Capital Income Taxation

8/7/2019 Capital Income Taxation

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PUBLIC ECONOMICS


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CAPITAL INCOME TAX

The basic model assumes:

2 times periods; t=1, 2.

Labour supply in both periods (L1, L2), with a wage of 1 (w=1).

A tax on labour in both periods (1, 2).

Some level of savings in the first period (k).

A capital income tax() the interest payments (rk) from the savings, taken in thesecond period.

The consumer is maximising utility U(c1,c2, L1, L2), where marginal utility ofconsumption in both periods is > 0 and the marginal utility of Labour in both

periods is < 0.

The Government has an exogenous revenue requirement and can transfer taxrevenue from one period to the other at the rate (1+r).

The next slide derives the intertemporal budget constraintfor the consumer.

The Government Budget constraint is given below:

1122 1 LrLrkG XXU !


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CAPITAL INCOME TAX

? A ? A 2111

11:ConstraintBudgetPeriod2nd

11

:ConstraintBudgetPeriod1st

222

222

111

Lkrc

Lrkkrc

kLc

XU

XU

X

!

!

y

!

y

Rearrange (2) to get the savings

function:

rL

r

ck

U

X

U

!

11

1

11

222

Substitute into (1) and divide through by

1-1:

? A ? A

! rLLr

cc

UX

X

UXX 111

1

11111

2

21

1

2

1

1

This is the INTERTEMPORAL BUDGET

CONSTRAINT. It can be re-written as:

? A 3

111

1

1

2

212211

!

r

LLcqcq

UX

X

This looks similar to the normal budgetconstraint, where the goods are datedgoods. Because it looks similar, we aregoing to use a result from OptimalCommodity Taxation (see topic 1 slides),

pertaining to uniform commodity taxation


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CAPITAL INCOME TAX

Commodities 1 and 2 are taxed uniformly ifthe ratio of consumer to producer prices isconstant over time.

P1 =1, P2 = 1/(1+r)

Therefore, =0. If>0, q2/p2 > q1/p1, hencefuture consumption is taxed at a higher ratethan present.

Uniform commodity taxation occurs when thepercentage change in consumer prices is equalacross all goods (see bottom equation).

The next slide derives the condition when wehave uniform commodity taxation.

We will be using the Ramsey Rule (Topic 1),where the change in compensated demandshould be equal across all goods.

To calculate the compensated demand, wewill solve the EXPENDITURE MINIMISATIONPROBLEM; minimising the budget

constraint subject to the utility function.

? A0

111

1

1

1

11

2

2

1

1

!

!

!

U

UXX

iff

rr

p

q

p

q

2

2

1

1

:..

q

dq

q

dq

TCU

!


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CAPITAL INCOME TAX

Additively separable utility

function, where >0:

2121

2121

,

,,,

LLvcucu

LLccU

!

F

Expe it re Mi .:

? A

? A

3'

'

';'

:..

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q

cu

cu

cuqcuq

CsOF

LLvcucuU

rLLcqcq

cc

!@

!!

!

F

EFE

FE

UXX

NN

N

4'

''

'

''

wheniffOptimal..

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dddRHS

dyFdxFYXdF yx

!@

!!y

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y

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FF

Take differentials ofboth sides

Ramsey

Rule


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CAPITAL INCOME TAX

Equation 4 therefore leads us to the following theorem:

A zero rate of capital income tax is optimal iff a uniform commodity tax is optimal.This will occur when either:

i. THEREISASTEADY STATEATTHEOPTIMUM (c1=c2)

ii. THESUB UTILITY FUNCTIONISOF THE FORM:

Part i is usedfor almost all types ofdynamic models, part ii suggests that theelasticity ofthe utilityfunction is constantacross all dates.

A 0 rate of capital income tax suggests that the planner is INDIFFERENT between:

Indifferent in transferring capital today to the future.

Indifferent in transferring taxburden from one time to another. These conditionsensure that the Government finds it optimal to keep a 0 rate of capital income tax

indefinitely.

0, "

!

WW

Wccu


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CAPITAL INCOME TAX

Robustness of result:

Because we assume additive separabilityof consumption and leisure, we arrived atthe iso-elastic utility function, however:

If not additively separable in

consumption and leisure, then anecessary and sufficient condition isfor homogeneity in consumption andleisure.

