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Housing Prices and Residential Land Use under Job Site Uncertainty
JOURNAL OF HOUSING ECONOMICS ARTICLE NO.
7, 1–20 (1998)
HE980222
Housing Prices and Residential Land Use under Job Site Uncertainty Geoffrey K. Turnbull Department of Economics, Louisiana State University, Baton Rouge, Louisiana 70803-6306 Received March 21, 1997
This paper examines the effects of job site uncertainty in a multiple center city. It shows that workers sort themselves spatially according to their job site stability. Risk averse workers also respond to the spatial variation in commuting cost risk which offsets the tendency to sort by job site stability. The theory predicts less spatial sorting by commuting costs than the certainty or risk-neutral models predict. House bid prices capitalize the spatial variations in expected travel cost and its variance, leading a flatter price surface than under certainty. In addition, risk aversion generates price peaks between employment centers. 1998 Academic Press
I. INTRODUCTION
Many urban areas have more than one concentrated business district (CBD).1 Motivated by this observation, White (1976), Wieand (1987), Yinger (1992), and Turnbull (1993) extend the canonical monocentric urban land use theory to investigate the spatial complications introduced by multiple CBDs. Like much of the monocentric theory, individuals know with certainty the values of key variables before committing to their consumption and location decisions in these models.2 One aspect ignored by these papers, however, is that when jobs are distributed across several CBDs there is the likelihood that workers may not know with certainty in which of the CBDs their future employment will be. Employers may move from one site to another or an individual worker may be laid off and take another job in a different CBD. Housing and location are long lived investments that are costly to adjust to short term changes in family circumstances and, in the context of the multiple CBD city, the worker’s job site is one such circumstance that might change. This suggests that at least some individuals make their housing and location decisions before knowing with certainty 1 This paper denotes all concentrations of business activity, hence employment, as CBDs, whether the traditional central business districts of monocentric theory or suburban beltlines. 2 See Turnbull (1995) for a complete treatment of monocentric uncertainty models.
1 1051-1377/98 $25.00 Copyright 1998 by Academic Press All rights of reproduction in any form reserved.
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where their job site will be throughout the housing consumption period. The question of interest is: how does this job site uncertainty affect the urban housing market? Crane (1996) presents a model of residential location choice in a long narrow two-CBD city under job site uncertainty. In his model workers are initially assigned a job in one of the CBDs, after which they make their housing and location decisions, committing themselves to their housinglocation consumption bundle and the housing bid price. Once the consumption decisions and housing price commitments are made, the worker finds out if his or her job site is going to change to the other CBD in the urban area. The model is not based in neoclassical uncertainty theory. Instead it represents an attempt to extend the certainty urban land use model by assuming that workers base their housing bid prices on expected commuting costs instead of the ex post costs as in the certainty models. Unfortunately, it can be shown that no bid rent exists in Crane’s model because it does not take into account the variability in the worker’s ex post consumption that arises when his job site changes.3 This paper contributes to urban land use theory under uncertainty in general and to the multiple CBD city theory in particular, applying neoclassical uncertainty theory to evaluate the impact of job site uncertainty on housing prices and residential land use patterns. Section II explains the model structure and derives a bid-price function under risk neutrality. Section III then uses the model to analyze the implications of job site uncertainty on bid rents and land use patterns. This model of job site uncertainty provides interesting new results even though it assumes risk neutrality. However, it also raises an additional question: Given the effects of job site uncertainty on prices and land use under risk neutrality, how do risk averse workers alter the model predictions? Section IV examines this question, extending the model to evaluate the effects of risk aversion on housing bid prices and location demand. Section V presents the concluding remarks.
II. THE UNCERTAINTY MODEL UNDER RISK NEUTRALITY
Consider a linear city extending over the unit interval [0, 1] with an employment center at each end. Let t denote the distance of a residence location from the first business district. Thus, all jobs are located in t 5 0 and t 5 1 and the commute to a job in the first center requires travelling distance t while the commute to a job in the second center requires travelling distance 1 2 t. The commuting cost from location t to job site 1 is therefore 3
A demonstration of this proposition is available from the author upon request.
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3
c1(t) while the commuting cost from location t to job site 2 is c2(1 2 t), with greater distance requiring greater commuting costs in each case: c9i . 0. Further, c1(0) , c2(0) and c2(1) , c1(1), that is, it always costs more to commute from one job site to the other than it does to not commute at all. Workers do not know with certainty where their jobs will be when making their housing and location decisions.4 The probability of a given worker’s job site being located in the first center is u. All workers are identical and, for simplicity, all wages are invariant across job sites. Workers are identified by their types u, their probability of being employed in site 1. The expected commuting cost for worker type u is a function of t in general, and is denoted e(t; u ) 5 uc1(t) 1 (1 2 u )c2(1 2 t).
(1)
Consumer utility is given by the well behaved expected utility function E[u(h, y)], where housing consumption is h and the consumption of all other goods, the numeraire, is y. The consumer’s money budget is m. Consumer choice under net income uncertainty implies that either h or y or both must be uncertain ex ante, exhibiting possible ex post variation around the chosen planned values. In general, it turns out that the structure of the housing and location choice (hence the bid price) problem implies that housing consumption is nonstochastic while spending on all other goods, being the residual of the money budget left after paying the bid price for housing (which is decided ex ante) and travel costs, is the stochastic or risky variable in the consumer choice model (Turnbull, 1995, Chap. 4 and 5). The worker chooses the bid price, the quantity of housing and a planned level of spending on other goods, x, before observing where his job site will be. After making his price bid and committing to his housing consumption, he observes his job site. Thus, his or her actual consumption is therefore the planned level of housing h, the required spending on commuting to the job site, and all other spending—what is left over after paying for housing and commuting. For the worker who is ultimately employed at site i, the ex post spending on nonhousing consumption is y 5 m 2 ci 2 ph. Denoting planned nonhousing spending as x, where x 5 m 2 e 2 ph, ex post or realized nonhousing spending is y 5 x 1 «,
(2)
where « [ he(t, u ) 2 c1(t), e(t, u ) 2 c2(1 2 t)j with probabilities hu, 4
Alternatively, the workers select their residence locations given their initial job sites, but they face a probability that their job site will change in the future without being able to instantaneously adjust their residence locations. Adding an initial period of certainty to the model complicates the notation but does not alter the conclusions.
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(1 2 u )j. Clearly, E[ y] 5 x. Using this notation, the bid price function under uncertainty is defined as the following: p(t, u; u ) ; maxhh,xjh[m 2 x 2 e(t; u )]/h
E[u(h, x 1 «)] $ uj.
s.t.
Under risk neutrality the expected utility can be expressed as a function of planned housing and nonhousing consumption, h and x, alone. The bidprice function under uncertainty is defined as p(t, u; u ) ; maxhh,xjh[m 2 x 2 e(t; u )]/h
s.t.
f (h) 1 g(h)x $ uj. (3)
To begin, recast (3) into a dual formulation that is easier to work with. This task is simplified by the fact that E[u(h, x 1 «)] ; u(h, x) under risk neutrality, which implies that (3) can be cast as a certainty problem in terms of expected values (of travel costs and nonhousing consumption). Define the Hicksian (compensated) demands for housing and planned nonhousing consumption as the expected expenditure minimizing combinations that yield expected utility u: hh( p, u), x( p, u)j ; argminh ph 1 x
s.t.
f (h) 1 g(h)x $ uj. (4)
Use (4) to define the consumer expected expenditure function conditional on t (via p) as e( p, u) ; ph( p, u) 1 x(p, u).
(5)
As shown in the Appendix, this function has many of the same properties as its counterpart under certainty, including Shephard’s lemma ep 5 h( p, u). The Appendix also demonstrates that the bid-price function under risk neutrality (3) is implicitly defined by the dual relationship e( p(t, u; u ), u) ; m 2 e(t; u )
(6)
over all t [ [0, 1]. This equation, with (1), is the key to deriving the model implications.
III. BID PRICES AND LAND USE
Differentiating (6) with respect to t and using Shephard’s lemma, we obtain the risk neutral uncertainty version of Muth’s (1969) equation for household location equilibrium,
JOB SITE UNCERTAINTY
h( p, u)pt 5 2et .
5 (7)
This result shows that the slope of the bid price function is determined solely by expected commuting costs, just as the price gradient follows commuting costs in the certainty theory: pt v 0 as et b 0.
(8)
Unlike the certainty theory, however, a monotonic ci does not by itself ensure that the bid price function for worker type u is monotonic when job sites are uncertain. To sort out the possibilities in this regard, define the parameters a and b such that a 5 c92(0)/[c91(0) 1 c92(0)]
(9)
b 5 c91(1)/[c91(1) 1 c92(1)].
