Equity Considerations in Economic and Policy Analysis

Jean-Paul Chavas

There is little doubt that equity considerations play some role in human decision making. This can be seen through statements that "we are all created equal," or that "we have equal rights ." Thi s can also be seen through some of the transfer mechanisms that societies are willing to put in place for the benefits of the poor or the malnourished . In spite of this evidence, the role of equity in resource allocation remains poorly understood. Most economic analyses tend to focus on efficiency issues and avoid ad­dressing distribution issues. As argued by Sen (1987, 1992), as long as human behavior is af­fected by equity considerations, it seems that economics could be made more productive by paying greater attention to welfare distribution. At the center of equity is whether and how to distinguish between people in terms of their needs, their abilities, their contributions, and their individual rights. People being different, at least some of their differences seem to be relevant to the economic evaluation of equity issues. At this point, it remains somewhat un­clear how to approach such issues.

The current debate about equity has been pro­foundly influenced by Rawls's Theory of Jus­tice. Rawls's theory is based on the concept of justice as fairness. I It relies on a so-called "original position" where individuals do not yet know their exact position in society. Rawls sug­gests that a society is just if its current mem­bers would find the "original position" fair. Rawls's views have been influential, yet con­troversial. Part of the debate comes in part from some antagonism between equity, and rights (e.g., Nozick). And private property rights are

Waugh lec[ure. Jean-Paul ehavas is professor of agric ultural eco nomics, Uni ver· s il Y of Wi sconsin, Madison.

Thi s research was supported in par! by a Ha[ch grant from [he College of Agriculture and Life Sciences, University of Vv'i sconsin .

I Aherna[ive definitions of fairness have been proposed in [he IitcralUre (e.g., see Varian). A common and intuitive definition is [0 equate fairness wi lh [he absence of envy, where no individual would prefer [0 have what another individual has.

often seen as contributing to inequalitie s of wealth in modern society.

The objective of thi s paper is to explore eq­uity issues, with a special focus on interactions between efficiency, equity, and the role of in­formation in resource allocation. In the next section, we first review a characterization of Pareto-efficient2 allocations under uncertainty. We then introduce equity considerations in the analysis . Being motivated in part by Rawls's work, this is done by relying on a fairness crite­rion, as a complement to Pareto optima lity. The usefulness of our approach is then illus trated by a few applications to economic and policy analysis. The applications emphasize the influ­ence of information and property rights in the evaluation of efficiency and equity issues . They also provide useful insights in the role of mar­ket and non market mechanisms in resource al­location .

Efficient Allocation

Consider a social group constituted of n indi­viduals . These n individuals make decisions concerning production, time allocation, and consumption activities under uncertainty. The uncertainty is represented by states of nature. Let ej denote the jth state of nature, and E = {e l ,

... , eM} be the finite set of mutually exclusive states of nature, where M is the total number of possible states. The states being mutually ex­clusive, only one state will eventually occur. However, it is typically not known ahead of time which state the n individuals will encoun­ter. Throughout the paper, the M states repre­sent all possible sources of risk facing the group, including both production-related uncer­tainties as well as consumption-related uncer­tainties. Also, they can represent lack of infor-

, Throughout the paper, economic efficiency is equated with Pareto optimality. An allocation is said to be Pareto optimal (or Pareto effici e nt) if no individual can be made beller off wilhout making someone else worse off.

Amer. J . Agr. Econ. 76 (December 1994): 1022-1033 Copyright 1994 American Agricultural Economics Association

, ,

Chavas Eqairy Considerations in Economic and Policy Analysis 1023

mation about the characteristics of the individu­als in the group, FinaJiy, in a dynamic setting, the states can reflect learning as new informa­tion becomes available over time,

We will be interested here in partitions of the set E = (el, "" eM) as a basis for evaluating the information used to evaluate decision-making and its welfare implications, Define a partition P of (e l , "" eM ) as a set composed of 5 ele­ments {PI' " " Ps}, 1:S: 5:S: M, such that

Ps n Ps' = 0 for all s *- s', uslPs: s = 1, ",,5) = lei' "" eM)'

For a given partition P = (PI' "" Ps), 5 can be interpreted as tbe total number of observable signals or messages, These signals provide in­formation on the states of nature, In particular, observing the sth signal means knowing that the true state is in p" the sth element of the par­tition P, but without being able to tell which state in Ps is the actual one . At one extreme, 5 = M is the case of perfect information where each element of P corresponds to a different state of nature, At the other extreme, 5 = I corresponds to "no information" where the only element of P is the set E itself. In general, the quality of information improves (deteriorates) as the par­tition P becomes finer (coarser), This can be formalized as follows,

DEFINITION 1. An information structure P' is at least as fine as P if, for every PEP and p' E

P', eitherp'kP orp np'= 0 ,

Intuitively, a finer partition is associated with the observation of new signals providing addi­tional information about which states might oc­cur within each element of the partition ,

The economic decisions are as follows, The production decisions involve labor and capital allocation used in the production of consumer goods , Let x be the vector of capital goods used in the production process , and t = (t u; i = 1, .. " n; j = 1, .. " J) be the vector of labor inputs, where tu denotes the Jabor spent by the ith indi­vidual on the jth activity, j = 1, .. " J, J being the total number of labor activities, Also, de­note by hi the amount of leisure used by the ith individual , i = 1, .. " n, Finally, let Yi be the vec­tor of consumer goods consumed by the ith in­dividual. These consumer goods Y = (YI' .. " Yn) are outputs of a production process using capi­tal inputs x and labor inputs t. The allocation thus involves choosing the vector z == (y. h, t, x), The vector z can be alternatively expressed as z = (Z,' .. " ZK ), where Zk is the kth decision

variable, k = 1, .. " K, K denoting the total num­ber of decision variables,

We consider the general case where each de­cision variable may be based on a different in­formation structure, This can be due to asym­metric information across individuals and/or learning over time , Let Pk be the information partition associated with the kth decision Zk' k = 1, .. " K. In general, denote by Zk(em) the kth de­cision made in the mth state of nature, k = 1, .. " K, and m = 1, .. " M, This allows for the use of different amounts of information about the state of nature for different economic decisions, The information partition Pk = (Pkl> Pk2> .. , ) imposes the following restriction on behavior:

