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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
- 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.

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) = limt0{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

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*)

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