Optimal Input Choice and Cost Functions. The engineering department of IMA Ltd estimated that

the production function for automatic packaging machines is of the form: Q= L^3/5K^2/5 Capital (K) and

labor (L) are the only two inputs. The firm's cost of labor (w) is equal to $6 while the cost of capital (r)

is equal to $4.

a) What type of production function is this? Determine the marginal rate of technical substitution

and provide an economic definition for it.

b) In the short run, capital is fixed at 243 units. What is the amount of labor needed to fulfill a

demand of 72 automatic machines for packaging? What is the firm's short-run total cost?

c) Compute the firm's optimal choice of factor inputs in the long run when its target remains an

output level equal to Q = 72. Illustrate your result in a graph.

d) What is the total budget needed to produce Q = 72 at the given input prices?

e) How much would the firm produce in equilibrium if it were given a budget equal to 720? Do you

need any calculations to answer this question? Explain.

f) What are the firm's long run total costs of producing a generic quantity Q? That is, what is its

total cost function? What are the firm's average and marginal cost functions?