Bounds on the willingness to pay for non-traded goods

Journal

of Public Economics

BOUNDS

30 (1986) 267-272.

North-Holland

ON THE WILLINGNESS TO PAY FOR NON-TRADED GOODS A Possibility Theorem Jon R. NEILL*

Department

of Economics, Western Michigan University, Kalamazoo, Received January

1985, revised version

received

MI 49008-3899,

USA

May 1986

This note contains a proof of the following theorem. If an agent’s preferences for traded goods are known, and if the quantity of a non-traded good provided him alters his demand for traded goods, then it is possible to place bounds on his willingness to pay for increments in this quantity.

Years ago, Maler (1971) demonstrated that when preference orderings have a certain property, it is possible to determine willingness to pay for nontraded goods from perferences for traded goods. More recently, Bradford and Hildebrandt (1977) demonstrated this possibility under weaker conditions. This note consists of a proof of the following theorem: Theorem. Suppose strictly convex, transitive preferences for bundles of traded and non-traded goods, with these bundles in I-l correspondence with the set of non-negative real n-tuples. Further suppose that these preferences can be represented by a continuous function, that agents are rational, and that there is non-satiation. Then if an agent’s preferences for traded goods are known and the quantity of a non-traded good provided him alters his demand for traded goods, it is possible to place bounds on his willingness to pay for sufficiently small increments in this quantity. Since only bounds are proved to be identifiable, this finding is weaker than Bradford and Hildebrandt’s. Nonetheless, it is of interest since it applies to a different class of preference orderings than that to which the BradfordHildebrandt result applies. Their proof requires that for any utility level there is a vector of prices at which the non-traded good has zero marginal utility, *I am grateful to Professors David Bradford and Robert Willig for their comments on drafts of this paper. I would also like to thank an anonymous referee for suggesting ways to improve the exposition of my tinding. 0047-2727/86/$3.50

0

1986, Elsevier Science Publishers

B.V. (North-Holland)

regardless of the quantity of the non-traded good consumed.’ This need not be true for the conclusion of our theorem to hold. Thus, our finding could be applied to problems which do not meet the conditions of the BradfordHildebrandt theorem. Moreover, our conclusion obtains from a relationship which can be empirically verified. It does not appear that the assumption upon which the Bradford-Hildebrandt theorem rests can be validated on the basis of the agent’s market behavior. In short, our theorem is a meaningful extension of Maler’s proposition. Proof: If changes in the quantity of the non-traded cause the agent to alter his consumption of traded goods, then it is possible to identify a more preferred commodity bundle containing more of the non-traded good but quantities of the traded goods which cost less than the quantities originally consumed. Therefore, the agent would be willing to pay at least the difference between his actual income and the cost of this bundle of traded goods. Fig. 1 illustrates this proposition for the two traded goods’ case.

Let

etc. denote bundles of traded goods. Let z,zo,zl, of a non-traded good. By assumption, the sets

x,x0,x1,

quantities

etc. denote

((x,zo)E c x co, a):(x, zo) 7 (x0,zo))

Fig. 1. x=x(p, y, ze), the traded goods purchased when the agent has income y and consumes zO units of the non-traded good. x’=x(p,y,z,) is the bundle of traded goods purchased when income is y but zr >zO units of the non-traded good are consumed. So, the agent would be good. willing to pay at least y-y, for zr - z,, additional units of the non-traded

‘Maler’s proof requires that the marginal utility of the non-traded agent is consuming zero units of a specific traded good.

good

be zero when

the

J.R. Neill, Bounds on willingness

269

to pay

are known for all (x0, zO) E C x [0, CD).~ Here C denotes the traded commodity space (the non-negative orthant of R.-l). Let x(p, y, z) represent agent i’s demand for traded goods when (p, y, z), where p is a vector of prices and y is i’s income. By assumption, if z0 # zr, x(p, y, z,-J # x(p, y, zl). Let z1 > zO. Then since P.X(P, Y, 4

convexity,

rationality,

=Y = P’ X(P, Y, z,),

and non-satiation

MP, Y, Zl), Zl) ! (X(P>Y>4,

Now, the Consequently,

indifference class the solution to

imply that

Zl) \ MP, Y>z,), zo).

to which

(x(p, y,z,), zt)

belongs

is known.

minp.x XGC s.t.

