Costs--Where S(P) comes from

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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

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Costs--Where S(P) comes from

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Costs--Where S(P) comes from

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.
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) = limt0{C(q+t) - C(q)}/t
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.

multiply by q(q+1)

and simplify

U Shaped Picture

AC

MC

AVC

\$/unit

Q

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

p

a

0

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

Discrete Approx. Algebra
• Revenue = p q
• p = p q - C(q) is profit
• We will show (within the limits of discrete approximation) that “flat spots” in the p(q) function occur where p = MC(q)
Making one less unit
• Now p(q*-1) - p(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 p(q*+1) - p(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, p(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*)
Picture and Talk

MC

\$/unit

MC-P

p

P-MC

q BIG

q SMALL

q*