benefit analysis

JOURNAL

OF ECONOMIC

Hedonic

THEORY

37, 55575 (1985)

Prices and Cost/Benefit

Analysis

SUZANNE SCOTCHMER Department

qf Economics, Received

Harvard February

University,

Cambridge,

21, 1984; revised

April

Massachusetts

02138

15, 1985

Hedonic prices have been used to evaluate the willingness to pay for attributes. We reformulate the notion of hedonic price from a composite price on housing to a unit price on traded quantities, in conformity with long run competitive equilibrium theory. This formulation was suggested (but not developed) by Rosen (5. Poiit. Econ. 82, No. 1 (1974) 34-35). By first characterizing an efficient allocation of consumers to space, we show that hedonic unit prices can be understood as a bid-rent function which supports the efficient allocation. This is despite the fact that the lots over which consumers bid are themselves endogenous. We show that unit hedonic prices reveal preferences in a manner different from composite hedonic expen11’ ,985 ditures. Journal o/‘Economic Literature Classification Numbers: 021, 024. Academic

Press. Inc.

1. INTRODUCTION In cost/benefit analysis it is necessary to evaluate willingness to pay for improvements. One technique to do this is to exploit the fact that prices of land at different locations reflect the value of differences in amenities associated with the land. Almost all of the literature uses as a theoretical foundation a model of Rosen [22] in which each consumer buys one unit of housing with a vector of characteristics Z.’ In equilibrium there is an hedonic expenditure function P(Z) which describes the price of one unit of type-Z housing. Quantity characteristics such as lot size and number of bedrooms are included in the characteristics Z. The budget constraint of a consumer is 1’ = m + P(Z), where m represents composite private good. In the same paper, Rosen offers a scond formulation of hedonic price for which the consumer’s budget constraint is y = m + q(a). The type-u good is, for example, land with a vector of amenities a. The consumer chooses I See. for example, Quigley 1201, Nelson [IS], Brown and Rosen [S], Witte, Sumka, and Erekson [ZS]. and Kanemoto and Nakamura [12] to mention some recent papers. Papers such as Ridker and Henning [21], Anderson and Cracker [l] predate Rosen, but are similar in spirit. 55 0022-0531/85

$3.00

CopyrIght te 1985 by Academic Press. Inc All nghl, of reproduction I” any lorm reserved

56

SUZANNE

SCOTCHMER

one type of land, a, at the exclusion of all others, and consumes it in quantity s. The second formulation uses the linear structure of prices which we associate with (long run) equilibrium: Total expenditures are linear in traded quantities.’ The relationship between these two formulations is as follows. 2 can be disaggregated into [a, s, A,], where u represents amenities associated with the location, s represents lot size (space), and h, represents quantity of type-c housing. The vector a would include weather, the school system, road maintenance, air quality; in short, any attribute which is attached to the location, and which the consumer cannot change. h, represents housing characteristics which (in the long run) can be chosen by the consumer at any location. In long run competitive equilibrium, the hedonic expenditure function P(Z) would have the form sp(a) + h,.p(c). That is, competitive prices are linear on traded quantities. It is our view that a natural meaning to the term “hedonic price” is the unit price p(a). In the long run3 we assume that each housing type can be built at each location at the same cost p(c). Thus we can lump constructed housing into the composite private good, and end up with the budget constraint )’ = m + sp(a). Because the consumer chooses one location at the exclusion of all others, it should be possible to think of p(a) as a bid-rent function, just as P(Z) is often thought of as a bid-rent function. In the short run, the housing stock, including its characteristics Z, is fixed. The equilibrium P(Z) is based on an assignment of individuals to houses supported by bid-rent prices P(Z). Interpreting p(a) as a bid-rent function is more challenging because there are not n well-defined objects over which people bid. Since lot sizes are endogenous, the number of type-u lots is itself endogenous. In this paper “linear” sometimes means that p(a) is linear in u. This is ’ In the hedonic price literature, rto/ what we mean by linearity. By linearity we mean precisely what is implied by the budget constraint y=nt +.?~(a). We mean linearity in traded quantities, as usual in competitive equilibrium. The amenities (I can be thought of as indexing locations, If there are two those locations will have the same price. geographic locations with the same amenities, provided amenity is defined broadly enough to include attributes like “access to New York City.” Thus, (I indexes location, and land at location (I is a traded quantity. One can add housing characteristics by formulating the budget constraint y=m+.sp(a)+ h, p(c). h, is a quantity of type-c housing with price p(c). h,p(c) will usually have more linearities than indicated here. The cost of carpets per square foot does not depend on the amount of landscaping. ‘The objection is sometimes made that long-run equilibrium is a very strong assumption. Indeed it is. It may be reasonable to argue that the economy will not immediately achieve a new long-run equilibrium after a change, but this is a different statement than that the initial cross-section data was not generated by a long-run equilibrium. The initial cross section exhibits great variance in population densities. Long-run equilibrium is an explanation for this variation.

HEDONIC

57

PRICES

we show that one can nevertheless think of p(a) as a bid-rent function. This paper generalizes the urban literature which constructs a similar equilibrium for a model with one attribute of land: distance from a city center.s In Sections 2 and 3 we present a model of space allocation and characterize eflicient assignments of people to space. We show that the efficient allocation can be characterized as one in which consumers are assigned to land according to a “highest benefit” function. In Section 4 we show that this “highest benefit” function can be converted into a “bid-rent” function. The efficient allocation is thus a competitive equilibrium for an appropriate allocation of endowments. We use the same formulation to show existence of a competitive equilibrium. We view this technique as the “natural” way to show the welfare properties and existence of competitive equilibrium, since bid-rent functions are the “natural” notion of price when consumers choose one location at the exclusion of all others. The general equilibrium location problem differs from the Arrow-Debreu problem through the nonconvexity of location choice.5 It is this nonconvexity of choice which implies here should be a “bid-rent” interpretation of land price. Existence of a (compensated) locational equilibrium was shown in Schweizer, Varaiya, and Hartwick [23]. They do not characterize efficient allocations or give the welfare theorems. Our technique depends explicitly on the relationship between efficient (or cost-minimizing) allocations and competitive equilibria. For each prespecified vector of utilities, there is a distribution of endowment which supports a cost-minimizing allocation’ as an equilibrium. In this a This Polinsky these

literature includes Beckman [3.4]. and Rubinfeld [Ih, 171. Polinsky

papers.

notably

Polinsky

and

Mirrlees [ 141. and Shave11 [IX.

