Public Finance, 2nd Edition

T  the lump-sum payment required to Wnance the deWcit.

M(

~

q;

U

0

)  the amount consumers are willing to pay for the new

prices,

~

q.

SAMUELSONIAN NONEXCLUSIVE GOODS

Chapter 6 developed the standard pareto-optimal decision rule for a non-

exclusive good in a Wrst-best environment,

P

H

h1

MRS

h

 MRT, but did not

consider the incidence of the good. The incidence is the gain in welfare to each

consumer from being able to consume the good at its optimal level, less the

loss in welfare from having to Wnance the good.

As a Wrst step in deriving an incidence measure, recall that all government

decisions with respect to Wnancing and providing the good are lump-sum

events from any one consumer's point of view. Since the market system

completely breaks down because of the revelation problem, the government

has no choice but to select a given quantity of the good that will be available

in equal amounts to all consumers, hope that it satisWes the

P

MRS  MRT

rule, and then Wnances its purchases with lump-sum taxes to preserve

eYciency in all other markets.

For the purposes of this discussion, assume that the government has

selected the optimal quantity, so that

P

H

h1

MRS

h

 MRT. Assume further

that production of the nonexclusive good and all other goods and services

exhibits either CRS or linear technology.

Consumers react in two ways to the existence of a nonexclusive good. On

the one hand, the good enters each consumer's utility function directly as one

of the arguments, although the sign of the argument is uncertain. Some

consumers may view it as a ``good,'' others as a ``bad,'' especially at the

margin. On the other hand, consumers may well adjust their own goods

demands and factor supplies in response to the nonexclusive good. That is,

the nonexclusive good may be a substitute for or complement to other goods

and factors.

A representation of the consumer's indirect utility function that captures

these features is

V

~

q;

I; e



 UX

i

~

q;

I; e



;e



(17:7)

with

qV

qe



P

N

i1

qU

qX

i

qX

i

qe



qU

qe

(17:8)

where:

~

q  the vector of consumer prices.

17. EXPENDITURE INCIDENCE AND ECONOMY-WIDE INCIDENCE STUDIES 579

X

i

 good (factor) i demanded (supplied) by the consumer.

I  a source of lump-sum income other than proW ts from production,

assumed constant unless taxed by the government.

e  the quantity of the nonexclusive good selected by the government.

Two results useful for the measure of incidence follow directly from the

Wrst-order conditions of utility maximization. First, diVerentiate the budget

constraint with respect to e to obtain:

P

N

i1

q

i

qX

i

qe

 0 (17:9)

From the primal of the consumer problem,

qU

qX

i

 l q

i

i  1, ... , N (17:10)

Substituting Eq. (17.10) into (17.9) yields:

1

l

P

N

i1

U

i

qX

i

qe

 0 (17:11)

Thus, Eq. (17.8) simpliWes to:

qV

qe



qU

qe

(17:12)

The change in utility from a marginal change in the nonexclusive good

equals its direct marginal eVect on utility. Although consumers may change

their other purchases and factor supplies in response to the change in e, these

changes have no further eV ect on utility.

Second, Eq. (17.12) implies that the marginal rate of substitution be-

tween e and ith good or factor, MRS

e

,X

i

,isdeWned exactly as it would be for

any exclusive good:

MRS

e

,

X

j



qU

qe

qU

qX

i

(17:13)

If good i is the numeraire, then

MRS

e

,

X

j



1

l

qU

qe



dU

de

dU

dI

dI /de

UU



(17:14)

Thus, the marginal rate of substitution establishes the value of a marginal

increase in the public good to the consumer, as it does for any good.

The value of a Wnite amount of the public good can be derived from the

consumer's expenditure function. In the presence of a nonexclusive good, the

dual to the standard consumer problem is

580 SAMUELSONIAN NONEXCLUSIVE GOODS

min

X

i



P

N

i1

q

i

X

i

s:t: U  U

~

X; e



The Wrst-order conditions yield compensated demand (supply) functions of

the form:

X

comp

i

 X

i

~

q;

U

~

X; e

hi

i  1, ..., N (17:15)

and the expenditure function:

M

~

q;

U

~

X; e

hi



P

N

i1

q

i

X

comp

i

~

q;

U

~

X; e

hi

(17:16)

Thus, even though the consumer does not purchase e, the expenditure func-

tion has e as an argument because e appears in the utility function, which is

being held constant. All we need establish, then, is that qM/qe 6 0, so that as

e changes the income required to keep the consumer at the same utility level

also changes.

qM

qe



P

N

i1

q

i

qX

comp

i

~

q;

U

~

X; e

hi

qe

(17:17)

Substituting Eq. (17.10) into (17.17) yields:

qM

qe



1

l

P

N

i1

qU

i

qX

i

qX

comp

i

~

q;

U

~

X; e

hi

qe

(17:18)

But U 

U(

~

X; e). Thus,

P

N

i1

qU

i

qX

i

qX

comp

i

qe



qU

qe

 0(17:19)

if utility is held constant, or:

P

N

i1

qU

i

qX

i

qX

comp

i

~

q;

U

~

X; e

hi

qe



qU

qe

(17:20)

Hence:

qM

qe



1

l

qU

qe



dI

de









UU

(17:21)

As expected, the derivative of the expenditure function with respect to

the nonexclusive good yields the change in lump-sum income that makes the

consumer indiVerent to a change in the nonexclusive good. From Eq. (17.19),

17. EXPENDITURE INCIDENCE AND ECONOMY-WIDE INCIDENCE STUDIES 581