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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

Costs--Where S(P) comes from

© 1998,2007 by Peter Berck

the cost function c q
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
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
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)

AFC

Q

marginal cost
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
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
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
MC crosses AC at its minimum
  • Whenever AC is increasing, MC is above AC.

multiply by q(q+1)

and simplify

u shaped picture
U Shaped Picture

AC

MC

AVC

$/unit

Q

firm s output choice
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
Theorem
  • Firm’s either produce nothing or produce a quantity for which MC(q) = p
necessary and sufficient
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
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
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
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
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
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
Picture and Talk

MC

$/unit

MC-P

p

P-MC

q BIG

q SMALL

q*

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