Rubinfeld

and

Polinsky

Harrison and 193. Rubinfeld and

Shavell.

Rubinfeld [23]. construct

[I I]. Some of hedonic

prices as a function of distance from city center and use the fact that air quality is Increasing with distance. These papers are a bridge between the urban literature that delines a “rent function” on distance from urban center, and the hedonic price literature that uses the hedonic expenditure defined function

function

P(Z).

on a more of distance

abstract unless

This

paper

generahzes

space of attributes than all amenities are monotone

that

bridge. distance. functions

bid-rent

function

Land prices of distance.

The

are not in which

can

be

merely a case the

willingness to pay for direrent amenities cannot possibly be sorted out. ’ An Arrow;‘Debreu-like treatment of nonconvex choices which has become standard is to convexity the economy. That is. an artiticial economy is created in which equilibrium exists. and it is then shown that there is an allocation in the actual economy which is described by the artificial equilibrium. This method was employed by Ellickson chosen not to employ such methods because the straightforward much simpler and lends itself to an intuitive interpretation. ’ A cost minimizing allocation achieving specified utilities. An crthauats the resources.

minimizes efficient

[6], for example. method presented

the resource cost (in composite allocation is a cobt-minimizing

private allocation

We have here is good) of which

58

SUZANNE

SCOTCHMER

equilibrium the bid-rent nature of p(a) is central. This construction of competitive equilibrium from an efficient allocation assigns to each utility vector a distribution of endowment.’ The existence proof consists (as usual) in showing that the image of this map includes all distributions of social endowment. Therefore, for prespecilied endowments, there is a corresponding efficient allocation and bid-rent function. We give a fixed point argument on utilities.’ The formulation of “hedonic price” as a competitive unit price is important for a reason other than aesthetic: If one wants to use hedonic prices to reveal preferences, one needs enough variation in the data that, respectively, P(Z) or p(u), is identified. One wants to observe sample points on the entire domain of Z or of a. But as we show in Section 5, in the long-run competitive equilibrium lot size will be a function of amenities for each type of consumer. Z has the form [a, s(a)]. For each a, there is no variation in lot size.’ We show in Section 5 that the manner in which preferences are packaged (if at all) in the hedonic unit price p(a) differs from the packaging (if at all) in the hedonic expenditure function P(Z). There must be a correction for the population density, or lot size.

2.

THE

MODEL

There is a continumm of sites and a continuum of consumers. Sites are distinguished by attributes and there are finitely many consumer types. A measure space of attributes (A, M, H) describes how much land of each type here is. An element a E A is a list of attributes of land. When A is finite, h(a) is a number designating the amount of type-u land. When A is atomless, h(u) is a density. An element of M is a subset of A, designating several types of land. The amount of such land is the sum (integral) of the corresponding h(.). We prove the results for finite A, but this should be viewed as an expedient rather than a necessity. If a dimension of A is distance to a city center as in the standard urban model, we are assuming there is a finite number of concentric rings. ’ It is not unique. ‘The proof that this map covers all distributions of social endowment is the continuity proof of Lemma I below. The general technique of using a fixed point argument on utilities is discussed in Arrow and Hahn [2]. ‘) If consumers with different tastes are segregated on different types of land in equilibrium, type-u land will have uniform density of population. Of course, consumers with different tastes may share type-a land, but will usually occupy different lot sizes. In this case there will be some variation of lot size. but it corresponds to variation in taste. introducing yet another identification problem.

HEDONIC

PRICES

59

There are K types of people. The measure of type-k people is N,, k = l,..., K. The arguments of utility are attributes of the lot occupied, the lot size, and consumption of one composite private good. Each type-k person earns jk exogenous income and receives rent income of Okr/N,, where r is the total rent earned by land and ok is the fraction of land owned by class k. Thus we assume that each class owns the same fraction of each type of land. Again, this is an expedient rather than a necessity. We could have defined K measures on A, describing different ownership endowments of the K classes. An allocation is a dispersion of population and a distribution of private good. A dispersion of population is {s,(u), g,(a)), where g,(a) is the fraction of type-u sites coccupied by class k. If type k occupies type-u land, the population density at a point of land is l/s,(u). The price system assigns price one to private good, and prices p(u) to land. Ownership of land is part of the endowment, and the total rent r becomes part of endogenous incomes. r=

p(U) dH(U) i .A

(2.1) (2.2)

Competitive equilibrium (CE) is defined by the condition that consumers are maximizing in their budget sets and there is material balance, as well as land balance with supply denoted by the measure H. Assumptions A.l. A is finite.” H is exogenous. H(A) > 0. (The finiteness of A is convenient for simple proofs of existence, but is inessential.) The following definitions will be useful in the proof of existence. U/, is normalized such that zero is the minimum utility in the consumption set and t, is the maximum utility attainable by class k when u, = 0, j # k, and the total resource endownment is z N, jk. Assumption A.2 says that no matter how undesirable the attributes a, if the consumer resides there, he needs some amount of land in order to get 14k> 0.

A.2. Utility functions U,(u, s, m): A x R” -+ R are C’. The consumption set of a type-k consumer is {(a, s, m) 1a E A, s 3 0, m 3 0 ). At “‘Even though it is convenient for simple proofs to assume A is finite. we shall retain the general (Stieltjes) integral notation. This has the advantage that summations are always over consumer types and integrals are always over land. Otherwise there would he two summations.