Ifnot additively separable across time,but across leisure, then utility must behomogenous across consumption inboth periods.

If no additive separability at all, utilityshould be homogenous acrossconsumption in both periods andleisure.

Elasticity of savings has nothing to do withthe zero-tax result. The result derives fromthe optimality of uniform commoditytaxes, and is not influenced by other

factors as long as it holds.

Judd (1985) finds that the result holds forheterogeneous individuals. This isbecause the conditions permitting uniformcommodity taxation cause the many person Ramsey formula collapse to thesingle-person Ramsey formula, which hasno redistributive/equity component.

Therefore, no agents will chooseredistributive capital income taxation,regardless of their initial stock of wealth.

Three broad classes of models also lead tothe result:

Infinite Horizon Dynastic Models (Judd(1985), Chamley (1986)) STEADYSTATE.

Finite Horizon Models w/ iso-elastic (orhomogenous) utilities (Atkinson +stiglitz (1972)) UTILITY FORM.

Models producing steady state growth(such as endogenous growth models)(Lucas (1990)) ONLY PREFERENCECLASS ABLE TO GENERATE STEADYSTATE GROWTH IS THE ISO-ELASTICUTILITY FUNCTION.


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CAPITAL INCOME TAX

Kydland and Prescott (1977) discovered the time-inconsistency problem in capital incometaxation:

Arises due to the delayed negative effects of capital income taxation (intertemporaldistortions in savings). The lump sum nature means the Government would rather relywholly on capital income tax, and set labour income tax to 0.

The Government sets a budget in discrete periods (i.e., at T1, T2 etc), which essentiallymeans that it re-optimises the problem.

Instead ofkeeping its promise to abolish capital income taxat T1, it re-optimises andkeeps the high tax rate, promising to abolish it at T2. This cycle continues such that the taxremains high in perpetuity.

However, a confiscatory capital income tax means that savings willfall to 0 in finite time.

The Government needs a commitment mechanism to prevent it from reneging on its priorcommitment.

Kydland and Prescott (1977) suggest Institutional arrangements that raise the time or political cost of reneging. For example, requiring a vote in Parliament, or instituting a lagbetween any changes any change would come into effect after 2 years.

Alternatively, another solution is to increase the length of the budget. If the re-optimisationoccurs after T1 (at H1), then the initial tax will be greater than 0, but less than 1 as it wasmeant to fall to 0 at T1. The longer the budget period, the lower the capital income tax ratewill be.


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CAPITAL INCOME TAX

tT1

1

0 T2H1

Time Inconsistency:Top line shows the path taken under time inconsistency; Government renegesat each re-optimising interval (T1, T2).

Longer Budgets (H1) lead to an indefinite equilibrium, where 0


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CAPITAL INCOME TAX

Zero-TaxEquilibrium:

In fact, the optimal path for a Government that cannot commit (the third-best scenario;zero capital income tax + positive other taxes is second-best, lump-sum is first-best) is aZEROTAXEQUILIBRIUM.

Go back to the pink and blue equations in slide 9; the infinite horizon models allow for theGovernment to not run a balanced budget in every time period (it can accumulate savings or

debt).

Thus, the Government could tax more in the initial period (keep all taxes (capital incomeand labour) as high as possible) to accumulate enough savings to finance futureexpenditure.

In subsequent periods, the Government is thus able to reduce all of its taxes to 0, relyingon the accumulated savings of the first period.

However, when the Government can commit to the optimal Ramsey rule, then Renstrom(1999) finds that, as long as leisure is a normal good, it is optimal to have a positive labourincome tax, and for a certain class of preferences, that tax is increasing over time.

If we assume indivisible labour (labour either is unemployed or in full-time work), a certainclass of preferences (with neutral leisure) also leads to the zero-tax equilibrium result in thelong-run.

Lucas (1990) finds that by abolishing capital income tax and increasing labour income tax,

there us a marked increase in capital stocks (32% - 55%), and a welfare gain of ~ 1% ofconsumption a FREE LUNCH.

Capital Income Taxation

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