Public Finance, 2nd Edition

THE SECOND-BEST ALLOCATION OF SAMUELSONIAN

NONEXCLUSIVE GOODS

The Wrst-best analysis of a Sameulsonian nonexclusive public good yielded

three speciWc policy prescriptions:

1. The government should provide the good such that

P

H

h1

MRS

h



MRT. The government has to provide the good because the incentive to free

ride prevents the market system from allocating nonexclusive goods.

2. If the government happens to select the quantity that satisWes the

optimal decision rule, then it can Wnance the good with any lump-sum tax.

The lump-sum tax keeps the economy on the Wrst-best utility possibilities

frontier. Any unwanted distributional consequences of the tax are overcome

by the lump-sum taxes and transfers that satisfy the Wrst-best interpersonal

equity conditions for a social welfare maximum.

3. The competitive market economy can be counted on to generate the

pareto-optimal allocations of all the purely private goods and factors.

None of these prescriptions applies in a second-best environment in

general, although the ways in which the Wrst-best optimal decision rules

change depend upon the nature of the additional constraints placed on the

system. This is always true in second-best analysis. A natural way to pose a

second-best problem is to let the government freely choose the quantity of the

nonexclusive good but constrain it to Wnance the good with distorting unit

commodity taxes. This implicitly precludes lump-sum redistributions to sat-

isfy the Wrst-best interpersonal equity conditions by equalizing marginal

social utilities of income, because if lump-sum taxes could be used for

distributional purposes they should also be available to Wnance the public

good. Otherwise, assume that the economy is perfectly competitive with all

other goods (factors) being purely private. In other words, the need to use

distorting taxes is the only constraint that makes the analysis second best.

Given this particular second-best environment, there are two compelling

policy questions to be asked:

1. How does the required distorting taxation aVect the optimal

decision rule for providing the public good?

2. How does the presence of the public good aVect the optimal tax

rules when revenue is raised for its own sake?

These questions can be addressed with a general equilibrium model that

is a straightforward extension of the many person model used in Chapter 14

to analyze optimal commodity taxation under general technology.

Preferences and Social Welfare

Let e stand for the nonexclusive good, deWned in units such that its price

equals 1. Since the government is selecting the quantity of e, consumers treat e

660 THE SECOND-BEST ALLOCATION OF SAMUELSONIAN NONEXCLUSIVE GOODS

as a parameter even though e enters their utility functions. Therefore, each

individual solves the following utility maximization problem:

max

X

hi



U

h

X

hi

;e

s:t:

P

N

i1

q

i

X

hi

 I

h

where:

q

i

 the consumer price of good (factor) i, i  1, ...,N

X

hi

 good (factor) i consumed (supplied by) person h, i  1, ...,N

h  1, ...,H.

I

h

 the Wxed amount of lump-sum income for person h, which

the government cannot change through lump-sum

redistributions.

The consumer's maximization problem leads to demand (factor supply)

functions of the form:

X

hi

 X

hi

~

q;

I

h

;e



i  1, ...,N; h  1, ..., H (20:1)

and indirect utility functions:

U

h

X

hi

~

q;

I

h

;e

hi

 V

h

~

q;

I

h

;e



h  1, ..., H (20:2)

Social welfare, then, is

W* U

h

X

hi

~

q;

I

h

;e

hi

 WV

h

~

q;

I

h

;e

hi

(20:3)

where W( ) is the Bergson±Samuelson individualistic social welfare function.

Production and Market Clearance

e must also enter the aggregate production function F because it uses

real resources.

1

Therefore, write the aggregate production function implicitly

as:

FX

i

;e0i 1, ..., N (20:4)

where X

i

is the aggregate demand (supply) for good (factor) i. Assume that

F( ) exhibits constant returns to scale so that there are no pure proWts in

the economy. Finally, incorporate market clearance directly into the aggre-

gate production function:

F

P

H

h1

X

hi

~

q;

I

h

;e



;e



 0

1

e could be privately produced, as missiles and military aircraft are.

20. EXTERNALITIES IN A SECOND-BEST ENVIRONMENT 661