Costs--Where S(P) comes from
© 1998,2007, 2010 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.
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.
C, AC and MC in a Chart
Q
C(Q)
0
1
2
3
4
5
AC(Q)
0
1
4
9
16
25
MC
1
2
3
4
5
1
3
5
7
9
11
C(Q) = Q2. A Diagram
30
$ or $/unit
25
20
C(Q)
AC(Q)
MC
15
10
5
0
0
2
4
Q
6
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
C (q  t )  C (q)
MC  lim t 0
t
C
t
C(q+t)-C(q)
q
q+t
MC: Slope of Tangent Line
C (q  t )  C (q)
MC  lim t 0
t
C
q
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
• SO AC and AVC get close together with large q.
• 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.
C ( q  1) C ( q )
AC ( q  1)  AC ( q ) 

q 1
q
C ( q )  MC ( q ) C ( q )


multiply by q(q+1)
q 1
q
and simplify
1
MC (q)  AC (q)

q 1
AC(q+1)-AC(q)=(1/q+1) MC-AC
• AC increasing means MC above AC
• AC decreasing means MC below AC
• So AC constant means MC = AC
U Shaped Picture
MC
AC
$/unit
AVC
Q
Profit
• Profit = P q – C(q)
–
= Revenue - Cost
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
Candidates for Optimality
p
0
a
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
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 ….)
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*, but need calculus to show that)
Picture and Talk
$/unit
MC
MC-P
p
P-MC
q SMALL
q*
q BIG