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# Costs and Such - PowerPoint PPT Presentation

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

• 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.

• 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

• 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

• 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) = Q2. A Diagram

• 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

q

q+t

MC: Slope of Tangent Line

t

C(q+t)-C(q)

q

q+t

MC: Slope of Tangent Line

• 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

• Whenever AC is increasing, MC is above AC.

multiply by q(q+1)

and simplify

AC

MC

AVC

\$/unit

Q

• 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.

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

• 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 ….)

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

• 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)

• 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*

• 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

• 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*)

MC

\$/unit

MC-P

p

P-MC

q BIG

q SMALL

q*