A Unified Theory of Consumption and Travel 1

Transcription

1 A Unfed Theory of Consumpton and Travel by Alex Anas Department of Economcs State Unversty of New York at Buffalo Amherst, New York 460 USA February 8, 006 ABSTRACT The mcroeconomc theory of demand explans the purchasng decsons of consumers gnorng that most consumpton requres travel. Travel demand theory, strongly nfluenced by econometrcs, has developed largely ndependently of mcroeconomc theory and gnores consumpton decsons. treat consumpton and travel n a unfed manner so that the consumer allocates hs tme and ncome among dscretonary trps n order to purchase a varety of consumpton goods. The focus s the complementarness and substtuton between travel and consumpton. formulate a model n whch the consumer decdes for a perod of tme the frequency of the trps to make to each store and how much to buy on each trp. then modfy the model to examne how the consumer should mx separate trps to two stores, wth chaned trps n whch he vsts both stores. Under the preferences specfed, travel cost saved by trp-channg goes nto more travel so that the total travel expendture remans unchanged. pont out how the model developed can be generalzed n a number of ways to mprove ts scope and applcablty. Keywords: Consumer theory, travel demand theory, trp-channg, taste for varety. JEL classfcaton: D, J9, R4. ntroducton The theory of the consumer n mcroeconomcs gnores that most consumpton cannot be realzed wthout ncurrng travel or communcaton costs. Manstream economcs generally gnores spatal aspects of realty regardless of ther mportance. The economc theory of the allocaton of tme (Becker, 965) overlooks explct treatment of travel tself as an actvty, but travel s ntmately related to both consumpton and the allocaton of tme among dscretonary actvtes. The theory of travel demand wthn transportaton scence, on the other hand, has developed largely ndependently of standard mcroeconomc theory. Vrtually all travel demand Ths artcle was presented at the Department of Economcs Unversty of Calforna at rvne, February 3-4, 006 at the conference n honor of Kenneth A. Small. thank the partcpants for ther comments and, especally, Kenneth Tran for pontng out the usefulness of the theoretcal approach developed n the paper for formulatng econometrc versons.

2 ALEX ANAS A Unfed Theory of Consumpton and Travel /8/006 analyss has ts roots n appled econometrc technques and especally n dscrete choce models that, snce the contrbuton of McFadden (973), have advanced understandng of many aspects of travel. Econometrcs, more than mcroeconomc theory, s nspred by the study of real problems on whch data are avalable and whch are of polcy nterest. t s not surprsng that the study of modal choce (e.g. the choce between auto and publc transt) and of commutng have n one way or another domnated conventonal travel demand analyss. Modal choce s mportant because of the competton between hghways and transt systems for passengers and for resources. Commutng s mportant because t competes wth lesure and work for the tme of consumers. But both commutng and modal choce are narrow aspects of the total travel experence. The travel demand problem can be and should be defned more generally by: how many trps and what knd of trps to make over a perod of tme and to whch destnatons? Commutng s generally (but not entrely) determned by pror decsons such as the choce of resdence and of workplace. What remans s largely dscretonary travel ncludng trps for shoppng, recreaton, socalzng, eatng out and other actvtes. The demand for dscretonary trps cannot be consdered separately from the consumpton benefts of such travel. From ths perspectve, standard consumer theory and travel demand should be nseparable. Although they have n fact evolved largely separately, n the present artcle brng them together. The queston defnng the combned dscretonary travel and consumpton problem s: how many trps and what knd of trps to make and to whch destnatons, and how much goods and servces to purchase on those trps? Such a defnton of travel theory appears natural and approprate to the casual observer, but most urban economcs and locaton theory models have gnored ths way of lookng at travel. For example, n conventonal locaton theory t s assumed that the consumer travels to the nearest store or destnaton. n urban economcs, t s standard to assume that the consumer commutes but non-standard n the extreme to assume that the consumer travels n any other way. Equlbrum models that combne both commutng and dscretonary travel n order to analyze land use patterns and the locaton of employment are recent (Anas and Xu, 999). The feld of transportaton analyss, on the other hand, has recognzed the mportance of the related problem of complex travel patterns and actvty schedulng from a conceptual and emprcal perspectve. Ths has gven rse to a rch and growng body of appled studes of travel actvty patterns whch take ther cue drectly from econometrcs and not from mcroeconomc theory. See Ettema and Tmmermans, (997); Ben-Akva and Bowman (998) and Bhat and Koppelman (999) for surveys.

3 ALEX ANAS A Unfed Theory of Consumpton and Travel /8/006 Accordng to Gordon and Rchardson (000), from 969 to 995 the average commutng tme n the U.S. fell somewhat from mnutes to 0.7 mnutes, presumably because decentralzng jobs mproved the proxmty between homes and workplaces, whle the total vehcle mles traveled (VMT) grew much faster than the land area of the 65 largest urbanzed areas. Some of the hgher VMT may have come about because people traveled longer dstances at hgher speeds on ther commutes, but most of t s lkely to have come from a larger number of dscretonary trps. Table, borrowed from Nelson and Nles (000), tells the story of the percentage change n vehcle travel and n trp lengths by purpose and per person from (and ). TABLE The bggest percentage changes have occurred n shoppng and n other famly and personal busness trps. n addton to beng the fastest growng, ths category was also the sngle largest category of trps consstng of 3.6% of all trps n 995. The authors also reported ther other observatons from the Natonal Personal Transportaton Survey: 3/4ths of person trps and 4/5ths of vehcle trps n the US are for non-work purposes and that non-work trps are a major travel reason even durng peak perods. The reason for the prolferaton of non-work dscretonary trps s easy to conjecture. Frst of all jobs and resdences have decentralzed, reducng the average dstance between homes and employment concentratons to whch non-work trps are made. ncomes have ncreased and as ncomes ncrease the demand for product varety grows and consumers seek a larger dversty of opportuntes to shop, purchase servces and engage n recreaton or lesure-related actvtes. As 3

