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Lecture # 12a
Costs and Cost Minimization
Lecturer: Martin Paredes
1. Long-Run Cost Minimization (cont.)
 The constrained minimization problem
 Comparative statics
 Input Demands
2. Short Run Cost Minimization
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Example: Linear Production Function
 Suppose:
Q(L,K) = 10L + 2K
 Suppose:
Q0 = 200
w=€5
r=€2
 Which is the cost-minimising choice for the firm?
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Example (cont.):
 Tangency condition
 MRTSL,K = MPL = 10 = 5
MPK
2
 w = 5
r
2
 So the tangency condition is not satisfied
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K
Example: Cost Minimisation: Corner Solution
Isoquant Q = Q0
L
5
K
Example: Cost Minimisation: Corner Solution
Isoquant
Isocost line
L
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K
Example: Cost Minimisation: Corner Solution
Direction of decrease
in total cost
L
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K
Example: Cost Minimisation: Corner Solution
Cost-minimising choice
•
A
L
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 A change in the relative price of inputs changes
the slope of the isocost line.
 Assuming a diminishing marginal rate of
substitution, if there is an increase in the price of
an input:
 The cost-minimising quantity of that input
will decrease
 The cost-minimising quantity of any other
input may increase or remain constant
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 If only two inputs are used, capital and labour, and
with a diminishing MRTSL,K :
1. An increase in the wage rate must:
a. Decrease the cost-minimising quantity of labor
b. Increase the cost-minimising quantity of capital.
2. An increase in the price of capital must:
a. Decrease the cost-minimising quantity of capital
b. Increase the cost-minimising quantity of labor.
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K
Example: Change in the wage rate
Q0
0
L
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K
Example: Change in the wage rate
•
A
Q0
-w0/r
0
L
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K
Example: Change in the wage rate
•
B
•
-w1/r
0
A
Q0
-w0/r
L
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 A change in output moves the isoquant
constraint outwards.
 Definition: An expansion path is the line that
connects the cost-minimising input combinations
as output varies, holding input prices constant
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K
Example: Expansion Path with Normal Inputs
TC0/r
•
Q0
TC0/w
L
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K
Example: Expansion Path with Normal Inputs
TC1/r
TC0/r
•
•
Q0
TC0/w
Q1
TC1/w
L
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K
Example: Expansion Path with Normal Inputs
TC2/r
TC1/r
TC0/r
•
•
•
Q2
Q0
TC0/w
Q1
TC1/w
TC2/w
L
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K
Example: Expansion Path with Normal Inputs
TC2/r
Expansion path
TC1/r
TC0/r
•
•
•
Q2
Q0
TC0/w
Q1
TC1/w
TC2/w
L
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 As output increases, the quantity of input used
may increase or decrease
Definitions:
 If the cost-minimising quantities of labour and
capital rise as output rises, labour and capital are
normal inputs
 If the cost-minimising quantity of an input
decreases as the firm produces more output, the
input is an inferior input
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K
Example: Labour as an Inferior Input
TC0/r
•
Q0
TC0/w
L
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K
Example: Labour as an Inferior Input
TC1/r
TC0/r
•
Q1
•
Q0
TC0/w
TC1/w
L
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K
TC1/r
TC0/r
Example: Labour as an Inferior Input
Expansion path
•
Q1
•
Q0
TC0/w
TC1/w
L
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Definition: The input demand functions show the
cost-minimising quantity of every input for
various levels of output and input prices.
L = L*(Q,w,r)
K = K*(Q,w,r)
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Definition: The input demand curve shows the costminimising quantity of that input for various
levels of its own price.
L = L*(Q0,w,r0)
K = K*(Q0,w0,r)
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K
Example: Labor Demand
Q = Q0
0
L
w
L
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K
Example: Labor Demand
•
0
Q = Q0
w1/r
L
w
w
•
1
L1
L
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K
Example: Labor Demand
•
•w /r
2
0
Q = Q0
w1/r
L
w
w
•
2
w
•
1
L2
L1
L
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K
Example: Labor Demand
•
w3/r
•
•w /r
2
0
Q = Q0
w1/r
L
w
w
•
•
3
w
2
w
•
1
L3
L2
L1
L
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K
Example: Labor Demand
•
w3/r
•
•w /r
2
0
Q = Q0
w1/r
L
w
w
•
•
3
w
2
w
•
1
L3
L2
L1
L*(Q0,w,r0)
L
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Example:
 Suppose:
Q(L,K) = 50L0.5K0.5
 Tangency condition
 MRTSL,K = MPL = K = w
MPK L
r
=> K = w . L
r
=> This is the equation for expansion path
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Example (cont.):
 Isoquant Constraint:
 50L0.5K0.5 = Q0
=> 50L0.5(wL/r)0.5 = Q0
=>
( )
( )
L*(Q,w,r) = Q . r
50 w
K*(Q,w,r) = Q . w
50
r
0.5
0.5
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Example (cont.):
 So, for a Cobb-Douglas production function:
1. Labor and capital are both normal inputs
2. Each input is a decreasing function of its own
price.
3. Each input is an increasing function of the
price of the other input
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Definition: The firm’s short run cost minimization
problem is to choose quantities of the variable
inputs so as to minimize total costs…
 given that the firm wants to produce an
output level Q0
 under the constraint that the quantities of
some factors are fixed (i.e. cannot be changed).
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 Cost minimisation problem in the short run:
Min TC = rK0 + wL + mM subject to: Q0=F(L,M,K0)
L,M
where: M stands for raw materials
m is the price of raw materials
 Notes:
L,M are the variable inputs.
wL+mM is the total variable cost.
K0 is the fixed input
rK0 is the total fixed cost
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 Solution based on:
 Tangency Condition:
 Isoquant constraint:
MPL = MPM
w
m
Q0=F(L,M,K0)
 The demand functions are the solutions to the
short run cost minimization problem:
Ls = L(Q,K0,w,m)
Ms = M(Q,K0,w,m)
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 Hence, the short-run input demands depends on
plant size (K0).
 Suppose K0 is the long-run cost minimizing level
of capital for output level Q0…
… then, when the firm produces Q0, the short-run
input demands must yield the long run cost
minimizing levels of both variable inputs.
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1. Opportunity costs are the relevant notion of
costs for economic analysis of cost.
2. The input demand functions show how the cost
minimizing quantities of inputs vary with the
quantity of the output and the input prices.
3. Duality allows us to back out the production
function from the input demands.
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4. The short run cost minimization problem can be
solved to obtain the short run input demands.
5. The short run input demands also yield the long
run optimal quantities demanded when the
fixed factors are at their long run optimal levels.
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