Welfare Loss of Taxation Li Gan September 2008 There are two methods to calculate welfare loss – the approximation method and the exact method. The approximation method: Tradeoffs between Equity and Efficiency (Browning and Johnson, 1984, JPE) A hypothetical example: Three households:

A B C No of people in the households 1.5 3 3 Effective marginal tax rates 40% 40% 40% Labor supply elasticity 0.15 0.15 0.15

A one percentage increase in marginal tax rate to finance equal per capita transfers Net after tax wage rate 60% 59%, a decrease of 1.66% With elasticity, labor supply 0.15*1.66% = 0.25% Earnings (and labor supply) is reduced by 0.25% (see column 3)

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Previous tax revenue: 5000*0.40 = 2000 Current tax revenue: (5000-12.50)* 0.41 = 2044.87 Additional tax revenue: 2044.87 – 2000 = 44.9 Similarly for other rows (see column 4) Transfer: total increase in tax revenues: 405 Per capita transfer: 405/7.5 = 54 The first family: 54*1.5 = 81 (see column 5) The second family: 54*3 = 162 The third family: 54*3 = 162 Example: what if elasticity = 1.5, and consider a 1% decrease in marginal tax rate to 39%. Earnings increases by 1.667% * 1.5 = 2.5%. Increased Earnings:

5000 * 2.5% = 125 15000 * 2.5% = 375 25000 * 2.5% = 625 Tax revenue: (5000 +125) * 0.39 = 1999 Original tax revenue: 2000 (15000 + 275)*0.39 = 5996 6000 (25000 + 625) *0.39 = 9994 10000 Roughly the same tax revenue as before. Example 2: 1.667 * 3 = 5% Increase earnings: 250 / 750 / 1250 Tax revenue: (5000+250)*.39 = 2047 (15000+750) *.39 = 6142.5 (25000+1250)*.39 = 10237.6 How to empirically identify these? By simulation. Data: 1975 March CPS, 47,114 US households. The following table gives the basic summary statistics. Interestingly, the average tax rate increases as income rises but the marginal tax rates show a U-shaped curve.

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Household labor supply response: ( ) ( ) 011 hATRMTRh ⋅⋅+⋅−= αβ β (non-negative) – substitution effect. The smaller the value of β, the more labor supply responds to changes in marginal tax rate. α (non-negative) – captures the income effect. Labor supply increases when average tax reduces. Compare this with the typical labor supply model: ( ) incomenonlaborbMTRawh −+−= *1 Note in this setup, a is positive and b is negative. Higher non-labor income, lower the amount of work.

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Two measures are used here to measure the efficiency of the tradeoffs: The first is

the ratio of combined income losses for the quintiles that gain. For our benchmark case A, the ratio is 9.51, implying that at the margin, each dollar of disposable income for the lower two quintiles costs the upper three quintiles $9.51.

Another way is: ratio of the top 4 quintiles vs the first quintile, which is roughly 11.

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Browning (1987): On the marginal welfare cost of taxation

w – wage, m is marginal tax rate. S* is the compensated labor supply curve. S is the labor supply curve. The equilibrium labor supply is L2 at point A, and the tax revenue is mWL2, which is the rectangle of w-(1-m)w-A-C.

The total welfare cost is ACB, because at point B, the utility remain the same, but labor supply increased to L1.

This total welfare cost equals to the increase in earnings if the marginal tax rate is

reduced to zero (but with worker kept on the same indifference curve), CBL1L2, less the value of leisure given up in generating that increment in earnings, ABL1L2.

The key point here is that the compensated labor supply curve is used here, which is necessary when evaluating welfare effects of the changes in labor supply.