4 ALEX ANAS A Unfed Theory of Consumpton and Travel /8/006 ncomes ncrease, car ownershp also ncreases and the avalablty of multple prvate vehcles or of more persons wth access to a prvate vehcle stmulates more travel and dscretonary moblty. To close the gap between theory and observaton we need to develop models that treat the complementarness and substtuton between consumpton and travel, recognzng that whle travel s necessary for consumpton t also competes wth consumpton for ncome and tme, that there are many alternatve shoppng destnatons avalable to modern consumers, that such destnatons are substtutes and that the degree of substtutablty vares. A proper theory would have to take nto account the mportant roles of ncome, the value of tme whch s related to ncome, and the substtutablty among alternatve trp destnatons. t would treat properly both the ncome and substtuton effects that arse when travel cost s reduced generally, for the commute or for a partcular trp. Also, the complexty of travel has ncreased by the prolferaton of trp-chans or mult-stop trps often called tours n the transportaton feld. These orgnate and end at home, at work or at some other startng pont. The fact that consumers can combne dfferent trps n a sngle tour or trp-chan s a major challenge for theory. We do not yet know whether channg trps reduces overall travel mles or whether the ablty to chan trps together frees up enough tme that s n turn allocated n part or n whole to the makng of more chaned and unchaned trps. n secton, lay out a model of the consumpton and travel actvtes of a sngle consumer based on a non-trval extenson of the Dxt-Stgltz (977) utlty functon subject to a nonconvex budget set combnng travel and consumpton expendtures. show how ths model can be solved usng a two-stage procedure and analyze the most salent propertes of the resultng demand equatons. n secton 3, examne the model s soluton for a smple hypothetcal context n whch stores are symmetrcally located wth respect to the consumer. n ths smple settng show how the consumer can optmally determne the choce set of stores that he vsts. The analytcs of trp channg are consdered n secton 4, usng an approprate extenson of the basc model of secton. show under what condtons a consumer wll prefer to chan some of hs trps and under what condtons trp channg wll not be practcal. Addtonal extensons of the model are brefly sketched out n secton 5. Whle n the basc model of secton, the consumer s so varety-hungry that he vsts all stores that are known to hm, the extended model of secton 4 shows how the range of the consumer s travel can be lmted by hs tastes and the travel costs and prces assocated by the varous stores. n secton 5 also brefly comment on how to extend the approach presented here to nclude the dscretonary tme spent n actvtes ncludng shoppng, and how to treat hgher level decsons such as modal choce and locaton. 4

5 ALEX ANAS A Unfed Theory of Consumpton and Travel /8/006. Consumpton and Trp-makng: Separate Trps vew the consumer as havng to travel to acqure goods and servces to consume. Ths approach recognzes the basc complementarness between travel and consumpton gnored n both consumpton theory and travel demand theory but also treats the substtuton between the cost or length of a trp and the quantty purchased per trp, as well as the substtuton of trps and purchases among dfferent trp destnatons. Ths perspectve has wde applcablty to a varety of travel contexts, but to keep a smple story n mnd n ths exposton, t s convenent to magne a consumer who, over a perod such as a month, a season or a year, makes separate vsts to each of =... stores. These trps are denoted by the non-negatve vector n ( n n ),...,. Although trps must be ntegers n realty, wll treat them as contnuous non-negatve varables for analytcal purposes, commentng as needed on the types of complcatons that arse when trps are ntegers. Durng a partcular vst to store, the consumer buys quantty z. The non-negatve vector z ( z z ),..., denotes the quanttes bought from each store per trp made. The consumer s budget constrant should nclude travel expendtures as well as purchases at the stores. Suppose that the unt prce of the goods sold at store s p and that the opportunty cost of travelng to and from the store s wt + c, where w s the consumer s wage rate or hs value of tme, t s the two-way travel tme and c s the two-way monetary travel cost. The consumer has a tme endowment H that he can allocate n part to earnng a wage w, whch requres commutng, and n part to dscretonary travel whch conssts of travel to the stores. The consumer s two-way commute (from home to work and back) takes T hours and also ncurs a monetary cost C. Hence, wt + C s the consumer s full opportunty cost of the daly commutng travel. d s the number of commutng days over the year. n the present, wll assume that the consumer s resdence and job locatons are pre-determned. Hence, n the dscusson that follows, the commute tme, T, and ts monetary cost, C, are treated as nondscretonary. They are, therefore, fxed costs. They have been predetermned by the choce of resdence and workplace whch am not treatng n the present. The budget constrant can now be wrtten as (a) below, wth earned money ncome on the left sde and all monetary expendtures related to commutng and shoppng on the rght sde. n Anas and Xu (999), a general equlbrum model of urban land use, a consumer chooses among the dscrete combnatons of workplace-resdence pars, whle evaluatng a number of contnuous choces for each such dscrete combnaton. Among these contnuous choces are the frequences of travel from the resdence locaton to a varety of shoppng destnatons. 5

6 ALEX ANAS A Unfed Theory of Consumpton and Travel /8/006 w H dt tn = dc + cn + npz, (a) = = = Ths can be rewrtten by gatherng all of the choce-related opportunty costs on the rght sde and the endowment ncome and all the fxed costs related to commutng on the left: ( ) (b) M wh d( wt + C) = wt + c n + n p z = = n the (b)-way of wrtng the budget, the left sde s the consumer s potental ncome avalable for purchases and for travel, whch s that ncome the consumer would have f he dd not shop at all but were to spend all hs tme net of commutng earnng a wage. From an analyss standpont, (b) s the more convenent way to wrte the budget although (a) s more natural to look at. Note that the budget constrant s lnear n product prces, travel costs and travel tmes but nonlnear n the choce varables n = ( n,..., n ) and z ( z,..., z ) homogeneous of degree zero n, w C, c = ( c c ) and p =( ),..., =. The budget constrant s p,..., p. Property.Non-convex budget set: The budget constrant may be wrtten as = ( ) [ wt + c n + n p z ]. t follows that the set of affordable trp-consumpton bundles (n, z) s not a convex set. The non-convexty of the budget set reflects a fundamental scale economy n shoppng. The expendture per trp made to store conssts of a fxed cost wt + c whch does not depend on quantty purchased and a varable cost pz whch ncreases wth the quantty purchased per trp at the store, z. Now suppose that the consumer s utlty functon for vstng all stores over the year s Dxt- Stgltz (977) C.E.S. as n (a) below, wth the elastcty of substtuton. U ( n, z ) = u( n, z), 0< <, (a) = where u( n, z ) s the sub-utlty derved from n vsts to store,quantty z beng bought durng each vst. wll assume that zero utlty s derved unless somethng s bought, u ( n,0) = 0, and that each sub-utlty functon s strctly concave and ncreasng n z gven n > 0. Hence, usng the upper prme to denote a dervatve wth respect to z, u ( n, z ) > 0and u ( n, z ) < 0 for 6

