ECON 802 Study Guide - Final Guide: Indirect Utility Function, Lagrange Multiplier, Expenditure Function

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Read each question carefully and try to use all of the information provided. Consider a competitive firm with one output and n inputs. Assume all functions are differentiable, all solutions are interior, and all questions refer to the long run. Prove mathematically that the following statements are true. (a) (b) (c) (a) (b) (c) (a) (b) The firm"s profit function is convex as a function of the output price when the input price vector is held constant. If the input vector x* maximizes profit and the resulting profit is positive, then the firm has locally decreasing returns to scale at x*. The lagrange multiplier in the firm"s cost minimization problem is marginal cost. Consider a competitive firm with one output and 2 inputs. Assume the functions in parts (a) and (b) are differentiable, all solutions are interior, and all questions refer to the long run. Prove mathematically that the following statements are true.

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