(1) Zk(e",) = Zk(e",.) if both em and em' are in Pks

for all m, m' = 1, .. " M; for all Ph E Pk ; and for all k = 1, .. " K, Equation (1) shows how the choice Zk depends on the information partition Pk ava ilable at the time of the decision, Since imperfect information means that it is not pos­sible to distingui sh between the states within each element Pks of the partition Pk, equation (1) states that the corresponding decision Zk cannot depend on the specific states in P ks ' At one ex­treme, perfect information corresponds to the partition Pk = Ie" .. " eM), implying that the de­cision Zk is made ex post after the actual state is observed, In this case, equation (1) imposes no restriction on Zkm' m = 1, .. " M, At the other ex­treme, no information corresponds to the parti­tion Pk = IE), implying that the deci sion Zk is made ex ante , before any learning takes place , In thi s case, equation (1) requires Zk to be the same for all poss ible states of nature em' m = 1, .. " M, In intermediate situations, the partition Pk is finer than IE} but not as fine as {el ' .. " eM}' In such a case, partial learning takes place, which allows Zkm to vary across elements Pks of the partition, However, equation (1) restricts behavior Zk to be the same across states within each element Ihs of the partition Pk'

Let tJ = (P" .. " PK) denote the information structure associated with the choice of z = (z" .. " ZK), where Pk is the partition associated with the kth decision variable, Zk ' The issue consid­ered here is how to choose the decision vector z = (Z, ' .. " ZK) == (Y. h, t, x), along with the infor­mation structure tJ, Endogenizing the informa­tion structure tJ will allow LIS to investigate be­low the role of information and learning in re­source allocation,

First, the choice of z and tJ must be feasible, This is characterized by a feasible set repre­senting the technology and constraints facing

1024 De cember 1994

the group of n individuals. Under the mth state of nature, this feas ible set is denoted by F(e",), where

(2) (So, z) E F(e",) .

The set F is assumed to be nonempty and com­pact . It can represent a general multi-input multi-output aggregate technology. The state of nature e", in (2) reflects possible uncertainty in the underlying production process . In the case where the vector z includes information gather­ing activities (e.g., through the time allocation t), the feasible set F a lso represents the learning process. This allows for a joint technology, where outputs yare jointly produced with the information structure tJ (e.g ., as would be the case under "learning by doing").

Assume that the ith individual has risk pref­erences represented by the ex ante utility func­tion denoted by

where Yi == [Yi(e,), ... , Yi(e M)], Yi(e",) being the vector of consumption decisions made in the mth state, Hi = [h/e,), ... , h;(eM)], h;Ce",) denot­ing leisure time in the mth state, and the states of nature e = (e" ... , eM) represent uncertainty influencing the welfare of the ith individual, i = 1, ... , n. The utility function Ui(Yi, H" e) is ordi­nal and is assumed to be continuous and quasi­concave in (Yi , H) It provides a fairly general characterization of ex ante preferences under risk .)

Consider choosing the allocation of resources z along with the information structure tJ in a way consistent with the following maximiza­tion problem:

(4) W(a)

=maJ<.{iai Ui(r; ,Hi, e): equations (1) and (2)} 50 ,2 ;=1

where Z = [z( e ,), ... , z(eM )] , z(en,) being the de­cision vector in the mth state, z = (z" ... , ZK) = (y, h, I , x), Y == (YI, ... , Yn), h = (h" ... , h,,), t = {tij; i = I , ... , n;j = I, ... , J.} The parameters a

3 As <1 specia l case, the ex allle utility fun ction V;( .) could be the ex pected utilit y function

U,(- ) = 2:.,. 1. ... '" prob(e",) V,(y;, II;, em)

where Vi (.) is the ilh jndividu Cl l's von Neumann-Morgenstern ut i lil y function. i = I, .... n.

Amer. 1. Agl: Econ.

== (at, ... , an) in (4) are chosen such that a i > 0, i = I, ... , n, and Li ai == 1. They can be inter-preted as individual welfare weights. 4 Denote the solution to the optimization problem (4) by tJ * (a) and T'(a) = [Y*(a) , H *(a) , pea), X*(a)], where rea) = [z*(e"a) , ... , z*( eM , a)], z*(e IlP a) being the decision made in the mth state. Let Ui*( a) = UJY/( a) , Hi*( a), e] be the corresponding utility level for the ith indi­vidual , i = I, ... , n.

In general, the allocation [tJ *(a), Z*(a)] de­pends on a. For each a, expression (4) gener­ates a solution such that no individual can be made better off without making someone el se worse off. It follows that [tJ*(a), Z*(a») is a Pareto optimal allocation.

PROPOSITION 1. As the a's vary between 0 and 1, the optimal solulions [p*(a), Z*(a)] to th e maximization problem (4) generate the set of Pareto optimal allocations, while the utility lev­els U *(a) = [U /' (a) , ... , U,,*(a)} trace out Ihe Pareto utility frontier.

The Pareto utility frontier expre ss es utility trade-offs under efficient allocation as welfare distribution varies across individuals (Samuelson; Graham). Proposition 1 shows that equation (4) provides a broad framework for the analysis of efficiency in resource allocation z and information tJ.

