Consider a firm with the following production function :

f(x1, x2) = min {2x1, x2} ,

where x1 and x2 are the amounts of input 1 and input 2, respectively. Let the amount of output be denoted by y while the prices of input 1 and input 2 be denoted by w1 and w2, respectively. Assume the firm is able to choose the amount of either input (i.e., it is in the long-run circumstance).

a) State the firm's cost minimization problem.

b) Derive the equations for isoquants and isocost curves. Show them on a diagram.

c) Derive the firm's conditional input demands. d) Derive the firm's total cost function.

Suppose now that the firm is in a short-run circumstance. In particular, the amount of input 2 that can be used for production is fixed at K > 0, i.e., x2 = K.

e)Derive the firm's short-run total cost function.

f)Show the firm's long-run and short-run total cost functions on the same diagram (put

the amount of output produced, y, on the horizontal axis).