Public Finance, 2nd Edition

(1  l )

P

N

i2

t

i

M

ik

 lM

k

 0k 2, ..., N (14:31)

Rearranging terms:

P

N

i2

t

i

M

ik

M

k



l

1  l

 Ck 2, ..., N (14:32)

with C independent of k. Equations (14.32) are identical to Eqs. (13.30). They

imply that the pattern of optimal taxes depends only upon the compensated

demand (factor supply) derivatives qX

comp

i

/qq

k

. Moreover, the equal percent-

age change interpretation applies to Eq. (14.32) exactly as it applies to Eq.

(13.30).

8

The assumption of CRS, then, greatly simpliWes the application of

second-best results. Admittedly Eq. (14.32) would be diYcult to apply in

practice given our limited econometric knowledge of the relevant Slutsky

substitution terms, but at least the general equilibrium supply responses to

the tax can be ignored.

That the assumption of CRS for private production simpliWes the opti-

mal commodity tax rules is not unique to that problem. CRS tends to simplify

all second-best results in both tax and expenditure theory. Whether CRS is an

appropriate assumption for the US economy is an open question. CRS is

often assumed for aggregate production in empirical analysis, but this is

mostly because aggregate production data are collected by the government

and are constructed under the assumption of CRS (exhaustion of the prod-

uct). The same is true of production analysis at the two-digit industry level.

Another diYculty in applying this, and other, second best results is the

market assumption of perfect competition. It is certainly violated for a

number of important goods and services. At the same time, perfect competi-

tion is the natural default assumption for second-best public sector analysis.

One could hardly attempt to model the various forms that market imper-

fections take industry by industry throughout the economy.

MANY-PERSON ECONOMIES: FIXED PRODUCER PRICES

Social Welfare Maximization Versus Loss Minimization

Before 1970, public sector economists chose to analyze second-best tax

theory almost exclusively within the context of one-consumer economies to

8

The independence of the optimal tax rules to supply responses with constant returns-

to-scale production was Wrst pointed out to us by Paul Samuelson in a set of unpublished class

notes. Refer to J. Stiglitz and P. Dasgupta, ``DiVerential Taxation, Public Goods, and Economic

EYciency,'' Review of Economic Studies, April 1971, for an alternative derivation of this result.

14. THE SECOND-BEST THEORY OF TAXATION WITH GENERAL PRODUCTION 461