7 ALEX ANAS A Unfed Theory of Consumpton and Travel /8/006 u n > 0. n what follows, assume that trps are goods : > 0 for each store. also assume n that f a trp s not made to a store, then no utlty can be derved from the store. Thus, u (0, z ) = 0 for z 0. 3 now consder a fundamental property of the utlty functon and a basc trade-off n the utlty maxmzaton problem. Property. Extreme taste for shoppng varety: An extreme taste for shoppng varety exsts U because ( ) U = u( n, z) and lmu 0 u = +. Ths means that f the consumer derves u no utlty from a partcular store because he s not shoppng there, he gets extremely desrous for what that store has to offer. U U u Proof: By the assumpton that u( n,0) = 0, t follows that lm z 0 = =+. Smlarly, z u z U U u by u(0, z ) = 0, lmn 0 = =+ n u n A consequence of ths property s that f the consumer patronzes some stores and not others, he wll seek to ncrease utlty by shftng hs tme and money to stores that are not patronzed. Ths taste for varety weakens to the pont of vanshng as and the stores become vewed as perfect substtutes wth an elastcty of substtuton that tends to nfnty. n ths case, the consumer wll normally patronze only one store. At the other extreme, as 0, and the elastcty of substtuton tends to unty, (a) becomes Cobb-Douglas n the subutltes and all stores are consdered essental. Let us now specfy the sub-utlty as follows: u ( n, z ) = nu ( z ) whch means that the total utlty derved from patronzng the store n tmes s the sum of the utlty derved from the store each tme t s patronzed. Then, the utlty functon (a) becomes: / U = ( nu ( z) ), 0< <. (b) = The consumer maxmzes the utlty functon (b) subject to the budget constrant (b) by choosng smultaneously the trp vector n =( n,..., n ), and the vector of quantty purchases per trp made, z = ( z z ). Let us suppose ntally that the values of the trp vector n are,..., 3 Ths does not rule out forms of shoppng that do not nvolve trps (by mal, telephone, nternet etc.) because these may be consdered as nvolvng vrtual trps wth lower access costs than physcal travel. 7

8 ALEX ANAS A Unfed Theory of Consumpton and Travel /8/006 restrcted so that each element can only be a non-negatve nteger. Then, we can dentfy the followng property. Property 3. Trade-off between travel cost and shoppng varety: Whle the consumer lkes to shop at all the stores to satsfy hs taste for shoppng varety, he saves transport cost from not vstng a partcular store and can allocate ths saved travel cost among the other stores to ncrease utlty. Proof: There s no loss of generalty n demonstratng ths property wth just two stores. Suppose that M s the consumer s potental ncome avalable for shoppng travel and purchases. Suppose that a wt + c,,. mn a + a, a + a > M > a + a. = Assume ( ) Ths means that the consumer has enough potental ncome to afford one vst to each store but not enough to afford a second vst to ether one of the two stores. t s now easy to see that the consumer has three choces: a) Vst once and buy only from store ; b) Vst once and buy only from store ; c) Vst each store once and buy from each store. n each of these cases the ncome avalable for purchases s M a for n = (,0), M a for n = (0,) and M a a for n = (,). Gven ths constraned trp pattern, the consumer maxmzes (b) subject to (b) treatng z and z as the decson varables. The value of each case to the consumer s measured by the ndrect M a utlty n each case. These are V () = p patronzng both stores, M a V () = p,, and for the case of α α α α V(& ) = p + p ( M a a ). By nspecton or by pluggng n values for the parameters verfy that ( ) max V(), V() > V(& ) s possble. Ths can occur, for example, when travel to the stores s suffcently dear that t makes sense to sacrfce store varety n order to save the cost of travel to the more naccessble store, allocatng ths savng towards more purchases n the nearer store. Although n must consst of ntegers, for convenence, wll hereafter follow the approxmaton and treat ts elements as real numbers. Ths wll be a good approxmaton provded the total trps made over the year are numerous so that any fractonal trps to a partcular store can be rounded off to the nearest nteger wthout any sgnfcant loss of accuracy. Such a condton 8

9 ALEX ANAS A Unfed Theory of Consumpton and Travel /8/006 can be had by assumng that potental ncome, M, s suffcently large relatve to the purchase prces and the travel costs of each store. Both n and z can have only non-negatve elements. 4 Now consder how such a problem can be solved. To get a specfc soluton, wll assume that u ( z ) = z α,0< α <. The majorty mght solve ths problem by formulatng a Lagrangan and then fndng all of the frst order condtons wth respect to (n,z) as n a textbook. nstead of emulatng ths standard and dull approach, n ths artcle we wll consder two alternatve twostage soluton procedures that gve the same result as the textbook method and at once are more ntutvely appealng and yeld more nsght than the textbook method. Frst two-stage procedure: Purchases condtonal on a trp pattern Our frst two-stage procedure s for the consumer to frst determne a feasble trp vector n, such that M (a,n) M an > 0, where a = (,..., ) = a a, and a wt + c. Then, the consumer maxmzes the utlty functon (b) wth respect to z and subject to the budget constrant = p nz = M an, takng n as gven. Dong so determnes the Marshallan = demands for quanttes purchased at each store, condtonal on n and p and a. The soluton s (3a), assumng as stated earler, that u ( z ) = z α,0< α <. z (p, n M(a,n)) = α α p n α ( α ) α α pj nj j= k k = ( M an), =,...,. (3a) k n the next stage, the consumer plugs the above z nto the utlty functon (b) to get the followng ndrect utlty whch s condtonal on n: α α ( α ) α U (p,n M(a,n)) /α α α = p n M an = =. (3b) Ths can now be maxmzed wth respect to n to obtan the optmal trp pattern whch s: 4 n emprcal study, the data obvously records trps as ntegers and consumers are observed to choose among all possble combnatons of ncome-feasble trp patterns, n. Usng the modern econometrc technques of estmaton by smulaton (e.g. Tran, 003), maxmum lkelhood estmates of probablstc choces over such trp combnatons are easy to obtan. 9

