Frank C

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icroeconomics

Microeconom

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Exercise 4.12

MICROECONOMICSMICROECONOMICSPrinciples and AnalysisPrinciples and Analysis

Frank CowellFrank Cowell

November 2006 November 2006

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Ex 4.12(1) Question

purposepurpose: to derive solution and response functions for quasilinear : to derive solution and response functions for quasilinear preferencespreferences

methodmethod: substitution of budget constraint into utility function and then : substitution of budget constraint into utility function and then simple maximisationsimple maximisation

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Ex 4.12(1) Preliminary

First steps are as follows:First steps are as follows:

Sketch indifference curvesSketch indifference curves Straightforward – parabolic contoursStraightforward – parabolic contours

Write down budget constraintWrite down budget constraint Straightforward – fixed-income caseStraightforward – fixed-income case

Set out optimisation problemSet out optimisation problem

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0

0 1 2

Ex 4.12(1) Indifference curves

x1

x2

Could have x2 = 0

Could have x2 = 0

Slope is vertical here

Slope is vertical here

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Ex 4.12(1) Budget constraint, FOC

Budget constraint:Budget constraint: Substitute this into the utility Substitute this into the utility

function:function: We get the objective function:We get the objective function:

FOC for an interior solution:FOC for an interior solution:

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Ex 4.12(1) Using the FOC

Remember that person might consume zero of commodity 2Remember that person might consume zero of commodity 2 consider two cases consider two cases

Case 1: Case 1: xx22** > 0 > 0

From the FOC: From the FOC:

But, to make sense this case requires:But, to make sense this case requires:

Case 2: Case 2: xx22** = 0 = 0

We get We get xx11** from the budget constraint from the budget constraint

xx11** = = yy / / pp11

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Ex 4.12(1) Demand functions

We can summarise the optimal demands for We can summarise the optimal demands for the two goods thus the two goods thus

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Ex 4.12(1) Indirect utility function

Get maximised utility by substituting Get maximised utility by substituting xx** into the utility into the utility function function VV((pp11, , pp22, , yy) = ) = UU((xx11

**, , xx22**) )

= = UU((DD11((pp11, , pp22, , yy), ), DD22((pp11, , pp22, , yy))))

Case 1: Case 1: pp11 > >pp11

Case 2: Case 2: pp11 ≤ ≤pp11

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Ex 4.12(1) Cost function

Get cost function (expenditure function) from the indirect Get cost function (expenditure function) from the indirect utility functionutility function maximised utility is maximised utility is = = VV((pp11, , pp22, , yy))

invert this to get invert this to get yy = = CC((pp11, , pp22, , ))

Case 1: Case 1: pp11 > >pp11

Case 2: Case 2: pp11 ≤ ≤pp11

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Ex 4.12(2) Question

purposepurpose: to derive standard welfare concept: to derive standard welfare concept methodmethod: use part 1 and manipulate the indirect utility function : use part 1 and manipulate the indirect utility function

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Ex 4.12(2) Compute CV Get compensating variation (1) from indirect utility functionGet compensating variation (1) from indirect utility function

before price change: before price change: = = VV((p1, , p2, , y)) after price change: after price change: = = VV((p1', , p2, , y − CV))

Equivalently (2) could use cost function directlyEquivalently (2) could use cost function directly CV = = CC((p1, , p2, , )) − CC((p1', , p2, , ))

In Case 1 above we haveIn Case 1 above we have

Rearranging, we find:Rearranging, we find:

EquivalentlyEquivalently

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Ex 4.12(3)

In case 1 we have In case 1 we have xx11** = [ = [½½ pp2 2 / / pp11]]22

So demand for good 1 has zero income effectSo demand for good 1 has zero income effect Therefore, in this case CV = CS = EVTherefore, in this case CV = CS = EV

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Ex 4.12: Points to remember

It’s always a good idea to sketch the indifference It’s always a good idea to sketch the indifference curvescurves in this case the sketch is revealing…in this case the sketch is revealing… ……because of the possible corner solutionbecause of the possible corner solution

A corner solution can sometimes just be handled A corner solution can sometimes just be handled as two separate casesas two separate cases

There’s often more than one way of getting to a There’s often more than one way of getting to a solutionsolution in this case two equivalent derivations of CVin this case two equivalent derivations of CV