(10)
PROPOSITION 1. For c0i $ 0: (i) u . a ⇒ pt (t, u; u ) , 0 ;t [ [0, 1]: The housing bid price function is monotonic decreasing in t for all worker types u . a. (ii) u , b ⇒ pt (t, u; u ) . 0 ;t [ [0, 1]: The housing bid price function is monotonic increasing in t for all worker types u , b. Proof. Differentiate (1) to find et 5 uc91 2 (1 2 u )c92
(11)
ett 5 uc01 1 (1 2 u )c02
(12)
etu 5 c91 1 c92 . 0
(13)
Substitute (9) into (11) to find et (0; a) 5 ac91(0) 2 (1 2 a)c92(0) 5 0 so that (13) ensures et (0; u ) . 0 for all u . a. Under the conditions of the proposition ett $ 0 which in turn implies et (t; u ) . 0 for all t [ [0, 1]. From (8), this further implies pt , 0 for all t, which is result (i). Substitute (10) into (11) to find et(1; b) 5 bc91(1) 2 (1 2 b)c92(1) 5 0 so that (13) ensures et (1; u ) , 0 for all u , b. Under the conditions of the proposition ett $ 0 which in turn implies et (t; u ) , 0 for all t [ [0, 1]. From (8), this further implies pt . 0 for all t, which is result (ii). Q.E.D. These results can be further refined for linear commuting cost functions.
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FIGURE 1
PROPOSITION 2. For c0i 5 0: (i) u . a ⇒ pt (t, u; u ) , 0 ;t [ [0, 1]: The housing bid price function is monotonic decreasing in t for all worker types u . a. (ii) u , a ⇒ pt (t, u; u ) . 0 ;t [ [0, 1]: The housing bid price function is monotonic increasing in t for all worker types u , a. (iii) u 5 a ⇒ pt (t, u; u ) 5 0 ;t [ [0, 1]: The housing bid price function is horizontal for worker type u 5 a. Proof. Under the assumption of linear commuting costs ett 5 0 so that et (t; a) 5 0 for all t by (9). Thus etu . 0 implies et (t; u ) v 0 for u v a. Relation (8) therefore implies pt b 0 for u v a, which is the proposition. Q.E.D. Consider the special case with identical linear commuting costs for each job site. Applying Proposition 2, a 5 As in this case. Workers most likely to end up with jobs at t 5 0 (that is, worker types u . As) have negatively sloped bid price functions. Workers most likely to end up with jobs at t 5 1 (that is, worker types u , As) have positively sloped bid price functions. Workers with the greatest job site instability (type u 5 As) have horizontal bid price functions. Figure 1 illustrates the case in which there are three
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worker types, u . As, u 5 As, and u , As. In equilibrium, the workers with the greatest probability of working at a single job site will reside closest to that site, [0, t1] for u . As and [t2 , 1] for u , As, while workers with the greatest instability gravitate toward median locations [t1 , t2] between the alternative job sites. Although suggestive, Propositions 1 and 2 and Fig. 1 do not adequately reveal specific equilibrium residential land use patterns among populations comprising many u types. To study this question we need to examine how bid prices vary across worker types. Differentiate (7) with respect to u to find hp pu pt 1 hptu 5 2etu .
(14)
Differentiate (6) to get pu 5 2eu /h so that substituting with pt 5 2et /h from (7) into (14) and rearranging yields ptu 5 2etu /h 2 et eu hp /h3.
(15)
From (13) etu . 0 so that the first term in (15) is negative. From the substitution theorem in the appendix hp , 0 so that the second term takes the sign of et eu . Under our cost assumptions eu (0; u ) , 0 and eu (1; u ) . 0 so that etu . 0 ensures that there exists a unique t0 such that eu b 0 as t b t0 . For u . a, et . 0 throughout and et eu , 0 over t , t0 . In this case the second term in (15) reinforces the first term. For t . t0 , however, the et eu term is positive and the second term in (15) offsets the first term, leaving ptu ambiguous. A similar conclusion follows for worker types u , b. We can be more definite about u effects on bid prices when commuting costs are identical linear functions of travel distance. With c1 5 dt and c2 5 d (1 2 t) we have et eu 5 d 2(1 2 2u )(1 2 2t).
(16)
In symmetric equilibrium, worker types u . As will never occupy residential locations t . As. Therefore, we can reasonably restrict our attention to t [ [0, As) when examining bid price properties for worker types u . As. For u . As and t , As, (16) is negative. Thus the second term in (15) is negative and reinforces the first term. The bid-price function is more steeply (negatively) inclined for higher u workers than for lower u workers residing in this region of the city. Similar analysis applies to the bid prices of worker types u , As. These workers will never occupy residential sites t , As in symmetric equilibrium, so we can concentrate on the properties of their bid prices for t [ (As, 1]. For u , As and t . As, (16) is again negative. The second term in (15) is
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negative for worker types u , As, again reinforcing the first term in (15) and leading to ptu , 0 unambiguously. Therefore, because pt . 0 for worker types u , As, a greater probability of employment at site t 5 0 (that is, a lower probability of employment at site t 5 1) flattens the upward-sloped bid-price function. In summary, we have the following result. PROPOSITION 3. Assuming identical linear commuting costs for both potential job sites implies ptu , 0 for u + As: Workers with greater employment stability (in the sense that u is higher for u . As types and (1 2 u ) is higher for u , As types) have relatively steeper bid price functions than their lessstably employed counterparts. Proposition 3 specifies the properties of the bid price functions of the different types of workers. These bid price functions are constructed for comparison purposes and so assume the same level of expected utility across worker types. In the land market equilibrium, however, the expected utility levels might not be identical across worker types. Because a change in the expected utility for a single worker type generally affects the slope of the bid-price function, the equilibrium price function deserves careful attention at this point. In general equilibrium, the intersections of the bid-price functions of the different types of workers delineates the boundaries between the regions in which each group locates. At such intersections, the bid prices of the two groups are equal and the group with the steeper bid price obtains the location closer the relevant employment center. The next result establishes the relationship between worker types and the relative bid-price slopes at such intersections in equilibrium, a key result needed to explain the spatial sorting pattern of the various types of workers in the urban area. PROPOSITION 4. Let ˜ti j be a location at which the bid price functions for worker types ui and uj intersect: p(t˜i j , ui ; ui ) 5 p(t˜i j , uj ; uj ) where ui and uj are the equilibrium expected utility levels for the two types of workers, respectively. Let v(u ) denote the (Marshallian) income elasticity of land demand for worker type u at ˜t in equilibrium. Assume that v(u )/[m 2 e(t˜, u )] , 2/ d. Then ˜ti j is unique when commuting costs are linear and identical for both potential job sites. In addition, [ pt (t˜i j , ui ; ui ) 2 pt (t˜i j , uj ; uj )][ui 2 uj ] , 0.
(17)
That is, at the intersection of the bid price functions, ˜ti j , the workers with greater employment stability (in the sense that u is higher for u . As types and (1 2 u ) is higher for u , As types) have relatively steeper bid price functions than their less-stably employed counterparts.
JOB SITE UNCERTAINTY
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Proof. The proposition makes two claims: first, that the intersection of two bid prices occurs at only one location, ˜ti j , and second, that the relative slopes of the bid price functions at the intersection follows the pattern established in Proposition 4. To establish both claims it is sufficient to show that the slope of the bid price function for higher u type workers is shallower than for lower u type workers at ˜t, hd[ pt (t˜, u(u ); u )]/du jdp50 , 0,
(18)
where u(u) is the expected utility of worker type u in land market equilibrium. The above result is calculated for dp 5 0 because the bid price functions are equal at ˜t by the definition of the intersection of bid prices. Implicitly differentiate (6) at ˜t with the condition dp 5 0 to obtain du/du 5 2eu /eu .
(19)
Implicitly differentiate (7) at ˜t with the condition dp 5 0 to obtain (dpt /du )dp50 5 2etu /h 1 et eu hu (du/du )/h2.
(20)
Substitute (19) into (20) and simplify to get (dpt /du )dp50 5 2etu /h 2 et eu hu /h2eu .
(21)
Multiply the last term by u/u and e/e and rearrange to show hu /heu 5 (huu/h)(e/euu)/e. However, (huu/h)(e/euu) 5 v while e 5 m 2 e from (6) so that (21) becomes (dpt /du )dp50 5 2etu /h 2 et euv/(m 2 e)h, or, rearranging, (dpt /du )dp50 5 2[etu /et eu 1 h /(m 2 e)]et eu /h.
(22)
Given uetu /et euu $ 2/ d for all t and u, it follows that v/(m 2 e) , 2/ d ensures that the first term within the brackets in (22) dominates. From (13) and (16), etu . 0 and et eu , 0 so that (22) is negative, which establishes (18). Q.E.D. According to this result, workers whose job sites are more stable reside closest to their highly probable job sites while workers with least stable job sites reside in median locations between their potential job sites. This
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FIGURE 2
is illustrated in Fig. 2 for the case where there are five worker types with u1 . u2 ? ? ? . u5 and u3 5 As. The resultant equilibrium housing price configuration is U-shaped, as depicted by the upper envelope of bid price functions in Fig. 2. The housing price is shallower than under job site certainty, as can be seen from the diagram by comparing the equilibrium price function (abcdef ) with that for the case in which the only two worker types are u 5 1 and u 5 0 (agf ). In addition, the uncertainty model generates flat portions in the price function around the median location— region [t2 , t3]—something not found in certainty models of multiple CBD cities. The bid price function is convex in commuting distance under linear commuting costs in certainty land use models. A similar result holds under job site uncertainty and risk neutrality. PROPOSITION 5. ett 5 0 ⇒ ptt . 0 ;u ? a: The bid-price function (3) is convex for linear commuting costs under risk neutrality. Proof. Differentiate (7) with respect to t to obtain hp p 2t 1 ptt 5 2ett .
