Cost Minimization and Cost Curves - Queen's Universityecon.queensu.ca/walras/custom/200/212/notes/13w/costmini.pdf · Cost Minimization and Cost Curves ... we solved that: L = (br

Cost Minimization and Cost CurvesEcon 212 Lecture 12

Tianyi Wang

Queen’s Univeristy

Winter 2013

Tianyi Wang (Queen’s Univeristy) Costs Winter 2013 1 / 37

Introduction

I We want to model how firms make use of technologies.I Take as given the objective of firms: profit maximization.I One period problem, no risks.I Opportunity cost, sunk cost and fixed cost.I Start with Cost Minimization, for

I itself is interesting.I can not max profit without minimize cost

I Profit = Revenue - Cost = P ∗Q − C (Q)I Suppost cost not minimized for a set of inputs, given Q , can inproveprofit by reducing cost.


Cost Minimization in Long-run

I Suppost firm has access to technology that relates Labor and Capitalto output.

I Ask what’s the cheapest way to produce a certain level (Q) ofoutput.

minL,K

wL+ rK

s.t. Q = Q(L,K )

I Long-run (less constrained) versus Short-run (more constrained).I Use Isocost lines to represent firm’s objective.

I Combinations of inputs that yield the same cost.


Figure: Isocost Line


Optimal Input Choices

I Graphically, start with arbitrary point (A) on Isoquant, find theIsocost that goes through A.

I Ask if can find a lower Isocost. Move to B.I Stop when Isocost just touches Isoquant (tangency). At C.I Read the optimal inputs.I Be aware of the difference from consumer’s utility maximizationproblem.

I we moved along BL there.







Optimality Condition

I Notice at optimum slope of Isoquant equals slope of Isocost(tangency).

I Slope of Isoquant = ∆K∆L = MRTSL,K = − MPL

MPKI Slope of Isocost = −wrI Then we have the following Optimality Condition:

I MPLMPK

= wr

I To understand, rewrite is as:I MPL

w = MPKr

I Marginal Product derived from last dollar equals.I "bang for the buck"


Optimality Condition (Con’t)

I Why bundles like A and B are not optimal?I For instance, at A Isoquant is steeper than Isocost:

−MPLMPK

≤ −wr, or

MPLw

≥ MPKr

I Last dollar on L produces more output than on K .I 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.


Optimality Condition (Con’t)

I Optimality Condition for Fixed Proportion technology aL = bK .I Corner solution for Perfect Substitute technology. Compare slopes or"bang for the buck".

I Compare MPLw and MPKr .

I Produce given output by using only the higher factor.I Indifferent if equal.

I Note: this is minimization instead of maximization due to the natureof the problem.

I See class notes for example.


Comparative Statics

I Want to know how optimal choices change when exogenous variablechanges. We will change input prices and output level.

I For Cobb-Douglas technology, we solved that:

L∗ = (βrαw)

αα+βQ

1α+β

K ∗ = (βrαw)−β

α+βQ1

α+β

C ∗ = Q1

α+βwβ

α+β rα

α+β [(α

β)−α

α+β + (α

β)−β

α+β ]

I Note: these are Conditional Input Demands, conditional on outputlevel.

I Suppose wage increases from w1 to w2, else being equal. FactorDemand.

I Suppose output increases from Q1 to Q2, else being equal. ExpansionPath.


Comparative Statics



L∗ = (βrαw)

αα+βQ

1α+β


α+βQ1

α+β

C ∗ = Q1

α+βwβ

α+β rα

α+β [(α

β)−α

α+β + (α

β)−β

α+β ]





Comparative Statics



L∗ = (βrαw)

αα+βQ

1α+β


α+βQ1

α+β

C ∗ = Q1

α+βwβ

α+β rα

α+β [(α

β)−α

α+β + (α

β)−β

α+β ]





Comparative Statics



L∗ = (βrαw)

αα+βQ

1α+β


α+βQ1

α+β

C ∗ = Q1

α+βwβ

α+β rα

α+β [(α

β)−α

α+β + (α

β)−β

α+β ]





Comparative Statics



L∗ = (βrαw)

αα+βQ

1α+β


α+βQ1

α+β

C ∗ = Q1

α+βwβ

α+β rα

α+β [(α

β)−α

α+β + (α

β)−β

α+β ]










Input Demands

I We can vary wages to get optimal labor demands. Downward sloping.

I We can vary rental rates to get optimal capital demands. Downwardsloping.

I See class notes for graphs.

I Price elasticity of demand ε =∆LL

∆ww= dL

dwwL


Input Demands

I We can vary wages to get optimal labor demands. Downward sloping.I We can vary rental rates to get optimal capital demands. Downwardsloping.



