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 fi xed amount of labor that is required before the fi rst unit of output can be produced (like a research cost). We assume that Y = 0 if L 6 F. Each unit of labor L costs the wage w to hire.

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

(b) More generally, how much does it cost to produce any arbitrary amount of output, Y? That is, fi nd 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 fi rst unit is produced).

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

(e) Show that if the fi rm charges a price P equal to marginal cost, its profi ts, defi ned as p = PY - C(Y), will be negative regardless of the level of Y.