Chapter 20
Tax Inefficiencies and Their Implications for
Optimal Taxation

Social efficiency is maximized at the competitive
equilibrium (in the absence of market failures).


Taxes entail a deviation from competitive, frictionless
markets. Consequently, taxing market participants creates
deadweight loss.
So, we will look at




Taxation and economic efficiency
Optimal commodity taxes
Optimal income taxes, and
Tax-benefit linkages in financing social insurance programs.
Figure 1
Imposing a Tax on Producers
Price per
gallon (P)
S2
S1
The tax creates
deadweight loss.B
P2 = $1.80
DWL
P1 = $1.50
A
$0.50
The tax on gasoline shifts
the supply curve.
C
D1
Q2 = 90 Q1 = 100
Quantity in billions
of gallons (Q)
DWL When Supply is Infinitely
Elastic

DWL=.5*b*h

d 
.5*t*dx
p dx
 (dp) x
 dx 
x dp
p
d
q=p+t
DWL
p
0
0.5(t ) d (dp) x
x
so DWL 
 0.5(t ) 2  d
p
p

x0
So in this simple example, DWL is
proportional to the square of the tax rate
and the demand elasticity.
Taxation and economic efficiency
Elasticities determine tax inefficiency



The efficiency consequences are identical, regardless
of which side of the market the tax is imposed on.
Just as price elasticities of supply and demand
determine the distribution of the tax burden, they
also determine the inefficiency of taxation.
As shown in the following figure, higher elasticities
imply bigger changes in quantities, and larger
deadweight loss.
Figure 2
Deadweight Loss Increases with Elasticities
Demand is fairly inelastic,
and DWL is small.
(a) Inelastic Demand
P
P
S2
(b) Elastic demand
Demand is more
elastic, and DWL
is larger.
S1
S2
S1
B
P2
B
DWL
P1
P2
P1
A
50¢
Tax
C
DWL
A
50¢
Tax
C
D1
D1
Q2 Q1
Q
Q2
Q1
Q
Taxation and economic efficiency
Determinants of deadweight loss

This formula for deadweight loss:
 S D
2 Q
DWL  
 
2(S   D )
P
1
2 Q
When S  , DWL     D   
2
P

Deadweight loss rises with the elasticity of demand.


The appropriate elasticity is the Hicksian compensated elasticity, not
the Marshallian uncompensated elasticity.
Deadweight loss also rises with the square of the tax rate.

That is, larger taxes have much more DWL than smaller ones.
The “marginal” DWL increases as taxes increase.
Figure 3
S3
P
D
P3
B
P2
P1
S2
S1
The next $0.10 tax creates
a larger marginal DWL,
TheBCDE.
first $0.10 tax creates
little DWL, ABC.
A
C
$0.10
E
$0.10
D1
Q3 Q2 Q1
Q
Taxation and economic efficiency
Deadweight loss and the design of efficient tax
systems

The more one loads taxes onto one source (and consequently,
the higher the rate), the faster DWL rises.


Efficient tax systems spread the burden broadly. Thus, efficient tax
systems have a broad base and low rates.
The fact that DWL rises with the square of the tax rate also
implies that government should not raise and lower taxes, but
rather set a long-run tax rate that will meet its budget needs
on average.

For example, to finance a war, it is more efficient to raise the rate by
a small amount for many years, rather than a large amount for one
year (and run deficits in the short-run).

This notion can be thought of as “tax smoothing,” similar to the notion of
individual consumption smoothing.
Optimal commodity taxation
Ramsey rule



Optimal commodity taxation is choosing tax rates across goods to
minimize the deadweight loss for a given government revenue
requirement.
MDWLi

   
The Ramsey Rule is:
MRi
D
It sets taxes across commodities so that the ratio of the
marginal deadweight loss to marginal revenue raised is
equal across commodities.


The goal of the Ramsey Rule is to minimize deadweight loss of
a tax system while raising a fixed amount of revenue.
8 measures the value of having another dollar in the government’s hands
relative to the next best use in the private sector.

