Public Finance, 2nd Edition

P

H

h1

b

h

 l

P

H

h1

P

N

i1

q

i

 t

i



qX

hi

qI

(14:70)

If consumers are on their budget constraints,

P

N

i1

q

i

qX

hi

qI

 1h 1, ..., H (14:71)

Hence,

P

H

h1

b

h

 lH  l

P

H

h1

P

N

i1

t

i

qX

hi

qI

(14:72)

But

g

h

 b

h

 l

P

N

i1

t

i

qX

hi

qI

the full social marginal utility of income. Therefore:

l 

P

H

h1

g

h

H

(14:73)

the average full social marginal utility of income, as in Eq. (14.52). Conse-

quently, the many-person optimal tax rules continue to have a simple covar-

iance interpretation.

THE SOCIAL WELFARE IMPLICATIONS OF ANY GIVEN CHANGE

IN TAXES

Once the optimal commodity tax problem had been fully developed by Dia-

mond and Mirrless in the late 1960s, public sector economists turned their

attention to more realistic forms of restricted taxation. This opened up two

new major lines of reasearch in the 1970s. One group of economists adopted

the basic model for optimal commodity taxation and attempted to develop

theorems on optimal changes (or levels) of taxes for a subset of the goods and

factors (e.g., Dixit, Guesnerie, and Hatta).

22

A second group, following the

lead of James Mirrlees and Ray Fair in 1971, concentrated speciWcally on

optimal income taxation (e.g., Mirrlees, Fair, Sheshinski, Atkinson, Stiglitz,

22

The seminal articles were A. Dixit, ``Welfare EVects of Tax and Price Changes,'' Journal

of Public Economics, February 1975 (also A. Dixit, and K. Munk, ``Welfare EVects of Tax and

Price Changes: A Correction,'' Journal of Public Economics, August 1977); R. Guesnerie, ``On the

Direction of Tax Reform,'' Journal of Public Economics, April 1977; R. Guesnerie, ``Financing

Public Goods with Commodity Taxes: A Tax Reform Viewpoint,'' Econometrica, March 1979;

and T. Hatta, ``A Theory of Piecemeal Policy Recommendations, Review of Economic Studies,

February 1977.

478 THE SOCIAL WELFARE IMPLICATIONS OF ANY GIVEN CHANGE IN TAXES

Sadka, Stern, and Seade).

23

The chapter concludes with a general example

representative of the Wrst line of research. Chapter 15 discusses optimal income

taxation.

The general method for analyzing restricted tax changes can been seen by

considering the social welfare implications of a marginal change in a single

tax, or of substituting one vector of tax rates for another, equal-revenue

vector of rates in the context of a many-person, general technology econ-

omy.

24

Begin by totally diVerentiating the social welfare function W* 

W[V

h

(

~

q; I

h

)] with respect to prices and income. Using Roy's Identity and

the deWnition of social marginal utility of income b

h

:

dW 

P

H

h1

P

N

i1

qW

qV

h

a

h

X

hi

dq

i



P

H

h1

qW

qV

h

a

h

dI

h

(14:74)

dW 

P

H

h1

P

N

i1

b

h

X

hi

dq

i



P

H

h1

b

h

dI

h

(14:75)

Next, totally diVerentiate the production-price frontier F(

P

H

h1

X

hi

)  0, in

which the market clearance equations have been used to substitute consumers'

demands and factor supplies for the production aggregates Y

i

:

P

N

i1

F

i

P

H

h1

dX

hi

 0 (14:76)

Assuming perfect competition, p

1

 1, and that the identity of person h is

irrelevant to production, Eq. (14.76) becomes:

P

N

i1

p

i

P

H

h1

dX

hi

 0 (14:77)

But, q

i

 p

i

 t

i

, for i  1, ..., N. Multiplying each price by dX

hi

, and sum-

ming over all goods and people yields:

23

The seminal articles were J. Mirrlees, ``An Exploration in the Theory of Optimum Income

Taxation,'' Review of Economic Studies, April 1971 (the seminal article); J. Mirrlees, ``Optimal

Tax Theory: A Syntheses,'' Journal of Public Economics, November 1976; R. Fair, ``The Optimal

Distribution of Income,'' Quarterly Journal of Economics, November 1971; E. Sheshinski, ``The

Optimal Linear Income Tax,'' Review of Economic Studies, July 1972; A. Atkinson, ``How

Progressive Should Income Tax Be?,'' in M. Parkin, Ed., Essays in Modern Economics, Longman

Group, Ltd., London, 1973; A. Atkinson and J. Stiglitz, ``The Design of Tax Structure: Direct vs.

Indirect Taxation,'' Journal of Public Economics, July/August 1976; E. Sadka, ``On Income

Distribution, Incentive EVects, and Optimal Income Taxation, Review of Economic Studies,

June 1976; N. Stern, ``On the SpeciWcation of Models of Optimum Income Taxation,'' Journal

of Public Economics, July/August 1976; J. Seade, ``On the Shape of Optimal Tax Schedules,''

Journal of Public Economics, April 1977; D. Bradford and H. Rosen, ``The Optimal Taxation of

Commodities and Income,'' American Economic Association Papers and Proceedings, May 1976.

24

The analysis in this section draws heavily from R. Boadway, ``Integrating Equity and

EYciency in Applied Welfare Economics,'' Quarterly Journal of Economics, November 1976.

14. THE SECOND-BEST THEORY OF TAXATION WITH GENERAL PRODUCTION 479

P

N

i1

P

H

h1

q

i

dX

hi



P

N

i1

P

H

h1

p

i

 t

i

dX

hi

(14:78)

which, from Eq. (14.78), becomes:

P

N

i1

P

H

h1

q

i

dX

hi



P

H

i1

P

H

h1

t

i

dX

hi

(14:79)

Next, totally diVerentiate each consumer's budget constraint and sum over

all consumers to obtain:

P

H

h1

dI

h



P

H

h1

P

N

i1

q

i

dX

hi



P

H

h1

P

N

i1

X

hi

dq

i

(14:80)

Combining Eqs. (14.79) and (14.80) yields:

P

H

h1

dI

h



P

H

h1

P

N

i1

X

hi

dq

i



P

H

h1

P

N

i1

t

i

dX

hi

(14:81)

Thus, Eq. (14.75) can be written as:

dW 

P

H

h1

P

N

i1

b

h

X

hi

dq

i



P

H

h1

b

h

dI

h



P

H

h1

dI

h



P

H

h1

P

N

i1

X

hi

dq

i



P

N

i1

P

H

h1

t

i

dX

hi

(14:82)

Finally, the dX

hi

in the last term of Eq. (14.82) can be eliminated by noting

that:

X

hi

 X

hi

~

q; I

h



h  1, ...,H; i 1, ..., N (14:83)

Totally diVerentiating Eq. (14.83) yields:

dX

hi



P

N

j1

qX

hi

qq

j

dq

j



qX

hi

qI

h

dI

h

h  1, ...,H

i  1, ...,N

(14:84)

Substituting Eq. (14.84) into (14.82) and combining terms yields:

dW 

P

H

h1

b

h

 1 

P

N

i1

P

H

h1

t

i

qX

hi

qI

h



dI

h



P

H

h1

1  b

h



P

N

i1

X

hi

dq

i



P

H

h1

P

N

i1

t

i

P

N

j1

qX

hi

qq

j

dq

j

(14:85)

Equation (14.85) highlights the importance of CRS in second-best analysis.

With general technology, pure proWts or losses can occur in production,

thereby changing the pattern of lump-sum incomes (I

1

, ...,I

H

) received by

480 THE SOCIAL WELFARE IMPLICATIONS OF ANY GIVEN CHANGE IN TAXES