∆ww= dL

dwwL


Input Demands




∆ww= dL

dwwL


Input Demands




∆ww= dL

dwwL


Cost Minimization in Short-run

I In the short-run, capital is fixed at some predetermined level, firm’sproblem becomes,

Cs (Q, K̄ ) = minLwL+ r K̄

s.t. Q = Q(L, K̄ )

I Simple: constraint determines optimal amount of Labor.I Note long-run cost can be written as C (Q) = Cs (Q,K ∗(Q))

I LR cost equals SR cost when capital is fixed at the optimal level.


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 fixed, we can write Cs (Q,K ) = C (Q,K ).I Note some costs depend on output others not.

I C (Q,K ) = Cv (Q,K )︸︷︷︸+ F︸︷︷︸Varaible Fixed

I See class note for graph



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 fixed, we can write Cs (Q,K ) = C (Q,K ).I Note some costs depend on output others not.





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 fixed, we can write Cs (Q,K ) = C (Q,K ).I Note some costs depend on output others not.





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 fixed, we can write Cs (Q,K ) = C (Q,K ).

I Note some costs depend on output others not.





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 fixed, we can write Cs (Q,K ) = C (Q,K ).I Note some costs depend on output others not.














Average Cost Curve

I Two characteristics of cost function, AC and MC.

I AC (Q,K ) = C (Q ,K )Q = Cv (Q ,k )

Q + FQ = AVC (Q,K ) + AFC (Q,K )

I AFC starts at infinity and declines as Q increases.I AVC is U-shape in general, but many other shapes are possible.I See class notes for examples.


Average Cost Curve

I Two characteristics of cost function, AC and MC.I AC (Q,K ) = C (Q ,K )

Q = Cv (Q ,k )Q + F

Q = AVC (Q,K ) + AFC (Q,K )

I AFC starts at infinity and declines as Q increases.I AVC is U-shape in general, but many other shapes are possible.I See class notes for examples.


Average Cost Curve


Q = Cv (Q ,k )Q + F

Q = AVC (Q,K ) + AFC (Q,K )I AFC starts at infinity and declines as Q increases.

I AVC is U-shape in general, but many other shapes are possible.I See class notes for examples.


Average Cost Curve


Q = Cv (Q ,k )Q + F

Q = AVC (Q,K ) + AFC (Q,K )I AFC starts at infinity and declines as Q increases.I AVC is U-shape in general, but many other shapes are possible.

I See class notes for examples.


Average Cost Curve


Q = Cv (Q ,k )Q + F

Q = AVC (Q,K ) + AFC (Q,K )I AFC starts at infinity and declines as Q increases.I AVC is U-shape in general, but many other shapes are possible.I See class notes for examples.





Marginal Cost Curve

I MC (Q,K ) = ∆C (Q ,K )∆Q = C ′(Q,K )

I Note MC (Q,K ) = C ′(Q,K ) = C ′v (Q, k) + F′ = C ′v (Q, k)

I Relationship b/w MC and AVC

I MC of the first unit is the same as the AVC of that unit.I When AVC decreases, MC must be less than AVC.I When AVC increases, MC must be greater than AVC.I MC intersects AVC at the minimum of AVC.

I Similar relationship b/w MC and AC.


Marginal Cost Curve

I MC (Q,K ) = ∆C (Q ,K )∆Q = C ′(Q,K )






Marginal Cost Curve

I MC (Q,K ) = ∆C (Q ,K )∆Q = C ′(Q,K )






Marginal Cost Curve

I MC (Q,K ) = ∆C (Q ,K )∆Q = C ′(Q,K )


I Relationship b/w MC and AVCI MC of the first unit is the same as the AVC of that unit.

I When AVC decreases, MC must be less than AVC.I When AVC increases, MC must be greater than AVC.I MC intersects AVC at the minimum of AVC.



Marginal Cost Curve

I MC (Q,K ) = ∆C (Q ,K )∆Q = C ′(Q,K )


I Relationship b/w MC and AVCI MC of the first unit is the same as the AVC of that unit.I When AVC decreases, MC must be less than AVC.

I When AVC increases, MC must be greater than AVC.I MC intersects AVC at the minimum of AVC.



Marginal Cost Curve

I MC (Q,K ) = ∆C (Q ,K )∆Q = C ′(Q,K )


I Relationship b/w MC and AVCI MC of the first unit is the same as the AVC of that unit.I When AVC decreases, MC must be less than AVC.I When AVC increases, MC must be greater than AVC.