(23)
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By (B.5) hp , 0 under job site uncertainty, so that ett 5 0 implies ptt . 0. Q.E.D. Finally, consider the housing consumption patterns across the urban area. Use h( p(t, u; u ), u) and differentiate to obtain dh/dt 5 hp pt . Once again, hp , 0 implies dh/dt v 0 as pt b 0
(24)
and the housing consumption gradient is monotonic for all worker types except u 5 As when commuting costs are linear and identical. Therefore, the cities depicted in Figs. 1 and 2 would exhibit equilibrium housing consumption patterns that differ significantly from those found under certainty; for example, the flat housing consumption profile observed over [t1 , t2] in Fig. 1 contrasts with the inverted V-shaped housing pattern for the same interval under certainty.
IV. THE EFFECTS OF RISK AVERSION
One important feature of decision-making under uncertainty has been neglected thus far: risk aversion. The expected commuting cost model, by assuming risk neutrality, basically assumes that workers ignore higher order moments of the distribution of possible commuting cost outcomes at each residential location. This section formally introduces risk aversion into the model to allow for these higher-moment effects. Let s (t) denote the variance of « at t,
s (t) 5 u [c1(t) 2 e(t; u )]2 1 (1 2 u )[c2(1 2 t) 2 e(t; u )]2. The spatial variation in this variance is
st 5 2u(1 2 u )(c91 1 c92)(c1 2 c2).
(25)
We begin with the risk-neutral expected utility function and then introduce risk aversion to examine how the introduction of risk aversion alters previous conclusions. Letting r parameterize risk aversion in the utility function, we use the general form E[u(h, x)] 5 E[ f (h) 1 cy 1 rf( y)] 5 f (h) 1 cx 1 rE[f( y)],
(26)
where f9 . 0 and f0 , 0. Under risk neutrality r 5 0, in which case the expected utility is linear in planned nonhousing consumption x:
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E[ f (h) 1 cy] 5 f (h) 1 cx. Consider an increase in r that increases the concavity of utility function in y while not changing total expected utility or the marginal expected utility of either h or x. Using (26) and (2) this introduction of risk aversion requires, when evaluated at hh, xj,5 dc/dr 5 2E[f(x 1 «)]/x.
(27)
To derive needed results, define the Hicksian housing and planned nonhousing demands under uncertainty, modifying the risk neutral problem (4) for risk aversion: hh( p, u; r, s), x( p, u; r, s)j ; argminh ph 1 x s.t.
f (h) 1 cx 1 rE[f(x 1 «)] $ uj.
(28)
The parameter r represents the index of risk aversion in the utility function, where c changes with r to satisfy (27) evaluated at x( p, u; r, s). Let h be the multiplier for the constrained optimization problem in (28); the Kuhn–Tucker conditions are necessary and sufficient for the solution (28), and reduce to6 p 2 hf 9 5 0
(29)
1 2 h(c 1 rE[f9]) 5 0
(30)
u 2 f (h) 2 cx 2 rE[f] 5 0.
(31)
Totally differentiating the system with respect to r using (27) and solving yields hr 5 2h(E[f]/x 2 E[f9])(c 1 rE[f9]) f 9/uJu . 0,
(32)
where the concavity of the expected utility function ensures that the Jacobian determinant of (28)–(30), uJu, is negative and that (E[f]/x 2 E[f9]) . 0. Now define the expected expenditure function V( p, u; r, s) ; ph( p, u; r, s) 1 x( p, u; r, s).
(33)
The index of absolute risk aversion is R 5 2uyy /uy 5 2rf0/ f9 so that dR/dr . 0: Increases in r increase the index of absolute risk aversion at every given hh, yj. 6 Use the nonsatiation properties of the expected utility to reduce the complementary slackness conditions to (31). 5
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The Appendix shows that Shephard’s lemma holds for expected expenditure functions under risk aversion, Vp 5 h. It also shows that expected expenditures do not change with a pure increase in risk aversion following (27) and that expected expenditures rise with greater «-riskiness: Vr 5 0
(34)
Vs 5 2rE[f9(x( p, u; r, s) 1 «)«] . 0.
(35)
Applying Young’s theorem to (34) and (35) reveals Vsr 5 0.
(36)
Now address the effects of risk aversion on the housing price. The bid price function p(t, u; u, r) under uncertainty is the implicit solution to the condition V( p(t, u; u, r), u; r, s (t)) ; m 2 e(t; u).
(37)
Differentiate (37) and use Shephard’s lemma to find the price slope pt 5 2et /h 2 Vsst /h.
(38)
The price function capitalizes both the spatial variation in expected commuting costs as well as the spatial variation in commuting cost variance. The first term captures the effects of spatial variation in expected commuting costs on housing price. The second term captures the effect of the spatial variation in commuting cost risk. Because Vs . 0 from (35), the second term in (38) takes the sign opposite that of st . How does risk aversion modify the bid price monotonicity results found under risk neutrality? Such comparisons are difficult to make in general. Recall that there exists a t0 such that c1(t0) 5 c2(1 2 t0). Clearly, the monotonicity of c1 2 c2 and (25) imply that st v 0 for t b t0 . It follows that for worker types u . a (for whom et . 0), the commuting cost risk effect term in (38) reinforces the expected commuting cost effect only for t . t0 . For t , t0 the risk effect offsets the expected commuting cost effect on housing price, leaving (38) ambiguous. A similar conclusion applies to worker types u , b; the bid price functions are unambiguously monotonic for t , t0 and ambiguous for t . t0 . Nonetheless, the effect of risk aversion on bid prices is rather dramatic for those workers for whom ett 5 0. PROPOSITION 6. For c0i 5 0 and t0 defined by c1(t0) 5 c2(1 2 t0), under risk aversion, u 5 a ⇒ pt (t, u; u, r) v 0 as t b t0 , ;t [ [0, 1]: For worker
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FIGURE 3
type u 5 a the housing bid price function is non-monotonic, increasing to a maximum at t0 and decreasing thereafter. Proof. Recall that for worker type u 5 a, et 5 0. (38) then implies pt 5 2Vs st /h v 0 as st b 0. By (23), st b 0 as t b t0 so that pt v 0 as Q.E.D. t b t0 , which is the result to be shown. Risk aversion fundamentally alters the bid price, hence the equilibrium housing price function, in the two-center city. Take the simplest case of identical linear commuting cost functions as an illustration. Worker type u 5 1 will have a monotonically decreasing bid-price function while worker type u 5 0 will have a monotonically increasing bid-price function. Worker type u 5 As, though, will have the inverted U-shaped bid-price function, as illustrated in Fig. 3, rather than the horizontal bid price under risk neutrality. Risk aversion increases the demand for median residential locations by the workers with the greatest employment uncertainty, so much so that the resultant equilibrium price function exhibits a peak at the median distance between the alternative job sites in the city. This result is intuitive once we recognize the implications of (38). Even though the expected commuting cost exhibits zero spatial variation for worker type u 5 As, the commuting
JOB SITE UNCERTAINTY
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cost variance is minimized at the median residential location; the commuting cost variance increases the farther away from the median location the worker resides, which by (38), lowers his bid price for those increasingly less desirable residential sites. The preceding comparison of risk-neutral and risk-averse cases is informative. In any event, we now more formally consider how the degree of risk aversion affects bid price functions for more general commuting cost functions. PROPOSITION 7. ptr v 0 as pt b 0: Stronger risk aversion flattens the bid price functions for all worker types u [ (0, 1). Proof. Differentiate (38) with respect to r, imposing (27) and using (36) to obtain ptr 5 2pt hr /h. From (32), hr . 0 so that the proposition follows immediately.
(39) Q.E.D.
Note that (38) implies that no worker type can have a horizontal bid price (that is, pt 5 0 only at isolated points or intervals). Thus, Proposition 8 applies to all worker types facing job site uncertainty, including type a when commuting costs are linear. The proposition reveals the pattern of spatial sorting of workers within each u group type by their strength of risk aversion. The more risk averse workers reside at locations with relatively lower commuting cost variances than their less risk-averse counterparts. This much is consistent with the uncertain transportation cost models of Papageorgiou and Pines (1988), Turnbull et al. (1991), and Turnbull (1995). They find that transportation cost uncertainty tends to increase the demand for residential sites with lower commuting cost (net income) variance relative to sites with greater variance. All of those models are single CBD models, however, so that the lower commuting cost variance sites are those closest to the single CBD. In the multiple CBD model here, in contrast, the lowest commuting cost variance sites are those nearest the location t0 between the two potential job sites. Therefore, risk aversion tends to increase all workers’ demands for median locations over those nearest each of the two CBDs, which is reflected as flatter bid price functions relative to the risk-neutral case. (Of course, the u 5 1 and u 5 0 workers’ bid prices are unaltered by risk aversion.) Finally, even though Proposition 7 holds for all commuting cost specifications, the identical linear cost case provides a sufficiently interesting illustration. Propositions 6 and 7 reveal an equilibrium housing price and land use pattern very different from those found under certainty or risk neutrality.