In order to gain insights in the role of infor­mation in resource allocation, it is useful to de­compose problem (4) into two s tages: fir s t, choose Z given tJ ; and second, choose So. For some feasible information structure tJ, the firs t stage is

(5) W(a , tJ)

== m:x{~ai Ui(r;,Hi, e): equations (1) and (2)}

Denote the solution to the optimization problem (5) by Z'!'( a , tJ) == [Y*(a, tJ), H *( a , tJ), pea, tJ), X*Ca, tJ)]. Let U;*(a, tJ) = UJY/,(a , tJ), H/,(a, tJ), e) be the corresponding utility level for the ith individual, i = I, ... , n. In general, the allocation Z*(a, tJ) depends on both a and tJ· Following the arguments presented in proposition 1, as the a 's vary between 0 and I, Z*(a, tJ) generates a set of constrained-Pareto

, A s s uc h, the function L;eX,V;(') could be interpre ted a s a B e(gson ~ Samue l son soc ial we lfare fun ction. This ra ises the issue of determining the indi vidual welfare we ights Ct.

Chavas Equity Considerations in Economic and Policy Analysis 1025

optimal allocations, while the utility levels U*(a, f.J) = [UI*(a, f.J), ... , UIl*(a, f.J)] trace out the constrained-Pareto utility frontier, the constraint being that the information structure f.J is treated as fixed. This provides a basis for investigating the issue of how the information structure f.J affects the utility frontier U*(a, f.J). For that purpose, consider two information structures f.J = (PI' ... , PK) and f.J' = (P/, ... , P/). We will say that f.J' is at least as fine as f.J if P/ is as least as fine as Pk for all k = 1, ... , K (see definition 1). Also, we will say that in­formation is costless if any information struc­ture f.J' that is finer than f.J remains feasible in the sense that {z: (z, f.J) E F(e",)} ~ {z: (z, f.J') E F(e",)}. This gives the following result. s

PROPOSITION 2. Assume that information is costless. If p / is as least as fine as p, then re­placing p by P / (i) either has no effect on the constrained-Pareto utility frontier, or (ii) is a Pareto-improving move that shifts out the con­strained-Pareto utility frontier.

Proposition 2 states that better (costless) infor­mation (as reflected by a finer information structure) tends to improve efficiency. This can be interpreted to mean that the social value of costless information is always nonnegative. At worst, improved information may not be used and would have no effect on resource alloca­tion . And if the information is used, it would be a Pareto improvement that shifts out the Pareto utility frontier. This corresponds to a situation where more costless information is necessarily better in the sense of improving the efficiency of resource allocation. Alternatively stated, lack of information would necessarily result in inef­ficient allocations. This is consistent with the extensive literature on the negative welfare ef­fects of moral hazard and adverse selection un­der imperfect information.

Proposition 2 suggests that, in general, there are economic incentives to obtain more costless information. However, it should be kept in mind that such results do not apply when infor­mation is costly. In this case, the evaluation of information involves comparing its cost and its benefit. If information cost is large, then ob­taining information may not be optimal. It is only when the benefit of information outweighs

, To prove proposilion 2, nOle Ih al p', being al leasl as fine as p, in equal ion (I) is al leasl as restrictive under p compared to p'. Ii follows Ihat, under coslless information, Ihe fea sible region for Z in Ihe maximizalion prob lem (5) is al Jeast as Jarge under p' than fP. The resu li s fo llow.

the cost that it is optimal to learn. The optimal learning is then represented by the optimal in­formation structure f.J *(a) obtained as the solu­tion to the optimization problem (4).

In general , both f.J*(a) and Z*( CY.) in (4) de­pend on a. This implies the existence of mul­tiple Pareto optimal allocations: one efficient allocation for each value of a. In this case, the Pareto-optimality criterion falls short of provid­ing precise guidance on which allocation should be recommended. It generates a class of efficient decisions, but with no basis for choos­ing among them. In such a situation, it is not possible to separate completely efficiency analysis from distribution issues: as a changes and one moves along the utility frontier, Pareto­efficient choices will also change. This is rather undesirable. Improving the normative role of economic analysis thus requires a joint investi­gation of distribution issues and efficiency is­sues. It indicates a need to go beyond Pareto optimality. This can be done by complementing the Pareto optimality criterion by some addi­tional criterion that provides a basis for choos­ing a single point on the Pareto-efficient utility frontier. More specifically, there is a need to in­troduce explicitly the evaluation of distribution issues in welfare analysis . This is done next by relying on a fairness criterion .

Fairness

Much research has been done exploring the concept of fairness in economics and welfare analysis (Foley; Rawls; Kolm; Varian; Feldman and Kirman; Pazner; Pazner and Schmeidler 1974, 1978; Crawford; Goldman and Sussangkarn; Thomson; Thomson and Varian; Baumol). The standard definition of fairness is presented next.

DEFINlTlON 2. Fairness is defined as the absence of envy. A group is said to be characterized by the absence of envy if no individual would pre­fer to have what another has.

The concept of fairness is appealing for several reasons: it provides an intuitive basis for ana­lyzing distribution issues, it exhibits symmetry across individuals, it is consistent with an ordi­nal representation of individual preferences, and it is free of interpersonal comparison of utility. Indeed, fairness only requires each indi­vidual to evaluate others' bundle using his/her own (ordinal) preferences. Using fairness as a reasonable equity criterion has thus appeared

1026 Decell1ber 1994

attracti ve in the analysis of alternati ve welfare and resource distributions.

Some difficulties with the concept of fair al­location have been pointed out in the literature. As shown by Pazner and Schmeidler (1974) and Goldman and Sussangkarn, the fairness crite-

AlI1er. 1. A g r. £ COI1..

characteristics are revealed and each individual becomes distinct.

Next, given (6), consider (5) under the addi­tional restrictions YI = Yi , and HI = Hi, i = 2, ... , n. In the presence of some transfer w, this yields the following hypothetical economy:G

(7) max{V(w, r;, HI> e): eq. (1) and (2); r; p.Z

V(W) 2, ... , n}

rion is not al ways consistent with the Pareto ef­ficiency criterion in a production economy. For example, this can happen when individuals with superior abil ity also exhibit a strong preference for leisure (see Pazner and Schmeidler 1974).