60

SUZANNE SCOTCHMER

each a, the indifference curve for uk = 0 is {S = 0, m 3 0 )-. At each a E A, C’, is increasing in s. At each a E A, and if s > 0, lJk is increasing in m. Thus U,(a, s, m) 3 0 =o

Assumption consumption.

forall iff

a~A,rn>O,s>O,

s =O.

(2.3)

A.3 says that consumers eventually become satiated in land

A.3. There exists .f such that provided U,(u, s^,m) < t,, (2.4)

For uk > 0, assumptions A.2 and A.3 allow definition of a C’ implicit function m,(a, s; uk) which is convex in S; m,,, shall mean the partial derivative of 11~~with respect to .F.A.2 and A.3 imply that for each ilk such that 0 < uk 6 t, lim -m,,(a, 5-o

.

s; u,)= c,

(2.5)

for

(2.6)

nz J a, s; Uk ) = 0

s > s^.

3. EFFICIENT DISPERSIONS OF POPULATION

Let n(s, g; U) be the social cost (in provate good) of achieving utilities u with population dispersion (s, g). Resource cost”:p(.s,

g;u)=jA

Zm,(a,s,(a);u,)~JH(a).

(3.1)

DEFINITION. For given utilities U, a cost-minimizing allocation is a {.y!Aa; u), g,(a; a)) such that ~(s, g; u)>,~((s~(u; u), g,(a; u); u)} for all (s, g) which satisfy (3.2) and (3.3). We name the constrained minimum Au).

Land constraint: C g,(a) d 1

for each

a EA,

(3.2)

0 < g,(a) < 1 for all a and all k; ” ,vi(cr; U) = 0 if and only if uk = 0, by assumption A.2. Then the cost-minimizing assigns m, =0 and we exclude this type of person from the summation in (3.1). matter how many such people we “account for” in the cost-minimizing allocation. are all costless.

allocation It does not since they

PRICES

HEDONIC

Population

h

if

constraint ’ ’ :

u,>O.

(3.3

An efficient allocation is a cost-minimizing allocation which uses all the resources C N, jk. It is on the utility frontier. The cost-minimizing aliocation can be characterized by the Kuhn-Tucker conditions associated with the following Lagrange function: L{(s,

g), I., ~~~=[~~rn,(u,s,(u);

i,

-c

u$+dH(u) A

-( 14 )[j

g,(a) --H(u)4 .PJU)

+ s, da: 21) [I

N,

g,(u)-

1 (3.4)

l] dH(u).

PROPOSITION 1 (See Appendix). Jf A.1 to A.2 hold, a minimum cost allocution exists for every u such that t, 3 uk > 0. A minimum-cost allocution (s,(u; u), g,(u; u)} suti$es (3..5-(3.7).

If u/, = 0, mk = 0 und sA= 0 and g, = 0 at all locutions u. Wed@ei,(u)=O.

(3.5)

Provided uk > 0, there exists,finite L,(u) > 0 such that -m,,J.(u, s,(u; u); Uk) s,(u; u) + m,(u, s,(u; 21);Uk) = ik(U). There e.uistsji:nite E(U;u) 3 0 such thut,,fbr all k /i,(U) -m,,Ju,

s,(u;

u);

Uk)

-

WZ,(U,

~(a; u) can be understood

24);

<

14 ) >

thut

uk

r:(u;

14)

>

0,

Ue)

= .s,(u;

tcith equulit_b’if ‘g, (a;

S,(U;

s14~h

(3.6)

u)

0.

(3.7)

as a “highest benefit” function. We define benefit of placing type-k on type-u

B, [u; i.k(u), Us] as the per-unit-land

land’“: ix(u)-m,(u,

s; ux)

B, [a; Ah(u)- uk] = max ~ \ 20

-

Ak(lf)-mk(u,

s

.~,(a; u); s,(u;

~4~).

u)

(3.8) Highest benefit function: ~(a; U) = maximum

Bk[u;

12The maxlmand of (3.8) is strictly quasiconcave in .Y. The maximum satisfying (3.6). since (3.6) is the lirst-order condition.

E.,(u), uk]. is achieved

(3.9) at .s~(o; u)

62

SUZANNESCOTCHMER

L,(u) is the social cost of a type-k person. Consider removing him from type-d land provided g,(ci; U) > 0. n,(u) is equal to the private good he consumed there, mk, plus the value to other people of land he occupied, -sm,,J.) = ss(b; u). ~(a; U) is the social benefit of a unit of type-u land. Suppose we introduce another unit of type-a land and transfer some type-k people to it.13 L,(u) - mk [a, s; uk] is the social saving if we give this person lot-size s. Divide by s, and we get the social saving per unit land. .~,(a; U) maximizes the social saving per unit land for type-k. But then we must choose k, the type of consumer, in order to maximize the social saving. This is (3.9) which merely restates first-order conditions (3.7).

4. COMPETITIVE DEFINITION.

EQUILIBRIUM

A CE is a {s(a), g(a), j, 6, p) such that (4.1)-(4.5)

are

satisfied: Optimal

lot-size on type-a land: At each a E A and for all k,

s 20.

Uk[a, s,(a), Yk-Sk(u) p(a)13 Uk[a, 3, Yk- VdaIl,

(4.1) Optimal

location:

If g,(u) > 0, then for all a’ in A,

UkC4 sk(4, yk - ~~mwi Spatial constraint: Population

3 Ukw, s,(a

Y, - ~~(4 ~w)i.

