Cost Minimization and Cost Curves
Econ 212 Lecture 12
Tianyi Wang
Queen’s Univeristy
Winter 2013
Tianyi Wang (Queen’s Univeristy)
Costs
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Introduction
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We want to model how …rms make use of technologies.
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Take as given the objective of …rms: pro…t maximization.
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One period problem, no risks.
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Opportunity cost, sunk cost and …xed cost.
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Start with Cost Minimization, for
I
I
itself is interesting.
can not max pro…t without minimize cost
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Pro…t = Revenue - Cost = P Q C (Q )
Suppost cost not minimized for a set of inputs, given Q, can inprove
pro…t by reducing cost.
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Cost Minimization in Long-run
I
Suppost …rm has access to technology that relates Labor and Capital
to output.
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Ask what’s the cheapest way to produce a certain level (Q) of
output.
min wL + rK
L,K
s.t. Q = Q (L, K )
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Long-run (less constrained) versus Short-run (more constrained).
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Use Isocost lines to represent …rm’s objective.
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Combinations of inputs that yield the same cost.
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Optimal Input Choices
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Graphically, start with arbitrary point (A) on Isoquant, …nd the
Isocost that goes through A.
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Ask if can …nd a lower Isocost. Move to B.
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Stop when Isocost just touches Isoquant (tangency). At C.
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Read the optimal inputs.
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Be aware of the di¤erence from consumer’s utility maximization
problem.
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we moved along BL there.
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Optimality Condition
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Notice at optimum slope of Isoquant equals slope of Isocost
(tangency).
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Slope of Isoquant =
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Slope of Isocost =
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Then we have the following Optimality Condition:
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MP L
MP K
=
∆K
∆L
w
r
= MRTSL,K =
MP L
MP K
w
r
To understand, rewrite is as:
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MP L
w
=
MP K
r
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Marginal Product derived from last dollar equals.
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"bang for the buck"
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Optimality Condition (Con’t)
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Why bundles like A and B are not optimal?
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For instance, at A Isoquant is steeper than Isocost:
w
, or
r
MPK
r
MPL
MPK
MPL
w
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Last dollar on L produces more output than on K .
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How to solve Cost Minimization Problem? In 3 steps:
1. Calculate MPL and MPK .
2. Use Optimality Condition to express L in terms of K or.
3. Sub into technology constraint to solve for K . Then get L from step 2.
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Optimality Condition (Con’t)
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Optimality Condition for Fixed Proportion technology aL = bK .
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Corner solution for Perfect Substitute technology. Compare slopes or
"bang for the buck".
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MP K
L
Compare MP
w and
r .
Produce given output by using only the higher factor.
Indi¤erent if equal.
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Note: this is minimization instead of maximization due to the nature
of the problem.
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See class notes for example.
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Comparative Statics
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Want to know how optimal choices change when exogenous variable
changes. We will change input prices and output level.
Tianyi Wang (Queen’s Univeristy)
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Comparative Statics
I
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Want to know how optimal choices change when exogenous variable
changes. We will change input prices and output level.
For Cobb-Douglas technology, we solved that:
L
K
C
Tianyi Wang (Queen’s Univeristy)
βr α+α β α+1 β
) Q
αw
βr α+ββ α+1 β
= (
) Q
αw
= (
β
1
α
α α
α β
= Q α+β w α+β r α+β [( ) α+β + ( ) α+β ]
β
β
Costs
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Comparative Statics
I
I
Want to know how optimal choices change when exogenous variable
changes. We will change input prices and output level.
For Cobb-Douglas technology, we solved that:
L
K
C
I
βr α+α β α+1 β
) Q
αw
βr α+ββ α+1 β
= (
) Q
αw
= (
β
1
α
α α
α β
= Q α+β w α+β r α+β [( ) α+β + ( ) α+β ]
β
β
Note: these are Conditional Input Demands, conditional on output
level.
Tianyi Wang (Queen’s Univeristy)
Costs
Winter 2013
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Comparative Statics
I
I
Want to know how optimal choices change when exogenous variable
changes. We will change input prices and output level.
For Cobb-Douglas technology, we solved that:
L
K
C
I
I
βr α+α β α+1 β
) Q
αw
βr α+ββ α+1 β
= (
) Q
αw
= (
β
1
α
α α
α β
= Q α+β w α+β r α+β [( ) α+β + ( ) α+β ]
β
β
Note: these are Conditional Input Demands, conditional on output
level.
Suppose wage increases from w1 to w2 , else being equal. Factor
Demand.