In this paper, we will rely on the concept of fairness-equivalence proposed by Pazner. Pazner (p. 463) defines an allocation to be fair­equivalent if there exists a fair allocation in some hypothetical economy in which each indi­vidual enjoys the same welfare level as that en­joyed by him/her at the allocation under consid­eration. This approach has several attractive features: (a) it is consistent with an ordinal rep­resentation of individual preferences; (b) it is free of interpersonal comparison of prefer­ences ; and (c) it is consistent with the Pareto optimality criterion (see below).

First, we need to define a hypothetical fair economy. Being motivated in part by Rawls 's work, we make two modifications to the analy­sis presented in the previous section. First, we introduce explicitly the possibility of transfers. We assume that these transfers take the form of ex ante monetary transfers. For the ith indi­vidual, denote a monetary transfer by the scalar Wi' where Wi > ° « 0) means an ex ante income transfer to (from) the ith individual, i = 1, ... , n. Second, we consider the case where the states (e l , ... , eM) include all the information on the characteristics of each individual, information that is necessary to distinguish between any two individuals. The ex ante utility function of the ith individual given in (3) then becomes

where we assume that V(w, Y, H , e) is jointly continuous in (w, Y, H) and stric tly increasing in w. This preference function, which now de­pends explicitly on possible transfers Wi' is a generic ex ante utility function for any indi­vidual before information becomes available about hi s/her individual characteristics. How­ever, as information is obtained, the individual

where Z = [z(e l ), ... , z(eM)], z(e".) being the de­cision vector in the mth state, z = (z I, ... , ZK) == (y, h, t, x), y = (YI' ... , Y,,), h = (hi' ... , h,,), t = (til; i = 1 ..... n;j = I .... , Jl. Denote by [&o a(w), ZO(w)] the solution to the maximization prob­lem (7). It is clear that the hypothetical alloca­tion Uoa(O), ZO(O») is fair. Indeed , in the ab­sence of transfers in (7) (Wi = 0, i = 1, ... , n), all the individuals receive the same utility-yielding goods (y. h), thus necessarily implying the ab­sence of envy within the group. It follows that, V(O) is the ex ante utility for any individual un­der a fair allocation.

We propose to use the hypo thetical fair allo­cation given in (7) to define fair-equivalent al­locat ion s. For that purpose. consider the ith individual' s ex ante willingness-to-receive wi', which sat isfies the following relationship:

(8) V(O, Yi , Hi' e) = V(w,'), i = I, ... , n.

Equation (8) defines W,* as the ex ante selling price of (f<)a, ZO), i .e., the smallest amount of money paid to the ith individual that would make him/her willing to give up the hypotheti­cal allocation (&0", Z") and replace it with some allocation generating (0, Yi, H,). Since the allo­cation (&0", ZO) is hypothetical, so is the will­ingness-to-receive Wi* defined in (8). We use this willingness-to-receive to define the fair­ness-equivalence of an actual allocation (&0 , Z).

DEF[NITION 3. An allocation ( p , Z) is said to be fair-equivalent if it satisfies (i) equations (J) and (2), and Oi) WI* = Wi'" , i = 2, ... , n. where Wi" is defined in (8).

Thus, an allocation is fair-equivalent if it is fea­sible [i.e. , satisfies equations (1) and (2») and if its benefits (as measured by the willingness-to-

• Note that eq uation (7) does not require cap il al x or la bor I 10

be equally distribUied among the 11 individuals. tn other words, we do not require an equa l distribution of resource s in our character­izalion of fai rness.

Chavas Equity Considerations in Economic and Policy Analys is 1027

receive w/) are equally distributed among the n individuals , using (p O, z a) as a fair hypotheti­cal reference point. From equation (8), the fair­equivalent allocation (P, Z) generates the same welfare levels as the hypothetical fair allocation given by (PO, za) and the willingnes s-to-re­ceive: WI* = w;*, i = 2, ... , n. This definition is general in the sense that it applies to any fea­sible allocation, including allocations that are not Pareto efficient. It also indicates that any situation where the wi*'s vary across individu­als is not fair-equivalent. 7

The next question concerns the existence of an allocation (p, Z) that is both efficient and fair-equi valent.

PROPOSITION 3. There always exists a utility vec­tor V * = (V, *, ... , V" *), unique up to positive monotonic transformations, that is on the Pareto utility frontier and corresponds to a fair- equivalent allocation.

The proof can be obtained by construction (see Pazner and Schmeidler 1978). First, consider the feasible allocation [pO(O), ZO(O)] just dis ­cussed; it is both feasible and fair. Except in the trivial case where it happens to be Pareto opti­mal, this allocation will generate a utility vec­tor U = (VI' ... , V,,) that is in the interior of the utility possibility set bounded by the utility frontier. Under continuity, it is clearly possible to move from this allocation toward the utility frontier. Since this move can be made in any positive direction (in the utility space), it can generate any distribution of relative individual net gains. One of these positive directions cor­responds to a fair allocation generating equal transfers w/ > 0 for all J1 individuals. If such a move is made all the way to the utility frontier, it necessarily generates a fair-equivalent and ef­ficient allocation. This implies that a fair­equivalent and efficient allocation always ex­ists. In other words, there is a point a* in (4) such that the corresponding allocation

(9a ) p+ = p *(a * )

(9b) Z + = Z *(a*)

, Thi s suggesis Ihe fo ll owi ng t~,irness index for Ihe ilh individual

, ; = [ IV;'· median (w,· ..... IV.:)].

i = I ..... II . Usin g Ihi s index. an a ll oc ulion is fa ir if I; = 0 for all i = I, .... II. And it is unfair in fa vor (again s t) the ith individual if I; > 0 « 0).

satisfies both the efficiency criterion and the fairness-equivalence criterion. And if equation (4) has a unique solution for p and Z, then p+ and Z+ in (9) are unique allocations. In this case, the efficient and fair-equivalent allocation (P+' Z+) provides a basis for making specific recommendations to policy makers. To the ex­tent that fairness-equivalence is a relevant con­cept, this can significantly improve the norma­tive usefulness of economic analysis.