C g,(u) < 1 for all a E A, 1 3 g,(u) 2 0.

constraint:

~

(4.3)

&Z(u) = N, provided sk(u) > 0

for some a E A14. Resource constraint:

(4.2)

(4.4) g,(a) Yk

-

fd”)

sk(u)

1 m

1 hrk jk, where yk iS defined by (2.1) and (2.;).

dH(u)

=

(4.5)

(4.1) and (4.2) state that the consumer makes his choice in two stages. I2 People are continuous, so we can transfer a divisible amount of type-k to this new unit of type-u land. By choosing the optimal lot size. we are choosing “how much of a person” to transfer to the new unit of type-a land. r4 By assumption A.2, if the consumer in competitive equilibrium has any income, he will get positive utility and occupy a positive amount of land. s,(a) > 0, no matter where he lives. If So is positive for any location, it is positive at all locations. But then type-k is using up space. Condition (4.4) states that all of type-k people are assigned somewhere. (4.4) in conjunction with (4.3) states that they do not use more space than exists.

HEDONIC PRICES

63

First he calculates the utility-maximizing lot chooses the location that affords the highest some type-k people actually choose to live on only if the maximum utility achievable there

size on type-a land. Then he utility. g,(a)>0 means that type-a land. They will do this is as great as elsewhere.

THEOREM 1. If A.1 to A.3 hold, every cost-minimizing allocation { (s, g), L, E,u )- is a CE,for appropriately chosen ( j, 0, p).

Proqf: We start with a cost-minimizing allocation and choose endowments (.t;, 0) and p(a). With these endowments and prices, the efficient allocation is a CE. We use the “bid-rent” nature of social benefits described by (3.9). Choose p(a) = ~(a; U) < co. Thus r(u) = jA ~(a; u) dH(a). Choose jk 3 0 and ek 3 0 such that ?fk = &(u):

jk = L,(U) - 8,r(u)INk

(4.6)

Ce,=i.

(4.7)

These choices ensure a material-balance equilibrium. A manipulation (3.7) shows that total consumption plus total rent collected equals

of

To show that the efficient aliocation is CE with these incomes and prices, we must show that the consumer achieves maximum utility at sites where g,(a; U) > 0 and does so by occupying lot size ~,(a; u). We show that by construction, p(a) is a bid-rent function. Type-k is the highest bidder for type-a land precisely when the efficient allocation would assign him there. Furthermore, his desired lot size is the efficient lot size. At utility uk, a consumer’s willingness to pay for type-a land (per unit) depends on the lot size. Define w,k[a,s;

)Ik,

Uk~=yk-mk(a~S;Uk)~

(4.9)

s

~‘~(0, S; y,, &) is quasiconcave in s.‘~ It is maximized by ~,(a; u), the efficient lot size, which is also the consumer’s maximizing choice with the income and prices chosen. We define type-k’s willingness to pay for type-a land (per unit) as the maximum: Wk[a;

” wk [u, .F: yL1 uk] is increasing

.vk,

ukl

=T:;

wk[a,

s;

yk,

for s < .~,(a; u) and decreasing

ukl

for s > .~,(a; u).

64

SUZANNE SCOTCHMER

It follows from the choice of incomes and prices and (3.9) that Bid-rent function: p(a) = maximum

WJU; y,, Uk).

(4.11)

k

it’s and W, are strictly decreasing in uk. The efficient allocation excludes type-k from type-u land when &(a; u))B~[cI; Rk(U), uk]. But then p(a)) W,(a; +I’~, uk) and the consumer will choose not to occupy type-u land because he gets less utility there than uk. The consumer can achieve uk at other locations: namely at the locations to which the efficient allocation has assigned him. At such locations, price equals the marginal rate of substitution. Thus. if g,(a; u) > 0, P(a)

THEOREM

2. A CE

is

=

-m.,.ktu,

~,(a;

u);

uk).

(4.12)

t#%eni.

Proof: We show that a Pareto-superior allocation is infeasible. Let ( g(u), s(u), m(u) I\ be an allocation which gives no less utility to any person than the CE, and gives greater utility to some. Then by revealed preference %(a)

+

p(a)

sk(u)

2

!‘k

(4.13)

with strict inequality for some k and some a. Multiply by g,(u)/s,(u) I6 and sum over k and A. Total rent r appears on both sides of the inequality and can be subtracted out. The summed inequality then shows that the total resource consumed exceeds the resource endowment. The better allocation is infeasible. End of proof. Figure 1 illustrates the CE and also the efficient allocation, since W, [a, yk, uk] = B,[u; 3+(u), uk]. &(a) is the highest utility achievable by type-2 on type-u land. It is less than u2 achievable elsewhere.

w2 [a; )‘2, UJ < w, [a; 1’2,Liz(U)]= w, [a; y,, Ull = p(a). The question of existence can be phrased as follows. Does there exist u for which, with fixed endowments (,,0),

(4.14) r(u) =

There is an efficient allocation

associated with u. Therefore,

there are

” If .~~(a) = 0. then the utility achieved by k is zero. Then the CE utility was also zero. Type k consumes no resources in either allocation, and we exclude that type from this discussion. We do not perform this calculation for such k; we merely exclude him from the discussion.

HEDONIC

65

PRICES

cb;u)=p(al

I s,(a;ui

FIG.

I.

Equilibrium

s,(a;u)

cm type-o

I

I

I \

5

$(a)

land

shadow prices L(u) and E(L~;u). If there is M satisfying (4.14), then the efficient allocation associated with it is a CE, as is shown in Theorem 1. With zero income a consumer can achieve his consumption set at s = 0, m =O. We wish the allocation to guarantee everyone a place to live. A.4 together with A.3 accomplishes this: A.4. jr > 0 for all k.” THEOREM

3 (Appendix).

5. PREFERENCE

Jf A.lLA.4 REVELATION

hold, CE exists.

AND

HEDONIC

PRICES

Thus, in competitive equilibrium there is a hedonic price function p(u) which consists of unit prices on land. It has the interpretation of being a bid-rent function, as well as having the usual marginal feature of optimal consumer choices: Bid-rent function: p(a) = maximum

W, [a; Jam, u,];

I)(LI) = --I??,.~ [a, .~,(a; u); u] if type-k occupies type-u land.

