A small private hospital performs eye surgeries. To have eye surgery performed, 2 input factors are needed, beds (K) and labor (L). If the hospital has K number of beds and L units of labor, the number of eye surgeries performed during a year, Q, is given by the product function:

Q = 20 K12 L12

a) Explain that this product feature has the following features:

i. The marginal productivity of K and L is positive and decreasing

ii. The isoquants are "curved towards the origin", ie the marginal technical

substitution fraction, MRTS, is decreasing

iii. The product function has a constant return with respect to. scale

b) Assume that the price the hospital must pay per. bed is r and that each unit of labor costs w.

First formulate the hospital's cost minimization problem if it wants to perform Q operations per. years as reasonably as possible.

Then set r = 360 and w = 640 and determine the optimal solution if the hospital wants to perform 240 operations per. year. Also illustrate the solution graphically.

c) The hospital wants to expand to perform 400 operations a year. Explain that this can be realized by setting either:

i. K = 16 & L = 25

ii. K = 25 & L = 16

but that none of these is the cost-minimizing solution. Then formulate the optimal solution (when Q = 400, r = 360, w = 640).

How does the optimal solution come out compared to options i) and ii) above)? Illustrate all points graphically.

d) The authorities refuse the hospital to expand the number of beds and K can not exceed 16. What will the hospital do if it still wants to perform 400 operations a year?