Frank C

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icroeconomics

Microeconom

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Exercise 2.10

MICROECONOMICSMICROECONOMICSPrinciples and AnalysisPrinciples and Analysis

Frank CowellFrank Cowell

November 2006 November 2006

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Microeconom

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Ex 2.10: Question

purposepurpose: to derive and compare short-run and long-run responses.: to derive and compare short-run and long-run responses. methodmethod: derive AC, MC, supply in original and modified models : derive AC, MC, supply in original and modified models

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icroeconomics

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Ex 2.10(1): Preliminary steps

Put the production function in a more manageable form

A quick check on the isoquant for m = 2:

Clearly isoquants do not touch the axes

Solution cannot be at a corner

z1

z2

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Ex 2.10(1): Cost minimisation The Lagrangean:

Differentiate w.r.t. zi to find the FOCs

Rearrange to get:

(the Lagrange multiplier) is an unknown

We need to We need to eliminate iteliminate it

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Ex 2.10(1): Finding

Use the production functionUse the production function

And substitute in for And substitute in for zzii:: wherewhere

From this we find thatFrom this we find that

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Ex 2.10(1): The cost function

can be simplified tocan be simplified to

Substitute into expression for Substitute into expression for zzii; get optimal input demands; get optimal input demands

So minimised costs expressed as a function of So minimised costs expressed as a function of ww and and qq are are

This can be written as This can be written as BqBq1/1/wherewhere

Differentiating this w.r.t. Differentiating this w.r.t. qq, MC is, MC is

So MC is increasing in So MC is increasing in qq if if < 1 < 1

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icroeconomics

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Ex 2.10(2): Preliminary In the “short run” the amounts of inputs In the “short run” the amounts of inputs kk+1,…,+1,…,mm are fixed are fixed

So, define the termSo, define the term (constant in the short run)(constant in the short run)

The production function can be written:The production function can be written:

TThis is the only part that is variable in the short run. his is the only part that is variable in the short run.

We see that the problem has We see that the problem has exactlyexactly the same structure as before the same structure as before but with different parameters.but with different parameters.

Therefore the solution has the same structure as beforeTherefore the solution has the same structure as before but with different parameters.but with different parameters.

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icroeconomics

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Ex 2.10(2): Short-run input demand

We can proceed by analogy with the long-run caseWe can proceed by analogy with the long-run case Cost-minimising input demands must be:Cost-minimising input demands must be:

where we have definedwhere we have defined

Multiplying each input demand by Multiplying each input demand by wwii and summing will give and summing will give

short-run variable costsshort-run variable costs

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icroeconomics

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Ex 2.10(2): Short-run costs Define short-run fixed costsDefine short-run fixed costs

the amounts of inputs the amounts of inputs kk+1,…,+1,…,mm are fixed are fixed

Then short-run total costs are given byThen short-run total costs are given by

Substituting in for Substituting in for zzii** costs in the short run are: costs in the short run are:

Clearly this expression has the form:Clearly this expression has the form:

Differentiate costs w.r.t. Differentiate costs w.r.t. qq and we find short-run MC: and we find short-run MC:

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Ex 2.10(3): short run supply

From the SRMC we get the short-run supply curveFrom the SRMC we get the short-run supply curve The condition “MC = price” givesThe condition “MC = price” gives

Solving this for Solving this for qq the supply function is the supply function is

The elasticity of supply is The elasticity of supply is

Clearly the elasticity falls if Clearly the elasticity falls if kk falls falls By definition of By definition of kk it must fall if it must fall if k k is reduced is reduced

Frank C

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Frank C

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icroeconomics

Microeconom

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Ex 2.10: Points to remember

Get the constraint into a convenient formGet the constraint into a convenient form Get a simple view of the problem by deriving ICsGet a simple view of the problem by deriving ICs Use a little cunning to simplify the FOCsUse a little cunning to simplify the FOCs Re-use your solution for other problems that have Re-use your solution for other problems that have

the same structurethe same structure