Under efficiency and fairness-equivalence, p+ gives the optimal information structure among the n individuals. It is worth pointing out here that the endogeneity of the information structure can be quite useful. For example , equation (2) can be interpreted as reflecting how information p is produced as the outcome of a learning process. In this context, learning cost involves the opportunity cost of time used in information gathering activities among the n individuals . Being efficient, p+ in (9a) must maximize the net social value of information (i.e ., information benefit minus its cost). This is consistent with the extensive literature on the economics of information. And being fair­equival ent, p+ in (9a) must be such that the willingness-to-recei ve Wi * is identical across individuals. This means that the information gathering activities must contribute to equaliz­ing the w;* 's, suggesting that the learning pro­cess plays a significant role in the evaluation of fairness . This stresses the importance of infor­mation management in collective decisions (e .g ., negotiations in bargaining, or lobbying in policy making) .

Since efficiency and fairness-equivalence can always be made consistent with each other (from proposition 3), this suggests the possibil­ity of combining them together in economic analysis. This can be done as follows:

(0) max {min{nwi } : V(O , 1';, Hi ' e) P. z. W I

V(wi ) , i 1, .. . n; eq. (1) and (2») .

where W = (WI' .. . , w,, ), and Z = [z(e l ),

z(e",,)], z(e m ) being the decision vector in the mth state, z = (ZI, ... , ZK) == (y, h, t, x) , Y = (Yl, .. . , y,,), h = (hi' ... , h"), t = {tij; i = 1, .. . , n;j = I , .. . , J}. The solution of the optimization problem (10) always generates the efficient and fair-equiv a­lent allocation, p+, Z+ and w+. Indeed, by defi­nition of a maximum, the solution is necessar­ily on the Pareto utility frontier. And it must be fair-equivalent and satisfy WI = Wi' i = 2, ... , n , since any other efficient allocation could be

102 8 December /994

subject to an efficiency-preserving move along the Pareto utility frontier toward the individual with lowest Wi that would increase the value of the objective f unct ion . The optimization prob­lem (10) thu s provides a convenient basis for conducting welfare analysis under the joint re­quirements of obtaining an efficient and fair­equivalent allocation. 8 Note that, (min; {n w; j) in (10) can be (loosely) interpreted as the fair ag­gregate va lue of information associated with re­placing the hypothetical economy (P', z a) by the actual economy (P+' Z+) .

The maximin characterization (10) exhibits a number of close s imil ar itie s with Rawls's Theory of Justice. Indeed , Rawls proposed a similar maximin criterion as a measure of so­cial welfare . Ex press ion (10) shows that a maximin principle can lead to a Pareto-efficient allocation. It also suggests that the Rawlsian "original position" corresponds to the ex ante situation where there is no information a llow­ing a distinction among individu als. Thi s I S

consistent with Rawls 's " veil of ignorance."

Some Illustrations

The model presented in the previous sections is quite general. As such, it can be appl ied to a va­riety of situations. In this section, we explore its usefulness through several applications.

An Agricultural Economy

We consider first a simple agricultural economy where the production decisions involve labor and land allocation used in the production of food. We will assume that land is a necessary input so that no food can be produced without land. Food production can take place under al­ternative property rights. Here , we consider a simple case of land rights , where property rights for land are either pri va te property (where land is privately owned by individuals in the group) , or collective property (where land is co))ecti vely owned).9 The issue is then to choose how much land to aUocate to each in­dividual as private property, and how much land to reserve as collectively owned . Let the vector of capital goods be x = (XI' ... , X"' xn+ l ) ;:::

0 , where X ; denotes the quantity of land as­signed pri vate ownership to the ith indi vidual, i = 1, ... , n, and where X ,,+ I is the quantity of land

tI See Chavas a nd Coggins for a more detailed discussion of the maxim in cr ilerio n (10) in Ihe a ll oca lion of public goods.

Ame l: 1. Agl". Econ.

assigned collective ownership to the group. De­note by L > 0 the total amount of land avail­able. The allocation of land rights mus t sa ti sfy the constraint

n +1

( 11 ) L, Xi ~ L. ;=1

As a result of land allocation , we thus have (n + 1) possible farms : one collective farm as­sociated with the collective land X,,+I, and 11 pri­vate farms , the ith private farm being associ­ated with X;, the land owned by the ith indi­vidual, i = 1,2, .. . , n .

The production decisions also involve labor allocation on the (n + 1) farms. For the ith indi­vidual , this means choosing between working on his/her own farm, working on someone else's farm, or working on the collective farm. Denote by ti) ;::: 0 the amount of labor spent by the ith individual working on the jth farm, j = 1, ... , n + 1. Then, t;; corresponds to "family la­bor" on the ith farm, while eL;,,} t;) is "hired la­bor" on the jth farm. Also, denote by tj ,,+2 the amount of time spent by the ith individ'ual on information gathering activities, i = 1, ... , n.lo

Let to > 0 be the total amount of time available to the ith individual , i = 1, ... , n. Then, labor al­location must satisfy the time constraint:

11+2

(12) L, t i) + h; ~ to' } =I

where h; ;::: 0 is the amount of Leisure time spent by the ith individual, i = 1, .. . ,11.