Tianyi Wang (Queen’s Univeristy)
Costs
Winter 2013
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Comparative Statics
I
I
Want to know how optimal choices change when exogenous variable
changes. We will change input prices and output level.
For Cobb-Douglas technology, we solved that:
L
K
C
I
I
I
βr α+α β α+1 β
) Q
αw
βr α+ββ α+1 β
= (
) Q
αw
= (
β
1
α
α α
α β
= Q α+β w α+β r α+β [( ) α+β + ( ) α+β ]
β
β
Note: these are Conditional Input Demands, conditional on output
level.
Suppose wage increases from w1 to w2 , else being equal. Factor
Demand.
Suppose output increases from Q1 to Q2 , else being equal. Expansion
Path.
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Input Demands
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We can vary wages to get optimal labor demands. Downward sloping.
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Input Demands
I
We can vary wages to get optimal labor demands. Downward sloping.
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We can vary rental rates to get optimal capital demands. Downward
sloping.
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Input Demands
I
We can vary wages to get optimal labor demands. Downward sloping.
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We can vary rental rates to get optimal capital demands. Downward
sloping.
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See class notes for graphs.
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Input Demands
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We can vary wages to get optimal labor demands. Downward sloping.
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We can vary rental rates to get optimal capital demands. Downward
sloping.
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See class notes for graphs.
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Price elasticity of demand e =
Tianyi Wang (Queen’s Univeristy)
∆L
L
∆w
w
Costs
=
dL w
dw L
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Cost Minimization in Short-run
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In the short-run, capital is …xed at some predetermined level, …rm’s
problem becomes,
Cs (Q, K̄ ) = min wL + r K̄
L
s.t. Q = Q (L, K̄ )
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Simple: constraint determines optimal amount of Labor.
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Note long-run cost can be written as C (Q ) = Cs (Q, K (Q ))
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LR cost equals SR cost when capital is …xed at the optimal level.
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Cost Curve in Short-run
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We derived cost function algebrically, let’s now represent it graphically.
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Cost Curve in Short-run
I
We derived cost function algebrically, let’s now represent it graphically.
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Cost function in general can be written as C (Q, w , r ).
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Cost Curve in Short-run
I
We derived cost function algebrically, let’s now represent it graphically.
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Cost function in general can be written as C (Q, w , r ).
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Since taken prices as given, it can usually be written as C (Q ).
Tianyi Wang (Queen’s Univeristy)
Costs
Winter 2013
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Cost Curve in Short-run
I
We derived cost function algebrically, let’s now represent it graphically.
I
Cost function in general can be written as C (Q, w , r ).
I
Since taken prices as given, it can usually be written as C (Q ).
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In the short-run capital is …xed, we can write Cs (Q, K ) = C (Q, K ).
Tianyi Wang (Queen’s Univeristy)
Costs
Winter 2013
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Cost Curve in Short-run
I
We derived cost function algebrically, let’s now represent it graphically.
I
Cost function in general can be written as C (Q, w , r ).
I
Since taken prices as given, it can usually be written as C (Q ).
I
In the short-run capital is …xed, we can write Cs (Q, K ) = C (Q, K ).
I
Note some costs depend on output others not.
Tianyi Wang (Queen’s Univeristy)
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Cost Curve in Short-run
I
We derived cost function algebrically, let’s now represent it graphically.
I
Cost function in general can be written as C (Q, w , r ).
I
Since taken prices as given, it can usually be written as C (Q ).
I
In the short-run capital is …xed, we can write Cs (Q, K ) = C (Q, K ).
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Note some costs depend on output others not.
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C (Q, K ) = Cv (Q, K ) + |{z}
F
| {z }
Varaible
Tianyi Wang (Queen’s Univeristy)
Fixed
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Cost Curve in Short-run
I
We derived cost function algebrically, let’s now represent it graphically.
I
Cost function in general can be written as C (Q, w , r ).
I
Since taken prices as given, it can usually be written as C (Q ).
I
In the short-run capital is …xed, we can write Cs (Q, K ) = C (Q, K ).
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Note some costs depend on output others not.
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C (Q, K ) = Cv (Q, K ) + |{z}
F
| {z }
Varaible
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Fixed
See class note for graph
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Average Cost Curve
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Two characteristics of cost function, AC and MC.
Tianyi Wang (Queen’s Univeristy)
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Average Cost Curve
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Two characteristics of cost function, AC and MC.
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AC (Q, K ) =
C (Q ,K )
Q
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=
C v (Q ,k )
Q
+
Costs
F
Q
= AVC (Q, K ) + AFC (Q, K )
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Average Cost Curve
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Two characteristics of cost function, AC and MC.
