/5
In short run, K = 27.
(27)1/3L1/2 = q
3L1/3L1/2 = q
3 x L5/6 = q
L5/6 = q / 3 = 0.33q
Taking (6/5)th root,
L = (0.33)6/5q6/5 = 0.27 x q6/5
K = 27
Total cost (TC) = L x wL + K x wK = 3L + 2K = 3 x (0.27 x q6/5) + 2 x 27
TC = 0.81 x q6/5 + 54
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