10 ALEX ANAS A Unfed Theory of Consumpton and Travel /8/006 n (a, p ( α)m ) = j= p α α a α ( α ) p j a j ( α) M, =,...,. (3c) Second two-stage procedure: trps and purchases condtonal on store-specfc expendtures Another ntutvely appealng and conceptually smpler way to solve the problem and obtan exactly the z and n gven by (3a) and (3c) s by means of an alternatve two-stage procedure. wll refer to the stages as the nner and outer stages. wll ntally approach ths procedure usng u ( z ) but wll later specalze to u ( z ) = z α,0< α <. At the outer stage, the consumer optmally allocates postve expendtures e = ( e,..., e ) to each store, so that the allocaton s feasble, summng up to the total ncome avalable for purchases and travel: e = wh d( wt + C) (4a) = At the nner stage, and for each store, the number of vsts and the purchase per vst that maxmzes the sub-utlty from that store are found. Ths nner stage problem for the th store s: Max, u n u ( z ) (4b) n z subject to an + npz = e. (4c) The budget constrant (4c) s non-lnear n the choce varables n and z. n fact the budget constrant s negatvely sloped and strctly convex to the orgn n ( z, n ) non-convex budget set (see Fgure ). FGURE ABOUT HERE Ths s seen drectly from the frst and second dervatves: n z n ( ) 0 = ep a+ pz < 3 = ep ( a + pz ) > 0 z -space and defnes a, (5a). (5b) Ths observaton reflects the basc non-convexty n travelng to a store. The cost of travel, per unt of the quantty purchased decreases f more s purchased per vst. The next two noteworthy facts are about the ndfference curves of the sub-utlty 0

11 ALEX ANAS A Unfed Theory of Consumpton and Travel /8/006 nu ( z ). Frst, recall that u( z) does not cut the axes of the (, ) s strctly concave n z. Second, note that the ndfference curve z n space. Fgure puts together these facts. The fact that the budget constrant and the ndfference curve are each strctly convex to the orgn and the fact that the budget does but the ndfference curves do not cut the axes mply that there s an nteror soluton. Corner solutons are not possble. Therefore, no matter how much or lttle ncome s allocated to store n stage one, the consumer wll choose to vst that store and to make a postve purchase. 5 Ths s clearly as t should be because, n a model wthout uncertanty, we do not want a soluton where the store s vsted a postve number of tmes but no purchases are made, or a soluton n whch purchases are made wthout vstng the store. 6 derve the frst order condtons wth respect to n and z, and dvde one nto the other. Dong so yelds: u( z) a + pz =. (6a) u ( z ) p The above says that the rato of the margnal utlty of a trp to the margnal utlty of the good purchased on the trp s equal to the rato of the cost of one trp (nclusve of the cost of purchases on the trp) to the prce of the good that s purchased. Dvdng both sdes wth z, (6a) s rewrtten as follows: u( z)/ z a = + (6b) u ( z ) pz Note that the left sde of (6b) s greater than unty by the concavty property. The condton says that for each store, the rato of the average utlty to the margnal utlty of the quantty purchased s equal to one plus the rato of the opportunty cost of the trp to the cost of purchases per trp at the destnaton. Yet another useful way to wrte (6b) s: u( z) a z =. (6c) u ( z ) p (6c) gathers on the left all the terms nvolvng the quantty purchased on the trp, whle on the rght s the exogenous rato of the margnal cost of a trp to the margnal cost of the quantty purchased. Note that the soluton to (6c) gves the quantty demanded per trp, z, and pluggng 5 Of course there must be enough expendture allocated to enable at least one vst. We are crcumventng ths problem n ths theoretcal treatment by gnorng the nteger nature of trps as explaned earler. 6 Tele-shoppng s not fundamentally dfferent because the communcaton cost replaces the travel cost. The essental dfference s that the communcaton cost s lower than the correspondng travel cost.

12 ALEX ANAS A Unfed Theory of Consumpton and Travel /8/006 ths nto the budget one gets the number of trps demanded by the consumer, n. These two demands are condtonal on knowng the expendture e from the outer stage and are related to each other by the followng property. Property 4a. Condtonal trp and quantty demands are trp-cost substtutes: Trps, n, and the quantty purchased on a trp, z, condtonal on expendture e are substtutes wth respect to the opportunty cost of travel, a wt + c. Proof: Cross-multplyng (6c) and totally dfferentatng the resultng equaton wth respect to z and a, we get: (4c) and usng dz u ( z) = > 0. Then, totally dfferentatng the budget constrant da u ( z )( z + a ) dz da n t we get that u ( z) p dn ( a + pz ) u ( z ) = n < 0. da a + p z Property 4b. Condtonal trp and quantty demands are prce-substtutes (complements) f the quantty purchased s prce-elastc (nelastc). Trps, n, and the quantty purchased on a trp, z, condtonal on expendture e are substtutes wth respect to the product prce p only f the elastcty of the demand for product z, Ε z, p<. Proof: Followng the same procedure as n the proof of Property 4a, we can derve the followng: dz u( z) zu ( z) = < 0 and dp u ( a + p z ) dn Ε z, p = n <=> 0 as Ε z,. p <=> dp z ( a + p z ) Thus, as a trp to a store gets more (less) expensve (hgher travel cost for whatever reason ncludng dstance), the consumer travels there fewer (more) tmes but buys a larger (lower) quantty on every trp. Smlarly, as the prce at a store gets lower (hgher), the consumer buys more (less) at that store but travels there less (more) often. We wll now specalze to u ( z ) = z α,0< α <. Then, the demand functon for the quantty to purchase on a sngle trp s: α a z( p, a) =, (7a) α p whch s ndependent of the expendture and homogeneous of degree zero n wc,, p. Cross multplyng, we see that the purchase expendture per trp s proportonal to the cost of the trp. f,