The production technology involves using the land resource L along with the labor provided by the n individuals to produce food . For s im­plic ity, assume tha t the food output is a stan­dard product across farms.!' Denote by q( p , t, x, em) ;::: 0 the aggregate production function for food produced in all (n + 1) farms in the mth s tate of nature. Farm output is di stributed for consumption y; ;::: 0 among the n individu als, implying the followin g constra int:

(13) L, y; ~ q(AO, t, X, e",). i==1

9 Examples of coJl ecli ve land ow nerShip include experimenla l farms on agricullUral researc h slations (w here Ihe primary objec ­rive is 1O generate informat ion abou t new technologies); kibbut zim, Israel's colleclive farm s; or g raz ing land in th e Sw iss Alp s.

10 This corresponds 10 having J = II + 2 in eq uali oll (4). 11 For simplicity. we limit our di scuss io n [Q a s ing le food QU I­

pu!. The eXlellsion 10 multipl e oUlputs is sira igilifo rward.

Chavas Equity Consideration s in Economic and Policy Analysis J 029

The feasible set in (2) then takes the form

(14) F(e",)

= {(P, z): eq. (1J), (12), and (13») .

It follows that the fair and efficient allocation given by equation (10) becomes

(15) max{min{nw}: V(O, Yi, Hi' e) = V(w), p,Z,1Y i l

i = 1, ... , n; eq. (1), (1), (12), and (13») .

(l7a) min max {n[lL,o,Y,P,t fl +2 ,(em)] A.D.r p.Y.H.T .. ,.w

M M

- L IL", L Yi(en,) + L o",L m=1 ;=1 m=1

At "

+ L L Yim[tO - ti.fl +2 (e",) - hi(e",)] 111=1 ;=1

+ min{nwi ) : V(O, 1';, Hi' e) i

= V(wi ), i = I, ... , n;

eq. (1) ;1L ~ 0,8 ~ 0, Y ~ O)

Under some weak regularity conditions,12 the where constrained optimization problem (15) can be equivalently expressed in terms of the follow-ing saddle-point problem:

M

(16) min max{I ILm(q(p, t(e",), x(e",), e",) - I Yi(em)] A,8.y 30 ,2,11-' 111=1 ;=1

M ,,+1 /1 M 11+2

+ I O",[L - I xi(em)] + I .I Yim[tO - L tij(em) - hi (em )] + rnfn{nwJ V(O, 1';, Hi' e) = V(w,) , m=1 ;=1 ;=1 m=l )=1

i = 1, ... , n ; eg. (1) ;1L ~ 0,8 ~ O,y ~ 0)

where the 8's, is, and lL's are the Lagrange multipliers associated with constraints (11), (12), and (13), respectively. Denote by IL*, 0*, and y* the optimal values of these Lagrange multipliers in (16) . The objective function in (15) being expressed in monetary units, these Lagrange multipliers have the standard inter­pretation of measuring the shadow price of the corresponding constraints. For example, at the optimum in (16), IL",* measures the shadow price of food in the mth state of nature . Simi­larly, 0",* is the shadow price of land in the mth state of nature. And Y;"'* is the shadow price of time for the ith individual in the mth state of nature.

Note that expression (16) can be alternatively expressed as

12 The ex istence of a saddle-point in (16) is guaranteed if Slater's condition is satisfied . such that there exis ts a fea s ibl e point for which the nonlinear constraints are nonbinding. and if appro­priate convexity conditions are satisfied so that the Lagrange mul­tipliers (0, y, A) can defin e a sepa rating hyperplan e (see Takaya ma).

M

= max {IIL",q[p,t(e",) ,x(e", ),em] X.T, .... ,TH1 111-==1

M 11+1 1/ M n+l

- LO", Ix/em)-LLYi'" Ltij(e",): eq. (1») 111=1 )=1 i=lm=1 )=1

with tfl+2 = {t i.n+2: i = 1, ... , n} and Tj = (tij(e",): i = 1, ... , n ; m = 1, ... , M}, j = 1, ... , n + 2. The lL 's, 8's , and is being shadow prices , the func­tion n(-) defined in (17b) can be interpreted as an aggregate profit function associated with the production inputs (x, (t i/ i = 1, ... , n;j = 1, ... , n + 1)). Equation (17b) holds under general conditions that allow for a joint production pro­cess as represented by equation (13).

Equation (16) or (17) generate a general equi­librium efficient allocation under fairness­equivalence among the n individuals. And equation (17a) has the intuitive interpretation of providing a monetary evaluation of economic activities. The objective function in (17a) de­composes economic value into five additive

1030 December 1994

terms: (a) the aggregate profit function n(-) given in (17b); (b) the negative of aggregate consumer expenditures, -2. A 2. y(e ). (c) aggregate land value , 2.m 0::,' L:" (d) a'gg;~e'gate value of labor, 2.m .2.; 11m [to - 1;.,,+2(e/ll) - h;(e/ll)]; and min; {nw;), the aggregate value of informa­tion associated with replacing the fair hypo­thetical allocation (SO", z a) by the efficient and fair-equivalent allocation (tJ+, Z+) .

A Special Case

How does equation (16) or (17) relate to more traditional economic analysis? In order to in­vestigate this ques tion, con s ider the special case where equation (13) takes the form

(18) n+1

s; I q j (ll j ' . . " tlli , Xj' em)' j =1

tJ g(t l •II +2 , . .• , tll .II +2)

qitlj ' ... , tllj' Xj ' em) being the food production function on the jth farm in the mth state, and get!. n+2, ... , 111 • 11+2) reflecting how the information gathering activities (t1 . 1I+2, ... , til. ,1+2) influence the information structure tJ. Compared to (13), the production process represented by (18) is restrictive in several ways: . 1. The production process in (18) is nonjoint In the sense that food output and information output are produced independently of each other. This restricts food production and learn­ing to be separate processes (e.g., excluding the possibility of "learning by doing").

2. In (18), property rights x are assumed to have no influence on the information structure tJ· This neglects possible interaction effects between social organization (e.g., the nature of property rights) and the information used in re­source allocation.

3. In (18), the production function of each farm is independent of decisions made on other farms. This rules out the presence of possible externalities across farm s.

