A film studio in Hollywood produces movies according to the

A film studio in Hollywood produces movies according to the function (yes, they can also produce fractions of movies... Think of half a movie as a B-movie or so.) q = F(K, L) = K0.5L 0.5 /100.(reads as K to the power of 0.5 times L to the power of 0.5 divided by 100) In the short run, capital (studios, gear) is fixed at a level of 100. It costs $4,000 to rent a unit of capital and $1,000 to hire a unit of labor (actors, stuntmen, camera crew etc.).

(a) What is the fixed cost? What is the variable cost as a function of output q?

(b) What is the marginal cost (MC) and the average cost (AC) of a movie? What is the average variable cost and average fixed cost?

(c) Where do the average and marginal cost curves intersect? What is the derivative of the AC curve and what value does it take at the intersection? What does it tell you about minimum average cost?

Solution

a) Fixed cost (FC) = Expenditure on Capital = 100 x 4000 = $ 400000

Variable cost = Expenditure on Labour = 1000L

q = 10 x L^0.5/100 = L^0.5/10

L = 100q²

Variable cost (VC) = 100000q²

b)

Total Cost (TC) = FC + VC = 400000 + 100000q²

Marginal cost = d(TC)/dq = 200000q

Average cost (ATC) = TC/q = 400000/q + 100000q

AVC = VC/q = 100000q

AFC = FC/q = 400000/q

c) MC and ATC would intersect when MC = ATC

400000/q + 100000q = 200000q

q = 2

Derivative of AC = d(ATC)/dq = -400000/q² + 100000

Value when q = 2:

Derivative of AC = 0

This means that Average cost is minimum when marginal cost curve intersects the average cost curve.