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Exercise 13.7

MICROECONOMICSMICROECONOMICSPrinciples and AnalysisPrinciples and Analysis

Frank Cowell Frank Cowell

March 2007 March 2007

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Ex 13.7(1): Question

purposepurpose: A simplified model of optimal income tax: A simplified model of optimal income tax methodmethod: Use tax parameters to determine individual budget constraint. Solve for : Use tax parameters to determine individual budget constraint. Solve for

individual optimisation. Put solution from this into government maximisation problem. individual optimisation. Put solution from this into government maximisation problem.

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dis

pos

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inco

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x

y0

y0

pre-tax incomey

Ex 13.7(1): tax-transfer system(pre-tax, disposable) income space

the no-tax line

No tax: x = y

disposable-income schedule induced by the tax

x = [1 t][y y₀] + y₀

marginal retention rate

break-even point

minimum-guaranteed income

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Ex 13.7(2): Question

methodmethod: : Standard optimisationStandard optimisation Take account of corner solutionTake account of corner solution Adapt from solution to Ex 5.7 (repeated here)Adapt from solution to Ex 5.7 (repeated here)

SkipEx 5.7 stuff

SkipEx 5.7 stuff

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Ex 13.7(2): Worker’s problem (no tax) In absence of tax constraints on worker areIn absence of tax constraints on worker are

xx is consumption is consumption yy is non-labour income is non-labour income ww is wage rate is wage rate ℓ ℓ is labour supplyis labour supply

Worker’s problem can therefore be written asWorker’s problem can therefore be written as

found by substituting from above into utility functionfound by substituting from above into utility function

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Ex 13.7(2): Worker’s optimum (no tax) Take log of maximand to getTake log of maximand to get

log(log(wwℓℓ + +yy) + [1 ) + [1 ] log(1 ] log(1 ℓℓ))

DifferentiateDifferentiate with respect to ℓ with respect to ℓ

This is zero ifThis is zero if wwℓℓ + + ww + [1 + [1 ]]yy = 0 = 0 which implies which implies ℓℓ = = + [1 + [1 ]]y / wy / w

But this only makes sense if But this only makes sense if ℓ is non-negativeℓ is non-negative requires requires ww ≥ ≥ [1 [1 ]]y / y / so optimal labour supply isso optimal labour supply is

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Ex 13.7(2): Worker’s optimum (with tax) Net wage is nowNet wage is now

[1 [1 tt]]ww rather than rather than ww

Non-labour income is nowNon-labour income is now tyty00 rather than rather thanyy

So we can modify previous result So we can modify previous result to give optimal labour supply:to give optimal labour supply:

wherewhere

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Ex 13.7(3): Question

methodmethod: : Use labour-supply function from part 2Use labour-supply function from part 2 Combine with a “break-even” condition for the governmentCombine with a “break-even” condition for the government

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Ex 13.7(3): Breakeven

To ensure that everyone worksTo ensure that everyone works must set tax parameters so that must set tax parameters so that ww00 > > ww11 requires requires yy00 > > yy11 where where yy11 is given by is given by

Net revenue raised in this system is given byNet revenue raised in this system is given by

If the tax is purely redistributive, then this should be zeroIf the tax is purely redistributive, then this should be zero If everyone works then this condition and If everyone works then this condition and ℓℓ** formula give: formula give:

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Ex 13.7(3): Breakeven (more)

Take the breakeven conditionTake the breakeven condition::

Simplify to giveSimplify to give

Use the definition of the mean of the distribution Use the definition of the mean of the distribution FF: :

Choosing Choosing tt fixes guaranteed income fixes guaranteed income tyty00 that can be afforded that can be afforded

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Ex 13.7(4): Question

methodmethod: : Find poorest person’s disposable income using solution to part 3Find poorest person’s disposable income using solution to part 3 Find Find t t that maximises this using standard FOC that maximises this using standard FOC

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Ex 13.7(4): income of poorest person

The after-tax income of the poorest person is given by The after-tax income of the poorest person is given by [1[1tt]]ww00ℓ + ℓ + tyty00

Using the expression for Using the expression for ℓℓ** this becomes this becomes (if the person works)(if the person works) α[1α[1tt] ] ww00 +α +αtyty00

In view of the net-revenue constraint this becomesIn view of the net-revenue constraint this becomes

This is then the objective function This is then the objective function government with "Rawls" type objectivesgovernment with "Rawls" type objectives max the min incomemax the min income

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Ex 13.7(4): optimal t Take the objective functionTake the objective function

Differentiate with respect to Differentiate with respect to tt and set equal to zeroand set equal to zero

We may eliminate We may eliminate ww00 to get to get

This yields the quadraticThis yields the quadratic

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Ex 13.7(4): optimal t (more) Take the quadratic for the optimal Take the quadratic for the optimal tt

Use standard algorithm to getUse standard algorithm to get

Rearrange and ignore the irrelevant root:Rearrange and ignore the irrelevant root:

Optimal tax rate increases with γ: Optimal tax rate increases with γ: the larger is the mean wage (relative to the lowest wage)…the larger is the mean wage (relative to the lowest wage)… … … the more well-off people there are to pay for transfersthe more well-off people there are to pay for transfers

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Ex 13.7: Points to note

Optimal income tax problem based on standard labour-Optimal income tax problem based on standard labour-supply modelsupply model

IngredientsIngredients individual utility functionindividual utility function SWFSWF distributional assumptionsdistributional assumptions government budget constraintgovernment budget constraint

Note simplification introduced byNote simplification introduced by linear tax functionlinear tax function max-min social welfare functionmax-min social welfare function