Gi ven (18) , the aggregate profit function nO in (17b) becomes

n+1

where

Am er. J. Ag r. Econ .

M

= x Tma~. {IAmq)l lj (em),··· , l"j (em),x j (e/ll),em] ) ' / 1· · · ·' / .,u l n/=I

M n M

- I o/llxj(em ) - I I Yi",!U(em) : eq . (I») 111 = 1 i =1 JI/=I

is the profit function for the jth farm, j = 1, ... , n + 1. Note that, in the absence of uncertainty (where there is only one state of nature , M = 1), equation (19b) becomes the standard profit maximizing condition for each farm under com­petitive markets, taking the prices (AI' 0 1, YI) as exogenous. In such a situation, provided that (AI ' 01, YI) = (AI *' 0 1*, yn, it is both efficient and fair-equivalent for each farm to maximize profit. However, this result obtained under (18) does not hold in general under (13) . This is fur­ther discllssed below.

Some Implications

Our analysi s has some interesting implications. First , it provides a basis for answerin a the

. b

questIOn: Can a market mechanism lead to an efficient and fair-equivalent allocation? A gen­eral an swer is yes, it can . This would be the case if competitive markets for food , land , and labor generate actual market prices equal to the shadow prices (A*, 0"" 1"), and if the allocation (tJ , Z) is chosen in a way consistent with (16) or (l7). However, the conditions for this to oc­cur are fairly stringent. For example, it would require that market prices are state-dependent and that they vary across states of nature ac­cording to the distribution (AI *' ... , AM*; 01"', •• • ,

OM*; YII * ' ... , Y-,M*)· A complete set of competi­tive commodity and risk markets can in prin­ciple meet this requirement. This is the Arrow­Debreu world of complete competitive markets. However, in the real world, risk markets are typically incomplete , implying the absence of prices in the missing markets . In such a situa­tion, it is not likely that a marke t mechanism will generate an efficient and fair-equivalent al­location. Thi s is consistent with Newbery and Stiglitz 's analys is . Newbery and Sti g litz have shown that, in general under ri sk avers ion. commodity markets alone do not generate a Pareto -optimal allocation of resources. Since their results apply for any efficient allocation , they also apply to our allocation (that is both efficient and fair-equivalent). However, our analysis goes beyond Newbery and Stiglitz's in the sense that there exist market allocations that

Chavas EquilY Consideralions in Economic and Policy Analysis 1031

are efficient but unfair. In this case, advocating a "market allocation" on the ground of efficiency alone (e.g. , in pricing or trade policy) can have adverse distributional impacts leading to unfair allocations. This does not mean that market allo­cations are undesirable. Rather, this suggests that, if fairness is a relevant concept, one would not want to rely exclusively on a market mechanism to allocate resources. This is illustrated in our above example where we endogenized the distri­bution of land rights and generated an allocation that is both fair-equivalent and Pareto-efficient.

Second, our analysis suggests possible direc­tions to improve economic analysis. For ex­ample, we have noted that (l7b) is not always consistent with farm-level profit maximization given in (19). This can happen in the presence of externalities across farms, a subject that is receiving growing attention (e.g., soil erosion, water pollution). It can also happen when the information structure tJ is influenced by social interactions among individuals. The investiga­tion of such issues deserves more attention. In particular, economists would greatly benefit from a better understanding of the linkages be­tween property rights and information, along with their implications for efficiency and fair­ness. By endogenizing both information and property rights, the model presented above should prove useful for that purpose.

Third, our analysis provides some insights into the following issue: can a nonmarket mechanism generate an efficient and fair­equivalent resource allocation? Again, a general answer to this question is yes , it can . In a sense, this is directly implied by expressions (15) or (16), where no explicit market mechanism was assumed. In this case, the optimal allocation (&D+, Z+) in (15) or (16) can be interpreted as being associated with "optimal contracts" among the n indi viduals. Such contracts can be "private contracts" between individuals (e.g ., within a firm or household) as well as "social contracts" (e.g., corresponding to government policy). This is consistent with the Coase theo­rem, which states that contracts can generate an efficient allocation of resources (Coase). Our results can be interpreted as an extension of the Coase theorem in two ways. First, in contrast with Coase's analysis, we allow for imperfect information, transaction costs (e .g., information cost), as well as nonzero income effects. Sec­ond, we address explicitly distribution issues through our fairness criterion. Our results thus suggest that, in principle, appropriately de­signed contracts can generate a fair-equivalent and efficient allocation. However, this would

require that the contracts be state-dependent and satisfy Zk = Zk*(e",), k = 1, .. . , K, m = 1, ... , M . In the real world, the ability of contracts to reflect many states of nature is typically lim­ited, reflecting in general the cost of processing information. In such a situation, it seems un­likely that contracts alone can generate an effi­cient allocation. Again, this should not be inter­preted to mean that contracts are undesirable. Rather, this suggests that one should not rely exclusively on contracts alone as a basis for al­locating resources .

By endogenizing both information &D and re­source allocation Z, our approach can provide useful insights into the interactions between in­formation, efficiency, and fairness in contract design. In bargaining as well as policy making, the precise way in which information &D is cho­sen can be of considerable interest. For ex­ample, lobbying activities are intended to influ­ence directly the information used in policy de­cisions . In general, additional information can contribute to improved efficiency (see proposi­tion 2). But it also affects fairness through the knowledge of distribution of net benefits asso­ciated with a particular action.

