frank cowell: microeconomics exercise 2.10 microeconomics principles and analysis frank cowell...
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Frank C
owell:
Frank C
owell: M
icroeconomics
Microeconom
ics
Exercise 2.10
MICROECONOMICSMICROECONOMICSPrinciples and AnalysisPrinciples and Analysis
Frank CowellFrank Cowell
November 2006 November 2006
Frank C
owell:
Frank C
owell: M
icroeconomics
Microeconom
ics
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
Frank C
owell:
Frank C
owell: M
icroeconomics
Microeconom
ics
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
Frank C
owell:
Frank C
owell: M
icroeconomics
Microeconom
ics
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
Frank C
owell:
Frank C
owell: M
icroeconomics
Microeconom
ics
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
Frank C
owell:
Frank C
owell: M
icroeconomics
Microeconom
ics
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
Frank C
owell:
Frank C
owell: M
icroeconomics
Microeconom
ics
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.
Frank C
owell:
Frank C
owell: M
icroeconomics
Microeconom
ics
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
Frank C
owell:
Frank C
owell: M
icroeconomics
Microeconom
ics
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:
Frank C
owell:
Frank C
owell: M
icroeconomics
Microeconom
ics
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
owell:
Frank C
owell: M
icroeconomics
Microeconom
ics
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