Question

Consider a firm that runs a medical clinic where it uses the services of labor (L) and equipment (K) to produce healthcare services. The clinic technology is summarized by the production function q=0.5 L1/3 K1/3 where q is the number of patients treated each day, L is labor measured in daily number of full working shifts, and K is equipment measured in number of pieces used each day. Currently, the clinic is renting and/or owns 512 pieces of equipment. 1. What is the clinic's short run production function? 2. Compute the expression for the clinic's marginal product of labor. Does the clinic's short run production function exhibit diminishing marginal returns to labor? 3. Starting from the short run production function, write the expression for the clinic's short run conditional demand for labor L∧=L(q) 4. What is the firm's short run cost function? Now, consider the long run production function. 5. Does the long run production function exhibit increasing, decreasing, or constant returns to scale? Suppose that the firm wished to double the number of patients its doctors visited each day. 6. Should the firm increase the size of the current clinic or should it open a second clinic of the same size? Briefly discuss.

Answer 1

The short run production function incorporates the fixed number of equipment units into the production function. The marginal product of labor helps determine if there's diminishing marginal returns, and the firm's decision to expand or replicate depends on the returns to scale in the long run.

Step-by-step explanation:

Let's start by addressing the clinic's use of labor (L) and equipment (K) to produce healthcare services. The production function provided is q = 0.5 L1/3 K1/3. With 512 pieces of equipment being held constant in the short run, the short run production function is defined by inserting the fixed amount of equipment into the formula, giving us q = 0.5 L1/3 (512)1/3. The marginal product of labor (MPL) is the increase in the quantity of output produced by using one more unit of labor, holding other factors constant, and can be calculated by taking the partial derivative of q with respect to L. To determine whether there are diminishing marginal returns to labor, we should look at the MPL function. If MPL decreases as L increases, then we have diminishing marginal returns to labor.

For the long run production function where both factors are variable, to identify returns to scale, we need to increase all inputs by the same percentage and check if the output increases by a higher, lower, or the same percentage. This determines if the production function has increasing, constant, or decreasing returns to scale, respectively. Regarding the firm's decision between expanding the current clinic or opening a new one, it depends on the returns to scale. If increasing all inputs by a certain percentage leads to a more than proportional increase in output, the firm should expand the current clinic. If results are proportional, opening a second clinic might be recommended for logistical or market reach reasons.