The above discussion suggests that both mar­ket and nonmarket mechanisms can contribute to an efficient and fair allocation. In general, assessing the relative advantage of each mecha­nism will depend on the particular situation considered. We would like to argue here that our endogenous treatment of information can give valuable insights into this issue. In gen­eral, a high cost of obtaining and processing in­formation has negative effects on efficiency. First, the resources used in the learning process are typically diverted from other productive ac­tivities, thus contributing to an inward shift in the Pareto utility frontier. Second, if some in­formation is deemed too costly to obtain, this reduces the ability to make state-dependent de­cisions, which also leads to an inward shift in the utility frontier (see proposition 2) . Thus, a desirable characteristic of an allocation is to obtain information at a relatively low cost. The relative efficiency of a mechanism will typi­cally depend on the nature of its information technology, i.e., on its ability to generate a fine information structure tJ at low cost. If there are economies of scale across individuals in obtain­ing and processing information, then a central­ized mechanism (e .g., government policy) would be appropriate. Examples include re­search, price information, national security, etc. Alternatively, in the absence of economies of scale in information generation, a decentralized

1032 December 1994

mechanism would likely be more desirable (Hayek). This includes private contracts as well as market exchanges. Private contracts would appear appropriate when efficiency-improving contracts involve only a few individuals. Mar­ket exchanges would likely be preferable when standard contracts can be developed at low cost among a large number of individuals. In each case, the relative efficiency of a particular mechanism will be heavily influenced by its ability to process and respond to information in allocation decisions. And the information avail­able about the distribution of net benefits across individuals will affect the fairness of the allocation.

SustainabililY

We now turn to the issue of sustainabi lity asso­ciated with long-term resource management. Such sustainability issues have been linked with both efficiency and intergenerational eq­uity (e.g., Howarth and Norgaard). To the ex­tent that equity across generations can be cap­tured by fairness-equivalence, then our ap­proach can provide a basis for analyzing sus­tainable management.

Consider the model developed above, where the n individuals belong to different genera­tions. Under fairness-equivalence, we defined a fair hypothetical economy [represented by (7)] where each generation enjoys the same ex ante welfare level as that enjoyed under the current allocation. The hypothetical economy is based on the information structure tJ o' where indi­viduals do not yet know which generation they belong to . Being fair, the associated allocation 2n(0, PrJ in (7) necessarily implies the absence of envy across generations. The efficient and fair-equi valent allocation given in (10) can then provide a basis for evaluating sustainability is­sues.

First, the model presented above may require some modifications. We assumed that w in (6) reflects monetary transfers. Because of infla­tion and the absence of long-term capital and risk markets , it is not clear that money remains the most effective way of transferring resources across generations . It may well be that some nonmonetary transfers become more effective in that context. Then w in (6) and the willing­ness-to-receive w;* in (8) would need to be in­terpreted in terms of these nonmonetary transfers.

Second , while all evaluations are made ex ante (i .e., based on the information available in the current generation) , in the presence of

Amer. 1. A g r. Econ .

learning future generations will have access to better information. This amounts to evaluating now some of the options that will become avail­able in the future. Some of forthcoming infor­mation will be "good news" (e.g ., the discovery of new technologies), and some will be "bad news" (e.g., the extinction of some species , soil erosion associated with violent rains, the spread of pesticide-resistant pests , etc.) . One could think of an "optimistic scenario" where techno­logical progress will be rapid and new tech­nologies will provide effective solutions to whatever economic problem arises. One could also think of a "pessimistic scenario" where re­source depletion and a growing human popula­tion, coupled with some natural disaster (e.g., a widespread drought) would generate mass star­vation and challenge man's survival. In the op­timistic scenario, rapid technological progress will tend to make future generations better off. In such a situation, it may appear both efficient and fair-equivalent to exploit inten s ively cur­rent resources , without much conservation ef­fort. In the pessimistic scenario, resource con­servation would be a very important part of cur­rent policies, in an attempt to give future gen­erations a chance of obtaining similar welfare levels as the current generation. In any case, fu­ture options and the effects of future uncer­tainty will be influenced by R&D policy (de­signed to stimulate future technological progress) as well as resource conservation policy. We can interpret sllch policies as being motivated by both efficiency and intergenera­tiona I equity considerations, trying to cope with future uncertainty by keeping future options opened and improving man's ability to deal with unforeseen contingencies. To the extent that fairness- equivalence is relevant, our model allows for a joint analysis of learning over time and resource allocation. By reflecting both effi­ciency and equity considerations across genera­tions , it can provide a useful basis for analyzing resource sustainability issues.

Concluding Remarks

We have presented a general model that incor­porates both efficiency and equity consider­ations in economic analysis. Efficiency is based on the Pareto optimality criterion, an intuitive concept that is widely accepted among social scienti sts and policy makers. Our treatment of equity is based on the intuitive notion of fair­ness, where each individual fails to envy any other individual. Following Pazner and

Chavas Equity Considerations in Economic and Policy Analysis 1033

Schmeidler, we proposed a fairness-equivalent criterion as the basis to evaluate distribution is­sues. Fairness-eq u i valence prov ides a nice complement to efficiency in the sense of always being consistent with Pareto optimality. It in­volves a hypothetical fair economy in which no information is available about each individual's characteristics. This is similar to Rawls's "ini­tial position." The model provides useful in­sights into the role of information and property rights in resource allocation. This is illustrated in the context of an agricultural economy and in the investigation of sustainable resource management.

The relevance of our model can be compared with other models addressing both allocation and distribution issues. A leading competing model is Nash's bargaining model (e.g., Nash, Harsanyi). Note that the Nash bargaining model requires cardinal preferences, while our ap­proach allows for more general ordinal prefer­ences. Another advantage of our approach may be that the intuitive notion of fairness is easier to communicate to policy makers.

FinaJly, it should be noted that not all situa­tions call for fairness . In general, fairness seems more likely to be appropriate in groups exhibiting advanced forms of a social contract. However, in the absence of such a contract, fairness may not be a relevant criterion (e.g ., the case of conflicts or wars). At this point, there is clearly a need for empirical assess­ments of our proposed model. It is hoped that our approach will help stimulate such endeav­ors and improve economic analysis through the explicit incorporation of equity considerations.

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