Public Finance, 2nd Edition

equal zero for some i or even most i. All that is required for the existence of a

consumption externality is that one person's utility be a function of one other

person's consumption (supply) of something.

THE WORST OF ALL WORLDSÐALL GOODS (FACTORS) ARE

PURE PUBLIC GOODS (FACTORS)

In the worst of all worlds, all goods (factors) are pure public goods (factors).

2

For policy purposes, this is really a horrendous situation as the government

can hardly interpret what the proper decision rules mean let alone have any

hope of implementing them. The government's problem is

max

X

ik



WU

h

X

ik





s:t: F

P

H

i1

X

ik



 0

with the understanding that each utility function U

h

( ) contains all NH

elements, X

ik

i  1, ...,H;k 1, ...,N.

The corresponding Lagrangian is

max

X

ik



L  WU

h

X

ik





 lF

P

H

i1

X

ik



Before proceeding, notice how deceptively similar this problem is to the

problem of social welfare maximization when there are only pure private

goods. In our notation, the pure private goods case is represented as:

max

X

hk



WU

h

X

hk





s:t: F

P

H

h1

X

hk



 0

In each case, maximization occurs with respect to HN goods and factors, the

N goods and factors consumed and supplied by each of H people. The

diVerence is that in the worst of all worlds all HN variables appear in each

utility function, whereas in the pure private goods (factors) world only N

variables appear in each utility function. This diVerence matters, because the

policy implications are enormously diVerent. In the latter case, the competi-

tive market can achieve full pareto optimality. In the former case, the market

cannot be expected to achieve eYciency, and the government is virtually

powerless to act in an optimal manner. The problems for the government

in the pure public goods case are self-evident upon examination of the

2

Ng provides an alternative model of this worst of all worlds in Y-K Ng, ``The Paradox of

Universal Externalities,'' Journal of Economic Theory, April 1975.

6. CONSUMPTION EXTERNALITIES 157

Wrst-order conditions for social welfare maximization, both the interpersonal

equity conditions and the pareto-optimal conditions.

Interpersonal Equity Conditions

Recall that the interpersonal equity conditions are obtained by comparing

the Wrst-order conditions for any one good (factor) consumed (supplied) by

any two people, say X

j1

and X

i1

. The Wrst-order conditions are

X

j1

:

P

H

h1

qW

qU

h

qU

h

X

j1

lF

1

(6:4)

X

i1

:

P

H

h1

qW

qU

h

qU

h

X

i1

lF

1

(6:5)

From conditions (6.4) and (6.5),

P

H

h1

qW

qU

h

qU

h

X

i1

lF

1

all i  1, ...,H (6:6)

The interpretation of the interpersonal equity conditions is identical to that

of the standard model in Chapter 2, which contained only pure private goods:

The government should redistribute good 1, lump sum, until social welfare is

equalized on the margin across all individuals. This task, diYcult enough

with pure private goods, is now hopelessly complex, however. When the

government gives (takes) an extra unit of good (factor) 1 to (from) person i,

it must know how all people react to that transfer (tax) on the margin, not just

how person i's utility is aVected, and similarly for units transferred to or from

any other person. This is clearly an impossible task, one the government

could not even hope to approximate.

Pareto-Optimal Conditions

The pareto-optimal conditions also diVer considerably from their counter-

part in a world of pure private goods, both in form and interpretation. Recall

that the pareto-optimal conditions are obtained from the Wrst-order condi-

tions of any two goods consumed (factors supplied) by any one person, say

X

ik

and X

i1

. The Wrst-order conditions are

X

ik

:

P

H

h1

qW

qU

h

qU

h

X

ik

lF

k

(6:7)

X

il

:

P

H

h1

qW

qU

h

qU

h

X

il

lF

1

(6:8)

Dividing Eq. (6.7) by (6.8) yields:

158 THE WORST OF ALL WORLDSÐALL GOODS (FACTORS) ARE PURE PUBLIC GOODS