Transport Cost and Optimal Number of Public Facilities

Transcription

1 Transport Cost an Optima Number of Pubic Faciities Kazuo Yamaguchi Grauate Schoo of Economics, University of Tokyo June 14, 2006 Abstract We consier the number an ocation probem of pubic faciities without obigation of use. We characterize the efficient provision conition an the reationship between transport cost an the optima number of faciities. Keywors: Pubic faciities without obigation of use; Transport cost; Congestion JEL Cassification: R53; H41 Mai aress: Hongo, Bunkyo-ku, Tokyo , Japan; Te.: ; E-mai aress: 1

2 1 Introuction It is common that pubic faciities, such as parks an ibraries, are provie by centraize authorities to avoi severa probems arising from their non-rivary an non-excuabiity. However, in most communities, it is aso common that the construction of ow benefit-to-cost pubic faciities have been carrie out usuay. Recenty, especiay in Japan, this ba habit grauay tens to be revise upwar by the tren of pubic opinion, an the authorities are require to anayze the cost-benefit of any project before the construction in orer to aocate imite tax revenue efficienty. Because consumers bear transport costs to trave to a pubic faciity, an in aition, they may not enjoy by a ense of crow, it may be ess appeaing or worthess for some consumers even if the faciity itsef is beneficia. Thus, spatia characteristics, such as infrastructure for transportation an istribution of consumers, are important in etermining socia vaue of a pubic faciity. To accompish provision of pubic faciities efficienty, the authority shou eiberate how many faciities it provies an where to construct each faciity at the expense of imite resources with taking these characteristics into consierations. However, the characterization of the optima number an ocation of pubic faciities has not been investigate competey. 1 In this paper, using a variant of the spatia ifferentiation moe of Hoteing 1929 where transport cost of a consumer who uses a pubic faciity epens on the istance between their ocations, we consier the optima number an ocation probem of pubic faciities without obigation of use. Especiay, we investigate the effect of spatia characteristics such as infrastructure for transportation an istribution of consumers to the optima number of pubic faciities. The remainer of this paper is organize as foows: Section 2 escribes the moe; Section 3 characterizes the resut; Section 4 concues. 1 A recent work by Beriant et a tacke the probem by a genera equiibrium wefare anaysis. 2

3 2 The moe There are avaiabe homogeneous pubic faciities without obigation of use within a city [0, ] where > 0. The quantity of the pubic faciity at each ocation is zero not buit or one buit. Denote a faciity ocate at x as faciity x. A set of pubic faciities, where [0, ] an < +, stipuates the number an ocation of faciities provie. It takes socia cost C to provie a set of faciities, where C : R + R is a C 2 function of n which satisfies im n 0+ C n n = 0 < C n, C n. Consumers are uniformy istribute with ensity f > 0 aong the city. Denote a consumer ocate at i as consumer i. Given a set of faciities, the consumers use one or zero unit of the faciity. Let i Σ i { } be the faciity consumer i uses for any i means that she uses nothing. Let Σ Π i Σ i be a profie such that the faciity consumer i uses is i for any i. Let m x be a measure of consumers using the faciity x given a profie. If a consumer uses a pubic faciity, she erives a surpus which epens on how many consumers access it, but bears her own transport cost which epens on the istance to it: if i given a profie, consumer i obtains v m i where v is a C 2 function of m which satisfies v m, v m 0 < v 0, but bears her own transport cost t i i 2 where t > 0. If she uses nothing, her utiity is 0. To sum up, given a set of faciities an a profie Σ, the utiity of consumer i is given by u i ; 0 if i { } v m i t i i 2 if i. Socia vauation of a set of faciities with a profie is given by V ; 0 u i ; fi 1 3

4 an socia wefare of a set of faciities with a profie is given by W ; V ; C. 2 In this paper, we assess socia wefare of a set of faciities at a Nash equiibrium ˆ Σ in the game given the set of faciities. The probem is now to maximize 2 with respect to ;. 3 The resut To begin with, it is convenient to characterize the properties of a Nash equiibrium of the game. Consier a set of pubic faciities. For any x, enote a set of consumers using faciity x given a profie as D x { i [0, ] i = x }. Denote a eft-han bounary of consumers using faciity x as x inf D x an a right-han bounary of consumers using faciity x as x sup D x. Let i, ˆ i be a profie such that the faciity consumer i uses is i an the faciity consumer j uses is ˆ j for any j except i. Lemma 1. Take any an any Nash equiibrium ˆ Σ. Take any i, j [0, ] with i < j, ˆ i an ˆ j. Then, we have ˆ i ˆ j. Proof. Suppose, on the contrary, ˆ j < ˆ i. Consier a eviation of consumer i from ˆ i to ˆ j. Then, we have u i j, ˆ i = v mˆ j j, ˆ i t i ˆ j = v mˆ j j, ˆ j t i ˆ j v mˆ i i, ˆ j t j ˆ i > v mˆ i i, ˆ i t i ˆ i t j ˆ j 2 t i ˆ j 2 2 = u i. Hence, she can increase her payoff by the eviation, which is a contraiction. 4

