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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 l.jpg

Costs--Where S(P) comes from

© 1998,2007 by Peter Berck


The cost function c q l.jpg
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 l.jpg
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 l.jpg
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 l.jpg
AFC(Q)

AFC

Q


Marginal cost l.jpg
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 q q 2 a diagram l.jpg
C(Q) = Q2. A Diagram


Towards a better definition of mc l.jpg
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


Mc slope of tangent line l.jpg

C

q

q+t

MC: Slope of Tangent Line

t

C(q+t)-C(q)


Mc slope of tangent line11 l.jpg

C

q

q+t

MC: Slope of Tangent Line


U shaped costs l.jpg
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 l.jpg
MC crosses AC at its minimum

  • Whenever AC is increasing, MC is above AC.

multiply by q(q+1)

and simplify


U shaped picture l.jpg
U Shaped Picture

AC

MC

AVC

$/unit

Q


Firm s output choice l.jpg
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 l.jpg
Theorem

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


Necessary and sufficient l.jpg
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 l.jpg
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 l.jpg
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 l.jpg
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 l.jpg
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 l.jpg
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 l.jpg
Picture and Talk

MC

$/unit

MC-P

p

P-MC

q BIG

q SMALL

q*


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