Costs--Where S(P) comes from © 1998,2007, 2010 by Peter Berck

The Cost Function C(q) Output. Product firm sells Input. Goods and services bought by firm and used to make output. includes: capital, labor, materials, energy C(q) is the least amount of money needed to buy inputs that will produce output q.

Fixed Costs FC are fixed costs, the costs incurred even if there is no production. FC = C(0). Firm already owns capital and must pay for it Firm has rented space and must pay rent

Average and Variable Costs VC(q) are variable costs. VC(q) = C(q) - FC. AC(q) is average cost. AC(q) = C(q)/q. AVC is average variable cost. AVC(q) = VC(q)/q. AFC is average fixed cost. AFC(q) = FC/q. limits: AFC(0) infinity and AFC(inf.) is zero.

AFC(Q) AFC Q

Marginal Cost MC(q) is marginal cost. It is the cost of making the next unit given that q units have already been produced MC(q) is approximately C(q+1) - C(q). Put the other way, C(q+1) is approximately C(q) + MC(q). The cost of making q+1 units is the cost of making q units plus marginal cost at q.

C, AC and MC in a Chart Q

C(Q) = Q 2. A Diagram

Towards a better definition of MC Per unit cost of an additional small number of units Let t be the number of additional units could be less than 1 MC(q) approximately {C(q+t) - C(q)}/t MC(q) = lim t  0 {C(q+t) - C(q)}/t

MC: Slope of Tangent Line q q+t C C(q+t)-C(q) t

MC: Slope of Tangent Line q q+t C

U Shaped Costs Now let’s assume FC is not zero AC(0) = AVC(0) + AFC(0) is unbounded AC(infinity) = AVC(infinity) + 0 SO AC and AVC get close together with large q. Let’s assume MC (at least eventually) is increasing. Fact: MC crosses AVC and AC at their minimum points

MC crosses AC at its minimum Whenever AC is increasing, MC is above AC. multiply by q(q+1) and simplify

AC(q+1)-AC(q)=(1/q+1) MC-AC AC increasing means MC above AC AC decreasing means MC below AC So AC constant means MC = AC

U Shaped Picture AC AVC MC Q $/unit

Profit Profit = P q – C(q) – = Revenue - Cost

Firm’s Output Choice Firm Behavior assumption: Firm’s choose output, q, to maximize their profits. Pure Competition assumption: Firm’s accept the market price as given and don’t believe their individual action will change it.

Theorem Firm’s either produce nothing or produce a quantity for which MC(q) = p

Candidates for Optimality  0 a b Profits could be maximal at zero or at a “flat place” like a or b. Thus finding a flat place is not enough to ensure one has found a profit maximum

Necessary and Sufficient When Profits are maximized at a non zero q, P = MC(q) P = MC(q) is necessary for profit maximization P = MC(q) is not sufficient for profit maximization (Is marijuana use necessary or sufficient for heroin use? Is milk necessary ….)

Discrete Approx. Algebra Revenue = p q  = p q - C(q) is profit We will show (within the limits of discrete approximation) that “flat spots” in the  (q) function occur where p = MC(q)

Making one less unit Now  (q*-1)  (q*) = { p (q*-1) - c(q*-1)}- { pq* - c(q*) } = -p + [ c(q*) - c(q*-1) ] = - p + mc(q*-1) so -p + mc(q*-1) is the profit lost by making one unit less than q*

Making one more unit... Now  (q*+1)  (q*) = { p (q*+1) - c(q*+1)}-[pq* - c(q*)] = p + [ c(q*) - c(q*+1) ] = p - mc(q*) so p - mc(q*) is the profit made by making one more unit

Profit Max If q* maximizes profits then profits can not go up when one more or one less unit is produced so,  (q) must be “flat” at q* No profit from one more: p - mc(q*)  0 No profit from one less: - p + mc(q*-1)  0 p- mc(q*-1)  0  p - mc(q*) since mc increasing, p-mc must = 0 between q*-1 and q* (actually happens at q*, but need calculus to show that)

q*q SMALL q BIG p MC Picture and Talk P-MC MC-P $/unit