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FIGURE 4
In Fig. 4 the positive and negative sloped portions of the equilibrium price function over [0, t2] and [t3 , 1] are flatter than under risk neutrality, while the u 5 As bid price creates a trimodal equilibrium price surface not found in the other models.
V. CONCLUSION
This paper developed a model of the spatial housing market under the assumption that some workers face job site uncertainty. The risk neutral model shows that under job site uncertainty workers spatially sort themselves according to their job site stability. Job site stability increases the demand for residential locations nearest the expected job sites, while instability increases the demand for median residential locations between alternative job sites. Risk aversion introduces an additional worker response to the spatial variation in commuting cost risk across the urban area, which offsets the effect of job site stability under risk neutrality. The theory with risk aversion therefore predicts less spatial sorting by either realized commuting costs or expected commuting costs than the certainty or risk-
17
JOB SITE UNCERTAINTY
neutral models predict. The risk neutrality model presented here predicts more commuting than the certainty model. In addition, though, risk aversion provides another rationale for why the simple certainty model predicts significantly less ex post commuting than observed empirically.7 The effects of job site uncertainty on the equilibrium housing price surface depends upon whether workers are risk neutral or risk averse. House bid prices capitalize the spatial variations in both expected travel cost and travel cost variance, leading to an equilibrium housing price surface more irregular than found under certainty. Under risk neutrality, job site uncertainty flattens the price function throughout the region between CBDs and introduces a completely horizontal housing price surface in the median region between potential job sites when commuting costs are linear. Under risk aversion, however, there is no extensive horizontal portion in the equilibrium housing price function. The housing price surface is flatter than under certainty, but risk averse behavior also generates price peaks at residential locations between employment centers, a feature not found under certainty or even uncertainty with risk neutrality.
APPENDIX
A. Derivation of the Dual Expenditure Minimization Problem (16) for the Risk-Neutral Bid-Price Problem (3) We begin by exploiting the fact that the risk-neutral problem can be cast as problem (3) in the paper. Following Wheaton (1977), the bid-price problem can be recast as a dual utility maximization problem, which can then be used to derive the expenditure function version used in the paper. Define the Marshallian (ordinary) demands for housing and spending on other goods for given t as hh( p, m 2 e), x( p, m 2 e)j ; argmaxhu(h, x)
s.t.
m 2 e 5 ph 1 xj. (A.1)
Use (A.1) to define the indirect utility function conditional on t (via p and e) as V( p, m 2 e) ; f (h( p, m 2 e)) 1 g(h( p, m 2 e))x( p, m 2 e).
(A.2)
For utility opportunity cost u0 in the risk-neutral problem (3), use (A.2) 7
See Hamilton (1982), White (1988). Thurston and Yezer (1991), Small and Song (1992), and Crane (1996) for the treatment of several aspects of the so-called ‘‘wasteful commuting’’ debate.
18
GEOFFREY K. TURNBULL
to show that the bid-price function p(t, u0; u ) for type u workers is implicitly defined by V( p(t, u; u ), m 2 e(t; u )) ; u
(A.3)
for all t [ [0, 1]. Condition (A.3) is the utility maximization dual to problem (3) under risk neutrality. It turns out to be easier to deal with the expenditure function. To convert (A.3) to an expenditure function equivalent form, note that V is monotonically increasing in net money income m 2 e. Thus, the indirect utility function can be inverted so that (A.3) yields V21( p(t, u; u ), u) ; m 2 e(t; u ),
(A.4)
or, using V21( p(t, u; u ), u) 5 e( p(t, u; u ), u), where e( p, u) is the expected expenditure function. Substituting into (A.4) yields e( p(t, u; u ), u) ; m 2 e(t; u ),
(A.5)
which is (6) in the paper. B. The Substitution Theorem for Housing under Uncertainty The substitution theorem asserts a negative own-price effect on the Hicksian (compensated) demand for housing, hp( p, u; s) , 0. To prove this result under uncertainty, define the general expected utility function E[u(h, x 1 «)], where uh . 0, uhh , 0, uy . 0, and uyy # 0 allows for risk neutrality or risk aversion. The Hicksian demand vector under uncertainty is hh( p, u; s), x( p, u; s)j ; argminh ph 1 x
s.t.
E[u(h, x 1 «)] $ uj. (B.1)
Let l be the Kuhn–Tucker multiplier. The (weak) concavity of the expected utility in planned consumption ensures that the Kuhn–Tucker conditions, which are necessary and sufficient for (B.1) under these assumptions, simplify to p 2 lE[uh] 5 0
(B.2)
1 2 lE[uy ] 5 0
(B.3)
u 2 E[u(h, x 1 «)] 5 0.
(B.4)
Totally differentiate (B.2)–(B.4) and solve the resultant system for
19
JOB SITE UNCERTAINTY
hp 5 E[uy ]2 /uJu , 0,
(B.5)
where J is the Jacobian matrix of (B.2)–(B.4) and the sign of (B.5) follows from uJu , 0 due to the (weak) concavity of E[u(h, x 1 «)] in hh, xj. Equation (B.5) is the substitution theorem for housing demand. C. Properties of the Expected Expenditure function under Uncertainty Define the expected expenditure function as V( p, u; r, s) ; ph( p, u; r, s) 1 x( p, u; r, s),
(C.1)
where h and x satisfy (29)–(31). Totally differentiate (C.1) with respect to the risk aversion parameter r: Vr 5 phr 1 xr Equation (29) implies h f 9 5 p and (30) implies h(c 1 rE[f9]) 5 1, so that substituting into the above yields Vr 5 hh f 9hr 1 (c 1 rE[f9])xrj.
(C.2)
Differentiate (31) with respect to r to obtain f 9hr 1 (c 1 rE[f9])xr 5 2hx(dc/dr) 1 E[f]j.
(C.3)
Applying (27), however, leaves hx(dc/dr) 1 E[f]j 5 0, so that the left and side of (C.3) is zero and Vr 5 0 from (C.2). This is property (34) in the paper. To find the effects of a mean preserving increase in the variance of « on expected expenditures, replace s« for « in (29)–(31), with the parameter s serving as an index of uncertainty. The variance VAR(s«) 5 s 2VAR(«), so that dVAR(s«)/ds . 0. Given that E[«] 5 0 by construction, an increase in s therefore represents a mean preserving increase in «-risk. Differentiate (C.1) with respect to s and evaluate the result at s 5 1 to find Vs 5 phr 1 xr .
(C.4)
Equation (29) implies h f 9 5 p and (30) implies h(c 1 rE[f9]) 5 1, so that substituting into the above yields Vs 5 hh f 9hs 1 (c 1 rE[f9])xsj.
(C.5)
Differentiate (31) with respect to s to and evaluate at s 5 1 to obtain
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GEOFFREY K. TURNBULL
f 9hs 1 (c 1 rE[f9])xs 5 2rE[f9(x( p, u; r, s) 1 «)«].
(C.6)
Substitute (C.6) into (C.5) to obtain Vs 5 2rE[f9(x( p, u; r, s) 1 «)«] 5 2rCOV[f9, «],
(C.6)
where COV denotes covariance and the second equality follows from E[«] 5 0. By the properties of similar–dissimilar orderings, the covariance takes the sign of f0, which is negative under risk aversion. Thus, (C.6) is positive, which is (35).
REFERENCES Crane, R. (1996). ‘‘The Influence of Uncertain Job Location on Urban Form and the Journey to Work,’’ J. Urban Econ. 39, 342–356. Hamilton, B. (1982). ‘‘Wasteful Commuting,’’ J. Polit. Econ. 90, 1035–1053. Muth, R. F. (1969). Cities and Housing. Chicago: Univ. of Chicago Press. Papageorgiou, Y. Y., and Pines, D. (1988). ‘‘The Impact of Transportation Cost Uncertainty on Urban Structure,’’ Reg. Sci. Urban Econ. 18, 247–260. Small, K. A., and Song, S. (1992). ‘‘ ‘Wasteful’ Commuting: A Resolution,’’ J. Polit. Econ. 100, 888–898. Thurston, L., and Yezer, A. M. J. (1991). ‘‘Testing the Monocentric Urban Model: Evidence Based on Wasteful Commuting,’’ AREUEA J. 19, 41–51. Turnbull, G. K. (1993). ‘‘Nonparametric Location Theory,’’ J. Real Estate Finance Econ. 7, 169–184. Turnbull, G. K. (1995). Urban Consumer Theory. Washington, DC: Urban Inst. Press. Turnbull, G. K., Glascock, J. L., and Sirmans, C. F. (1991). ‘‘Uncertain Income and Housing Price and Location Choice,’’ J. Reg. Sci. 31, 417–433. Wheaton, W. C. (1977). ‘‘A Bid Rent Approach to Housing Demand,’’ J. Urban Econ. 4, 200–217. White, M. J. (1988). ‘‘Urban Commuting Journeys are Not ‘Wasteful’,’’ J. Polit. Econ. 96, 1097– 1110. Wieand, K. F. (1987). ‘‘An Extension of the Monocentric Urban Spatial Equilibrium Model to a Multicenter Setting: The Case of the Two-Center City,’’ J. Urban Econ. 21, 259–271. Yinger, J. (1993). ‘‘City and Suburb: Urban Models with More than One Employment Center,’’ J. Urban Econ. 31, 181–205.
