Question

Please answer Question F, G, H

Green is looking to establish a new market producing energy gummy bears. After intensive research Green has found the production function is given by q =75K^1/3*L^2/3, where q is thousands of pounds of energy gummy bears, known as GreenergiesTM, K is capital, L is labor, r is the price of capital, w is the price of labor, and q is per year.

a. Find the long-run input demand functions for K and L as a function of r,w and q.

b. Use K(q,w,r) and L(q,w,r) to find the long-run cost function C(q,w,r). Discuss why someone might want to know this functional relationship.

c. What returns to scale does C(q,w,r) exhibit? How do you know and why might the long-run returns to scale make sense in this case?

d. The price of capital is given by r= $100,000, the wage is w= $25,000, and market research indicates that the market will support q= 300. What are the long-run cost minimizing levels of K and L?

e. What is the cost of producing q= 300 units of output?

f. Green decides to invest and purchases the optimal level of capital, find the short-run production function q = f(L).

g. Use the short-run production function to find the short-run variable costs, fixed cost, total cost, marginal cost, average cost, and average variable cost functions?

h. What is the conceptual difference between the long-run and short-run total cost functions?

Answer 1

(Part f)

From previous parts, Optimal level of K = 1 unit

75K^1/3L^2/3 = q

75 x (1)^1/3L^2/3 = q

75L^2/3 = q, i.e.

q = 75L^2/3

(Part g)

Since q = 75L^2/3,

L^2/3 = (q/75)

L = (q/75)^3/2

(I) Short run variable cost (TVC) = wL = w x [(q/75)^3/2]

[When w = 25,000, TVC = 25,000 x (q/75)^3/2]

(II) Fixed cost = rK = r [since K = 1]

[When r = 100,000, FC = 100,000]

(III) Short run total cost (TC) = wL + rK = w x [(q/75)^3/2] + r

[When w = 25,000 and r = 100,000, TC = 25,000 x (q/75)^3/2 + 100,000]

(IV) MC = dTC/dq = w x (3/2) x (1/75)^3/2 x (q)^1/2  

[When w = 25,000, MC = 25,000 x (1/75)^3/2 x (q)^1/2]

(V) AC = TC/q = [w x (1/75)^3/2 x (q)^1/2] + (r/q)

[When w = 25,000 and r = 100,000, MC = [25,000 x (1/75)^3/2 x (q)^1/2] + (100,000/q)]

(VI) AVC = TVC/q = w x (1/75)^3/2 x (q)^1/2

[When w = 25,000, AVC = [25,000 x (1/75)^3/2 x (q)^1/2​​​​​​​]

(h)

In short run, capital being fixed, there is existence of fixed cost. But in long run, all costs are variable and so, fixed costs do not exist.