(5.1) (5.2)

We begin by discussing the simplest case, a homogeneous population. Then, with Z= [a, s], P(Z) is y -m[a, s; u]. For two vectors 2 and 2, P( 2) - P( 2) captures compensating variation, provided the population is

66

SUZANNE

SCOTCHMER

homogeneous. Our discussion reveals, however, that due to the functional relationship s(u; u), the data will generally not be rich enough to compute the compensating variations of interest. Suppose a city had initial amenities 8, and population density l/S. We wish to know the compensating variation for improving amenities by a nonmarginal amount from U, to UT. Assuming the population remains static subsequent to this improvement (which it may in the short run, but will not in the long run’X), we would want to calculate the compensating variation P[a, .s] - P[a*, S]. There is no P[a* , S] to observe. since ~(a*; U) will usually not be the same as .?= .~(a; 21).On the other hand, if the willingness to pay for space is independent of the willingness to pay for amenities, one should be able to disentangle the two, as we now show, exploiting the unit hedonic price p(u). We have

Differentiating t?p(u) -=?a,

p(u)‘” with respect to a,, the first amenity, a ?‘- m[u, s(u; 24);u] ~&(a; u) as .$a; u) aa,

m,, [a, s(u; 24);u] s(u; u) .

(5.4

The first term vanishes at ~(a; U) because of the consumer’s optimizing choice. Thus the partial derivative of p(a) is the marginal willingness to pay for amenity u, times the population density l/s(u; u). Assuming that the population remains static after the air quality has been improved from a, to a,* , ” the aggregate willingness to pay is the vertical area under the curve (l/s@,, [a; S; U] is Fig. 2. For purposes of this curve, the axis represents hypothetical air qualities for the city with fixed population. The other curve in Fig. 2 is the rate of change of the hedonic price function in the pre-improvement cross section. For purposes of the curve dp(u)/da 1, the axis represents air qualities in the pre-improvement cross section. Figure 2 illustrates how iip(u)/&~, relates to willingness to pay for an improvement. The “hedonic price estimate” p(a*) - p(G), which is the integral in Fig. 2, overestimates or underestimates the benefits of the Ix Long run versus short run benefits are themselves an interesting issue. See Scotchmer [25]. I’) In Section 4 we used tiniteness of A for easy proofs. This means amenities do not vary continuously. In order to show how the interpretation of hedonic unit prices p(u) differs from the interpretation of hedonic expenditure function P(Z), we nevertheless differentiate p(u). ?“This means the economy does not move to long run equilibrium. In a new long run equilibrium the dispersion of population will change.

HEDONIC

FIG.

2.

The value

67

PRICES

of a nonmarginal

improvement

improvement according to whether the population density I/s(a; U) is increasing or decreasing with a, in the pre-improvement equilibrium. There is a simple correction for this problem, since ~(a; u) is observable. At each QE A, one can simply multiply ~?~(a)/&, by the ratio of population densities: (~(a; u)/~)(+~(u)/&z,) = (l/-)s m,, [ .I, See (5.4). If m(.) is separable in a, and s, willingness to pay for the nonmarginal improvement (with fixed population) is therefore identified. The reason this multiplicative correction by population density is necessary is intuitively simple: Suppose in the cross section used to estimate p(u), land with better amenities has denser population. Then the wiilingness-to-pay for clean air per unit fund is greater because each unit of land contains more people. We now turn briefly to heterogeneity. Historically there has been a debate about whether personal characteristics belong in the hedonic price equation. Suppose in Eq. (5.1) we represent type-k’s willingness to pay as W[u; t?,], where Ox is a vector of personal characteristics like income and taste parameters. An equilibrium will often be separating in the sense that each type of land, a, is occupied by only one type of consumer. Then there is a function 0(u). Does it therefore follow that we should include personal characteristics as well as characteristics of land in the hedonic price equation. Since the derivatives of p(u) reveal preferences, the parameters of p(u) at each u E A depend on the people who live there. However, this does not mean that one can estimate a function of the form p(a, 0) = h(u)rc[O] or p(a, 0) = h(a) + n[tGJ]. These two functions are monotone in rc[0], and thus an equilibrium only occurs if rc[e] has the same value for all U. Otherwise (5.1) would be violated. Each type of person must be the highest bidder on the land he occupies. Personal parameters must be introduced in a manner which preserves this property. To our knowledge, this has not been formulated successfully in an applied model.

68

SUZANNE

SCOTCHMER

Clearly there is no method of exploiting cross-section data to estimate preferences or demands unless there is some homogeneity of tastes. A restriction often invoked is that people differ only by income. 0 = ~9. Then there may be a function ,~(a) which reveals how income classes separate themselves on different types of land. If several income classes occupy typea land, they will generally occupy different lot sizes .~,(a; u). Income cannot enter the hedonic price equation monotonically*’ for the reason given in the previous paragraph. For completeness we point out that there are two reasons hedonic prices may not reveal preferences. The first reason relates to heterogeneity of tastes. If tastes differ, different consumer types are likely to be segregated on different types of land. The price of type-u land is the willingness to pay of its own occupants. However those people were outbid for the amenities of sites they do not occupy. Thus, the difference in prices does not reveal accurately the compensating variation for the difference in amenities. This is a version of the usual problem with cross-section data: If there is no uniformity of tastes, cross-section data reveral very little about tastes.“* The second problem relates to whether p(a) can be known. Suppose a E R”. If data points cover all of R”, then p(u) can be known with reasonable accuracy. In practice, however, the data will be deficient. Good air quality may be highly correlated with good public parks. In that case, it is difftcult to determine whether the variation in land prices is determined by air quality or parks. This identification problem is less severe than the corresponding identification problem for the expenditure function P(Z) which includes housing characteristics. The latter is the subject of Brown and Rosen [5]. In a long run competitive equilibrium, the price p(c) reveals construction cost. That is, it reveals the supply function and not the demand function. One of the advantages of separating out the exogenous amenities, a, from the produced characteristics, c, is that there is no prima facie reason the variation in A would be restricted.23 2’ Rubinfeld 123 J uses income in the hedonic price equation in a manner which would have the high-income people outbidding low-income people on every site. In other words, highincome people can increase their utility by occupying sites assigned to low-income people. Consumers are not in equilibrium. 21 This is a well-known problem and is, for example, the subject of Kanemoto and Nakamura [ I?]. 22 There are other reasons the variation may be restricted. Areas with good air quality may also have good schools because they are both in the suburbs. This may occur because. for example, high income people occupy unpolluted land. High-income people may have a commensurately higher willingness to pay for education, and local governments respond by providing good education. Again. it is an endo~eneit~ of choice which inhibits identification of a(u). In this example, the attributes of land are not exogenously fixed. If there is perfect