7, 1–20 (1998)
HE980222
Housing Prices and Residential Land Use under Job Site Uncertainty Geoffrey K. Turnbull Department of Economics, Louisiana State University, Baton Rouge, Louisiana 70803-6306 Received March 21, 1997
This paper examines the effects of job site uncertainty in a multiple center city. It shows that workers sort themselves spatially according to their job site stability. Risk averse workers also respond to the spatial variation in commuting cost risk which offsets the tendency to sort by job site stability. The theory predicts less spatial sorting by commuting costs than the certainty or risk-neutral models predict. House bid prices capitalize the spatial variations in expected travel cost and its variance, leading a flatter price surface than under certainty. In addition, risk aversion generates price peaks between employment centers. 1998 Academic Press
I. INTRODUCTION
Many urban areas have more than one concentrated business district (CBD).1 Motivated by this observation, White (1976), Wieand (1987), Yinger (1992), and Turnbull (1993) extend the canonical monocentric urban land use theory to investigate the spatial complications introduced by multiple CBDs. Like much of the monocentric theory, individuals know with certainty the values of key variables before committing to their consumption and location decisions in these models.2 One aspect ignored by these papers, however, is that when jobs are distributed across several CBDs there is the likelihood that workers may not know with certainty in which of the CBDs their future employment will be. Employers may move from one site to another or an individual worker may be laid off and take another job in a different CBD. Housing and location are long lived investments that are costly to adjust to short term changes in family circumstances and, in the context of the multiple CBD city, the worker’s job site is one such circumstance that might change. This suggests that at least some individuals make their housing and location decisions before knowing with certainty 1 This paper denotes all concentrations of business activity, hence employment, as CBDs, whether the traditional central business districts of monocentric theory or suburban beltlines. 2 See Turnbull (1995) for a complete treatment of monocentric uncertainty models.
1 1051-1377/98 $25.00 Copyright 1998 by Academic Press All rights of reproduction in any form reserved.
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GEOFFREY K. TURNBULL
where their job site will be throughout the housing consumption period. The question of interest is: how does this job site uncertainty affect the urban housing market? Crane (1996) presents a model of residential location choice in a long narrow two-CBD city under job site uncertainty. In his model workers are initially assigned a job in one of the CBDs, after which they make their housing and location decisions, committing themselves to their housinglocation consumption bundle and the housing bid price. Once the consumption decisions and housing price commitments are made, the worker finds out if his or her job site is going to change to the other CBD in the urban area. The model is not based in neoclassical uncertainty theory. Instead it represents an attempt to extend the certainty urban land use model by assuming that workers base their housing bid prices on expected commuting costs instead of the ex post costs as in the certainty models. Unfortunately, it can be shown that no bid rent exists in Crane’s model because it does not take into account the variability in the worker’s ex post consumption that arises when his job site changes.3 This paper contributes to urban land use theory under uncertainty in general and to the multiple CBD city theory in particular, applying neoclassical uncertainty theory to evaluate the impact of job site uncertainty on housing prices and residential land use patterns. Section II explains the model structure and derives a bid-price function under risk neutrality. Section III then uses the model to analyze the implications of job site uncertainty on bid rents and land use patterns. This model of job site uncertainty provides interesting new results even though it assumes risk neutrality. However, it also raises an additional question: Given the effects of job site uncertainty on prices and land use under risk neutrality, how do risk averse workers alter the model predictions? Section IV examines this question, extending the model to evaluate the effects of risk aversion on housing bid prices and location demand. Section V presents the concluding remarks.
II. THE UNCERTAINTY MODEL UNDER RISK NEUTRALITY
Consider a linear city extending over the unit interval [0, 1] with an employment center at each end. Let t denote the distance of a residence location from the first business district. Thus, all jobs are located in t 5 0 and t 5 1 and the commute to a job in the first center requires travelling distance t while the commute to a job in the second center requires travelling distance 1 2 t. The commuting cost from location t to job site 1 is therefore 3
A demonstration of this proposition is available from the author upon request.
JOB SITE UNCERTAINTY
3
c1(t) while the commuting cost from location t to job site 2 is c2(1 2 t), with greater distance requiring greater commuting costs in each case: c9i . 0. Further, c1(0) , c2(0) and c2(1) , c1(1), that is, it always costs more to commute from one job site to the other than it does to not commute at all. Workers do not know with certainty where their jobs will be when making their housing and location decisions.4 The probability of a given worker’s job site being located in the first center is u. All workers are identical and, for simplicity, all wages are invariant across job sites. Workers are identified by their types u, their probability of being employed in site 1. The expected commuting cost for worker type u is a function of t in general, and is denoted e(t; u ) 5 uc1(t) 1 (1 2 u )c2(1 2 t).
(1)
Consumer utility is given by the well behaved expected utility function E[u(h, y)], where housing consumption is h and the consumption of all other goods, the numeraire, is y. The consumer’s money budget is m. Consumer choice under net income uncertainty implies that either h or y or both must be uncertain ex ante, exhibiting possible ex post variation around the chosen planned values. In general, it turns out that the structure of the housing and location choice (hence the bid price) problem implies that housing consumption is nonstochastic while spending on all other goods, being the residual of the money budget left after paying the bid price for housing (which is decided ex ante) and travel costs, is the stochastic or risky variable in the consumer choice model (Turnbull, 1995, Chap. 4 and 5). The worker chooses the bid price, the quantity of housing and a planned level of spending on other goods, x, before observing where his job site will be. After making his price bid and committing to his housing consumption, he observes his job site. Thus, his or her actual consumption is therefore the planned level of housing h, the required spending on commuting to the job site, and all other spending—what is left over after paying for housing and commuting. For the worker who is ultimately employed at site i, the ex post spending on nonhousing consumption is y 5 m 2 ci 2 ph. Denoting planned nonhousing spending as x, where x 5 m 2 e 2 ph, ex post or realized nonhousing spending is y 5 x 1 «,
(2)
where « [ he(t, u ) 2 c1(t), e(t, u ) 2 c2(1 2 t)j with probabilities hu, 4
Alternatively, the workers select their residence locations given their initial job sites, but they face a probability that their job site will change in the future without being able to instantaneously adjust their residence locations. Adding an initial period of certainty to the model complicates the notation but does not alter the conclusions.
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GEOFFREY K. TURNBULL
(1 2 u )j. Clearly, E[ y] 5 x. Using this notation, the bid price function under uncertainty is defined as the following: p(t, u; u ) ; maxhh,xjh[m 2 x 2 e(t; u )]/h
E[u(h, x 1 «)] $ uj.
s.t.
Under risk neutrality the expected utility can be expressed as a function of planned housing and nonhousing consumption, h and x, alone. The bidprice function under uncertainty is defined as p(t, u; u ) ; maxhh,xjh[m 2 x 2 e(t; u )]/h
s.t.
f (h) 1 g(h)x $ uj. (3)
To begin, recast (3) into a dual formulation that is easier to work with. This task is simplified by the fact that E[u(h, x 1 «)] ; u(h, x) under risk neutrality, which implies that (3) can be cast as a certainty problem in terms of expected values (of travel costs and nonhousing consumption). Define the Hicksian (compensated) demands for housing and planned nonhousing consumption as the expected expenditure minimizing combinations that yield expected utility u: hh( p, u), x( p, u)j ; argminh ph 1 x
s.t.
f (h) 1 g(h)x $ uj. (4)
Use (4) to define the consumer expected expenditure function conditional on t (via p) as e( p, u) ; ph( p, u) 1 x(p, u).
(5)
As shown in the Appendix, this function has many of the same properties as its counterpart under certainty, including Shephard’s lemma ep 5 h( p, u). The Appendix also demonstrates that the bid-price function under risk neutrality (3) is implicitly defined by the dual relationship e( p(t, u; u ), u) ; m 2 e(t; u )
(6)
over all t [ [0, 1]. This equation, with (1), is the key to deriving the model implications.
III. BID PRICES AND LAND USE
Differentiating (6) with respect to t and using Shephard’s lemma, we obtain the risk neutral uncertainty version of Muth’s (1969) equation for household location equilibrium,
JOB SITE UNCERTAINTY
h( p, u)pt 5 2et .
