Lecture 5: From Production Functions to Cost Functions
Economic Costs and Cost Functions
Cost Functions
Example of a Cost Function: TC(Q)=1000 + 50Q + 10Q2
Costs (FC) = 1000
Variable Costs (VC) = 50Q + 10Q2
Average Cost (AC) = TC(Q)/Q=1000/Q+50+10Q
Marginal Cost (MC) = 50+20Q
Minimum of average cost curve is where MC=AC
Average costs are minimized when Q=10.
MC
AC
$250
Q=10
output
Isocost Lines: combinations of Labor and
Capital where the total input cost is constant
Cost (C) = wL + rK
Capital
(K)
K= C/r - w/rL
The slope of the
isocosts lines is the
ratio of the price of
Labor to price of
Capital (-w/r)
Labor (L)
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Going from Production Functions to Cost
Functions (for the general case where inputs
have some degree of subsitutability)
Labor becomes
relatively more
expensive
Capital
(K)
K
#
K*
Q=60
L#
L*
The slope of this isocost
line indicates that the
price of labor is
relatively inexpensive:
Labor (L)
Mathematical Example:
Q=K.5L.5 (Production Function)
w=4, r=64
MRTS = w/r ⇒ K/L=4/64 ⇒ 64K=4L ⇒ L=16K
Q= K.5L.5 = K.5(16K).5 = 4K ⇒ K*=Q/4
L*=16K*=4Q
Cost (C) = 4L* + 64K* = 4(4Q)+64(Q/4) =32Q
⇒ CRS, Marginal cost = 32, average cost=32
From Production Functions to Cost Functions
Continued: A general result for the case when
Production function is Q=KaLb.
•When a+b=1, there are constant returns to scale, i.e., C(Q)=cQ,
where “c” is a constant. (Intuition is that when inputs double,
production doubles, and hence costs double.)
•When a+b<1, there are decreasing returns to scale. (Intuition is
that when inputs double, production increases by less than
100%; hence cost of increasing output by 100% requires more
than a 100% increase in inputs.)
When a+b>1, there are increasing returns to scale. (When
inputs double, production increases by more than 100%; hence
cost of increasing output by 100% requires less than a 100%
increase in inputs.)
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Economies of Scale
) occur when average costs decline with output
Economi
es
of
Scale
$
Consta
nt
Returns
To
Scale
Diseconomie
s
of
Scale
Output
Natural Monopolies (declining average costs) Water,
postal services, telecommunications, electricity
Economies of Scope – when it is cheaper for
one firm to produce both goods X & Y than for
two firms to produce the goods.
Formally, TC(X,Y)<TC(X,0) +TC(0,Y) or
TC(X,Y)-TC(0,Y)<TC(X,0)-TC(0,0)
Example from BDS: (set up costs)
TC(0,Y)=$100m+.2Y
TC(X,0)= $50m+.05x
TC(X,Y)=$120m+.2Y+.05X
Examples of Economies of Scope
Hub & Scope Networks in the Airline Industry
Break-up of AT&T -- One key issue was whether
AT&T’s cost function exhibited economies of scope (the
two products were local & long distance service.)
Break-up of Microsoft? – Does Microsoft’s cost
function exhibit economies of scope (the two products
are operating systems & applications software.)
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