Public Finance, 2nd Edition

p

1

g

Z

k

dZ

k

 p

k

dZ

k

 0k 2, ..., N (22:64)

implies

p

1

dZ

1

 p

k

dZ

k

 0k 2, ..., N (22:65)

SECOND-BEST PRODUCTION RULES WHEN EQUITY MATTERS

Assuming a one-consumer-equivalent economy in second-best analysis is

always somewhat contradictory. Unless consumers' tastes are severely re-

stricted, one-consumer equivalence implies that the government is optimally

redistributing income lump sum in accordance with the Wrst-best interper-

sonal equity conditions, thereby equilibrating social marginal utilities of

income. But if the government can do this, why would it ever have to use

distorting taxes?

The more natural approach in a second-best framework is to deny the

existence of optimal income redistribution and assume explicitly that social

marginal utilities of income are unequal. This means, however, that the

optimal shadow prices for government production decisions depend upon

both eYciency and equity considerations, just as the many-person optimal

tax and nonexclusive goods decision rules were seen to incorporate both

eYciency and equity terms. This is doubly discouraging for policy purposes,

as we noted when discussing those problems. Not only are optimal prices

further complicated by the addition of equity terms, but also society may not

agree on the proper equity weights for each individual. Thus, the analysis runs

the risk of becoming totally subjective, since diVerent sets of ethical weights

imply diVerent optimal shadow prices. Nonetheless, if society can agree on a

ranking of social marginal utilities of income (a big if ), then the proper shadow

prices for government production can be determined. Furthermore, the

shadow prices can be expressed as a simple combination of distinct equity

and eYciency eVects, at least for the particular government production deci-

sions and second-best distortions being considered in this chapter.

To relate the many-person results as closely as possible to the one-person

rules, we will assume away all sources of lump-sum income by requiring that

all factor supplies are variable and private production exhibits CRS. A

further assumption is that the government budget exactly balances. These

assumptions greatly simplify the analysis, while capturing the Xavor of many-

person second-best analysis.

8

The government's objective function, then, is:

W  WV

h

~

q



 V

~

q (22:66)

8

With only minor changes, the analysis of this section is taken directly from R. Boadway,

``Integrating Equity and EYciency in Applied Welfare Economics,'' Quarterly Journal of Eco-

nomics, November 1976.

22. GENERAL PRODUCTION RULES IN A SECOND-BEST ENVIRONMENT 709

where W is the agreed-upon individualistic Bergson±Samuelson social wel-

fare function whose arguments are the individuals' indirect utility functions

V

h

(

~

q). DiVerentiating totally,

dW 

P

H

h1

P

N

i1

qW

qV

h

qV

h

qq

i

dq

i



P

H

h1

P

N

i1

b

h

X

hi

dq

i

(22:67)

from Roy's Identity and the deWnition of an individual's social marginal utility

of income as b

h

 (qW/qV

h

)a

h

, where a

h

 the private marginal utility of

income for person h. It will be convenient to express the change in social

welfare in terms of Martin Feldstein's distributional coeYcient for X

i

:

R

i



P

H

h1

b

h

X

hi

X

i

i  1, ..., N (22:68)

to work with aggregate consumption.

9

R

i

is a weighted average of the

individuals' social marginal utilities of income, with the weights equal to

the proportion of good (factor) i consumed (supplied) by person h. Substi-

tuting Eq. (22.68) into (22.67) yields:

dW 

P

N

i1

R

i

X

i

dq

i

(22:69)

The problem is to deWne dW in terms of the government control variables

~

t  (t

2

, ...,t

N

) and

~

Z  (Z

2

, ...,Z

N

), given the following constraints:

1. Private production possibilities, F(Y

1

, ...,Y

N

)  0, assumed to

exhibit CRS.

2. The government production function, G(Z

1

, ...,Z

N

)  0, or

Z

1

g(Z

2

, ...,Z

N

), with inputs measured negatively.

3. The government budget constraint,

P

N

i2

t

i

X

i



P

N

i1

p

i

Z

i

 0.

4. N market clearance relationships, X

i

(

~

q)  Y

i

(

~

p)  Z

i

, for

i  1, ..., N. All markets clear in the actual general equilibrium.

5.

~

q 

~

p 

~

t, with q

1

 p

1

 1, and t

1

 0.

As always, the Wrst good serves as the untaxed numeraire.

The analysis proceeds much as in the one-consumer case. Begin by totally

diVerentiating the market clearance equations:

dX

i

 dY

i

 dZ

i

i  1, ..., N (22:70)

Multiply each equation by q

i

 (p

i

 t

i

) and sum over all N equations to

obtain:

9

M. Feldstein, ``Distributional Equity and the Optimal Structure of Public Prices,'' Ameri-

can Economic Review, March 1972. Also see our discussion of Feldstein's distributional coeY-

cient in Chapter 14.

710 SECOND-BEST PRODUCTION RULES WHEN EQUITY MATTERS

P

N

i1

q

i

dX

i



P

N

i1

p

i

 t

i

dY

i



P

N

i1

p

i

 t

i

dZ

i

(22:71)

Equation (22.71) can be simpliWed as follows. Totally diVerentiate the indi-

vidual consumers' budget constraints

P

N

i1

q

i

X

hi

 0, all h  1, ..., H, and

sum over all individuals to obtain:

P

N

i1

q

i

dX

i



P

N

i1

X

i

dq

i

(22:72)

Next, diVerentiate the aggregate private production possibilities F(

~

Y)  0,

P

F

i

dY

i

 0 (22:73)

But, if markets are perfectly competitive,

F

i

F

1



p

i

p

1

 p

i

,withp

1

 1i 2, ..., N (22:74)

Therefore:

P

N

i1

F

i

dY

i

 0  F

1

P

N

i1

p

i

dY

i

(22:75)

or

P

N

i1

p

i

dY

i

 0 (22:76)

Substituting Eqs. (22.72) and (22.76) into (22.71) yields:



P

N

i1

X

i

dq

i



P

N

i1

t

i

dY

i



P

N

i1

t

i

dZ

i



P

N

i1

p

i

dZ

i

(22:77)

Using Eq. (22.70), Eq. (22.77) can be expressed as:



P

N

i1

X

i

dq

i



P

N

i1

t

i

dX

i



P

N

i1

p

i

dZ

i

(22:78)

Substituting Eq. (22.78) into (22.69) yields:

dW 

P

N

i1

R

i

X

i

dq

i



P

N

i1

X

i

dq

i



P

N

i1

t

i

dX

i



P

N

i1

p

i

dZ

i

(22:79)

or

dW 

P

N

i1

1  R

i

X

i

dq

i



P

N

i1

t

i

dX

i



P

N

i1

p

i

dZ

i

(22:80)

Next, incorporate the government production function, Z

1

g(Z

2

, ...,

Z

N

), and note that t

1

 0, dq

1

 0, to rewrite Eq. (22.80) as:

22. GENERAL PRODUCTION RULES IN A SECOND-BEST ENVIRONMENT 711