Frank C

owell:

Frank C

owell: M

icroeconomics

Microeconom

ics

Exercise 9.1

MICROECONOMICSMICROECONOMICSPrinciples and AnalysisPrinciples and Analysis

Frank CowellFrank Cowell

March 2007 March 2007

Frank C

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Frank C

owell: M

icroeconomics

Microeconom

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Ex 9.1(1): Question

purposepurpose: Analyse consumption externality and efficiency: Analyse consumption externality and efficiency methodmethod: Solve for equilibrium prices and allocation using standard GE. Then examine source of inefficiency: Solve for equilibrium prices and allocation using standard GE. Then examine source of inefficiency

Frank C

owell:

Frank C

owell: M

icroeconomics

Microeconom

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Ex 9.1(1): incomes and demands The term The term xx11

aa is irrelevant to is irrelevant to bb-people's behaviour -people's behaviour they cannot do anything about it…they cannot do anything about it… ……although it affects their utilityalthough it affects their utility

Incomes are Incomes are yyaa = 300 = 300 pp11

yyaa = 200 = 200 pp22

Both types have Cobb-Douglas utility functionsBoth types have Cobb-Douglas utility functions so we could jump straight to demand functions…so we could jump straight to demand functions… ……skip the Lagrangean stepskip the Lagrangean step

We know that their demands will be given byWe know that their demands will be given by xx11

*a*a = = ½ ½ yya a / / pp11 , , xx22*a*a = = ½ ½ yya a / / pp22

xx11*b*b = = ½ ½ yyb b / / pp11 , , xx22

*b*b = = ½ ½ yyb b / / pp22

SkipLagrangean

SkipLagrangean

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Frank C

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icroeconomics

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Ex 9.1(1): Lagrangean method

Lagrangean for either type can be writtenLagrangean for either type can be written kxkx11

h h xx22hh + + [[yyh h pp11xx11

hh pp2 2 xx22hh ] ]

where where is a Lagrange multiplier is a Lagrange multiplier kk is a constant ( is a constant (kk =1 for type =1 for type aa , , kk =1/ =1/ xx11

aa for type for type bb))

FOC for an interior maximumFOC for an interior maximum kxkx22

hh pp1 1 = 0 = 0 kxkx11

hh pp2 2 = 0= 0 yyh h pp11xx11

hh pp2 2 xx22hh

= 0= 0

Substitute from FOC1, FOC2 into FOC3 to find Substitute from FOC1, FOC2 into FOC3 to find yyh h pp11[[pp2 2 /k/k]] pp22[[pp1 1 /k/k] = 0] = 0 = = ½½kykyh h /p/p11pp22

Substitute this value of Substitute this value of back into FOC2, FOC1 to get the demands: back into FOC2, FOC1 to get the demands: xx11

*h*h = = ½ ½ yyh h / / pp11 xx22

*h*h = = ½ ½ yyh h / / pp22

Frank C

owell:

Frank C

owell: M

icroeconomics

Microeconom

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Ex 9.1(1): Equilibrium price ratio Total demand for commodity 1 isTotal demand for commodity 1 is

N N [ [ xx11*a*a + + xx11

*b*b ] = ] = N N [ [ ½½ 300 + ⋅ 300 + ⋅ ½½ 200/⋅ 200/⋅ ] ] where where N N is the large unknown number of tradersis the large unknown number of traders and and := := pp11 / / pp22 only the price only the price ratioratio matters in the solution matters in the solution

There are 300There are 300NN units of commodity 1 units of commodity 1 So the excess demand function for commodity 1 isSo the excess demand function for commodity 1 is

EE11 = [150 + 100/ = [150 + 100/] ] NN 300 300 NN = [100/= [100/ 150] 150] NN

To find equilibrium sufficient to put To find equilibrium sufficient to put EE11 = 0 = 0 if if EE11 = 0 then = 0 then EE22 = 0 also = 0 also by Walras' Lawby Walras' Law

Clearly Clearly EE11 = 0 exactly where = 0 exactly where = = ⅔⅔ the equilibrium price ratiothe equilibrium price ratio

Frank C

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Frank C

owell: M

icroeconomics

Microeconom

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Ex 9.1(1): Equilibrium allocation Take the equilibrium price ratio Take the equilibrium price ratio = = ⅔⅔ Then, using the demand functions we findThen, using the demand functions we find

xx11*a*a = = ½½ ⋅⋅ 300300 = 150= 150

xx22*a*a = = ½½ 300⋅ 300⋅ = 100= 100

xx11*b*b = = ½½ 200 / ⋅ 200 / ⋅ = 150= 150

xx22*b*b = = ½½ 200⋅ 200⋅ = 100= 100

This is the equilibrium allocationThis is the equilibrium allocation

Frank C

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Frank C

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icroeconomics

Microeconom

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Ex 9.1(2): Question

methodmethod: : Verify that CE allocation is inefficient by finding a Verify that CE allocation is inefficient by finding a

perturbation that will produce a Pareto improvementperturbation that will produce a Pareto improvement

Frank C

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Frank C

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icroeconomics

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Ex 9.1(2): Source of inefficeincy

It is likely that the It is likely that the aa-people are consuming “too much” of -people are consuming “too much” of good 1good 1 there is a negative externality there is a negative externality in the CE this is ignored in the CE this is ignored

So try changing the allocation So try changing the allocation so that the so that the aa-people consume less of good 1 -people consume less of good 1 xx11

a a < 0 < 0 but where the but where the aa-people's utility remains unchanged-people's utility remains unchanged

The means that their consumption of good 2 must increaseThe means that their consumption of good 2 must increase given that, in equilibrium, given that, in equilibrium, = MRS, = MRS, required adjustment is required adjustment is xx22

a a = = −−xx11a a >0>0

Frank C

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Frank C

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icroeconomics

Microeconom

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Ex 9.1(2): Pareto-improving adjustment bb-people's consumptions move in the opposite direction-people's consumptions move in the opposite direction

(there is a fixed total amount of each good)(there is a fixed total amount of each good) xx11

b b = = −−xx11a a > 0> 0

xx22b b = = −−xx22

aa < 0 < 0

Effect on their utility can be computed thus:Effect on their utility can be computed thus:log log UUbb = = xx11

bb / / xx11bb + + xx22

bb / / xx22bb − − xx11

a a / /xx11a a

= [ = [ − − 1/150 + 1/150 + ⅔⅔(1/100) (1/100) − − 1/150] 1/150] xx11aa

= = − − xx11aa / 150 >0 / 150 >0

So it is possible to make a Pareto-improving perturbationSo it is possible to make a Pareto-improving perturbation move away from the CEmove away from the CE in such a way that some people's utility is increasedin such a way that some people's utility is increased no-one else's utility decreasesno-one else's utility decreases

Frank C

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Frank C

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icroeconomics

Microeconom

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Ex 9.1(3): Question and answer

Can this be done by just tweaking prices?Can this be done by just tweaking prices? increase relative price of commodity 1 for the increase relative price of commodity 1 for the aa-people…-people… ……relative to that facing the relative to that facing the bb-people? -people?

This will not work This will not work aa-people’s income is also determined by -people’s income is also determined by pp11 … … ……and their resulting consumption of commodity 1 is independent and their resulting consumption of commodity 1 is independent

of price of price

A rationing scheme may workA rationing scheme may work

Frank C

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Frank C

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icroeconomics

Microeconom

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Ex 9.1: Points to remember

Be careful to model what is under each Be careful to model what is under each agent’s controlagent’s control

Use common-sense to spot Pareto Use common-sense to spot Pareto improvementsimprovements