Power Point Slides to Accompany:
Public Finance
by John E. Anderson
Chapter 11
Incentive Effects of
Taxation
Introduction
The existence of taxes brings with it
incentives to change behavior.
 In this chapter we examine how taxes
provide incentives to change behavior in
three specific ways:

– Labor supply,
– Savings,
– Tax evasion.
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Taxation and Labor Supply

Labor-leisure choice problem.

People value both income and leisure.

In order to earn income, one must work.

The problem is to allocate time, a scarce
resource, to labor and leisure.
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Labor-Leisure Choice

Time endowment T.

Allocate time to leisure activity l or work
activity T-l.

Work pays wage of w per hour.

Wage income is y = w(T-l).

Optimal combination of labor and leisure is
that where the MRS between income and
leisure equals the wage.
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Figure 11.1: Labor/Leisure Choice
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Figure 11.2: Labor Supply Effects of a Wage Reduction
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Figure 11.3: Substitution and Income Effects of a Wage
Reduction
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Figure 11.4: Effects of a Wage Reduction When the
Income Effect Dominates
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Figure 11.5: The Laffer Curve
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Taxation and Saving/borrowing
Life cycle model is used.
 Suppose that life consists of two periods:
the present and the future.
 Income in the present is y0 while income
in the future is y1.
 The discounted present value of lifetime
income is then y0+y1/(1+r), where r is the
discount rate.

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Lifetime Consumption
Lifetime consumption can be written
similarly as c0+c1/(1+r).
 Over the lifetime, a budget constraint
requires that these two be equal.
 If we set them equal and solve for c1 as a
function of c0 we get.
 c1=[(1+r)y0+y1]-(1+r)c0.

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Figure 11.6: The Lifetime Budget Constraint
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Figure 11.7: Life Cycle Model of Consumption and Savings
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Figure 11.8: Effect of a Tax on the Lifetime Budget Constraint
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Figure 11.9: Saver Responds to a Tax by Saving Less
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Figure 11.10: Saver Responds to a Tax by Saving More
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Figure 11.11: Borrower Responds to a Tax by Borrowing More
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Taxation and Risk-taking

Taxation reduces the return to a risky
asset, hence it reduces the willingness to
take risk.

But, taxation also reduces risk through
loss deductions, hence it increases
willingness to take risk.

We must consider both aspects of the
influence of taxation on risk-taking.
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Expected Value of the
Consequence

Let c1 and c2 be the consequences of
taking risk under the two states of nature,
low return and high return, occurring with
probability 1- p and p.

Expected consequence is then:

E[c] = (1-p)c1 + pc2.
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Expected Utility of the Risky
Venture

The expected utility of the risky venture
with these two potential consequences is
the weighted average of the utilities that
would arise from the two potential
consequences:

EU[c] = (1-p)u(c1) + pu(c2)
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Marginal Rate of Substitution

The marginal rate of substitution between
the two consequences is given by:

MRS = [(1-p)mu(c1)] / [pmu(c2)].

This is the rate at which the person is
willing to trade one consequence for the
other, holding total utility constant, given
risk.
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Marginal Rate of Substitution
[continued]

Notice that we can write the MRS as:

MRS = [(1-p)/p][mu(c1) / mu(c2)].

The first term is the rate at which one can
trade in the market (1-p)/p, i.e. the odds of
a low return.

So, the MRS in the face of risk is the usual
MRS multiplied by the odds of a low return.
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Effect of a Tax
First, a tax reduces the size of both
consequences proportionately: (1-t)c1, (1t)c2.
 Since c1 < c2, we know (1-t)c1 < (1-t)c2.
 The expected after-tax consequence is
now:
 E[(1-t)c] = (1-p)(1-t)c1 + p(1-t)c2.
 Which is (1-t) E[c], or simply the after-tax
expected consequence.