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AC (Q, K ) =
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= Cv (QQ ,k ) + QF = AVC (Q, K ) + AFC (Q, K )
AFC starts at in…nity and declines as Q increases.
C (Q ,K )
Q
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Average Cost Curve
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Two characteristics of cost function, AC and MC.
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AC (Q, K ) =
I
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= Cv (QQ ,k ) + QF = AVC (Q, K ) + AFC (Q, K )
AFC starts at in…nity and declines as Q increases.
AVC is U-shape in general, but many other shapes are possible.
C (Q ,K )
Q
Tianyi Wang (Queen’s Univeristy)
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Average Cost Curve
I
Two characteristics of cost function, AC and MC.
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AC (Q, K ) =
I
I
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= Cv (QQ ,k ) + QF = AVC (Q, K ) + AFC (Q, K )
AFC starts at in…nity and declines as Q increases.
AVC is U-shape in general, but many other shapes are possible.
See class notes for examples.
C (Q ,K )
Q
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Marginal Cost Curve
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MC (Q, K ) =
∆C (Q ,K )
∆Q
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= C 0 (Q, K )
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Marginal Cost Curve
∆C (Q ,K )
= C 0 (Q, K )
∆Q
MC (Q, K ) = C 0 (Q, K ) = Cv0 (Q, k ) + F 0
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MC (Q, K ) =
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Note
Tianyi Wang (Queen’s Univeristy)
Costs
= Cv0 (Q, k )
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Marginal Cost Curve
∆C (Q ,K )
= C 0 (Q, K )
∆Q
MC (Q, K ) = C 0 (Q, K ) = Cv0 (Q, k ) + F 0
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MC (Q, K ) =
I
Note
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Relationship b/w MC and AVC
Tianyi Wang (Queen’s Univeristy)
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= Cv0 (Q, k )
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Marginal Cost Curve
∆C (Q ,K )
= C 0 (Q, K )
∆Q
MC (Q, K ) = C 0 (Q, K ) = Cv0 (Q, k ) + F 0
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MC (Q, K ) =
I
Note
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Relationship b/w MC and AVC
I
= Cv0 (Q, k )
MC of the …rst unit is the same as the AVC of that unit.
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Marginal Cost Curve
∆C (Q ,K )
= C 0 (Q, K )
∆Q
MC (Q, K ) = C 0 (Q, K ) = Cv0 (Q, k ) + F 0
I
MC (Q, K ) =
I
Note
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Relationship b/w MC and AVC
I
I
= Cv0 (Q, k )
MC of the …rst unit is the same as the AVC of that unit.
When AVC decreases, MC must be less than AVC.
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Marginal Cost Curve
∆C (Q ,K )
= C 0 (Q, K )
∆Q
MC (Q, K ) = C 0 (Q, K ) = Cv0 (Q, k ) + F 0
I
MC (Q, K ) =
I
Note
I
Relationship b/w MC and AVC
I
I
I
= Cv0 (Q, k )
MC of the …rst unit is the same as the AVC of that unit.
When AVC decreases, MC must be less than AVC.
When AVC increases, MC must be greater than AVC.
Tianyi Wang (Queen’s Univeristy)
Costs
Winter 2013
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Marginal Cost Curve
∆C (Q ,K )
= C 0 (Q, K )
∆Q
MC (Q, K ) = C 0 (Q, K ) = Cv0 (Q, k ) + F 0
I
MC (Q, K ) =
I
Note
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Relationship b/w MC and AVC
I
I
I
I
= Cv0 (Q, k )
MC of the …rst unit is the same as the AVC of that unit.
When AVC decreases, MC must be less than AVC.
When AVC increases, MC must be greater than AVC.
MC intersects AVC at the minimum of AVC.
Tianyi Wang (Queen’s Univeristy)
Costs
Winter 2013
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Marginal Cost Curve
∆C (Q ,K )
= C 0 (Q, K )
∆Q
MC (Q, K ) = C 0 (Q, K ) = Cv0 (Q, k ) + F 0
I
MC (Q, K ) =
I
Note
I
Relationship b/w MC and AVC
I
I
I
I
I
= Cv0 (Q, k )
MC of the …rst unit is the same as the AVC of that unit.
When AVC decreases, MC must be less than AVC.
When AVC increases, MC must be greater than AVC.
MC intersects AVC at the minimum of AVC.
Similar relationship b/w MC and AC.
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Cost curves in the Long-run
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All factors can be varied in the Long-run.
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Cost curves in the Long-run
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All factors can be varied in the Long-run.
