Public Finance, 2nd Edition

P

H

h1

qW

qU

h

qU

h

qX

ik

P

H

h1

qW

qU

h

qU

h

qX

i1



F

k

F

1

all i  1, ...,H (6:9)

any k  2, ...,N

The right-hand side of (6.9) has a standard interpretation, the marginal rate

of transformation (MRT) in production between goods (factors) k and 1. The

left-hand side has no standard interpretation, however. As written, it is a

ratio of marginal impacts on social welfare from consuming (supplying) the

two goods (factors), and there is no way to simplify the expression. In

particular, the social welfare terms, qW/qU

h

, do not cancel, so that the rule

is not really a pareto-optimal or eY ciency condition at all. Recall that pareto-

optimal conditions do not contain social welfare terms. In this worst of all

worlds, then, the model does not dichotomize into interpersonal equity and

pareto-optimal conditions, the only exception we will encounter in all of Part

II. All the decision rules are of the interpersonal equity type and can be

achieved only by lump-sum redistributions of all goods and factors, a truly

hopeless situation. Moreover, the competitive market system, which equates

marginal rates of substitution in consumption to marginal rates of transform-

ation, would be absolutely useless. Nothing short of a complete government

takeover of the economy would be capable of satisfying the Wrst-order

conditions for social welfare maximization, even in principle.

THE EXISTENCE OF AT LEAST ONE PURE PRIVATE GOOD

Fortunately, the real world is not so riddled with consumption externalities.

A large number of goods are pure private goods, or close enough to pure

private goods that a government would not consider intervening in their

markets. To keep the discussion as general as possible, however, let us assume

that there is only one pure private good in the economy, the Wrst. Formally,

qU

h

/qX

i1

 0, i 6 h. The other (N  1) goods and factors remain pure public

goods. As it turns out, only one private good is needed to resurrect the

dichotomy between the pareto-optimal and interpersonal equity conditions

that normally exists in Wrst-best analysis and to retain a role for the competi-

tive market system in allocating all the goods and factors.

With a single private good, the social welfare maximization problem

becomes:

max

X

ik

; X

h1



WU

h

X

ik

;X

h1





s:t: F

P

H

i1

X

ik

;

P

H

h1

X

h1



 0

6. CONSUMPTION EXTERNALITIES 159

where k  2, ..., N. Good 1 has been written separately to indicate speciW-

cally that it is a pure private good.

Interpersonal Equity Conditions

Consider the interpersonal equity conditions with respect to good 1, the pure

private good. The Wrst-order conditions are

3

X

h1

:

qW

qU

h

qU

h

X

h1

lF

1

(6:10)

X

i1

:

qW

qU

i

qU

i

X

i1

lF

1

(6:11)

or

qW

qU

h

qU

h

X

h1

lF

1

all h  1, ...,H (6:12)

Equation (6.12) is identical to the interpersonal equity conditions in the

standard model of Chapter 2. Assume the government can redistribute X

1

lump sum to achieve this condition as part of its Wrst-best policy strategy.

Pareto-Optimal Conditions

As above, consider the Wrst-order conditions with respect to two goods

(factors) consumed (supplied) by any one person i, say X

ik

and X

i1

. The

choice of k is arbitrary, but good 1, the private good, must be one of the two

goods chosen. The Wrst-order conditions are

X

ik

:

P

H

h1

qW

qU

h

qU

h

X

ik

lF

k

(6:13)

X

i1

:

qW

qU

i

qU

i

qX

i1

lF

1

(6:14)

Dividing Eq. (6.13) by (6.14) yields:

P

H

h1

qW

qU

h

qU

h

qX

ik

qW

qU

i

qU

i

qX

i1



F

k

F

1

, for k  2, ...,N (6:15)

Condition (6.15) can be simpliWed if the government has satisWed the

interpersonal equity conditions for good 1. The left-hand side is a summation

3

l is the Lagrangian multiplier associated with F ( ).

160 THE EXISTENCE OF AT LEAST ONE PURE PRIVATE GOOD

of social welfare terms over a common denominator,

qW

qU

i

qU

i

qX

i1

. But, if inter-

personal equity holds,

qW

qU

i

qU

i

X

i1

lF

1

all i  1, ...,H (6:16)

Selectively substitute for the denominator term by term, matching up the

social welfare terms, and write:

qW

qU

1

qU

1

qX

ik

qW

qU

1

qU

1

qX

11

 , ..., 

qW

qU

h

qU

h

qX

ik

qW

qU

h

qU

h

qX

h1

 , ..., 

qW

qU

H

qU

H

qX

ik

qW

qU

H

qU

H

qX

H1



F

k

F

1

,(6:17)

any k  2, ...,N

P

H

h1

qW

qU

h

qU

h

qX

ik

qW

qU

h

qU

h

qX

h1

2

6

6

6

4

3

7

7

7

5



F

k

F

1

all i  1, ...,H

any k  2, ...,N

(6:18)

The social welfare indexes, qW/qU

h

cancel term by term, yielding:

P

H

h1

qU

h

qX

ik

qU

h

qX

h1

2

6

6

6

4

3

7

7

7

5



F

k

F

1

(6:19)

The left-hand side of Eq. (6.19) has a standard pareto-optimal interpretation,

devoid of social welfare terms. It is a sum of marginal rates of substitution,

each person's marginal rate of substitution (MRS) between person i's con-

sumption of good k and her own consumption of the pure private good.

Thus, the rule can be written as:

P

H

h1

MRS

h

X

ik

; X

h1

 MRT

k; 1

for all i  1, ...,H

any k  2, ...,N

(6:20)

Note carefully that the ability to cancel the social welfare terms is not just

a formal ``trick.'' It implies an optimal Wrst-best policy action, a lump sum

redistribution that satisWes the interpersonal equity conditions for good

(factor) 1. Without the optimal redistribution, the terms would not cancel

and all the policy implications of the pareto-optimal conditions which we are

about to discuss become irrelevant. Conditions (6.19) would not be the

necessary conditions for a social welfare maximum. We will employ this

6. CONSUMPTION EXTERNALITIES 161