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A Second Effect
There is a second effect to consider.
 Since the tax system permits deduction
of losses, the riskiness of the project is
affected.
 Suppose that we measure risk using the
standard deviation in the returns.
 Consider the effect of the tax on the
standard deviation SD[c].

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A Second Effect
[continued]
Using the properties of the variance from
statistics, we know the relationship
between the variance of the after-tax
consequences and the before-tax
consequences.
 SD[(1-t)c] = (1-t)SD[c].
 Notice that the tax reduces the standard
deviation and thereby reduces risk.

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Expected Value of Venture

EV = 0.6(300) + 0.4(-100) = $140 million.

Suppose there is a 25% tax with full loss
offset.

EV = 0.6(225) + 0.4(-75) = $105 million.

The tax has reduced the expected return
from 40% to 5%.
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Riskiness of Project
The tax also changes the riskiness of the
project.
 Consider the standard deviation in the
outcomes as a measure of risk.
 Originally SD = $196 million.
 Now SD = $147 million.
 The tax reduces the risk of the project,
and does so by a factor of (1-.25).

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Government Role

In effect, the government is acting as a
silent partner with the business.

The tax system acts to reduce the
riskiness of the venture.

That encourages the business to take
more risk.
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A Portfolio Model

Consider a household with initial wealth
w that they want to divide between two
available assets.

The safe asset provides a certain return
of r.

A risky asset provides expected return 
over the interval (-1, +infinity).
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A Portfolio Model

Let a be the amount invested in the risky
asset that provides an uncertain return ,
either 1 where things go well, or 2 where
things go badly: 1 > 2.

Final wealth of the household is given by:
y = (w-a)(1+r) + a(1+).

Figure 12 illustrates.
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Figure 11.12: Portfolio Choice
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Taxation and Risk-taking
The proportion of the portfolio invested in
the risky asset is given by bc/bd.
 The effect of a wealth tax is to move
points b and d back toward the origin the
the budget line shifting in parallel fashion.
 The net effect of a tax on risk-taking is
ambiguous.
 Figure 13 illustrates.

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Figure 11.13: Effects of a Wealth Tax on Portfolio Choice
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Effects of an Income Tax
An income tax acts differently, simply
reducing the rate of return.
 Consider the simplest case where the
safe asset provides a zero rate of return.
 Figure 14 illustrates the effect of an
income tax.
 The income tax induces the family to take
more risk.

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Figure 11.14: Portfolio Choice With a Zero Return to the
Safe Asset and an Income Tax
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Tax Evasion
Tax evasion refers to illegal activity,
whereas tax avoidance is legal.
 Tax evasion is an application of portfolio
composition.
 Consider holding a tax reporting portfolio.
 Honestly reported income is the safe
asset with certain liability.
 Under-reported income is the risky asset
with uncertain liability.

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The Tax Evader’s Problem





Figure 15 illustrates the tax evader’s problem.
The honest taxpayer operates on the 45degree ray from the origin, at a point like b.
The tax evader operates off the 45-degree ray,
somewhere on the line segment bd.
Point d is the point of complete dishonesty,
where the taxpayer under-reports all of his
income.
At point c, for example, the tax evader is
evading tax on the share of income bc/bd.
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Figure 11.15: The Tax Evader’s Problem
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Limiting Tax Evasion

Increase the probability of detection.

That flattens the indifference curve by
reducing the MRS (due to a reduction in
the odds of not being caught).

The tax evader moves up the budget line
closer to point b where he evades tax on
less income.
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Limiting Tax Evasion
[continued]

Increase the surcharge applied to
detected evasion.

This makes the budget line steeper.

That forces the tax evader to move to a
steeper point on his indifference curve,
where he is evading less.
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Policy Study: Do Ira’s Increase
Private Savings?

One form of saving is the Individual
Retirement Account (IRA) that permits
savings at a tax-favored rate.

The question is whether people are
responsive to this type of incentive and
save more.
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