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