W(P,Y,

zcl),z1) = WG Zl)

is known. U( .) represents the agent’s utility function. Let y, =p’x, is the solution to the above minimization problem. Then y>y,. since if y, =y, convexity implies that x(p, y,z,) is the solution to

where x1 This is so

max U(x, zr) XEC st. y=p.x. That is, x(p, y,z,,) =x(p, y, z,), a contradiction. Therefore, i would to pay the positive amount y-y, for the increment z1 -ze.

be willing

In order to establish an upper bound, we need to identify a less preferred commodity bundle containing the larger quantity of the non-traded good. Of course, to be less preferred, this bundle would have to contain less of at least one traded good. Thus, a logical candidate for such a commodity bundle would be any bundle made up of zero units of a traded good that the agent was consuming a positive quantity of and the original quantities of the

‘The symbols -, }, and { are: ‘is indifferent to’, ‘prefers more than’, and ‘prefers less than’, respectively. They are defined using the agent’s reflexive, transitive, complete ordering of the commodity space, ‘prefers at least as much’. Thus, all three are transitive relations.

J.R. Neil, Bounds on

270

remaining sufficiently combination preferred:

willingnessto pay

n--2 traded goods. The following lemma shows that for z1 close to zO, these commodity bundles, as well as any convex of such a bundle and the agent’s original bundle, are less

Lemma. Let x0 be the agent’s demand for traded goods when p,y,,z,. Let xj* =(xgl,. . . ,x~~-~, O,X,~+ 1,. . . , xon- 1), where xok denotes the quantity of the kth traded good in the bundle x0. Then there exists z? > z0 such that

ab$,z)+(l -4bo,zO) {(XO,~O), L

for all a E (0,11,z 5 zT.

Proof:

Let pZ, y* be such that (x,, zO) is the solution

to

max U(x, z) (X32) s.t. y*=p.x+p,z. Define zT=(y*-p.xT)/pZ. with the strict inequality

Since y*-p.xTzy*-p.x,, it follows that zj*zzo, holding for some j. But since

p.((l-a)x,+axj*)+p,((l-a)z,+azj*)=y*, and (x,,z,) that

is the solution

(l-~)(XO~zO)+a(xj*,zj*){(x~,zO),

to the above

i

aE(O,l], maximization

ae(O,l].

Thus, by non-satiation,

which proves the lemma. Let zT > zO and let zT 2 z2 > z1 > zO. Define a* =(z,

-zO)/(z2-zO).

problem,

it must

be

J.R. Neill, Bounds on willingness to pay

271

By the lemma,

(1-a*)(%, zo)+ a*txi*, 4 { (%,%). i

Let yz =p.

x2, where x2 is the solution

to

minp.x XEC s.t.

UN

Clearly,

I-

a*)%

+

u*q,

(1 - u*)z(J + u*zJ

= U(x, (1 - u*)z, + u*z2).

y,
(X2> (1-a*)% + a*4 { (x,, %I) I

the agent would Therefore, ye-y, for the increment

not be willing to pay ye-y, for the increment a*(~* -ze). is an upper bound on ,the willingness of this agent to pay z1 -zO. Q.E.D.

In other words, the agent would willingly pay no more than the amount which, when taken from him, leaves him with only enough income to reach the indifference class containing the bundle with more of the non-traded good, but less good j in it than originally (see fig. 2). Note that the upper

Fig. 2. If z1 >zO is such that (x,zJ is more preferred than all convex combinations of (x’, z,). Thus, the agent would be willing to pay no more (x, q,). then U(x, zO) > U(x”, az,+(l-a)z,). than yO-y, for a(zl -zO) additional units of the non-traded good; a is that number less than one and greater than zero such that ax’ + (1- a)x =x”.

bound would depend on which traded good the agent is given less of. Presumably, the smallest upper bound identifiable via this procedure would be associated with the traded good having the smallest market share. Also note that the requirement that z1 be sufftciently close to zO is not particularly troublesome since by taking the limit of the upper bound as zr approaches zO, we obtain an upper bound on the agent’s marginal willingness to pay for the non-traded good. To conclude, we would add that although the above procedure requires a great deal of information about consumer behavior to implement, when additional restrictions on preferences are added, bounds can often be established from minimal information about an agent’s market behavior. As an example, suppose two traded and one non-traded goods. In addition, suppose that the two traded goods are substitutes in the Hicksian sense, and that

Then if the agent’s utility

u = WG

3

function

can be written

.I+,, z)),

with

the agent’s willingness to pay for an additional unit of the non-traded good will be at least as great as (ax,/az)/(ax,/ay).” Thus, the application of our theorem to specific policy questions is quite feasible. :f;,f,2, etc. denote of this claim.

first- and second-order

partial

derivatives.

See Neil1 (1984, 1985) for a proof

References Bradford, David F. and Gregory G. Hildebrandt, 1977, Observable preferences for public goods, Journal of Public Economics 8, 1 I l-131. Maler, Karl-Goren, 1971, A method of estimating social benefits from pollution control, Swedish Journal of Economics 75, 121-133. Neill, Jon, 1984, Using market behavior to determine marginal willingness to pay for non-traded goods, unpublished manuscript. Neil& Jon, 1985, Another theorem on using market behavior to determine willingness to pay for non-traded goods, unpublished manuscript.