Frank C
owell:
Frank C
owell: M
icroeconomics
Microeconom
ics
Exercise 2.9
MICROECONOMICSMICROECONOMICSPrinciples and AnalysisPrinciples and Analysis
Frank Cowell Frank Cowell
March 2007 March 2007
Frank C
owell:
Frank C
owell: M
icroeconomics
Microeconom
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Ex 2.9(1): Question
purposepurpose: demonstrate relationship between short and long run: demonstrate relationship between short and long run methodmethod: Lagrangean approach to cost minimisation. First part can be : Lagrangean approach to cost minimisation. First part can be
solved by a “trick”solved by a “trick”
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owell:
Frank C
owell: M
icroeconomics
Microeconom
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Ex 2.9(1): Long-run costs
Production function is homogeneous of degree 1Production function is homogeneous of degree 1 increase all inputs by a factor increase all inputs by a factor tt > 0 (i.e. > 0 (i.e. zz →→ ttzz)…)… ……and output increases by the same factor (i.e. and output increases by the same factor (i.e. qq →→ tqtq)) constant returns to scale in the long runconstant returns to scale in the long run
CRTS implies constant average costCRTS implies constant average cost CC((ww, , qq) / ) / qq = = AA (a constant) (a constant) so so CC((ww, , qq) = ) = AqAq differentiating: differentiating: CCqq((ww, , qq) = ) = AA
So LRMC = LRAC = constantSo LRMC = LRAC = constant Their graphs will be an identical straight lineTheir graphs will be an identical straight line
Frank C
owell:
Frank C
owell: M
icroeconomics
Microeconom
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Ex 2.9(2): Question
methodmethod: : Standard Lagrangean approachStandard Lagrangean approach
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owell:
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owell: M
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Ex 2.9(2): short-run Lagrangean In the short run amount of good 3 is fixedIn the short run amount of good 3 is fixed
zz33 = = zz33
Could write the Lagrangean asCould write the Lagrangean as
But it is more convenient to transform the problem thusBut it is more convenient to transform the problem thus
wherewhere
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owell: M
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z2
z1
Ex 2.9(2): Isoquants Sketch the isoquant mapSketch the isoquant map
Isoquants do not touch the axesIsoquants do not touch the axes So maximum problem must have an interior solutionSo maximum problem must have an interior solution
Frank C
owell:
Frank C
owell: M
icroeconomics
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Ex 2.9(2): short-run FOCs
Differentiating Lagrangean, the FOCS areDifferentiating Lagrangean, the FOCS are
This impliesThis implies
To find conditional demand function must solve for To find conditional demand function must solve for use the above equations… use the above equations… ……and the production functionand the production function
Frank C
owell:
Frank C
owell: M
icroeconomics
Microeconom
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Ex 2.9(2): short-run FOCs (more)
Using FOCs and the production function:Using FOCs and the production function:
This impliesThis implies
wherewhere
This will give us the short-run cost functionThis will give us the short-run cost function
Frank C
owell:
Frank C
owell: M
icroeconomics
Microeconom
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Ex 2.9(2): short-run costs By definition, short-By definition, short-
run costs are:run costs are:
This becomesThis becomes
Substituting for Substituting for kk::
From this we getFrom this we get
SRAC:SRAC:
SRMC:SRMC:
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q
Ex 2.9(2): short-run MC and AC
marginalcost
averagecost
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Ex 2.9(3): Question
methodmethod: : Draw the standard supply-curve diagramDraw the standard supply-curve diagram Manipulate the relationship Manipulate the relationship pp = MC = MC
Frank C
owell:
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owell: M
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Microeconom
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Ex 2.9(3): short-run supply curveaverage cost curvemarginal cost curve
supply curve
q
p
q
p
minimum average cost
Frank C
owell:
Frank C
owell: M
icroeconomics
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Use the expression for marginal cost:Use the expression for marginal cost: Set Set p p = MC for = MC for pp ≥≥ pp Rearrange to get supply curveRearrange to get supply curve
Differentiate last line to get supply elasticityDifferentiate last line to get supply elasticity
Ex 2.9(3): short-run supply elasticity
Frank C
owell:
Frank C
owell: M
icroeconomics
Microeconom
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Ex 2.9: Points to remember
Exploit CRTS to give you easy resultsExploit CRTS to give you easy results Try transforming the Lagrangean to make it easier Try transforming the Lagrangean to make it easier
to manipulateto manipulate Use MC curve to derive supply curveUse MC curve to derive supply curve