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Chapter 7: Costs and Cost Minimization
•Consumers purchase GOODS to maximize
their utility.
•This consumption depends upon a
consumer’s INCOME and the PRICE of the
goods
•Firms purchase INPUTS to produce
OUTPUT
•This output depends upon the firm’s
FUNDS and the PRICE of the inputs
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Chapter 7: Costs and Cost Minimization
In this chapter we will cover:
7.2 Isocost Lines
7.3 Cost Minimization
7.4 Short-Run Cost Minimization
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One of the goals of a firm is to produce
output at a minimum cost.
This minimization goal can be carried out in
two situations:
1) The long run (where all inputs are variable)
2) The short run (where some inputs are not
variable)
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Suppose that a firm’s owners wish to
minimize costs…
Let the desired output be Q0
Technology: Q = f(L,K)
Owner’s problem: min TC = rK + wL
K,L
Subject to Q0 = f(L,K)
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From the firm’s cost equation:
TC0 = rK + wL
One can obtain the formula for the ISOCOST LINE:
K = TC0/r – (w/r)L
The isocost line graphically depicts all
combinations of inputs (labour and capital) that
carry the same cost.
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K
Example: Isocost Lines
TC2/r
Direction of increase
in total cost
TC1/r
Slope = -w/r
TC0/r
TC0/w TC1/w TC2/w
L
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Isocost curves are similar to budget lines,
and the tangency condition of firms is also
similar to the tangency condition of
consumers:
MRTSL,K = -MPL/MPK = -w/r
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K Example: Cost Minimization
TC2/r
TC1/r
•
Cost inefficient point for Q0
Cost minimization point for Q0
TC0/r
•
Isoquant Q = Q0
TC0/w TC1/w TC2/w
L
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1) Tangency Condition
- MPL/MPK = w/r
-gives relationship between L and K
2) Substitute into Production Function
-solves for L and K
3) Calculate Total Cost
4) Conclude
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Q = 50L1/2K1/2
MPL = 25K1/2/L1/2
MPK = 25L1/2/K1/2
w = $5 r = $20
Q0 = 1000
1) Tangency:
MPL/MPK = w/r
(25K1/2/L1/2)/(25L1/2/K1/2)=w/r
K/L = 5/20
L=4K
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2) Substitution:
1000 = 50L1/2K1/2
1000 = 50(4K)1/2K1/2
1000=100K
K = 10
L = 4K
L = 4(10)
L = 40
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3) Total Cost:
TC0 = rK + wL
TC0 = 20(10) + 5(40)
TC0 = 400
4) Conclude
Cost is minimized at $400 when labour is 40 and
capital is 10.
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K Example: Interior Solution
400/r
Cost minimization point
10
•
40
Isoquant Q = 1000
400/w
L
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Q = 10L + 2K MP = 10
L
w = $5
MPK = 2
r = $2
Q0 = 200
1) Tangency Condition:
MPL/MPK = w/r
10/2=5/2
10=5????
Tangency condition fails!
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We can rewrite the tangency condition:
MPL/w = MPK /r
-the productivity per dollar for labour is equal to
the productivity per dollar for capital
-but here:
MPL/w = 10/5 > MPK /r = 2/2
…the “bang for the buck” in labour is ALWAYS
larger than the “bang for the buck” in capital…
So you would only use labor:
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2) Substitution (K=0)
Q = 10L + 2K
200 = 10L + 2(0)
20 = L
3) Total Cost
4) Conclude
Cost is minimized at
$50 when labour is
5 and capital is
zero.
TC0 = rK + wL
TC0 = 2(0) + 5(10)
TC0 = 50
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Example: Cost Minimization: Corner Solution
K
Isoquant Q = Q0
•
Cost-minimizing
input combination
L
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Comparative Statistics
•The isocost line depends upon input prices
and desired output
•Any change in input prices or output will
shift the isocost line
•This shift will cause changes in the optimal
choice of inputs
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A change in the relative price of inputs changes
the slope of the isocost line. If MRTSL,K is
decreasing,
An increase in wage:
-decreases the cost minimizing quantity of labour
-increases the cost minimizing quantity of capital
An increase in rent
-decreases the cost minimizing quantity of capital
-increases the cost minimizing quantity of labour.
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K
Example: Change in Relative Prices of Inputs
Cost minimizing input combination w=2, r=1
•
Cost minimizing input combination, w=1
r=1
•
0
Isoquant Q = Q0
L
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Originally, MicroCorp faced input prices of $10 for
both labor and capital. MicroCorp has a contract
with its parent company, Econosoft, to produce 100
units a day through the production function:
Q=2(LK)1/2
MPL=(K/L)1/2 MPK=(L/K)1/2
If the price of labour increased to $40, calculate the
effect on capital and labour.
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Originally :
MPL w