Smaller values of 8 mean additional government revenues have little value
relative to the value in the private market.
Optimal commodity taxation
Inverse elasticity rule

The inverse elasticity rule, based on the Ramsey result, allows
us to relate tax policy to the elasticity of demand.



The government should set taxes on each commodity inversely to the
demand elasticity.
Therefore, ignoring equity, less elastic items should be taxed at a
higher rate.
Two factors must be balanced when setting optimal (efficient)
commodity tax rates (again, ignoring equity):


The elasticity rule: Tax commodities with low elasticities.
The broad base rule: It is better to tax many goods at lower rates,
because deadweight loss increases with the square of the tax rate.

Thus, the government should tax all of the commodities that it is able to, but
at different rates.
OPTIMAL INCOME TAXES


Optimal income taxation is choosing the tax rates
across income groups to maximize social welfare
subject to a government revenue requirement.
Raising tax rates will likely affect the size of the tax
base. Thus, increasing the tax rate on labor income
has two effects:



Tax revenues rise for a given level of labor income.
Workers reduce their earnings, shrinking the tax base.
At high tax rates, this second effect becomes
important.

Thus, there are equity-efficiency tradeoffs in designing
income tax rates.
The Laffer curve motivated the supply-side
economic policies of the Reagan presidency
Figure 7
The Laffer curve
demonstrates that at some
point, tax revenue falls.
Tax revenues
right
side
0
wrong
side
τ*%
100% Tax rate
Optimal income taxes
General model with behavioral effects


The goal of optimal income tax analysis is to identify a tax
schedule that maximizes social welfare, while recognizing
that raising taxes has conflicting effects on revenue.
The optimal tax system meet the condition that tax rates are
set across groups such that: MU i

MRi


Where MUi is the marginal utility of individual i, and MR is the
marginal revenue from that individual.
As with optimal commodity taxation, this outcome
represents a compromise between two considerations:


Vertical equity
Behavior responses
Figure 8
MU/MR
Mrs. Poor
Mr. Rich
Optimal income taxation
equates the ratio of
(MU/MR) across individuals.
 MU 
 MU 
λ
 

 MR  poor  MR  rich
10%
20%
Tax rate
Low income may have higher MUc. A given tax will raise more for
a high income household (up to a point), so MR is higher.
TAX-BENEFITS LINKAGES AND THE
FINANCING OF SOCIAL INSURANCE
PROGRAMS

Tax-benefit linkages are direct ties between taxes paid
and benefits received.


Summers (1989) shows that such linkages can affect the equity
and efficiency of a tax. The link between payroll taxes and
social insurance benefits can lead the incidence to fall more fully
on workers than might be presumed.
The key point of Summers’ analysis is that with taxes alone, only
the labor demand curve shifts, but with tax-benefit linkages, the
labor supply curve shifts as well.

That is, workers are willing to work the same amount of hours at a lower
wage, because they get some other benefit as well, such as workers’
compensation or health insurance.
Figure 10
Tax-Benefit Linkages
Wage (W)
Wage (W)
C
Creating
smaller DWL.
S1
S1
Mandated
benefits
A shift
also
the supply
curve.
E
W1
W1
A
F
W
W
B
2
B
S2
2
W
D1
D
3
D1
D2
L2
L1
D2
Labor (L)
L2
L3
L1
Labor (L)
Wages adjust by more with the tax-benefit linkage, employment falls by less, and deadweight loss is smaller
than with a pure tax.
Tax-benefits linkages and the financing of
social insurance programs: The model


With full valuation, the cost of the program is fully shifted
onto workers in the form of lower wages, and there is no
deadweight loss or employment reduction.
This raises some issues with tax-benefit linkages, especially
with respect to employer mandates.

If there is no inefficiency, why doesn’t the employer simply provide
the benefit without government intervention?


Market failures, such as adverse selection, may be present. The employer that
provides a benefit such as workers’ compensation or health insurance may end
up with high risks.
When are there tax-benefit linkages?

They are strongest when the taxes paid are linked directly to a
specific benefit that workers can receive.