69

HEDONIC PRICES

6. CONCLUSION

There has been a discrepancy between the notion of hedonic price used in the literature and the natural notion of hedonic price in competitive equilibrium theory. The former is a composite price, while the latter is a unit price. This distinction was initially pointed out, but not developed, by Rosen [22]. The essence of the spatial allocation problem is that consumers choose one type of land at the expense of all others. It is natural in such a context to think of prices as a bid-rent function, even though lot sizes are endogenous and the number of lots on each type of land is determined by the equilibrium. We have given a proof of existence and welfare properties of equilibrium based on such a bid-rent function. This paper may be thought of as a generalization of the body of literature in urban economics which constructs bid-rent functions on a l-dimensional characteristic distance from a city center. The folk wisdom is that differences in hedonic prices reveal willingness to pay for amenities when tastes are homogeneous. This is misleading if one means by hedonic price the competitive equilibrium unit price. A simple difference between hedonic prices does not reveal willingness to pay for amenities even in the simplest case of homogeneous tastes. This discussion has not invalidated the interpretations of P(Z) adhered to in the literature. We have, however, pointed out that if Z includes characteristics chosen in equilibrium by utility-maximizing consumers (such as lot size), the data may not be rich enough to make use of Z’(Z). The reformulation of hedonic price as a unit price adds no information to that contained in P(Z). It merely presents a different package for it; one that conforms to the competitive equilibrium notion of expenditures which are linear in quantities. We show in Fig. 2 that when the willingness to pay for amenities in unaffected by the amount of space occupied, the rate of change of hedonic prices can be normalized by population density to reveal the willingness to pay of a static population for a discrete improvement to amenities.

APPENDIX

PROPOSITION 1. Jf’ A.I-A.3 hold, a minimum cost allocation exists ,fi)r rrtq* u such that tk > uk > 0. A cost-minimizing { .~,(a; u), gk(a; u)) can he characterixd h,v (3.5 )-( 3.7). correlation

between

the value of good marizes when there

air

quality

and

schools,

the

schools. Starrett [27] gives is enough variation in data

value

of air

B general to identify

quality “spanning” preferences.

cannot

be separated

condition

which

from sum-

70

SUZANNESCOTCHMER

That a constrained minimum to (3.1) exists follows from the fact that we are minimizing a continuous functional on a compact set of (s, g). (This is true when A is atomless also, but requires additional argument.) A.3 is required for compactness. We can restrict attention to sk such that 0 6 sk 0, the optimizing s,(a; u) cannot be zero, because s = 0 gives u/i = 0 for all (a, nzk). Thus (3.6) is satisfied as an equality and implies that n,(u) > 0 if ul, >O. Since the lefthand side of (3.6) is finite for positive s,(a; u), n,(u) is finite. If u/, = 0, gk(a; U) = 0. (These consumers get ~,(a; U) = 0, so they occupy no space.) ~(a; U) is zero if C g,(a; U) < 1. This means that all the land is not utilized and occurs only if the marginal willingness to pay for type-a land is zero for all k for which ux > 0. ~(a; U) is finite because the left-hand side of (3.7) is finite. If uk > 0, s,(a; U) > 0. THEOREM

3.

If

A.lLA.4 hold, CE exists.

Prooj We will make a fixed point argument on a simplex in RK which is homeomorphic to the utility possibility frontier. b(u) is the minimum private good required to achieve utilities u.

(A.1)

UER” 1/i(u)=&V&

F is homeomorphic to a simplex S= {X E R” 1xx/, = c, .Y~> 0 all k, c E R}. We can choose a homeomorphism h: S + F such that xk = 0 iff uk = h,(x) = 0 and .x/, = c iff uk = hk(x) = t,. F is closed and monotone. Closedness follows from (A.l) and the continuity of L,(u) and &(a; u) below. Monotonicity follows from the monotonicity of everyone’s preference for all goods. (Suppose you could increase uk without decreasing other utilities, and without increasing private good consumption. This could only be done by making space-utilization more efficient. But then it was not efficient initially.)

P{b,(a;

u), g,(a; u),; u) =c &i,(u)-jA

&(a; u)dH(u)=~N&.

(A.2)

(A.2) can be verified by substituting (3.6) into (3.1). We define the homeomorphism h by defining its inverse h^in (A.3). There is a continuous inverse h because the Jacobian for h^ is nonsingular on F. (There are not two u’s in F which are on a ray from the origin of RK.) h(x) maps to UE F such that uk/ui = x,/.u,, all j, k. /;( u ) = cu/c Uk,

u EF.

(A.3)

71

HEDONIC PRICES

A fixed point for (A.5) will give a CE. In the following expressions, am satisfies (2.2) and JJ,JU) satisfies (2.2) with r(u) = s,4 &(a; u) &J(a). h,(x) = xk + N,{ ykJr[h(x)] h: S-+ R”,

h(x)=

] - &[h(x)]

1 (below).