13 ALEX ANAS A Unfed Theory of Consumpton and Travel /8/006 for example, α = /, then the cost of the trp and the cost of the purchases on the trp are equal. f α > /the cost of the purchases on the trp are hgher than the trp s cost. f α < / the trp s cost exceeds the cost of the purchases on the trp. Note also that the elastcty of z( p, a) wth respect to p s mnus one. Therefore, accordng to Property 4b, the demand for trps s ndependent of p. Ths s ndeed verfed by pluggng (7a) nto the budget constrant (4c) and solvng the demand for the number of trps, e na (, ) ( ) e = α, (7b) a whch are homogeneous of degree zero n we,, c. Cross multplyng, ths shows that the expendture on travel to destnaton s a constant fracton, α, of the expendture allocated to that destnaton n the outer stage. Then, t follows mmedately (and can also be verfed from (7a) and (7b)) that the remanng fracton α of the expendture allocated to destnaton s spent on purchases there, so that pnz = αe. Substtutng (7a) and (7b) nto the sub-utlty functon for the th store, we get the ndrect sub-utlty whch s homogeneous of degree zero n p, c, w, e : ( α ) α α ( α ) v( p, a, e) = ( α) α p a e (7c) We are prmarly nterested n how ths ndrect sub-utlty depends on the expendture, e, allocated to. Snce 0 < <, the ndrect utlty (7c) s strctly concave n e. wrte t as v ( e) = Ae, where A ( α) α p a. ( α ) α α ( α ) We can now move to the outer stage of the consumer s utlty maxmzaton: / e (,..., ) = = Max U e e Ae ; 0< <, (8a) subject to : e = wh d( wt + C). (8b) = As promsed, the soluton to ths problem determnes the optmal way the consumer should allocate hs ncome among all the destnatons, to be expended for trps and purchases at those destnatons. (8a) s Dxt-Stgltz C.E.S. The ndfference curves are strctly concave to the orgn and are tangent to all axes. The budget constrant (8b) s lnear and symmetrc n e = ( e,..., e ) (see Fgure ). Therefore, the soluton s strctly an nteror one: the consumer wll FGURE ABOUT HERE 3

14 ALEX ANAS A Unfed Theory of Consumpton and Travel /8/006 allocate some ncome to each and every store because he has an extreme taste for varety n spendng at each store. He would rather spend even a small amount at each store than not vst any one store. Solvng, we get the Marshallan allocaton of ncome or the demand for expendture at store : A Aj j= [ ( )] e = wh d wt + C. (8c) From (8c) t s drectly verfed by summaton that the budget constrant s satsfed. t s revealng to rewrte the above as a proporton. Let M wh d( wt + C). Then, dvdng (8c) by M and substtutng for the A : e π = M A Aj j= = α ( α ) p a α ( α ) p j a j j=. (8d) (8d) says that the proporton, π,of ncome allocated to the th destnaton s gven by the logt model but, n ths case, the model s determnstc not the stochastcally derved one of appled travel demand analyss, and the dependent varable s not a probablty but a fracton of ncome. (8d) shows that ceters parbus more ncome s allocated to more accessble destnatons and less to destnatons wth a hgher unt prce. Notably, of course, the ndependence of rrelevant alternatves property of logt holds: the rato of ncomes allocated to two destnatons s ndependent of changes n prce or accessblty that can occur n any other destnaton. But obvously ths does not hold because of any assumpton about the dstrbuton of random utltes snce there aren t any n ths determnstc model. Property 5. Strct gross substtuton of store-expendtures: The expendtures optmally allocated to the alternatve stores satsfy strct gross-substtuton wth respect to product prces at the stores as well as the opportunty cost of travelng to the stores. Proof: From (8d) one can calculate the own- and cross- elastctes of the expendture allocated to a partcular store. These elastctes are: α Ε = π < e, ( ) 0 p ( α) and Ε e, ( ) 0 a = π < α Ε = π > e, 0 pj, j j and ( α) Ε e, 0 a = j, j π j> 4

15 ALEX ANAS A Unfed Theory of Consumpton and Travel /8/006 These elastctes also reveal the specal property that the expendture allocated to a store s more (less) elastc and more (less) cross-elastc wth respect to the prce for the goods sold at the store rather than wth respect to the opportunty cost of accessng the store f α > / ( α < /). We have completed both stages of the utlty maxmzaton analyss. Puttng the two stages together, th e full soluton gves the uncondtonal trp and quantty demands n, z for =,, whch are as follows: and j= z α a = (9a) α p α α a α ( α ) j a j p n = ( α) [ wh d( wt + C) ]. (9b) p The exogenous varables are,, whdtc,,,, and c = ( ) α and the vectors p = ( p p ) t = ( t t ),...,,,..., c,...,. c Both demands are homogeneous of degree zero n wc,, c, p. Note that (9b) s dentcal to (3c) derved earler va the prevous two-stage procedure. Property 6. Uncondtonal quantty and trp demands are prce-complements and trp-costsubsttutes: The consumer s optmally determned trps, n, and the quantty purchased on a trp, z, are complements wth respect to the product prce p and substtutes wth respect to the opportunty cost of travel, a wt + c. Proof: The dervatves of z are z p α = ap < 0, α z a α = p > 0. To see the α dervatves of n, wrte t as n = ( α) Mπ a. From ths, we get n p ( αα ) π ( π ) = M < 0, pa n ( α) M ( α) π = ( ) 0. + π < a a From these partal dervatves we can see that f the unt product prce ncreases, fewer trps are made to that store and less s purchased per trp. f the cost of travelng to a store ncreases, fewer trps are made to that store but more s purchased on each trp. t s also easy to see that the total 5