Assume linear in compensated labor supply curve, then the welfare cost, equals to ½ of CB x CA:

W = ½ (dL)wm Change of labor supply can be expressed as the inverse of the slope of the

compensated supply curve, dL/dw, times the change in the marginal wage rate, wm, so:

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( )

2

2

2

2

2

121

11

21

21

wLm

m

wLm

mL

mwdwdL

wmwmdwdLW

−=

−⎥⎦

⎤⎢⎣

⎡ −=

⎥⎦⎤

⎢⎣⎡=

η

where η is the compensated labor supply elasticity. In contrast to Harberger formula,

wLmW 2

21η=

It is easy to show that Harberger formula correctly evaluates the welfare cost if

we measure the compensated elasticity and the level of labor earnings at their undistorted levels, that is, at point B in the diagram. However, these values are not observable, and available estimates pertain to elasticities and earnings evaluated in the presence of distorting taxes, at point A in the diagram. Marginal welfare cost: is the ratio of the change in total welfare cost to change in tax revenues produced, or dW/dR, where W is the total welfare, and R total tax revenue.

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When m m’, L2 L3. The incremental welfare CDEA = dW, dividing this by the increase in tax revenues, which is not shows in the diagram. Assume the line between A and E is linear, then dW = 1/2 (wm + wm’)dL2 (for the whole area of CDEA) for DCAA’: mw*dL2 (the rectangular area) for AEA’: 1/2*(m’-m)w*dL2 (the triangular area) Sum up together, we get the formula.

Since m’ = m + dm, and dL2 = (ηL2/(1-m))dm, this can be written as:

dmwLm

dmmdW 215.0 η⎥⎦

⎤⎢⎣⎡

−+

=

The change in tax revenue depends on how the average tax rate changes and on

the change in actual labor income. dR = wL2 dt + wdL(m+dm) where dt is the change in the average tax rate evaluate at the initial level of earnings, wL2.

The first term gives the additional revenue produced if the average rate rises by dt

and labor income remains unchanged.

The second term gives the revenue lost when earnings fall by wdL. Note that dL need not be equal to L3-L2 in figure 2, because L3-L2 is the compensate change in labor supply while dL is the actual change in labor supply. Therefore,

( )dmmwdLdtwL

dmwLm

dmm

dRdW

++

⎟⎠⎞

⎜⎝⎛

−+

=2

215.0 η

In principle, previous equation can be used to evaluate marginal welfare cost for any discrete change in tax rates, but to do so requires understanding how actual labor earnings, the wdL term, will be affected.

Note that the marginal welfare cost of raising additional tax revenue does not

depend solely on the change in the tax system, but also on how the government spends the funds.

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The Exact Method Hausman (1981): Exact Consumer’s Surplus and Deadweight Loss Consider a basic maximization problem of a consumer choosing n goods:

( ) ∑=

≤⋅=n

iiix

yxpxptsxu1

..max

The solutions to this problem: uncompensated demand x(p, y). The indirect utility function: v(p, y) = u(x(p, y)).

The expenditure function, e(p, y), is the inverse of the indirect utility function.

Roy’s Identity:

( ) ( )( ) yypv

pypvypx j

j ∂∂

∂∂−=

/,/,

,

The Hicksian compensated demand curves is given by:

( ) ( )uphp

upej

j,,

=∂

∂

Exact consumer surplus: Holding nonlabor income constant at y0, the compensating variation ( )010 ,, yppCV is the minimum quantity required to keep the consumer as well of as he was in the initial state characterized by (p0, y0) as he is in the new state (p1, y0+CV). ( ) ( )CVypvypv += 0100 ,, . Using the expenditure function: ( ) ( ) ( )0001010 ,,,, upeupeyppCV −=

An alternative measure of the welfare change is the equivalent variation, ( )010 ,, yppEV , which is the minimum amount of money to keep the consumer as well of

as he is in the state characterized by (p1, y0). ( ) ( )0100 ,, ypvEVypv =+

Using the expenditure function:

( ) ( ) ( )1011010 ,,,, upeupeyppEV −=

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The following figure illustrates our points when the price is increased from p0 p1.

(1) The difference between the compensated Hicksian demand curve and the uncompensated Marshallian demand curve follows from Slutsky’s equation:

( ) ( ) ( )y

ypxxp

ypxp

uph∂

∂⋅+

∂∂

=∂

∂ 000 ,,,

Since it is typical that the ( ) yypx ∂∂ /, 0 is positive (if the good is normal good),

then the slope (with respect to p) of Hicksian demand curve is smaller in magnitude. In the graph above, the y-axis is price p --- therefore, the Hicksian demand curve looks steeper in this graph. (2) Consider the case that the price has changed from p0 p1. Note since the Hicksian demand curve is derived from taking derivative of expenditure function with respect to p. Therefore, we have:

( ) ( ) ( )

( ) ( )∫∫ =∂

∂=

−=1

0

1

0

00

0001010

,,

,,,,p

p

p

pdpuphdp

pupe

upeupeyppCV

From the graph, the CV is the area Bp1p0F. Note that the area is Bp1p0E is the increased revenue due to increasing price. The triangular BFE (green area) is the so-called Harberger triangular of the welfare loss, as discussed in the Browning (1987) paper.