5 (7)
This result shows that the slope of the bid price function is determined solely by expected commuting costs, just as the price gradient follows commuting costs in the certainty theory: pt v 0 as et b 0.
(8)
Unlike the certainty theory, however, a monotonic ci does not by itself ensure that the bid price function for worker type u is monotonic when job sites are uncertain. To sort out the possibilities in this regard, define the parameters a and b such that a 5 c92(0)/[c91(0) 1 c92(0)]
(9)
b 5 c91(1)/[c91(1) 1 c92(1)].
(10)
PROPOSITION 1. For c0i $ 0: (i) u . a ⇒ pt (t, u; u ) , 0 ;t [ [0, 1]: The housing bid price function is monotonic decreasing in t for all worker types u . a. (ii) u , b ⇒ pt (t, u; u ) . 0 ;t [ [0, 1]: The housing bid price function is monotonic increasing in t for all worker types u , b. Proof. Differentiate (1) to find et 5 uc91 2 (1 2 u )c92
(11)
ett 5 uc01 1 (1 2 u )c02
(12)
etu 5 c91 1 c92 . 0
(13)
Substitute (9) into (11) to find et (0; a) 5 ac91(0) 2 (1 2 a)c92(0) 5 0 so that (13) ensures et (0; u ) . 0 for all u . a. Under the conditions of the proposition ett $ 0 which in turn implies et (t; u ) . 0 for all t [ [0, 1]. From (8), this further implies pt , 0 for all t, which is result (i). Substitute (10) into (11) to find et(1; b) 5 bc91(1) 2 (1 2 b)c92(1) 5 0 so that (13) ensures et (1; u ) , 0 for all u , b. Under the conditions of the proposition ett $ 0 which in turn implies et (t; u ) , 0 for all t [ [0, 1]. From (8), this further implies pt . 0 for all t, which is result (ii). Q.E.D. These results can be further refined for linear commuting cost functions.
6
GEOFFREY K. TURNBULL
FIGURE 1
PROPOSITION 2. For c0i 5 0: (i) u . a ⇒ pt (t, u; u ) , 0 ;t [ [0, 1]: The housing bid price function is monotonic decreasing in t for all worker types u . a. (ii) u , a ⇒ pt (t, u; u ) . 0 ;t [ [0, 1]: The housing bid price function is monotonic increasing in t for all worker types u , a. (iii) u 5 a ⇒ pt (t, u; u ) 5 0 ;t [ [0, 1]: The housing bid price function is horizontal for worker type u 5 a. Proof. Under the assumption of linear commuting costs ett 5 0 so that et (t; a) 5 0 for all t by (9). Thus etu . 0 implies et (t; u ) v 0 for u v a. Relation (8) therefore implies pt b 0 for u v a, which is the proposition. Q.E.D. Consider the special case with identical linear commuting costs for each job site. Applying Proposition 2, a 5 As in this case. Workers most likely to end up with jobs at t 5 0 (that is, worker types u . As) have negatively sloped bid price functions. Workers most likely to end up with jobs at t 5 1 (that is, worker types u , As) have positively sloped bid price functions. Workers with the greatest job site instability (type u 5 As) have horizontal bid price functions. Figure 1 illustrates the case in which there are three
JOB SITE UNCERTAINTY
7
worker types, u . As, u 5 As, and u , As. In equilibrium, the workers with the greatest probability of working at a single job site will reside closest to that site, [0, t1] for u . As and [t2 , 1] for u , As, while workers with the greatest instability gravitate toward median locations [t1 , t2] between the alternative job sites. Although suggestive, Propositions 1 and 2 and Fig. 1 do not adequately reveal specific equilibrium residential land use patterns among populations comprising many u types. To study this question we need to examine how bid prices vary across worker types. Differentiate (7) with respect to u to find hp pu pt 1 hptu 5 2etu .
(14)
Differentiate (6) to get pu 5 2eu /h so that substituting with pt 5 2et /h from (7) into (14) and rearranging yields ptu 5 2etu /h 2 et eu hp /h3.
(15)
From (13) etu . 0 so that the first term in (15) is negative. From the substitution theorem in the appendix hp , 0 so that the second term takes the sign of et eu . Under our cost assumptions eu (0; u ) , 0 and eu (1; u ) . 0 so that etu . 0 ensures that there exists a unique t0 such that eu b 0 as t b t0 . For u . a, et . 0 throughout and et eu , 0 over t , t0 . In this case the second term in (15) reinforces the first term. For t . t0 , however, the et eu term is positive and the second term in (15) offsets the first term, leaving ptu ambiguous. A similar conclusion follows for worker types u , b. We can be more definite about u effects on bid prices when commuting costs are identical linear functions of travel distance. With c1 5 dt and c2 5 d (1 2 t) we have et eu 5 d 2(1 2 2u )(1 2 2t).
(16)
In symmetric equilibrium, worker types u . As will never occupy residential locations t . As. Therefore, we can reasonably restrict our attention to t [ [0, As) when examining bid price properties for worker types u . As. For u . As and t , As, (16) is negative. Thus the second term in (15) is negative and reinforces the first term. The bid-price function is more steeply (negatively) inclined for higher u workers than for lower u workers residing in this region of the city. Similar analysis applies to the bid prices of worker types u , As. These workers will never occupy residential sites t , As in symmetric equilibrium, so we can concentrate on the properties of their bid prices for t [ (As, 1]. For u , As and t . As, (16) is again negative. The second term in (15) is
8
GEOFFREY K. TURNBULL
negative for worker types u , As, again reinforcing the first term in (15) and leading to ptu , 0 unambiguously. Therefore, because pt . 0 for worker types u , As, a greater probability of employment at site t 5 0 (that is, a lower probability of employment at site t 5 1) flattens the upward-sloped bid-price function. In summary, we have the following result. PROPOSITION 3. Assuming identical linear commuting costs for both potential job sites implies ptu , 0 for u + As: Workers with greater employment stability (in the sense that u is higher for u . As types and (1 2 u ) is higher for u , As types) have relatively steeper bid price functions than their lessstably employed counterparts. Proposition 3 specifies the properties of the bid price functions of the different types of workers. These bid price functions are constructed for comparison purposes and so assume the same level of expected utility across worker types. In the land market equilibrium, however, the expected utility levels might not be identical across worker types. Because a change in the expected utility for a single worker type generally affects the slope of the bid-price function, the equilibrium price function deserves careful attention at this point. In general equilibrium, the intersections of the bid-price functions of the different types of workers delineates the boundaries between the regions in which each group locates. At such intersections, the bid prices of the two groups are equal and the group with the steeper bid price obtains the location closer the relevant employment center. The next result establishes the relationship between worker types and the relative bid-price slopes at such intersections in equilibrium, a key result needed to explain the spatial sorting pattern of the various types of workers in the urban area. PROPOSITION 4. Let ˜ti j be a location at which the bid price functions for worker types ui and uj intersect: p(t˜i j , ui ; ui ) 5 p(t˜i j , uj ; uj ) where ui and uj are the equilibrium expected utility levels for the two types of workers, respectively. Let v(u ) denote the (Marshallian) income elasticity of land demand for worker type u at ˜t in equilibrium. Assume that v(u )/[m 2 e(t˜, u )] , 2/ d. Then ˜ti j is unique when commuting costs are linear and identical for both potential job sites. In addition, [ pt (t˜i j , ui ; ui ) 2 pt (t˜i j , uj ; uj )][ui 2 uj ] , 0.
(17)
That is, at the intersection of the bid price functions, ˜ti j , the workers with greater employment stability (in the sense that u is higher for u . As types and (1 2 u ) is higher for u , As types) have relatively steeper bid price functions than their less-stably employed counterparts.
JOB SITE UNCERTAINTY
9
Proof. The proposition makes two claims: first, that the intersection of two bid prices occurs at only one location, ˜ti j , and second, that the relative slopes of the bid price functions at the intersection follows the pattern established in Proposition 4. To establish both claims it is sufficient to show that the slope of the bid price function for higher u type workers is shallower than for lower u type workers at ˜t, hd[ pt (t˜, u(u ); u )]/du jdp50 , 0,
(18)
where u(u) is the expected utility of worker type u in land market equilibrium. The above result is calculated for dp 5 0 because the bid price functions are equal at ˜t by the definition of the intersection of bid prices. Implicitly differentiate (6) at ˜t with the condition dp 5 0 to obtain du/du 5 2eu /eu .
(19)
Implicitly differentiate (7) at ˜t with the condition dp 5 0 to obtain (dpt /du )dp50 5 2etu /h 1 et eu hu (du/du )/h2.
(20)
Substitute (19) into (20) and simplify to get (dpt /du )dp50 5 2etu /h 2 et eu hu /h2eu .
(21)
Multiply the last term by u/u and e/e and rearrange to show hu /heu 5 (huu/h)(e/euu)/e. However, (huu/h)(e/euu) 5 v while e 5 m 2 e from (6) so that (21) becomes (dpt /du )dp50 5 2etu /h 2 et euv/(m 2 e)h, or, rearranging, (dpt /du )dp50 5 2[etu /et eu 1 h /(m 2 e)]et eu /h.