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Short-run costs must exceed Long-run costs, so do average costs.
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Cost curves in the Long-run
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All factors can be varied in the Long-run.
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Short-run costs must exceed Long-run costs, so do average costs.
I
C (Q )
C (Q, K )
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Cost curves in the Long-run
I
All factors can be varied in the Long-run.
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Short-run costs must exceed Long-run costs, so do average costs.
I
I
C (Q )
C (Q, K )
Pick a output level Q and let K = K (Q ) be the optimal capital
level.
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Cost curves in the Long-run
I
All factors can be varied in the Long-run.
I
Short-run costs must exceed Long-run costs, so do average costs.
I
I
C (Q )
C (Q, K )
Pick a output level Q and let K = K (Q ) be the optimal capital
level.
I
then C (Q ) = C (Q , K )
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Cost curves in the Long-run
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All factors can be varied in the Long-run.
I
Short-run costs must exceed Long-run costs, so do average costs.
I
I
C (Q, K )
Pick a output level Q and let K = K (Q ) be the optimal capital
level.
I
I
C (Q )
then C (Q ) = C (Q , K )
Short-run and Long-run average costs have the same property,
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Cost curves in the Long-run
I
All factors can be varied in the Long-run.
I
Short-run costs must exceed Long-run costs, so do average costs.
I
I
C (Q, K )
Pick a output level Q and let K = K (Q ) be the optimal capital
level.
I
I
C (Q )
then C (Q ) = C (Q , K )
Short-run and Long-run average costs have the same property,
I
AC (Q )
AC (Q, K )
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Cost curves in the Long-run
I
All factors can be varied in the Long-run.
I
Short-run costs must exceed Long-run costs, so do average costs.
I
I
C (Q, K )
Pick a output level Q and let K = K (Q ) be the optimal capital
level.
I
I
C (Q )
then C (Q ) = C (Q , K )
Short-run and Long-run average costs have the same property,
I
I
AC (Q ) AC (Q, K )
AC (Q ) = AC (Q , K )
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Cost curves in the Long-run
I
All factors can be varied in the Long-run.
I
Short-run costs must exceed Long-run costs, so do average costs.
I
I
then C (Q ) = C (Q , K )
Short-run and Long-run average costs have the same property,
I
I
I
C (Q, K )
Pick a output level Q and let K = K (Q ) be the optimal capital
level.
I
I
C (Q )
AC (Q ) AC (Q, K )
AC (Q ) = AC (Q , K )
short-run curves always lie above long-run curve except for one point.
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Figure: Envelope Theorem
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Long-Run Marginal Costs
I
Long-run marginal cost curve of any output level has to equal the
short-run marginal cost curve associated with the optimal level of
plant size to produce that level of output.
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Long-Run Marginal Costs
I
Long-run marginal cost curve of any output level has to equal the
short-run marginal cost curve associated with the optimal level of
plant size to produce that level of output.
I
This is known as the Envelope Theorem
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Figure: Envelope Theorem
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Alternative derivation of Cost function (OPTIONAL)
I
Find the long-run cost-minimizing value of costs. Solve
min wL + rK
K ,L
s.t. Q = K α Lβ
I
How to do? First solve L from technology constraint.
L = Q 1/β K (
I
then sub. into cost function to convert the multivariate minimization
problem into univariate minimization problem:
min wQ 1/β K (
K ,L
I
α/β)
α/β)
+ rK
Then solve cost-minimizing value of K from …rst-order-approach, and
get L from tech. constraint.
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Alternative derivation of Cost function (Con’t)
I
Note it is similar as …xing K in step 1 and …nd the optimal L for a
given value of capital, then sets capital equal to the value that would
have been chosen if it had not been …xed.
I
This is short-run cost computed at the optimal capital level!
I
So, long-run cost equals the short-run cost determined at the
’appropriate’capital stock
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Math derivation of Envelope Theorem (OPTIONAL)
I
By de…nition the following holds,
C (Q ) = C (Q, K (Q ))
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I
I
I
I
Di¤erentiating both side w.r.t. Q
∂C (Q, K (Q )) ∂C (Q, K (Q )) ∂K
dC (Q )
=
+
dQ
∂Q
∂K
∂Q
Suppose for a level of Q , K = K (Q ) is the optimal capital size.
At this point we know that, by the de…nition of optimality,
∂C (Q ,K )
= 0.
∂K
Therefore at this point,
dC (Q )
∂C (Q , K )
=
dQ
∂Q
Thus the slopes of Long-run and short-run cost curves (i.e. marginal
costs) are equal.
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