Public Finance, 2nd Edition

Thus, the government production rule is simply:

p

k

 p

1

g

Z

k

 0k 2, ..., N (22:29)

or

p

k

 p

1

g

Z

k

k  2, ..., N (22:30)

Equation (22.30) is the standard Wrst-best rule for production eYciency in

competitive markets. Alternatively,

p

k

p

j



g

Z

k

g

Z

j

 MRT

Z

k

; Z

j

k, j  2, ..., N (22:31)

with the government using the competitively determined producer prices as

shadow prices in its production decisions.

This may well be the most striking result in all of second-best public

expenditure theory, one of the precious few examples of a simple second-

best decision rule. It implies overall production eYciency for the economy

5

or that the economy should remain on its aggregate production-possibilities

frontier. Of course, with distorting taxation the economy cannot also be on its

Wrst-best utility-possibilities frontier. A Wnal implication in an intertemporal

context is that government investment decisions should use the private sector's

gross-of-tax returns to capital as the rate of discount in present value calcula-

tions (recall that p

k

is a gross-of-tax price for an input such as capital).

6

Since

the U.S. marginal corporate tax rate is 34 or 35% for most Wrms in the United

States, this implies a fairly high government rate of discount, the rate of return

the government must beat to justify public investment at the expense of private

investment. We will return to this point in Part IV when discussing the rate of

discount in cost±beneWt analysis.

PRODUCTION DECISIONS WITH NONOPTIMAL TAXES

The Diamond±Mirrlees problem provides a clear example of just how far

removed second-best theory often is from the complexities of the real world,

even though it contains elements that are more realistic than the traditional

Wrst-best assumptions. Taxes are distorting in this model, but assuming that

current tax rates are (even approximately) at their optimal values is every bit

as heroic as assuming that taxes are (approximately) lump sum, which Wrst-

best theory requires. We can move somewhat closer to reality by assuming

explicitly that the current rates are nonoptimal and asking how this aVects the

5

Recall that the private sector is assumed to be perfectly competitive and therefore Wrst-best

pareto eYcient.

6

Intertemporally, all budget constraints in the general equilibrium framework must balance

in terms of present value, not year by year, and there must be perfect capital markets for

borrowing and lending.

22. GENERAL PRODUCTION RULES IN A SECOND-BEST ENVIRONMENT 701

government's production decision rules. Formally, this assumption is equiva-

lent to adding further constraints to the original Diamond±Mirrlees problem

of the form that a subset of the tax rates are predetermined at nonoptimal

levels. Given these predetermined rates, the Wrst-order conditions of the new

problem indicate how the government can adjust its production decisions to

minimize loss.

Unfortunately, the resulting production rules are extremely complex.

They have a plausible interpretation, but it is doubtful whether any govern-

ment would have suY cient information to implement them. Furthermore,

this problem is still far removed from reality, for it retains the assumption of a

perfectly competitive CRS private production sector. Were we to introduce

monopoly elements in private production and/or decreasing or increasing

returns to scale with pure proWts or losses, the optimal production rules

would change once again. Consequently, the normative policy content of

this model is not especially compelling either. Nonetheless, it is instructive to

explore the production decision rules when taxes are nonoptimal if only to

give a Xavor for this kind of analysis.

To keep the notation as simple as possible, rewrite the loss function

entirely in vector notation as:

Lt;ZMq;

U

0



 t

0

M

i

 p

1

gZq  t

0

Z  p q  t(22:32)

Written in this form the loss function incorporates every relevant constraint

except for the market clearance equations, Eq. (22.6), expressed in vector

notation as:

M

i

q; U

0



 p

i

q  tZ (22:33)

The nonoptimal tax and production rules are derived by totally diVerentiat-

ing the loss function with respect to t and Z, and using Eq. (22.33) to simplify

the resulting expression:

dL t; ZM

0

i

qq

qt

dt  M

0

i

qq

qZ

dZ  M

0

i

dt  t

0

M

ij

qq

qt

dt

 t

0

M

ij

qq

qZ

dZ  p

1

g

Z

dZ  Z

0

qq

qt

dt  Z

0

dt  Z

0

qq

qZ

dZ

 q t

0

dZ  p

0

i

qq

qt

dt  p

0

i

qq

qZ

dZ  p

0

dt

(22:34)

From market clearance:

M

0

i

dt  p

0

i

dt  Z

0

dt (22:35)

M

0

i

qq

qt

dt  p

0

i

qq

qt

dt  Z

0

qq

qt

dt (22:36)

and

702 PRODUCTION DECISIONS WITH NONOPTIMAL TAXES