Continuity LEMMA

.KE s,

{h,(x).../?,(x)).

To apply Brouwer’s fixed point theorem, back into S. and that h is continuous. LEMMA

),

(A.4) (A.5 1

we need to show that h maps

i( u ) and &(a; u ) are continuous in II.

of h follows because Iz is a homeomorphism.

2. h(S) is contained in S.

Proof. We show that h maps into S at the boundaries of S. Therefore it is possible to choose a scale c for S such that b(S) is in S. ~,Jx) = ZQ= 0 iff .Y~= 0. Irk(x) = Us = t, iff .Y~= c. Every boundary point of S has .K~= 0 for some k and may have .Y~= c for one k. Let the notation (X / xk) denote s with .Y~ in the kth slot. It is enough to show that at a boundary point, h,(.u 1.Y~= 0) > 0 and h,(s 1.Y~= c) < 0. Then one can choose the simplex ( large enough that h(S) is in S. Recall that j.,( u 1Us = 0) = 0. Using assumption A.4,

For ZIE F, if nk = I,, then II, =0 for ,j#k. EN&&4 h

I t,)-&(u)

By (4.8). th)]=O.

(A.7)

Since E.,(u I 1,) = 0, 1

i#h

N,

I .r,(u

I f, I- j.,(zrI f, 113 c N, p, > 0 /iA

(A.81

and N,y,(uI

1,)-NN,i,(U(

f,)
(A.91

End of proof. LEMMA

1. E.(24) and E(U;u) m-e continuous in II.

Uiziqtrene.0

For each k, i.,(u) and s~(u; u) are unique. This implies &(a; u) and Y(U) are unique even though ~~(a; u) may not be.

72

SUZANNE

SCOTCHMER

Suppose they are not unique. That is, there are two distinct allocations B and R associated with U. j.‘(u) #n’(u). Define an index set T of consumer types for which i.:(u) < i”:(u), and the complement set J, with ir(u) 3 “,~(lf).

The left-hand side of (3.5) is nonincreasing in s. Thus, for all ,j in J, and all a, S: ,< s,“. For t in T, SF > .rf. Each type-t person gets more space in R than in B. By convexity, tin T,

-nz,,,(u,

sf(u; u); u,) < -nz ,(,,(a, sfya; u); u,)

all

-nz,,,(a,

sp(a; u); 24,)3 --nz ,\,,(a. .f(u; u); u,)

all j in J.

(A.lO) (A.1 1)

this implies and (3.61, that E g,B(a; u) > C ,$(a; u) BY With the same poplx s,fk zl) < c gp( a; u). But this is a contradiction: ulation in allocations R and B, it cannot be true that each type-t person gets more land in R than B, and at the same time there is a smaller percentage of total land allocated to types t. If UI, = 0,

i,(u)

=0

and

s,(u; u) = 0

forall

USA.

Continuit~~ We first consider

a neighborhood of uR > 0. are continuous in 11 iff {II/; 1 are continuous in u. Let By (3.51, (Q; (sR, gR, iR, cR) and (sB, gB. 3.‘, 6’) be allocations associated with uR and uB arbitrarily close to each other. If j.R and iB are not close, the contradiction which arises is the same as arises in the proof of uniqueness. We will use the following statements. [36>0s.t.

~(u;z4~)-~(u;~~)~6forsomeu~A]

[3q >O s.t. E.,(u’)-I,(u’)

>y for some t]

[3r > 0 s.t. .~,(a; u’) --~,(a; 2.P) > u for all UE A] [3/3 > 0 s.t. ~(a; uB) - ~(a; uR) < -b for some a E A] [gp>Os.t.;l,(UB)-%,(uR)< [3~1>0

s.t. ,~,(a; I/-,~~(a;

(A.12) (A.13) (A.14) (A.15)

--pforsomej]

(A.16)

uB) < --~~for all agAl.

(A.17)

It follows from continuity of m, and n~,,,~ in s and u and Eqs. (3.5) and (3.6). that (A.12) holds iff (A.13) holds. (A.13) holds iff (A.14) holds. Thus, any of (A.12), (A.13), (A.14) implies the other two. Any of (A.15), (A.16) (A.17) implies the other two. If E takes a jump at some UE A, then some ;Lk must take a jump because some sk takes a jump at that a (and vice versa.) First we show that if either (A.12) or (A.15) is true, the other must also be true. Suppose (A.12). (A.13), (A.14) are true but not (A.15), (A.16),

HEDONIC

73

PRICES

(A.17). Since (A.15) is not true, for all UEA, either (~(a; u~)-E(u;u~)[ is arbitrarily small or ~(0; u~)-E(u; Us)> (5. By (3.5) and (3.6) for all k and all a except where (A.14) applies, Isr(u, uR) -.~,(a; uB)I is arbitrarily small. Hence the total space occupied is greater under allocation R than B, a contradiction. The only remedy for this contradiction is for (A.1 5) (A.1 6). (A. 17) to hold as well. Together (A.14) and (A. 17) imply that each type-r person will occupy more space in R than B. Typej must occupy less. This requires that the percentage of land allocated to type t is larger in R than in B. of nz,,, in s and conBy (3.6) which determines g,, the monotonicity tinuity of UZ,,~ in U, this is not possible. (A.14) and (A.17) imply for t and ,j satisfying (A.13) and (A.16), [3q>Os.t.

(-mI:)-(-mfl,)>~~allcr~A]

[3r>Os.t.

(-m;,)

- (-wzf,)

< -T all aE A].

(A.18)

But then (3.6) implies c Rr(“; uR IF7

I< c g,(u:14

(A.19)

it 1

for all II E A, with strict inequality for some u E A. Type t cannot have a larger percentage of land in R than in B. Contradiction. Therefore, none of (A.12)-(A.17) hold for U’ arbitrarily close to uR. Now consider a sequence U” + uR with U; + 0. (Throughout the remainder of the Appendix, lim shall mean lim as ZP goes to Us.) Since we have defined ik [u 1uk = 0] = 0, we must show that for any such sequence lim I., (21”) = 0. For every finite u, &(a; U) < infinity. bounded and therefore lim -rn,.,(u,

By (3.7)

(A.20) -r?z,,.,(u, s,(u; u); uk)

.~,(a; 0”); u”) < infinity.

