Public Finance, 2nd Edition

The goal of this chapter is not to oVer a comprehensive discussion of

investment analysis under uncertainty. Instead, the chapter is centered

around a remarkable theorem by Arrow and Lind.

1

They proved that the

government can ignore risk under certain fairly broad conditions. That is, the

aggregate risk premium under these conditions is zero. Since the conditions

are not likely to apply to private investment projects, this is perhaps the one

instance in which cost±beneWt analysis may be conceptually easier than

private investment analysis.

THE ARROW-LIND THEOREM

Kenneth Arrow and Robert Lind proved that government policy analysts

can ignore risk under two conditions that may well be approximated by many

public projects:

1. The net beneWts of a government project are distributed

independently of national income.

2. The net beneWts of the project are each spread over a suYciently

large population.

Under these two conditions, the risk premium that society in the aggregate

would be willing to pay to convert a stream of uncertain returns into a certain

return goes to zero in the limit.

The Arrow±Lind theorem is especially powerful because it makes no

assumptions about the underlying market environment. For instance, it

need not be perfectly competitive. Also, there are no restrictions imposed

on the government other than the two conditions assumed about the net

beneWts of the government's projects. Hence, their result is applicable to a

wide range of second-best policy environments (as well as the Wrst-best

environment).

The only potentially unrealistic feature of their model concerns the

distribution of project costs and beneWts. Arrow and Lind guarantee that

the Wrst independence condition holds by assuming that the government pays

all the costs of the investment, receives all the beneWts, and then distributes

the net beneWts lump sum to each individual. They assume further that the

project's net beneWts are free of tax. Thus, there can be no further Wscal

repercussions of the project that could lead indirectly to a correlation be-

tween its net beneWts and each consumer's disposable income. These assump-

tions are clearly not meant to be realistic. Arrow and Lind use them merely as

an analytically convenient way of satisfying the independence condition.

They turn out not to be innocuous, however. L. Foldes and R. Rees contend

1

K. Arrow and R. Lind, ``Uncertainty and the Evaluation of Public Investment Decisions,''

American Economic Review, June 1970.

760 THE ARROW-LIND THEOREM

that it may not be possible to satisfy the independence condition for very

many projects in the context of an actual Wscal system.

2

The Foldes±Rees

objection follows the Arrown±Lind theorem.

Proof of the Arrow±Lind Theorem

The proof of the Arrow±Lind theorem requires nothing more sophisticated

than the deWnition of a derivative and some properties of the expected value

operator. Begin by assuming that society consists of N identical consumers,

each of whom has initial income equal to A, where A is a random variable. In

line with accepted practice in uncertainty analysis, assume further that each

consumer maximizes expected utility. (The assumption that the consumers

are identical is not necessary to the proof but greatly simpliWes the deriv-

ation.) Let B  the total net returns from some government project. Assume

B is also a random variable, equal to its expected value,

B, and a random

component X with zero mean:

B 

B  X (25:1)

with:

EX0

Finally, assume that B and A are independently distributed (the Wrst condi-

tion) and that each of the N identical individuals receives an equal share of

the returns B. Thus, each person's share is s  1/N.

Under these assumptions, an individual's income without the project is

A. With the project, the individual receives A  sB  A  s

B  sX. The

corresponding expected utilities with and without the project are

E[U(A  s

B  sX)] and E[U(A)]

DeWne each person's expected utility with the project as a function of s,

or

WsEUA s

B  sX



(25:2)

DiVerentiate W(s) with respect to s and evaluate the derivative at s  0:

W

0

sEU

0

A  sB  sX



B  X



(25:3)

Hence:

W

0

0EU

0

AB  X



 BE U

0

AEU

0

AX(25:4)

But, if A and X are independently distributed,

EU

0

AXEU

0

AEX0 (25:5)

2

L. Foldes and R. Rees, ``A Note on the Arrow±Lind Theorem,'' American Economic

Review, March 1977.

25. UNCERTAINTY AND THE ARROW±LIND THEOREM 761