A rm has the following long run production function x = a(K^1/2)(L^1/2)(P^1/4), where a > 0 is a constant and K, L , P are

A firm has the following long run production function x = a(K^1/2)(L^1/2)(P^1/4), where a > 0 is a constant and K, L , P are inputs of the three factors. The prices of K, L , P are Rs. 1 , Rs. 9 and Rs. 8 respectively.

a) Derive the firm’s long run total cost function , long run average cost function and long run marginal cost function. Show the workings in detail

b) In the short run P is fixed and K and L are variable. Derive the firms short run a) Total Cost Function b) Variable Cost Function c) Average Variable Cost Function d) Marginal Cost Function.

c) Obtain an equation of the form P = f(x) showing the optimum quantity of the fixed factor P for the firm to acquire as a function of the intended output x.