MPK r
K / L 10

L / K 10
KL
Q  2 KL
Q  2 KK
100  2 K
50  K
50  L
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After Change :
Q  2 KL
MPL w

MPK r
Q  2 4 LL
K / L 40

L / K 10
K  4L
100  4 L
25  L
100  K
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If the price of labour quadruples
from $10 to $40…
Labour will be cut in half, from 50 to 25
Capital will double, from 50 to 100
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An increase in Q0 moves the isoquant Northeast.
The cost minimizing input combinations, as Q0
varies, trace out the expansion path
If the cost minimizing quantities of labor and
capital rise as output rises, labor and capital are
normal inputs
If the cost minimizing quantity of an input
decreases as the firm produces more output,
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the input is called an inferior input
Example: An Expansion Path
K
TC2/r
TC1/r
Expansion path, normal inputs
•
TC0/r
•
•
Isoquant Q = Q0
TC0/w TC1/w TC2/w
L
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Example: An Expansion Path
K
TC2/r
TC1/r
Expansion path, labour is inferior
•
•
TC1/w TC2/w
L
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Originally, MicroCorp faced input prices of $10 for
both labor and capital. MicroCorp has a contract
with its parent company, Econosoft, to produce 100
units a day through the production function:
Q=2(LK)1/2
MPL=(K/L)1/2 MPK=(L/K)1/2
If Econosoft demanded 200 units, how would labour
and capital change?
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Tangency :
MPL w

MPK r
K / L 10

L / K 10
KL
Q  2 KL
Q  2 KK
200  2 K
100  K
100  L
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If the output required doubled from
100 to 200..
Labour will double, from 50 to 100
Capital will double, from 50 to 100
(Constant Returns to Scale)
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Input Demand Functions
•The demand curve for INPUTS is a
schedule of amount of input demanded at
each given price level
•This demand curve is derived from each
individual firm minimizing costs:
Definition: The cost minimizing quantities of
labor and capital for various levels of Q, w and r
are the input demand functions.
L = L*(Q,w,r)
K = K*(Q,w,r)
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K
Example: Labour Demand Function
•
W3/r
•
•
Q = Q0
W2/r
0
W1/r
L
w
•
L1
•
L2
When input prices (wage
and rent, etc) change, the
firm maximizes using
different combinations of
inputs.
As the price of inputs goes
up, the firm uses LESS of
that input, as seen in the
input demand curve
•
L3
L*(Q0,w,r)
L
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K
•
•
•
•
• •
0
w
Q = Q0
Q = Q1
L
This will result in a shift in
the input demand curve.
•
•
•
•
•
•
L1
A change in the quantity
produced will shift the
isoquant curve.
L2
L*(Q0,w,r)
L*(Q1,w,r)
L3
L
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1) Use the tangency condition to find the
relationship between inputs:
MPL/MPK = w/r
K=f(L) or L=f(K)
2) Substitute above into production function
and solve for other variable:
Q=f(L,K), K=f(L) =>L=f(Q)
Q=f(L,K), L=f(K) =>K=f(Q)
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Q = 50L1/2K1/2
MPL=25(K/L)1/2, MPK=25(L/K)1/2
1) Tangency Condition:
MPL/MPK = w/r
K/L = w/r
K=(w/r)L
or
L=(r/w)K
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2) Production Function
Q0 = 50L1/2K1/2
Q0 = 50L1/2[(w/r)L]1/2
L*(Q,w,r) = (Q0/50)(r/w)1/2
or
Q0 = 50 [(r/w)K]1/2K1/2
K*(Q,w,r) = (Q0/50)(w/r)1/2
• Labor and capital are both normal inputs
• Labor is a decreasing function of w
• Labor is an increasing function of r
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•Price elasticity of demand can be calculated
for inputs similar to outputs:
 