16 ALEX ANAS A Unfed Theory of Consumpton and Travel /8/006 quantty purchased from a store s Q n z = αmπ p. t s easy to see usng the foregong that Q Q < 0 and < 0. p a Property 7. Effect of wage on changes n expendtures by store dstance: Consder a sequence of many stores =, located at dfferent dstances from the consumer s home as shown n the mddle panel of Fgure, so that for any, t > + t and c+ > c a+ > a. () Substtuton effect: As the consumer s wage ncreases, keepng hs total ncome constant, he wll decrease hs expendtures on far away stores whle ncreasng hs trps and expendtures on nearby stores. () ncome effect: As the consumer wage ncreases, hs ncome ncreases and ths causes the consumer to want to ncrease hs expendtures n all the stores. Proof: From (8d), we can calculate that e π = + + w w [ wh d( wt C)] π ( H dt ). We can also calculate that t π ( α ) t j = π π j w a j= a j. The frst dervatve shows substtuton and ncome effects of the wage ncrease on store expendtures and the second s used to determne the sgn of the substtuton effect for each store, keepng ncome constant. () The bracket s zero for an average store n the sense that the tme cost of a trp to that store as a rato of the total trp cost s average when weghted by expendtures. For such an average store, the substtuton effect s exactly zero. For stores that are closer to the consumer than average n the above sense, the bracket s negatve and ths gves a postve substtuton, whle for stores that are more dstant than average n the above sense, the bracket s postve and the substtuton negatve. Thus, as the consumer s wage rses, ncome constant, he tlts hs expendtures n favor of nearby stores (and away from far away stores) because hs value of tme rses. () The ncome effect of the wage ncrease as measured by π ( H dt) s postve but falls wth store dstance snce, ceters parbus, the fracton of expendture allocated to a store falls wth dstance as we have seen. Thus, f the ncome effect of the wage ncrease s not very large, the substtuton effect wll domnate and the consumer wll spend more on travel and purchases on nearby and less on more dstant stores as hs wage rate ncreases and thus the opportunty cost of travel ncreases. But f the ncome effect s large enough, t s possble that the consumer wll spend more travelng to and purchasng from all stores as hs wage ncreases. 6

17 ALEX ANAS A Unfed Theory of Consumpton and Travel /8/ The Symmetrcal Case n order to gan nsght nto the economc propertes of the model, t s worthwhle to examne the symmetrcal case. Suppose that all stores are located symmetrcally wth respect to the consumer so that t and c are travel tme and travel cost to any store, and p s the unt prce at any store. The top panel of Fgure llustrates ths by puttng the consumer at the center of a crcle and locatng the stores on the crcle s permeter, wth the consumer connected to each store by a radal road. n ths hypothetcal stuaton, usng a = wt+ c, (9a) and (9b) smplfy to α a z =, (0a) α p and wh d( wt + C) n = ( α). (0b) a can now substtute (0a) and (0b) nto (b) to get the ndrect utlty of the consumer n the case of symmetry. Ths ndrect utlty, homogeneous of degree zero n p, w, c and M s, Ths leads to the followng observaton. α α α ( α) V( p, a, M, ) = ( α) α p a M. () Property 8. The margnal utlty of a store: when the consumer s symmetrcally stuated wth respect to the stores, then as the number of stores,, ncreases remanng symmetrc, the consumer s ndrect utlty s ncreasng and strctly convex, lnear or strctly concave wth respect to accordng to whether the elastcty of substtuton among stores,, s less than, equal to or greater than. Proof: Note that () s ncreasng strctly convex, lnear or strctly concave n accordng to whether >=<, hence accordng to whether <=> and, hence, accordng to whether <=> The ntuton behnd Property 9 s that f the elastcty of substtuton among the stores s hgh enough (greater than ) then they are suffcently close to the case of perfect substtutes that the taste for varety weakens to gve the effect of a postve but decreasng margnal utlty for an addtonal store. Property 9. nvarance of aggregates wth respect to the number of alternatves: When the consumer s symmetrcally stuated wth respect to all stores, then the total trps (TRPS) made by 7

18 ALEX ANAS A Unfed Theory of Consumpton and Travel /8/006 the consumer, the consumer s total travel tme (TTT), total travel cost (TTC) and the total quantty purchased on all trps (TPQ) all reman unchanged as the number of stores ncreases or decreases. Proof: Let us defne TRPS = n as the total number of trps. t s seen from (0b) that these trps are ndependent of the number of stores,. The consumer lkes varety. Therefore, any new stores that appear wll be vsted but the trps to the new stores come at the expense of vsts to exstng stores so that the total number of trps remans unchanged. The total travel tme, TTT = n t, total travel cost, TTC = na and the total quantty purchased on all trps, TQP = n z are also ndependent of the number of avalable stores. Snce, z, the quantty purchased per trp gven by (0a) s also ndependent of the number of stores, as more stores are added the consumer spreads hs trps and hs total purchase quantty among all stores wthout changng the total number of hs trps or the quantty he buys per trp. We can also see that the total number of trps ncreases as the wage rate, w, ncreases. Note that as the wage ncreases there are ncome and substtuton effects on the total trps (Property 8). The ncome effect s that the consumer sees the total trps as a normal good and wants to make more trps as hs wage and hence hs ncome ncreases. At the same tme as the wage ncreases, the tme cost of a trp becomes more onerous because of the opportunty cost of travelng. As a result of ths, each trp becomes more expensve and ths creates a substtuton effect n favor of makng fewer trps. n the present model, however, the ncome effect domnates the substtuton effect whch can be ascertaned by calculatng that, TRPS α = ( H dt) > 0. () w α Property 0. Optmal number of stores: f the elastcty of substtuton among stores s greater than, and the consumer ncurs an annual cost f for each store that he patronzes, then there exsts an optmal number of stores = ( ) M that unquely maxmzes the consumer s utlty f wth respect to the number of stores,. Proof: ncludng the annual cost ncurred for all stores, the ndrect utlty () becomes, V ( M f, ) = const. ( M f). Maxmzng ths wth respect to, we get = ( ) M, f and t can be shown that as long as > ( > ), order condton for a maxmum. V < 0 for all satsfyng the second 8

