Costs--Where S(P) comes from © 1998,2007, 2010 by Peter Berck.
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Transcript of Costs--Where S(P) comes from © 1998,2007, 2010 by Peter Berck.
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
C(Q) AC(Q) MC0 0 11 1 1 32 4 2 53 9 3 74 16 4 95 25 5 11
Q
C(Q) = Q2. A Diagram
0
5
10
15
20
25
30
0 2 4 6
Q
$ o
r $/u
nit
C(Q)AC(Q)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) = limt0 {C(q+t) - C(q)}/t
MC: Slope of Tangent Line
qq+t
C
C(q+t)-C(q)
t
t
qCtqCMC t
)()(lim 0
MC: Slope of Tangent Line
qq+t
C
t
qCtqCMC t
)()(lim 0
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.
)()(1
1
)(
1
)()(
)(
1
)1()()1(
qACqMCq
q
qC
q
qMCqC
q
qC
q
qCqACqAC
multiply by q(q+1)and simplify
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
AC
AVCMC
Q
$/unit
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 toensure 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)
q*q SMALL q BIG
p
MC
Picture and Talk
P-MC
MC-P
$/u
nit