(22)
Given uetu /et euu $ 2/ d for all t and u, it follows that v/(m 2 e) , 2/ d ensures that the first term within the brackets in (22) dominates. From (13) and (16), etu . 0 and et eu , 0 so that (22) is negative, which establishes (18). Q.E.D. According to this result, workers whose job sites are more stable reside closest to their highly probable job sites while workers with least stable job sites reside in median locations between their potential job sites. This
10
GEOFFREY K. TURNBULL
FIGURE 2
is illustrated in Fig. 2 for the case where there are five worker types with u1 . u2 ? ? ? . u5 and u3 5 As. The resultant equilibrium housing price configuration is U-shaped, as depicted by the upper envelope of bid price functions in Fig. 2. The housing price is shallower than under job site certainty, as can be seen from the diagram by comparing the equilibrium price function (abcdef ) with that for the case in which the only two worker types are u 5 1 and u 5 0 (agf ). In addition, the uncertainty model generates flat portions in the price function around the median location— region [t2 , t3]—something not found in certainty models of multiple CBD cities. The bid price function is convex in commuting distance under linear commuting costs in certainty land use models. A similar result holds under job site uncertainty and risk neutrality. PROPOSITION 5. ett 5 0 ⇒ ptt . 0 ;u ? a: The bid-price function (3) is convex for linear commuting costs under risk neutrality. Proof. Differentiate (7) with respect to t to obtain hp p 2t 1 ptt 5 2ett .
(23)
JOB SITE UNCERTAINTY
11
By (B.5) hp , 0 under job site uncertainty, so that ett 5 0 implies ptt . 0. Q.E.D. Finally, consider the housing consumption patterns across the urban area. Use h( p(t, u; u ), u) and differentiate to obtain dh/dt 5 hp pt . Once again, hp , 0 implies dh/dt v 0 as pt b 0
(24)
and the housing consumption gradient is monotonic for all worker types except u 5 As when commuting costs are linear and identical. Therefore, the cities depicted in Figs. 1 and 2 would exhibit equilibrium housing consumption patterns that differ significantly from those found under certainty; for example, the flat housing consumption profile observed over [t1 , t2] in Fig. 1 contrasts with the inverted V-shaped housing pattern for the same interval under certainty.
IV. THE EFFECTS OF RISK AVERSION
One important feature of decision-making under uncertainty has been neglected thus far: risk aversion. The expected commuting cost model, by assuming risk neutrality, basically assumes that workers ignore higher order moments of the distribution of possible commuting cost outcomes at each residential location. This section formally introduces risk aversion into the model to allow for these higher-moment effects. Let s (t) denote the variance of « at t,
s (t) 5 u [c1(t) 2 e(t; u )]2 1 (1 2 u )[c2(1 2 t) 2 e(t; u )]2. The spatial variation in this variance is
st 5 2u(1 2 u )(c91 1 c92)(c1 2 c2).
(25)
We begin with the risk-neutral expected utility function and then introduce risk aversion to examine how the introduction of risk aversion alters previous conclusions. Letting r parameterize risk aversion in the utility function, we use the general form E[u(h, x)] 5 E[ f (h) 1 cy 1 rf( y)] 5 f (h) 1 cx 1 rE[f( y)],
(26)
where f9 . 0 and f0 , 0. Under risk neutrality r 5 0, in which case the expected utility is linear in planned nonhousing consumption x:
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GEOFFREY K. TURNBULL
E[ f (h) 1 cy] 5 f (h) 1 cx. Consider an increase in r that increases the concavity of utility function in y while not changing total expected utility or the marginal expected utility of either h or x. Using (26) and (2) this introduction of risk aversion requires, when evaluated at hh, xj,5 dc/dr 5 2E[f(x 1 «)]/x.
(27)
To derive needed results, define the Hicksian housing and planned nonhousing demands under uncertainty, modifying the risk neutral problem (4) for risk aversion: hh( p, u; r, s), x( p, u; r, s)j ; argminh ph 1 x s.t.
f (h) 1 cx 1 rE[f(x 1 «)] $ uj.
(28)
The parameter r represents the index of risk aversion in the utility function, where c changes with r to satisfy (27) evaluated at x( p, u; r, s). Let h be the multiplier for the constrained optimization problem in (28); the Kuhn–Tucker conditions are necessary and sufficient for the solution (28), and reduce to6 p 2 hf 9 5 0
(29)
1 2 h(c 1 rE[f9]) 5 0
(30)
u 2 f (h) 2 cx 2 rE[f] 5 0.
(31)
Totally differentiating the system with respect to r using (27) and solving yields hr 5 2h(E[f]/x 2 E[f9])(c 1 rE[f9]) f 9/uJu . 0,
(32)
where the concavity of the expected utility function ensures that the Jacobian determinant of (28)–(30), uJu, is negative and that (E[f]/x 2 E[f9]) . 0. Now define the expected expenditure function V( p, u; r, s) ; ph( p, u; r, s) 1 x( p, u; r, s).
(33)
The index of absolute risk aversion is R 5 2uyy /uy 5 2rf0/ f9 so that dR/dr . 0: Increases in r increase the index of absolute risk aversion at every given hh, yj. 6 Use the nonsatiation properties of the expected utility to reduce the complementary slackness conditions to (31). 5
JOB SITE UNCERTAINTY
13
The Appendix shows that Shephard’s lemma holds for expected expenditure functions under risk aversion, Vp 5 h. It also shows that expected expenditures do not change with a pure increase in risk aversion following (27) and that expected expenditures rise with greater «-riskiness: Vr 5 0
(34)
Vs 5 2rE[f9(x( p, u; r, s) 1 «)«] . 0.
(35)
Applying Young’s theorem to (34) and (35) reveals Vsr 5 0.
(36)
Now address the effects of risk aversion on the housing price. The bid price function p(t, u; u, r) under uncertainty is the implicit solution to the condition V( p(t, u; u, r), u; r, s (t)) ; m 2 e(t; u).
(37)
Differentiate (37) and use Shephard’s lemma to find the price slope pt 5 2et /h 2 Vsst /h.
(38)
The price function capitalizes both the spatial variation in expected commuting costs as well as the spatial variation in commuting cost variance. The first term captures the effects of spatial variation in expected commuting costs on housing price. The second term captures the effect of the spatial variation in commuting cost risk. Because Vs . 0 from (35), the second term in (38) takes the sign opposite that of st . How does risk aversion modify the bid price monotonicity results found under risk neutrality? Such comparisons are difficult to make in general. Recall that there exists a t0 such that c1(t0) 5 c2(1 2 t0). Clearly, the monotonicity of c1 2 c2 and (25) imply that st v 0 for t b t0 . It follows that for worker types u . a (for whom et . 0), the commuting cost risk effect term in (38) reinforces the expected commuting cost effect only for t . t0 . For t , t0 the risk effect offsets the expected commuting cost effect on housing price, leaving (38) ambiguous. A similar conclusion applies to worker types u , b; the bid price functions are unambiguously monotonic for t , t0 and ambiguous for t . t0 . Nonetheless, the effect of risk aversion on bid prices is rather dramatic for those workers for whom ett 5 0. PROPOSITION 6. For c0i 5 0 and t0 defined by c1(t0) 5 c2(1 2 t0), under risk aversion, u 5 a ⇒ pt (t, u; u, r) v 0 as t b t0 , ;t [ [0, 1]: For worker
14
GEOFFREY K. TURNBULL
FIGURE 3
type u 5 a the housing bid price function is non-monotonic, increasing to a maximum at t0 and decreasing thereafter. Proof. Recall that for worker type u 5 a, et 5 0. (38) then implies pt 5 2Vs st /h v 0 as st b 0. By (23), st b 0 as t b t0 so that pt v 0 as Q.E.D. t b t0 , which is the result to be shown. Risk aversion fundamentally alters the bid price, hence the equilibrium housing price function, in the two-center city. Take the simplest case of identical linear commuting cost functions as an illustration. Worker type u 5 1 will have a monotonically decreasing bid-price function while worker type u 5 0 will have a monotonically increasing bid-price function. Worker type u 5 As, though, will have the inverted U-shaped bid-price function, as illustrated in Fig. 3, rather than the horizontal bid price under risk neutrality. Risk aversion increases the demand for median residential locations by the workers with the greatest employment uncertainty, so much so that the resultant equilibrium price function exhibits a peak at the median distance between the alternative job sites in the city. This result is intuitive once we recognize the implications of (38). Even though the expected commuting cost exhibits zero spatial variation for worker type u 5 As, the commuting
JOB SITE UNCERTAINTY
15
cost variance is minimized at the median residential location; the commuting cost variance increases the farther away from the median location the worker resides, which by (38), lowers his bid price for those increasingly less desirable residential sites. The preceding comparison of risk-neutral and risk-averse cases is informative. In any event, we now more formally consider how the degree of risk aversion affects bid price functions for more general commuting cost functions. PROPOSITION 7. ptr v 0 as pt b 0: Stronger risk aversion flattens the bid price functions for all worker types u [ (0, 1). Proof. Differentiate (38) with respect to r, imposing (27) and using (36) to obtain ptr 5 2pt hr /h. From (32), hr . 0 so that the proposition follows immediately.
