Charlie likes to make candy at his factory, but needs two inputs: (1) Labor in the form of OompaLoompas, which we denote with L, and (2) Capital which we denote with K.

The production function is y = (L^1/2 + K^1/2)²

(a) In the short run, Capital is fixed at K = 64 candy machines. As the manager for Charlie's candy factory, how much labor should you hire if Charlie asks for y units of candy?

(b) What are the short run costs to produce y, as a function of wL and wK? (In other words, what is C(y, wL, wK)?)

(c) In the long run, Charlie can alter his choice of Capital - in this case, what would the optimal input bundles be as a function of y, wL, and wK? In other words, what are the input demand functions L(y, wL, wK) and K(y, wL, wK)?

(d) For this question, assume wL = 4 and wK = 1. What are the long run total costs as a function of y?

(e) For this question, assume wL = 4 and wK = 1. What can you infer about the returns to scale from the long run total cost function? How does long run average total cost change as you increase output y?