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Winter 2024-2025 Econ 465, Problem Set 1

Problem 1 (2pt) As discussed in class, A firm’s cost minimization problem in the short run is as follows: for a given level of output y,

where V denotes a variable input and K denotes quantity of capital (fixed in the short run). F denotes fixed costs, and Q(·, ·) denotes the production function for the firm. P v and r are the prices of the variable inputs and capital, respectively.

(a) (1pt) In class we made the claim that the marginal cost at q is equal to the Lagrange multiplier for this cost minimization problem. Show why that is true. The TA will provide hints to this question on Friday in his session (Jan 17th, 2025).

(b) (1pt) Show that the markup, i.e., is equal to the elasticity of output with respect to the variable input times the reciprocal of the revenue share of variable costs, i.e.,