(39) Q.E.D.
Note that (38) implies that no worker type can have a horizontal bid price (that is, pt 5 0 only at isolated points or intervals). Thus, Proposition 8 applies to all worker types facing job site uncertainty, including type a when commuting costs are linear. The proposition reveals the pattern of spatial sorting of workers within each u group type by their strength of risk aversion. The more risk averse workers reside at locations with relatively lower commuting cost variances than their less risk-averse counterparts. This much is consistent with the uncertain transportation cost models of Papageorgiou and Pines (1988), Turnbull et al. (1991), and Turnbull (1995). They find that transportation cost uncertainty tends to increase the demand for residential sites with lower commuting cost (net income) variance relative to sites with greater variance. All of those models are single CBD models, however, so that the lower commuting cost variance sites are those closest to the single CBD. In the multiple CBD model here, in contrast, the lowest commuting cost variance sites are those nearest the location t0 between the two potential job sites. Therefore, risk aversion tends to increase all workers’ demands for median locations over those nearest each of the two CBDs, which is reflected as flatter bid price functions relative to the risk-neutral case. (Of course, the u 5 1 and u 5 0 workers’ bid prices are unaltered by risk aversion.) Finally, even though Proposition 7 holds for all commuting cost specifications, the identical linear cost case provides a sufficiently interesting illustration. Propositions 6 and 7 reveal an equilibrium housing price and land use pattern very different from those found under certainty or risk neutrality.
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GEOFFREY K. TURNBULL
FIGURE 4
In Fig. 4 the positive and negative sloped portions of the equilibrium price function over [0, t2] and [t3 , 1] are flatter than under risk neutrality, while the u 5 As bid price creates a trimodal equilibrium price surface not found in the other models.
V. CONCLUSION
This paper developed a model of the spatial housing market under the assumption that some workers face job site uncertainty. The risk neutral model shows that under job site uncertainty workers spatially sort themselves according to their job site stability. Job site stability increases the demand for residential locations nearest the expected job sites, while instability increases the demand for median residential locations between alternative job sites. Risk aversion introduces an additional worker response to the spatial variation in commuting cost risk across the urban area, which offsets the effect of job site stability under risk neutrality. The theory with risk aversion therefore predicts less spatial sorting by either realized commuting costs or expected commuting costs than the certainty or risk-
17
JOB SITE UNCERTAINTY
neutral models predict. The risk neutrality model presented here predicts more commuting than the certainty model. In addition, though, risk aversion provides another rationale for why the simple certainty model predicts significantly less ex post commuting than observed empirically.7 The effects of job site uncertainty on the equilibrium housing price surface depends upon whether workers are risk neutral or risk averse. House bid prices capitalize the spatial variations in both expected travel cost and travel cost variance, leading to an equilibrium housing price surface more irregular than found under certainty. Under risk neutrality, job site uncertainty flattens the price function throughout the region between CBDs and introduces a completely horizontal housing price surface in the median region between potential job sites when commuting costs are linear. Under risk aversion, however, there is no extensive horizontal portion in the equilibrium housing price function. The housing price surface is flatter than under certainty, but risk averse behavior also generates price peaks at residential locations between employment centers, a feature not found under certainty or even uncertainty with risk neutrality.
APPENDIX
A. Derivation of the Dual Expenditure Minimization Problem (16) for the Risk-Neutral Bid-Price Problem (3) We begin by exploiting the fact that the risk-neutral problem can be cast as problem (3) in the paper. Following Wheaton (1977), the bid-price problem can be recast as a dual utility maximization problem, which can then be used to derive the expenditure function version used in the paper. Define the Marshallian (ordinary) demands for housing and spending on other goods for given t as hh( p, m 2 e), x( p, m 2 e)j ; argmaxhu(h, x)
s.t.
m 2 e 5 ph 1 xj. (A.1)
Use (A.1) to define the indirect utility function conditional on t (via p and e) as V( p, m 2 e) ; f (h( p, m 2 e)) 1 g(h( p, m 2 e))x( p, m 2 e).
(A.2)
For utility opportunity cost u0 in the risk-neutral problem (3), use (A.2) 7
See Hamilton (1982), White (1988). Thurston and Yezer (1991), Small and Song (1992), and Crane (1996) for the treatment of several aspects of the so-called ‘‘wasteful commuting’’ debate.
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GEOFFREY K. TURNBULL
to show that the bid-price function p(t, u0; u ) for type u workers is implicitly defined by V( p(t, u; u ), m 2 e(t; u )) ; u
(A.3)
for all t [ [0, 1]. Condition (A.3) is the utility maximization dual to problem (3) under risk neutrality. It turns out to be easier to deal with the expenditure function. To convert (A.3) to an expenditure function equivalent form, note that V is monotonically increasing in net money income m 2 e. Thus, the indirect utility function can be inverted so that (A.3) yields V21( p(t, u; u ), u) ; m 2 e(t; u ),
(A.4)
or, using V21( p(t, u; u ), u) 5 e( p(t, u; u ), u), where e( p, u) is the expected expenditure function. Substituting into (A.4) yields e( p(t, u; u ), u) ; m 2 e(t; u ),
(A.5)
which is (6) in the paper. B. The Substitution Theorem for Housing under Uncertainty The substitution theorem asserts a negative own-price effect on the Hicksian (compensated) demand for housing, hp( p, u; s) , 0. To prove this result under uncertainty, define the general expected utility function E[u(h, x 1 «)], where uh . 0, uhh , 0, uy . 0, and uyy # 0 allows for risk neutrality or risk aversion. The Hicksian demand vector under uncertainty is hh( p, u; s), x( p, u; s)j ; argminh ph 1 x
s.t.
E[u(h, x 1 «)] $ uj. (B.1)
Let l be the Kuhn–Tucker multiplier. The (weak) concavity of the expected utility in planned consumption ensures that the Kuhn–Tucker conditions, which are necessary and sufficient for (B.1) under these assumptions, simplify to p 2 lE[uh] 5 0
(B.2)
1 2 lE[uy ] 5 0
(B.3)
u 2 E[u(h, x 1 «)] 5 0.
(B.4)
Totally differentiate (B.2)–(B.4) and solve the resultant system for
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JOB SITE UNCERTAINTY
hp 5 E[uy ]2 /uJu , 0,
(B.5)
where J is the Jacobian matrix of (B.2)–(B.4) and the sign of (B.5) follows from uJu , 0 due to the (weak) concavity of E[u(h, x 1 «)] in hh, xj. Equation (B.5) is the substitution theorem for housing demand. C. Properties of the Expected Expenditure function under Uncertainty Define the expected expenditure function as V( p, u; r, s) ; ph( p, u; r, s) 1 x( p, u; r, s),
(C.1)
where h and x satisfy (29)–(31). Totally differentiate (C.1) with respect to the risk aversion parameter r: Vr 5 phr 1 xr Equation (29) implies h f 9 5 p and (30) implies h(c 1 rE[f9]) 5 1, so that substituting into the above yields Vr 5 hh f 9hr 1 (c 1 rE[f9])xrj.
(C.2)
Differentiate (31) with respect to r to obtain f 9hr 1 (c 1 rE[f9])xr 5 2hx(dc/dr) 1 E[f]j.
(C.3)
Applying (27), however, leaves hx(dc/dr) 1 E[f]j 5 0, so that the left and side of (C.3) is zero and Vr 5 0 from (C.2). This is property (34) in the paper. To find the effects of a mean preserving increase in the variance of « on expected expenditures, replace s« for « in (29)–(31), with the parameter s serving as an index of uncertainty. The variance VAR(s«) 5 s 2VAR(«), so that dVAR(s«)/ds . 0. Given that E[«] 5 0 by construction, an increase in s therefore represents a mean preserving increase in «-risk. Differentiate (C.1) with respect to s and evaluate the result at s 5 1 to find Vs 5 phr 1 xr .
(C.4)
Equation (29) implies h f 9 5 p and (30) implies h(c 1 rE[f9]) 5 1, so that substituting into the above yields Vs 5 hh f 9hs 1 (c 1 rE[f9])xsj.
(C.5)
Differentiate (31) with respect to s to and evaluate at s 5 1 to obtain
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GEOFFREY K. TURNBULL
f 9hs 1 (c 1 rE[f9])xs 5 2rE[f9(x( p, u; r, s) 1 «)«].
(C.6)
Substitute (C.6) into (C.5) to obtain Vs 5 2rE[f9(x( p, u; r, s) 1 «)«] 5 2rCOV[f9, «],
(C.6)
where COV denotes covariance and the second equality follows from E[«] 5 0. By the properties of similar–dissimilar orderings, the covariance takes the sign of f0, which is negative under risk aversion. Thus, (C.6) is positive, which is (35).
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