Public Finance, 2nd Edition

22

GENERAL PRODUCTION RULES IN

A SECOND-BEST ENVIRONMENT

THE DIAMOND±MIRRLEES PROBLEM:

ONE-CONSUMER ECONOMY

Optimal Taxation

Optimal Government Production

PRODUCTION DECISIONS WITH NONOPTIMAL TAXES

Tax Rules

Production Rules

SECOND-BEST PRODUCTION RULES WHEN EQUITY MATTERS

Concluding Comments

Chapter 22 concludes our survey of second-best public expenditure

theory by exploring some fairly general propositions about government

production in an environment made second best because of distorting tax-

ation. A major goal of the chapter is to integrate our previous results on

second-best tax theory with second-best public expenditure theory. There-

fore, all the analysis in Chapter 22 employs essentially the same set of

assumptions regarding government activity and the underlying structure of

the private sector that we have been using time and again.

Regarding the government sector, the government is making a set of

production decisions under two constraints: (1) it must buy inputs and sell

outputs at the established private sector producer prices, and (2) it must cover

any resulting deWcit (surplus) with distorting commodity taxes levied on the

consumer. Otherwise, government production is fully general. The govern-

ment may buy or sell any inputs or outputs, including those traded in the

private sector, and there are no restrictions on the form of the aggregate

government production function other than the exclusion of externalities.

Following Chapter 21, the government's production function is speciWed as

G(Z)  0, or Z

1

g(Z

2

, ...,Z

N

), where Z spans (potentially) the entire set

of the economy's inputs and outputs. The only diVerence between the

693

speciWcation of the government sector in this chapter and the speciWcation

employed in the Boiteux analysis is that the government taxes (subsidizes) all

consumer transactions to cover its deWcits (surpluses), not just those between

the consumers and the government.

Regarding the private sector, all markets are assumed to be perfectly

competitive and private production exhibits general technology with con-

stant returns to scale (CRS). There can be no pure proWts or losses from

private production. We also assume that the consumers have no other sources

of lump-sum income. All income derives from the sale of variable factors.

These assumptions about the private sector are not necessary, but they

greatly facilitate the analysis. In sum, the only distortions in the economy

that render the analysis second best are the distorting commodity taxes used

to cover government production deWcits.

Given this analytical framework, the Wrst problem to be considered is the

so-called Diamond±Mirrlees problem, which Peter Diamond and James Mirr-

lees set out in their two-part article in the 1971 American Economic Review

entitled ``Optimal Taxation and Public Production.''

1

By 1968, when their

paper was drafted and widely circulated, the optimal tax rule for a one-

consumer (equivalent) economy was well known, but only under the assump-

tion that the government simply raised revenue to be returned lump sum to the

consumer. Diamond and Mirrlees added government production to the stand-

ard second-best general equilibrium tax model and asked two questions:

1. How does the existence of government production aVect the optimal

tax rule? In particular, if the revenue is raised to cover a government produc-

tion deWcit under the conditions set forth above, what form do the tax rules

take? They found that the optimal tax rule was unchanged. This result could

have been anticipated since it was well known by then that the tax rules as

originally derived did not contain any production terms even when private

production exhibited general technology.

2. Turning the question around: What eVect does distorting taxation

have on government production rules? Their answer to this question most

deWnitely was unanticipated. They proved that as long as the taxes are set

optimally, the government should follow the standard Wrst-best production

rules, using the private sector producer prices and equating these price ratios

to marginal rates of transformation. Distorting taxation necessarily forces

society underneath its utility-possibilities frontier, but it should remain on the

production-possibilities frontier. This is one of the strongest results in all of

second-best theory.

Having established the Diamond and Mirrlees production result, the

chapter then generalizes their analysis to consider government production

1

P. Diamond and J. Mirrlees, ``Optimal Taxation and Public Production'' (2 parts, Part I:

Production EYciency, Part II: Tax Rules), American Economic Review, March, June 1971.

694 GENERAL PRODUCTION RULES IN A SECOND-BEST ENVIRONMENT