First suppose that lim .~,(a; u”) =O. By assumption

(A.21 )

A.2, (A.21 ) implies

lim rn,( a, sk( u; u”); 24”) = 0 Now

is

(A.22)

suppose that lim sk(u, u”) = s* > 0. Since lim nz,( u, s; u”) = 0

for any

s>O,

(A.23)

74

SUZANNESCOTCHMER

it is only necessary to show lim m,,,(cc, .~~(a, u”); u”) = 0.

(A.24)

We need to show that for every 6 >O and for each h>O, there exists N* such that for n > N* (that is, uk getting smaller), -m,,,(a,

s*; u;l) d

m,(a, s* -11;

u;) - m,(a, s*; u’;-) < 6. h

(A.25)

Bt the first inequality follows from convexity of mk in s. The second inequality follows from the fact that nrk converges to zero at each s> 0. Therefore (A.24) holds. We have shown that {A,}, {sx), and E are continuous. Continuity of r follows from its definition:

r(u) = J‘ c(a;u) dH(a). .A

(A.26)

REFERENCES 1. J. ANDERSON, JK. AND T. D. CKOCKER, Air pollution and residential property values. Urhun Stud.. 8 No. 3 (1971). 171-180. 2. K. ARKOW AND F. HAHN, “General Competitive Analysis.” North-Holland, Amsterdam. Holland, 1971. 3. M. BECKMAN. On the distribution of urban rent and residential density, J. Eron. Theory I (1969), 60-67. 4. M. BECKMAN, Spatial equilibrium: A new approach to urban density, “Resource Alocation and Division of Space’ (T. Fugi and R. Sato, Eds.), Springer-Verlag, Berlin, 1977. 5. J. N. BKOWN ANU H. S. ROSEN, On the estimation of structural hedonic price models. Eumome~rica 50 ( 1982). 765-768. 6. B. ELLICKSON, Competitive equilibrium with local public goods, J. Gon. Theory 21 (1979). 46661. 7. M. FREEMAN. Air pollution and property values: A methodological comment, Rev. Econ. Sratisr. 53 (1971). 415416. 8. M. FKEEMAN, Air pollution and property values: A further comment, Rev. Econ. Stcrtist. 56 (1974). 554-556. 9. M. FREEMAN, On estimating air pollution control benefits from land value studies, J. Environ. GWL Manuge. I (1974) 74-83. 10. M. FKEEMAN, “The Benefits of Environmental Improvement: Theory and Practice.” Resources for the Future. inc.. Washington, D. C., 1979. Il. D. HAKKISON, JK. AND D. RUBINFELD. Hedonic housing prices and the demand for clean air, J. Em~iron. Ewn. Munup. 5 (1978). 81L102. I?. Y. KANEMOTO ANU R. NAKAMURA, “A New Approach to the Estimation of Structural Equations in Hedonic Models,” Discussion Paper 223, Institute of Socio-Economic Planning, University of Tsukuba. Sakura, 1984.

HEDONIC

PRICES

15

13. P. LINNEMAN. The demand for residence site characteristics, J. C’rharr Econ. 9 (1981 1, 129-148. 14. J. MIRRLEES, The optimum town, .S~&i.sh J. Econ. 00 (1972), 114-135. 15. J. NELSON, Residential choice, hedonic prices and the demand for urban air quality. J. U&m

Econ.

5 (1978),

357-369.

16. A. M. POLINSKY AND D. RUBINFELD, Property values and the benefits of environmental improvements: Theory and measurement, in “Public Economics and the Quality of Life” Wingo and Evans, Eds.), Johns Hopkins, Baltimore. Maryland, 1978. 17. A. M. POLINSKY ANU D. RIJBINFELD, The long-run effects of a residential property tax and local public services, J. Urban Econ. (1978). 241-262. 18. A. M. POLINSKY ANI) S. SHAVELL. The air pollution and property value debate, Rw. Econ. Siarisr. 57 (1975). 10@104. 19. A. M. POLINSKY ANII S. SHAVELL. Amenities and property values in a model of an urban area, J. Public. Econ. 5 (1976). 119-129. 20. J. QUIGLEY. Nonlinear budget constraints and consumer demand: An application to pulic programs for residential housing, J. Urhun Econ. 12 (1982), 177-201. 21. R. G. RIDKEK ANL> J. A. HENNING, The determinants of residential property values with special reference to air pollution, Rw. Ewn. Slafisf. 49 (1967). 246-257. 22. S. ROSEN, Hedonic prices and implicit markets: Product differentiation in pure competition, J. Poiir. Gvn. 82, No. I (1974). 34-55. 23. D. RUBINFELD, Market approaches to the measurement of the benefits of air pollution abatement, in (A. Friedlaender. Ed.), 24. U. SCHWEIZER. P. VAKAIJA. ANV HARTWICR. General equilibrium and location theory. J. Urban Econ. 3 ( l976), 285-303. 25. S. SCOTCHMER. “The Short Run and Long Run Benefits of Environmental Improvement.” Discussion Paper 1135, Harvard Institute of Economic Research, February 1985. 26. K. SMALL, Air pollution and property values: Further comment, Rev. EWE. S~urlsr. 57 (1975), 105-107. 27. D. STARRETT. Foundations of public economics, manuscript, Stanford University. 1985. 28. A. WITTE. H. SUMKA. AND H. EKEKSON. An estimate of a structural hedonic price model of the housing market: An application of Rosen’s theory of implicit markets, Econonzetric~u 47 (1969). 1151~1173.