Public Finance, 2nd Edition

3. They also considered the welfare loss of moving to uniform taxes from

the optimal tax rates, with the loss deWned as the amount that the government

would have to lower its revenue to maintain social welfare at its value with the

optimal rates. The welfare cost is negligible, only .17% of disposable income

for the baseline case and never higher than 2.28% across all values of R and e.

(The welfare cost is zero in the utilitarian case, e  0, since that is equivalent

to a one-consumer economy and the Stone±Geary utility function satisWes the

suYcient condition for uniform taxation to be optimal in the one-consumer

model.)

MANY-PERSON ECONOMY WITH GENERAL TECHNOLOGY

Synthesizing the separate analyses of a one-person economy with general

technology and the many-person economy with linear technology (constant

producer prices) is relatively straightforward, especially under the assump-

tion of CRS production.

Let us begin by considering the optimal commodity tax problem. We saw

that assuming CRS in the context of a one-consumer economy generates the

same optimal tax rules that result when production technology is character-

ized by Wxed producer prices. The key to this result is that there can never be

pure economic proWts or losses under CRS and perfect competition, so that

the value of the general equilibrium proWt function p(

~

p) is identically zero for

all values of the producer price vector

~

p.

The same correspondence exists in the many-person economy. As long as

we assume CRS, the original distribution of lump-sum incomes (I

1

, ...,I

H

)

remains unchanged as producer prices vary in response to taxation. Hence,

the many-person optimal tax rule is identical to its linear technology coun-

terpart. In fact, Diamond used a general-technology CRS model to generate

the many-person optimal tax rules.

A model appropriate for analyzing second-best tax (and expenditure)

problems in a many-person, general-technology economy has four compon-

ents: the social welfare function, consumer preferences, a general production

technology, and market clearance.

Social Welfare and Preferences

The object of government policy is to maximize a social welfare function

of the form:

WV

h

~

q; I

h



 V

~

q; I

1

, ...,I

H



speciWed in terms of consumer prices, exactly as in the many-person, linear-

technology case. (We drop the * on W here.)

474 MANY-PERSON ECONOMY WITH GENERAL TECHNOLOGY

Production Technology

Production must be speciWed in terms of prices and actual general equi-

libria to be compatible with social welfare. The speciWcation must also be

Xexible enough to allow for various kinds of technologies. But the general

technology production can no longer be speciWed by means of the generalized

proWt function, p(

~

p), as in the one-consumer economy, because social welfare

is not measured in terms of lump-sum income. Instead, the natural choice is

to return to an implicit aggregate production frontier of the form F(

~

Y)  0,

as in Wrst-best analysis, where

~

Y  the vector of aggregate goods supplies

(factor demands). Then, replace the quantities

~

Y with the general equilibrium

market supply (input demand) functions Y

i

 Y

i

(

~

p), i  1, ..., N (the same

functions that would result from a social planner maximizing aggregate

proWts at given competitive prices). The resulting function, F[

~

Y(

~

p)]  0,

which is called the production-price frontier, speciWes all relevant production

parameters assuming competitive market behavior.

Market Clearance

General technology requires explicit market clearance equations of the

form:

P

H

h1

X

hi

~

p 

~

t, I

h



 Y

i

(

~

p) i  1, ..., N (14:63)

to solve for the vector of producer prices given a vector of tax rates. All N

market clearance equations apply because the model describes an actual

general equilibrium, not a compensated general equilibrium. The pricing

identities

~

q 

~

p 

~

t then solve for the vector of consumer prices.

The Model

Thus, a full general equilibrium model useful for analyzing any problem

in the second-best theory of taxation can be represented as:

max

(

~

q,

~

t

,

~

p)

WV

h

~

q;

I

h

hi

s:t: F[

~

Y(

~

p)]  0

P

H

h1

X

hi

~

q; I

h



 Y

i

(

~

p) 1, ...,N

q

i

 p

i

 t

i

i  2, ...,N

q

1

 p

1

 1t

1

 0

As always, setting t

1

 0 ensures that the tax vector

~

t changes the vector of

relative consumer and producer prices and thereby generates distortions.

14. THE SECOND-BEST THEORY OF TAXATION WITH GENERAL PRODUCTION 475