I MC intersects AVC at the minimum of AVC.



Marginal Cost Curve

I MC (Q,K ) = ∆C (Q ,K )∆Q = C ′(Q,K )


I Relationship b/w MC and AVCI MC of the first unit is the same as the AVC of that unit.I When AVC decreases, MC must be less than AVC.I When AVC increases, MC must be greater than AVC.I MC intersects AVC at the minimum of AVC.



Marginal Cost Curve

I MC (Q,K ) = ∆C (Q ,K )∆Q = C ′(Q,K )


I Relationship b/w MC and AVCI MC of the first unit is the same as the AVC of that unit.I When AVC decreases, MC must be less than AVC.I When AVC increases, MC must be greater than AVC.I MC intersects AVC at the minimum of AVC.




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 C (Q) ≤ C (Q,K )

I Pick a output level Q∗ and let K ∗ = K (Q∗) be the optimal capitallevel.

I then C (Q∗) = C (Q∗,K ∗)

I Short-run and Long-run average costs have the same property,

I AC (Q) ≤ AC (Q,K ∗)I AC (Q∗) = AC (Q∗,K ∗)

I short-run curves always lie above long-run curve except for one point.



I All factors can be varied in the Long-run.I Short-run costs must exceed Long-run costs, so do average costs.

I C (Q) ≤ C (Q,K )I Pick a output level Q∗ and let K ∗ = K (Q∗) be the optimal capitallevel.

I then C (Q∗) = C (Q∗,K ∗)







I C (Q) ≤ C (Q,K )

I Pick a output level Q∗ and let K ∗ = K (Q∗) be the optimal capitallevel.

I then C (Q∗) = C (Q∗,K ∗)








I then C (Q∗) = C (Q∗,K ∗)








I then C (Q∗) = C (Q∗,K ∗)








I then C (Q∗) = C (Q∗,K ∗)








I then C (Q∗) = C (Q∗,K ∗)

I Short-run and Long-run average costs have the same property,I AC (Q) ≤ AC (Q,K ∗)

I AC (Q∗) = AC (Q∗,K ∗)






I then C (Q∗) = C (Q∗,K ∗)

I Short-run and Long-run average costs have the same property,I AC (Q) ≤ AC (Q,K ∗)I AC (Q∗) = AC (Q∗,K ∗)






I then C (Q∗) = C (Q∗,K ∗)

I Short-run and Long-run average costs have the same property,I AC (Q) ≤ AC (Q,K ∗)I AC (Q∗) = AC (Q∗,K ∗)





Figure: Envelope Theorem


Long-Run Marginal Costs

I Long-run marginal cost curve of any output level has to equal theshort-run marginal cost curve associated with the optimal level ofplant size to produce that level of output.

I This is known as the Envelope Theorem


Long-Run Marginal Costs

I Long-run marginal cost curve of any output level has to equal theshort-run marginal cost curve associated with the optimal level ofplant size to produce that level of output.

I This is known as the Envelope Theorem


Figure: Envelope Theorem


Alternative derivation of Cost function (OPTIONAL)

I Find the long-run cost-minimizing value of costs. Solve

minK ,L

wL+ rK

s.t. Q = K αLβ

I How to do? First solve L from technology constraint.

L = Q1/βK (−α/β)

I then sub. into cost function to convert the multivariate minimizationproblem into univariate minimization problem:

minK ,L

wQ1/βK (−α/β) + rK

I Then solve cost-minimizing value of K from first-order-approach, andget L from tech. constraint.


Alternative derivation of Cost function (Con’t)

I Note it is similar as fixing K in step 1 and find the optimal L for agiven value of capital, then sets capital equal to the value that wouldhave been chosen if it had not been fixed.

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


Math derivation of Envelope Theorem (OPTIONAL)

I By definition the following holds,

C (Q) = C (Q,K (Q))

I Differentiating both side w.r.t. Q

dC (Q)dQ

=∂C (Q,K (Q))

∂Q+

∂C (Q,K (Q))∂K

∂K∂Q

I Suppose for a level of Q∗, K ∗ = K (Q∗) is the optimal capital size.I At this point we know that, by the definition of optimality,

∂C (Q ∗,K ∗)∂K = 0.

I Therefore at this point,

dC (Q∗)dQ

=∂C (Q∗,K ∗)

∂QI Thus the slopes of Long-run and short-run cost curves (i.e. marginalcosts) are equal.


Cost Minimization and Cost Curves - Queen's Universityecon.queensu.ca/walras/custom/200/212/notes/13w/costmini.pdf · Cost Minimization and Cost Curves ... we solved that: L = (br

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