Note that CV is obtained by assuming the utility at the u0 level, similarly, EV will be obtained by assuming the utility at u1 level, which is basically the area Ap1p0D.

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( ) ( ) ( )∫∫ ∂

∂==

1

0

1

0

11010 ,,,,

p

p

p

pdp

pupedpuphyppEV

The Harberger triangle is, therefore, ACD (the yellow area). Note potentially Harberger triangle could be different depending on if EV or CV is used. In reality, whether EV or CV should be used depending on if we are calculating the welfare loss ex ante or ex post. If it is ex ante, then CV is probably more appropriate. More generally, I believe the benchmark should be the current situation. This is also the point of Browning (1987) when he modified the Harberger formula.

The Marshallian consumer surplus is given by Ap1p0F:

( ) ( ) ( )( )∫∫ ∂∂

∂∂−==

1

0

1

0 /,/,,,, 0

00010 p

p

p

pdp

yypvpypvdpypxyppA

For a single price change,

( ) ( ) ( )010010010 ,,,,,, yppCVyppAyppEV ≤≤

In Hausman (1981), instead of using the simple Harberger-Browning formula, he suggests that precise formula could be used. There are two approaches to get precise formula. One approach is to start with a utility function (CRRA or CES, for example), solve for the demand equation, the Marshallian expenditure function, the indirect utility function, the Hicksian compensated demand function, and others.

To estimate the parameters of the utility function, it is typical to estimate the Marshallian uncompensated demand equation.

Hausman suggests a different approach. Instead of starting from a utility function,

one would start with the demand equation. This makes sense because we do not typically know the functional forms of the utility function and the demand function. Therefore, it is natural to consider an approximation of the demand function to be estimated. This approximation could be parametric or non-parametric. After this demand equation is obtained, one can obtain the indirect utility function, the expenditure function, and even the direct utility function.

Now the question is: how to solve for the indirect utility function when given the Marshallian demand equation? Examples:

(1) Linear demand:

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( )( ) yypv

pypvzypx∂∂∂∂

−=++=/,/,γδα

Solve this linear partial differential equation by applying the method of characteristic curves which assures a unique solution, given an initial condition.

To make welfare comparison, we will want to be on a given indifference curve.

As the price changes, the equation v(p(t), y(t)) = u0 for some u0. Along the path of price change to stay on the indifference curve, we have:

( ) ( )( )( )

( ) ( ) ( )( )( )

( ) 0,,=⋅

∂∂

+⋅∂

∂dt

tdyty

tytpvdt

tdptp

tytpv

Rearrange this equation:

( ) ( )( ) ( )( ) ( )( ) ( )

( )( ) dttdp

dttdytytytpvtptytpv

//

/,/,

−=∂∂∂∂

Then using the implicit function theorem and Roy’s identity, we have:

( ) γδα zypdp

pdy++=

Solving this differential equation, we have the expenditure function:

( ) ⎟⎠⎞

⎜⎝⎛ ++−= γ

δαα

δδ zpcepy p 1 ,

where c, the constant of integration, depends on the initial utility level u0. In fact, that can be obtained:

To get indirect utility function:

( ) ⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ +++== − γ

δαα

δδ zpyeuypv p 1,

Expenditure function is the inverse of previous function:

( ) ⎟⎠⎞

⎜⎝⎛ ++−== γ

δαα

δδ zpueupey p 1,

The CV is given by:

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( ) ( ) ( )