% QInput
% PInPut
L / L
L ,W 
w / w
K / K
K , r 
r / r
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JonTech produces the not-so-popular J-Pod.
JonTech faces the following situation:
Q*=5(KL)1/2=100
MRTS=K/L.
w=$20 and r=$20
Calculate the Elasticity of Demand for Labour if
wages drop to $5.
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Initially:
MRTS=K/L=w/r
K=20L/20
K=L
Q=5(KL)1/2
100=5K
20=K=L
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After Wage Change:
MRTS=K/L=w/r
K=5L/20
4K=L
Q=5(KL)1/2
100=10K
10=K
40=L
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Price Elasticity of Labour Demand:
L / L
L ,W 
w / w
( 40  20) /( 40  20) / 2
L ,W 
(5  20) /( 5  20) / 2)
20 / 30
 250
L ,W 

 0.55
 15 / 12.5
450
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7.4 Short Run Cost Minimization
Cost minimization occurs in the short run when one input
(generally capital) is fixed (K*).
Total variable cost is the amount spent on the variable
input(s) (ie: wL)
-this cost is nonsunk (can be avoided)
Total fixed cost is the amount spent on fixed inputs (ie: rK*)
-if this cost cannot be avoided, it is sunk
-if this cost can be avoided, it is nonsunk
(ie: rent factory to another firm)
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Short Run Cost Minimization
Cost minimization in the short run is easy:
Min TC=wL+rK*
L
s.t. the constraint Q=f(L,K*)
Where K* is fixed.
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Short Run Cost Minimization
Example:
Minimize the cost to build 80 units if Q=2(KL)1/2 and
K=25.
Q=2(KL)1/2
80=2(25L)1/2
80=10(L)1/2
8=(L)1/2
64=L
Notice that price doesn’t matter.
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K
Short Run Cost Minimization
TC2/r
TC1/r
Long-Run Cost Minimization
•
K*
Short-Run Cost Minimization
•
TC1/w TC2/w
L
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Short Run Expansion Path
Choosing 1 input in the short run doesn’t depend on
prices, but it does depend on quantity produced.
The short run expansion path shows the increased
demand for labour as quantity produced increases:
(next slide)
The demand for inputs will therefore vary according to
quantity produced. (The demand curve for inputs
shifts when production changes)
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K
Example: Short and Long Run Expansion Paths
TC2/r
Long Run Expansion Path
TC1/r
TC0/r
K*
•
•
•
•
•
Short Run Expansion Path
TC0/w TC1/w TC2/w
L
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Short Run and Many Inputs
If the Short-Run Minimization problem has 1 fixed input
and 2 or more variable inputs, it is handled similarly to the
long run situation:
MPInput A
PInput A

MPInput B
PInput B
etc.
ie : You feed your worke rs :
MPL MPFood

w
PFood
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Chapter 7 Key Concepts
The Isocost line gives all combinations of
inputs that have the same cost
Costs are minimized when the Isocost line
is tangent to the Isoquant
When input costs change, the
minimization point (and minimum cost)
changes
When required output changes, the
minimization point (and minimum cost)
changes
The creates the expansion path
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Chapter 7 Key Concepts
Individual firm choice drives input demand
As input prices change, input demanded
changes
There are price elasticities of inputs
In the short run, at least one factor is fixed
Short run expansion paths differ from long
run expansion paths
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