Robert owns a clinic. The employs doctors (L) and equipment (K) to produce medical services. The weekly production function is q = L^(0.5)K 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.

h. What steps can Robert follow to find the clinic's long run cost function C(w,r,q)?

i. Find the clinic's conditional demand for doctors and the conditional demand for equipment.

j. Keeping in mind your findings from part i), find the clinic's total cost function.

k. If the clinic treated more patients, would the average cost of treating a patient stay the same, decline or increase?