( )0

100

010

0

010

00010

10

11

11

1,,,,

01

01

1

yzpzpye

yzpzpyee

yzpue

upeupeyppCV

pp

pp

p

−⎟⎠⎞

⎜⎝⎛ ++−⎥

⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ +++=

−⎟⎠⎞

⎜⎝⎛ ++−⎟⎟

⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ +++=

−⎟⎠⎞

⎜⎝⎛ ++−=

−=

−

−

γδαα

δγ

δαα

δ

γδαα

δγ

δαα

δ

γδαα

δ

δ

δδ

δ

Example 2: γδα zypx ++= lnlnln

( )δα

α δγ

−+

++

⋅−=−

11,

1ypeypv z

( ) ( )δα

γ

αδ

−+

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+

+−=1

11

11, peuupe z

Example 3: two goods (Feng, Fullerton and Gan) ),,( iii cWearVMTUU =

s.t. iiiiii

g rycWearqVMTMPG

p−=+−

Indirect utility function is given by:

( ) iiqiipi

pi

ii qpηxkyαV εαα

αγββ

β+−−−−−+−= )exp(1'exp1

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ηγβα +++−−+= ')ln( 1 xkβyqpααVMT iiqiip

iVi

( ) ηγβαααα +++−−++= '/ln)ln( 1 xkβyqpαWear iiqi

ip

ipq

iWi

Example 4: γδα zypx ++= lnln

( ) γδα

δα

zyepypv ++

−+

=1

,1

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Fullerton and Gan (2004): CV and EV under piecewise budget constraints.

Define CV and EV as:

( ) ( )

( ) ( ) '',',

',',00

000

uywvEVywv

CVywvywvu

==−

+==

Suppose k0 and k’ are total number of segments before and after a tax reform.

( ) (

( ) (∑∑

∑∑

==

==

+=

+=

0'

00

1

'''

1

''''

1

000

1

0000

,,

,,

k

jj

ajj

k

j

vjjj

k

jj

ajj

k

j

vjjj

HyuKywvSu

HyuKywvSu )

)

When the budget is not convex, we must consider the possibility that more than one budget segments is 1.

( ) ( )( ) ( )mKKHyuu

mSSywvv

jjjajj

jjvjjj

−+≡

−+≡

1,

1

where m is a large negative number. The utility level before and after a change in tax can be written as:

{ }{ }''''

0000

,...,1;,max

,...,1;,max

kjuvu

kjuvu

jjj

jjj

==

==

Additional complications arise because a lump-sum transfer of CV or EV could change a person’s entire budget set (because of a change in nonlinear income). Let ” represent variables after the transfer:

( ) ( )( ) ( mKKHEVyuu

mSSEVywvv

jjjajj

jjvjjj

''''00''

''''00''

1,

1,

−+−≡

−+−≡

)

Therefore, EV is the solution to: { }0'''' ,...,0;,max': kjuvuEV jjj

==

Welfare Loss based on stochastic simulations: Indirect and Direct Utility function:

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( )

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛ −⎟⎟⎠

⎞⎜⎜⎝

⎛

−

−+=

⎟⎟⎠

⎞⎜⎜⎝

⎛+−+=

ββα

βαβ

ββα

βαβ

//

exp,

, 2

j

j

jaj

jaj

jvj

wj

vj

HH

HsyHyu

swyewyv j

Hausman specification: ( ) ζγηβα ++++= zywh v

jj Triest (1990) specification: ζηγβα ++++= zywh v

jj

For each worker, we take I = 1000 draws of the error term. For the ith draw, we find the values of and . Then from previous optimal solutions, we find EV'0 , ijij SS '0 , ijij KK ij, and then the average over then 1,000 draws. We may obtain:

( ) ( )∑∑ Δ−−==I

iii

I

ii REV

IDWL

IDWLE 11

This procedure can also calculate the probability of moving from segment j0 to segment j’, and change of working hours. Example 1:

Consider one person, with fixed effect of working: ( )( ) ζγηβα ++−++= zFCywh v

jj

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Using Harberger DWL, the deadweight loss is $471, which is 26.0% of the tax revenue for this person. However, using the stochastic method, DWL is $1401 ($716), which is 75.5% of the tax revenue.

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Example 2: the Tax Reform Act of 1986 for married women.

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Example 3: Bush 2001 Tax reform

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Welfare gain is 3.5% of revenue in the old tax regime.

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