A small private hospital performs eye operations. To perform the eye operation they need two input factors, beds (K) and labour (L). If the hospital has K number of beds and L number labour, the number of eye operations that are performed during a year, Q, given by the product function: Q = 20K^1/2L^1/2

a. Explain that this product function has the following characteristics:

i) The Marginal productivity of K and L are positive and decreasing

ii) The isoquant is "curved towards origo", meaning the MRTS is decreasing

iii) The product function has a constant scale of return with respect to the scale

b. Assume that the price the hospital has to pay per bed is r and each unit of labour costs w. Formulate first the hospitals' cost-minimizing problem if they want to perform Q operations per year the most affordable way

Then put r = 360 and w = 640 and determine the optimal solution if the hospital wishes to perform 240 operations a year. Also, illustrate the solution graphically.

c. The hospital wishes to expand to perform 400 operations a year. Explain that this can be realised by putting either:

i. K = 16 and L = 25.

ii. K = 25 and L = 16.

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

How does the optimal solution do in comparison with the possibilities in i. and ii. above? Illustrate all points graphically

d. The authorities deny the hospital to expand the number of beds and K cannot exceed 16. What will the hospital then do if they still want to perform 400 operations a year?