Costs--Where S(P) comes from

© 1998,2007 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

C(Q) AC(Q) MC0 0 11 1 1 32 4 2 53 9 3 74 16 4 95 25 5 11

Q

C(Q) = Q2. A Diagram

0

5

10

15

20

25

30

0 2 4 6

Q

$ o

r $/u

nit

C(Q)AC(Q)MC

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) = limt0 {C(q+t) - C(q)}/t

MC: Slope of Tangent Line

qq+t

C

C(q+t)-C(q)

t

t

qCtqCMC t

)()(lim 0

MC: Slope of Tangent Line

qq+t

C

t

qCtqCMC t

)()(lim 0

U Shaped Costs

• Now let’s assume FC is not zero• AC(0) = AVC(0) + AFC(0) is unbounded• AC(infinity) = AVC(infinity) + 0

• 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.

)()(1

1

)(

1

)()(

)(

1

)1()()1(

qACqMCq

q

qC

q

qMCqC

q

qC

q

qCqACqAC

multiply by q(q+1)and simplify

U Shaped Picture

AC

AVCMC

Q

$/unit

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

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 ….)

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 toensure one has found a profit maximum

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*)

q*q SMALL q BIG

p

MC

Picture and Talk

P-MC

MC-P

$/u

nit