Problem 2 (Q3 From PSet 5) Cost, when all inputs are variable In class, we observed how the problems of the firm are isomorphic to the problems of the consumer - we use the same mathematical tools to solve both. In this exercise, you will consider a firm that whishes to find the cheapest technique to produce a target level of output and then evaluate how its cost of production would vary in response to changes in input prices and output. Formally (in math terms) the problem is similar to the problem of a consumer who wishes to find the cheapest bundle to enjoy a target standard of living (or utility) and then evaluate how his expenditure would vary in response to changes in prices and utility level. As you work through this exercise, take notice of the formal similarity and the conceptual difference between isoquants and indifference curves; marginal products and marginal utilities, the Marginal Rate of Technical Substitution and the Marginal Rate of Substitution; the cost function and the expenditure function. Robert owns a clinic. The employs doctors (L) and equipment (K) to produce medical services. The weekly production function is q = LOK where L is measured in number of doctors, K in pieces of equipment, and q is the number of patients treated each week. Each week doctors are paid a gross salary of $4,000 and the weekly rental cost of a piece of equipment is $8,000. a. Does the clinic's production function exhibit constant, increasing or decreasing returns to scale? b. What is the clinic's marginal product of labor? What is the clinic's marginal product of capital? At the current input prices, what is the lowest cost of seeing/treating 1,000 patients each week? b.1 Write the cost minimization problem of the clinic. b.2 Write the clinic's Lagrangian function. b.3 Compute the first order conditions of the cost minimization problem. b.4 Solve the first order conditions for the endogenous variables L and K. b.5 Keeping the result from part b.4. in mind, compute the cost of seeing 1,000 patients. c. Illustrate the cost minimization problem in an isoquant/isocost diagram. Now, analyze the clinic's cost of production. d. What share of the overall cost comes from doctors' wages and what share comes from renting equipment? What relationship do you find between these shares and the production function? Robert is making long term plans and he is interested in the way the clinic's weekly cost would change if they increased the number of patients treated each week. Recitation - Week 5 & 6