Question

Pricing with increasing returns to scale. Consider the following production function (similar to that used earlier for ColdAway):

Y = 100 * (L - F),

where Y is output, L is labor input, and F is a fixed amount of labor that is required before the first unit of output can be produced (like a research cost). We assume that Y = 0 if L < F. Each unit of labor L costs the wage w to hire.

(a) How much does it cost (in terms of wages) to produce five units of output?

(b) More generally, how much does it cost to produce any arbitrary amount of output, Y? That is, find the cost function C(Y) that tells the minimum cost required to produce Y units of output.

(c) Show that the marginal cost dC/dY is constant (after the first unit is produced).

(d) Show that the average cost C/Y is declining.

(e) Show that if the firm charges a price P equal to marginal cost, its profits, defined as π = PY - C(Y), will be negative regardless of the level of Y.

Answer 1

Solution:

Production function: Y = 100*(L - F)

So, L = Y/100 + F

a) Let the fixed cost (like research cost) be normalized to 1. Then, Total cost = F + w*L

Using above equation, Cost = F + w*(Y/100 + F)

Cost for producing 5 units of output (that i s with Y = 5), Cost = F + w*F + w*5/100

Cost = F(1 + w) + (1/20)*w

b) The general form we have already derived in part (a), so, C(Y) = F(1+w) + (Y/100)*w

c) Marginal cost, MC(Y) =

MC(Y) = w/100, which is a constant (for Y > 0), as wage, w, is fixed

d) Average cost, AC(Y) = C(Y)/Y = ( F(1+w) + (Y/100)*w)/Y

AC(Y) = F(1+w)/Y + w/100

Notice that AC is sum of two parts: one is a constant, w/100; and other is decreasing in Y since Y is in the denominator. So, as Y increases, average cost is declining or decreasing.

e) If P = MC = w/100

Profit = (w/100)*Y - ( F(1+w) + (Y/100)*w)

Profit = w*Y/100 - F(1+w) - w*Y/100

So, profit = -F(1+w), which is negative, irrespective of level of Y (since, independent of Y)