19 ALEX ANAS A Unfed Theory of Consumpton and Travel /8/006 The cost f can be nterpreted as the opportunty cost of gettng to know a store. Suppose that a consumer does not vst a store unless he knows t and that, n the symmetrcal stuaton, t costs f to get to know a store. Once such a cost s ncurred, the consumer knows a store perfectly and wll patronze t. Under ths assumpton, the consumer wll not want to patronze more than stores (gnorng the nteger nature) because the margnal utlty of an addtonal store (Property 8) s lower than f the margnal cost of knowng t. Note that after ncurrng the annual cost of gettng to know the optmal number of stores, the consumer has M of hs potental ncome avalable and wll allocate a fracton α of ths to purchases and a fracton α to trps, whch follows from secton. 4. Trp Channg: A Smple Case assumed that the consumer travels to each store on a separate trp. The model extends nontrvally to combned trps commonly known as trp chans or tours. Because trp-chans can be complex, wll here examne only the smplest case of just two stores. These could be located on the crcumference of the crcle n the upper panel of Fgure, on the lne n the mddle panel, or on the Eucldan plane shown n the bottom panel. FGURE 3 ABOUT HERE The arrows n each pattern show the nature of a tour. f on the crcle, the consumer travels along a radus to a store on the crcumference, then along an arc to another store and back home along another radus. t s ntutvely clear that f the arc dstance between the two stores s large, then t s better not to trp chan than to trp chan. By trp channg the consumer saves the cost of two one-way trps along a radus. By not trp-channg, two round trps are made, one to each store, and the cost of a one-way travel along the crcle arc s saved. Therefore, on the crcle, trp channg costs less n travel as long as the one-way cost of travelng along the arc s less than the two way cost of reachng the perphery. n the case of the lne, the consumer travels out from home to the most dstant store and back but on hs way stops and shops n a store of ntermedate dstance. n ths case, the consumer saves the cost of a two-way trp to the nearest of the two stores. f ceters parbus the two stores are very close to each other, the savng s bg and there s a very strong ncentve to trp chan. But f store s very close to home, the travel cost savng becomes neglgble. On the Eucldan plane, assume that all ponts can be traveled to as the crow fles. The consumer would travel from hs home to store, then to store and then back home. Then the savng can be vewed as beng smlar to what happens on the crcle. What s saved s the cost of a 9

20 ALEX ANAS A Unfed Theory of Consumpton and Travel /8/006 one-way trp to each store. The consumer s travel cost s reduced f the one-way cost of travel between stores and s less than hs the sum of the one-way cost of travelng to each store. Proposton.Trp-channg nequalty: The consumer can ncrease hs potental ncome a a avalable for purchases and trps provded that + s where a s the cost of two-way travel from the consumer s home locaton to store =,, and s s the one-way cost of travel on the arc (or dstance) between the two stores. n the case of the two stores beng located on the lne, the savng from trp channg s never negatve. Proof: The cost of separately travelng once to each store and back s a + a, whle the cost of a a channg the two stores n a sngle tour s + + s. Therefore, trp channg creates a savng a a a as long as a + + s a + a + s. n the case of the lnearly arranged a a stores, s = and the nequalty always holds n each of the above geometres, the travel cost savngs from trp-channg dentfed n Proposton should be balanced aganst the loss of utlty from commttng to vst each of the two stores the same number of tmes. Snce such an equalty n the number of trps s not optmal n general, t does not make sense for the consumer to so restrct hmself and, n general, the consumer can supplement chaned-trps wth addtonal unchaned trps. Let us now set up a utlty maxmzng analyss of trp-channg for the lnear stuaton n the mddle panel of Fgure. t readly generalzes to any geometry. Store two s farther away from the consumer than store so that t > t and c > c. Hence, a. > a f the consumer makes only one trp-chan stoppng at each store, then hs total travel tme and cost are equal to what he would have ncurred f he had vsted only store two. Thus by trp-channg, the cost of a second trp to store (the nearer store) s saved. The problem of utlty maxmzaton for ths smple case can be stated as follows, where am usng an upper ~ to dstngush between the n n ths secton and those of sectons and 3. do not use a ~ on the z snce how much one buys at a store does not depend on whether one got there on a chaned or unchaned trp: subject to: (( ) ( ) ( ) ( ) ) Max U = n + n u z + n + n u z nn,, n, z, z na ( + pz + pz ) + n ( a + pz) + n ( a + pz ) = M / (3a) 0

21 ALEX ANAS A Unfed Theory of Consumpton and Travel /8/006 where n are the chaned trps and n, n are addtonal separate trps to each store. The other symbols are as defned earler. Addng na to both sdes, get: ( ) ( ) n + n ( a + p z ) + n + n ( a + p z ) = M + na, (3b) whch clearly shows on the rght sde, the economc cost savng due to the potental trp channg. Ths problem can be solved correctly n three stages whch easly generalzes to stores located on the crcle or the plane. n the nnermost stage, treat n, the number of trp chans, as a parameter, recognzng that ts ultmate optmal value could be zero or postve. Gven n, n the nnermost stage, also take the allocaton of expendture to each store as gven and followng a procedure that s smlar to that of secton, solve for the n, n whch are optmal condtonal on n and e, e respectvely. n the mddle stage, the problem s evaluated usng the functons n ( n, e ), n ( n, e ) mpled by the nner stage and, smlar to secton, the expendtures are then allocated optmally among the stores condtonal on n. Fnally, n the thrd stage, the consumer optmzes wth respect to n. Note that the problem can nclude corner solutons. One s the soluton wth n = 0 and n, n > 0 whch means that there are no trp chans and the two stores are vsted n separate trps. Ths corresponds to the outcome where the separate trps strategy assumed n secton wns over any mxed strategy nvolvng trp chans. Ths wll defntely be the case when the trp-channg nequalty of Proposton does not offer an advantage n favor of channg trps. A second corner soluton can occur so that all trps are chans: n > 0, n = n = 0. For ths to be the case, t s necessary but not suffcent that the trp channg nequalty holods. Fnally, there can also be solutons where n > 0 and n = 0, n > 0 or where n > 0, n = 0. Before lookng the above problem whch allows for mxng of chaned and unchaned trps, wll frst consder only the two extremes whch s helpful for our ntuton. Suppose that the consumer s comparng whether to chan all hs trps to the two stores ( n = n = 0 ) or not to chan them at all ( n = 0). The latter case was analyzed n secton, and what need now s to evaluate the ndrect utlty (optmzed (b)) from secton, for the case of just two stores. Makng the requred substtutons, ths ndrect utlty functon s: α ( α ) α ( α ) α α = = α α + V( a, a, p, p, M n 0) ( ) p a p a M (4) The case of purely trp-channg ( n = n = 0 ) requres solvng the followng problem: nz, ( ), z = ( ) + ( ) Max U n u z u z

