Public Finance, 2nd Edition

3. We saw that public policy is problematic in the case of individualized

consumption externalities because the government must design a set of H

corrective taxes, one for each of H people consuming the good. In contrast,

when the external eVect depends only on aggregate consumption a single tax

paid by all consumers can achieve the pareto-optimal conditions. The same

distinction applies for production externalities.

Because of these similarities, Chapter 7 presents only the aggregate

production externalities model. The aggregate model is by far the one most

widely used in policy applications, and it provides a simple analytical frame-

work for considering a number of policy implications that could have been

discussed in the preceding chapter but are especially intuitive in a production

framework. Most of the policy examples in Chapter 7 center on pollution

control, as industrial pollution is a particularly appropriate and important

application of the aggregate production externalities model. Chapter 8 then

discusses U.S. antipollution policy as an extended example incorporating

both production and consumption externalities.

Having analyzed aggregate production externalities and noted their

similarities with aggregate consumption externalities, the reader should have

no diYculty modeling other types of production externalities. The other pro-

duction cases are also closely analogous to their consumption counterparts.

THE CONDENSED MODEL FOR PRODUCTION EXTERNALITIES

The analysis of consumption externalities used a condensed version of the

general equilibrium model in Chapter 2 for its analytical framework of

the form:

max

X

ik



WU

h





s:t: F

P

H

i1

X

ik



 0

where X

ik

was deWned as the consumption of good k by person i. The way in

which the X

ik

entered each person's utility function determined the appropri-

ate policy response by the government.

Production externalities can also be analyzed with a condensed version

of the full general equilibrium model, the only diVerence being that the model

must highlight possible interdependencies in production rather than in con-

sumption. To achieve this, we will ignore once again any notational distinc-

tion between goods and factors but deWne the arguments, X, in terms of

production. Let X

ji

 good (factor) i supplied (demanded) by Wrm j, with

factors measured negatively, j  1, ...,J and i  1, ..., N. There are J

Wrms and N goods and factors.

206 THE CONDENSED MODEL FOR PRODUCTION EXTERNALITIES

Since we are now interested in production interrelationships, writing

production as a single production-possibilities frontier is no longer use-

ful. The model must retain the individual-Wrm production functions. De-

Wne f

k

  0 as the implicit production function for Wrm k, k  1, ...,

J.Write:

f

k

X

ji



 0k 1, ...,J (7:1)

as the most general notation. This allows for the worst possible case of

individualized externalities, in which each of the J production relationships

has JN arguments: The production (use) of any of the N goods (factors) by

any of the J Wrms in the economy aVects every Wrm. In this model, each Wrm

could produce multiple outputs, rather than a single output as in the Chapter

2 model. The model also permits each good and factor to be produced,

although this is not necessary. J can be larger or smaller than N.

1

Analogous with consumption externalities, deWne a pure public good

(factor) as one for which:

qf

k

qX

ji

 f

k

ji

6 0 all k, j  1, ...,J (7:2)

That is, production (use) of good (factor) i aVects all production relation-

ships on the margin no matter where activity i occurs. This is the worst case

described above. Similarly, a pure private good (factor) is one for which:

qf

k

qX

ji

 f

k

ji

 0k6 j(7:3)

Firm k's use or production of i aVects only itself on the margin. Production

with private goods and factors is represented notationally as f

k

X

ki

0,

analogous with the notation of Chapter 6.

The condensation occurs in the household sector of the Chapter 2 model.

Interrelationships among consumers are irrelevant to the study of production

externalities, so that it is no longer necessary to retain a many-consumer

economy along with the social welfare function to resolve distributional

questions. These could be retained, to be sure, but the existence of production

externalities does not alter any of the pareto-optimal consumption conditions

or the interpersonal equity social welfare conditions that are necessary for

reaching the Wrst-best bliss point. No loss of generality occurs, then by

assuming a one-consumer-equivalent economy in which the consumer

supplies all factors of production and receives all the produced goods

and services, providing it is understood that one-consumer equivalence arises

because the government is optimally redistributing lump sum to satisfy the

interpersonal equity conditions of social welfare maximization. Without this

1

J is much larger than N in actual economiesÐthe number of Wrms far exceeds the number

of goods and factors.

7. PRODUCTION EXTERNALITIES 207