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Exercise 2.9

MICROECONOMICSMICROECONOMICSPrinciples and AnalysisPrinciples and Analysis

Frank Cowell Frank Cowell

March 2007 March 2007

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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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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

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Ex 2.9(2): Question

methodmethod: : Standard Lagrangean approachStandard Lagrangean approach

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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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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

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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

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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

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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

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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

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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

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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