22 ALEX ANAS A Unfed Theory of Consumpton and Travel /8/006 subject to: na + np z + np z = M. (5) t helps to recognze that (5) can also be wrtten as p z + p M z = a. Now maxmzng n utlty wth respect to ths budget condtonal on n wll gve the quanttes that should be purchased from each store durng the chaned trp. Usng the sub-utlty functon u ( z ) = z α, these condtonal Marshallan demands are: α p α α α α + p z = ( n ); =,, (6a) p M They can also be obtaned from (3a) by settng n = (,), where n ( ) a. Now, the nner n stage condtonal ndrect utlty s: α α α α α α = = = + V( n, ( n ), p, p a, n n 0) n ( n ) p p (6b) n the outer stage, ths s maxmzed wth respect to n, the number of chaned trps to get: wh d( wt + C) = ( ). a α (6c) n Now evaluatng (6b), the condtonal ndrect utlty of purely trp-channg, by usng (6c), we get: α α α α ( α) α α α = = = α α + V( a, p, p, M n n 0) ( ) a p p M. (6d) The consumer wll prefer to trp-chan than not to trp-chan whenever, gven the access costs, the store prces and the other parameters, (6d) s greater than (4). An mportant consderaton here s the proxmty of the two stores n terms of travel cost. Assume that store s closer to the consumer than s store. Then, a a. The nequalty that tells us whether trp-channg s preferred to not trp-channg, (6d) > (4), can be wrtten as: α ( α ) α α α a p p < + a p p. (7)

23 ALEX ANAS A Unfed Theory of Consumpton and Travel /8/006 Note frst that f store s located next to the consumer s home, then a = 0 and the consumer wll not trp chan because the left sde shoots out to nfnty. The consumer wll prefer to make separate trps to store and adjust the number of these trps optmally as n secton, snce t costs nothng to make such trps. Suppose, next, that the two stores are located at the same place away from the consumer s resdence. Then the left sde s unty. f the prces are also equal, then ( α ) α the rght sde becomes >. n ths case the consumer wll always chan all trps p than not chan at all. Next consder the stuaton where the prce rato >. f ths s p suffcently bgger than unty, the consumer wll want to travel only a few tmes to store and more tmes to store and ths effect s stronger the closer s to one and, hence, the closer the two goods are to perfect substtutes. n ths stuaton, trp-channg all trps s hghly undesrable. Let us now return to the utlty maxmzaton problem (3a) where the mxed choce of trp channg some trps and supplementng them wth separate trps s not ruled out. t s easy to prove the followng general theorem. Theorem. Trp-channg domnance: Provded the nequalty of Proposton s satsfed, a consumer can always mprove hs utlty by trp channg some or all trps. Gven that n, n are the optmal trps n the case of separate trp-makng, the trp-channg soluton wll be such that n mn( n, n ) and supplemental separate trps wll satsfy nn = (one or both are zero). 0 Proof: Suppose that ntally force the consumer not to trp chan and usng the utlty maxmzaton problem of secton, calculate the optmal separate trps to be n, n > 0. Snce the fundamental trp channg nequalty holds (see Proposton ), the consumer can now choose n = mn( n, n ) and n = n mn( n, n ) for =,. Dong so saves expendture wthout reducng the utlty acheved by the earler n, n > 0 n the case of separate trps. Ths saved expendture can be allocated to makng more separate trps or to makng more chaned trps. Clearly, however, f the expendture were to be entrely or partly allocated to makng more trps to the store wth n = 0, then repetton of the argument shows that addtonal savngs wll accrue by channg such a trp wth a trp to the other store. (The reader s remnded that we are gnorng the nteger nature of trps) There are mportant questons pertanng to trp-channg that can be llumnated wth the type of behavorally consstent theoretcal model have developed here. Trp-channg s favored socally because of the presumpton that t reduces total travel mles by combnng trps and 3

24 ALEX ANAS A Unfed Theory of Consumpton and Travel /8/006 shortenng trp dstances. There s, however, an ncome-and-tme effect from decdng to chan trps compared to not channg them. f planners rearrange the dstrbuton of stores and thus successfully nduce more trp-channg, they cause consumers to save ncome and tme but they also ncrease the attractveness of travel and could nduce more trp chans, more and shorter separate trps and more travel on aggregate. n the present model, t s easy to see that the total cost of travel remans unchanged when the consumer chans trps than when he makes separate trps. To see ths, can verfy the followng equalty by substtutng n from (6c) for n and for the separate trps n, n from (9b): ( ). (8) na = na + na = α M Snce the consumer saves tme and money by trp-channg, t s clearly the case that, n the present model, those savngs go nto more trp-chans (and more store vsts). So whle the consumer benefts there s no reducton n aggregate travel expendtures. Ths result s of potental consequence to planners who are concerned about total travel expendtures, vehcle mles traveled and emssons from personal travel. However, the result s not as dsmal as t appears. f were to extend the model to nclude tme allocaton to lesure (defned as home actvtes that do not nvolve travel), then some of the tme savngs from trp-channg would be allocated to such lesure and so that would tend to work toward some reducton n total travel expendtures. There s, however, a substtuton effect that comes from channg trps that nduces more trps and more trp channg. Therefore, t s unclear whether the consequences of total travel (that can be defned n dollars, mnutes, mles or total emssons) ncreases or decreases and whether theoretcally consstent models wth more general functonal forms than the one used here would reveal a dfferent result when tested. Ths s an open queston that can beneft from emprcal scrutny. 5. Extensons. Store-specfc effects: n the foregong gnored that the consumer may feel dfferently about dfferent stores. But ths s easy to take care of by assumng that the utlty from the th store s u ( z ) = θ z α wth 0 < < and θ > 0 parameters that can be used to calbrate the α relatve mportance to the consumer of the varous stores and the goods sold there.. The lmted range of travel: Another mportant extenson s obtaned by modfyng the utlty functon so that the consumer does not want to vst all the stores that are avalable and, furthermore, s less lkely on a ceters parbus bass to vst more dstant stores. already ponted out the exstence of corner solutons nvolvng lmted travel, namely patronzng a subset of all 4