Now consider Marie’s problem in the long run, where her stock of cattle may vary. Set up her long run cost minimization problem and find her labor ℓ c (q) and capital k c (q) demands and cost function C(q).
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Suppose that Marie produces milk q using her own labor l and cattle k using the production function
q = f(k, l) = (k^(2/3))*(ℓ^(1/3))
Although Marie does not need to pay anyone to use either input, the
(a) Suppose that Marie’s stock of cattle is fixed at k0 = 8. Set up her short run cost minimization problem and find her labor demand ℓ(q) and cost function SC(q).
(b) Find Marie’s short run marginal cost SMC(q) and average cost SAC(q) functions and find the quantity at which short run average cost is minimized.
(c) Set up Marie’s short run profit maximization problem and find her short run supply curve q(P).
d) Now consider Marie’s problem in the long run, where her stock of cattle may vary. Set up her long run cost minimization problem and find her labor ℓ c (q) and capital k c (q) demands and cost function C(q).
(e) Set up Marie’s long run profit maximization problem and find her long run supply curve q(P).
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