5 Lemma 2. Take any an any Nash equiibrium ˆ Σ. Then, D x is a non-empty interva for any x. Proof. First, we show that D x is not empty. Suppose, on the contrary, D x =. Consier a eviation of consumer x from ˆ x to x. Then, we have u x x, ˆ x = v mx x, ˆ x t x x 2 = v 0 > u x. Hence, she can increase her payoff by the eviation, which is a contraiction. Next, we show that D x is an interva. If x = x, the proof ens. Suppose x x. Take any i x, x [. Note that there exists j x, i an k i, x ] such that ˆ j = ˆ k = x. Suppose x i. Given the other consumers strategies, if consumer i uses faciity x, her utiity is u i x, ˆ i = v mx x, ˆ 2 i t i x = v m x x, ˆ k t i x 2 > v m x x, ˆ k t k x 2 k = u u k, ˆ k = u i, ˆ i. Therefore, we have ˆ i. In the case where i < x, the simiar argument aso hos. By Lemma 1, we concue that ˆ i = x. By Lemmas 1 an 2, for any = x j n j=1 where x 1 < < x n an any Nash equiibrium ˆ Σ, we have 0 x1 x1 xn xn 5

6 an a consumers in x, x use faciity x. In aition, socia vauation 1 of a set of pubic faciities with a Nash equiibrium ˆ Σ can be rewritten as V ; = x = x x x { v m x t i x 2} fi [ v m x m x 1 { 3 tf x 3 x x }] 3 x with m x = x x f. In the foowing, we characterize the optima number an ocation of pubic faciities. The characterization consists of two steps: first, we consier the optima ocation of pubic faciities given the number of them provie; next, given optima ocation, we etermine the optima number of faciities. 3.1 Optima ocation given the number of pubic faciities In this subsection, we consier the optima ocation of pubic faciities given the number of them provie. Take any n N an consier a set = x j n j=1 of pubic faciities where x 1 < < x n. Given t, v an f, et ɛ > 0 be the unique number which satisfies v 2ɛf = tɛ 2. Note that if a ength of the interva of consumers using a certain faciity is greater than 2ɛ, the farthest consumer who uses the faciity erives a surpus ess than or equa to v 2ɛf but must bear transport cost greater than tɛ 2, which impies that her utiity is negative. Thus, 2ɛ correspons to a potentia maxima ength of the interva a singe faciity can attract. Hereafter, we assume an aitiona conition on congestiabiity of faciities, summarize as: Assumption 1. v m m m=2ɛf > 0. Note that v m m represents consumers surpus from a faciity if a measure of the consumers using it is m an satisfies v m m 0. Noting aso that 2ɛf is a potentia maxima measure 6

7 of consumers using a faciity, this assumption assures that congestion is not so tragic such that the pie from a faciity is amage by congestion in a Nash equiibrium. For iustrative purposes, the foowing two exampes, where the proofs are in Appenix, are now in orer: Exampe 1. Suppose n < 2ɛ. If x 1 0, x n ɛ an x j x j 1 2ɛ for any j {2,, n}, for any Nash equiibrium ˆ Σ, we have xj = x j ɛ xj = x j + ɛ for any x j. Moreover, we have V ; = 4 3 tɛ3 fn. Exampe 2. Suppose n 2ɛ. If x 1 0, x n = an x j x j 1 = n for any j {2,, n}, for any Nash equiibrium ˆ Σ, we have xj = x j xj = x j + for any x j. Moreover, we have V ; = v f n t2 1 f. 2 Our first resut on the efficient provision probem of pubic faciities is that the ocations in above exampes are optima: Proposition 1. Take any n N an consier a set = x j n j=1 of pubic faciities where x 1 < < x n. Then, it is optima given the number n of pubic faciities if an ony if the foowing conitions are met: if n < 2ɛ, we have x 1 0, x n ɛ an x j x j 1 2ɛ for any j {2,, n}; otherwise, we have x 1 0, x n = j {2,, n}. an x j x j 1 = n Proof. Take any with = n an any Nash equiibrium ˆ Σ. Let m for any P x m x 7

8 an x x + x 2. Note that V ; v m x m x 1 x 3 tf x 3 x x x with eq. iff x = x for any x = v m x m x 1 t 12 f 2 m x 3 x n v m m 1 t 12 f 2 m 3 with eq. iff m x = m for any x m 2f = n v m ti 2 fi. m 2f Suppose n < 2ɛ. Noting that v 2ɛf = tɛ2, we have m 2f n v m ti 2 ɛ fi n v 2ɛf ti 2 fi with eq. iff m = 2ɛf m 2f ɛ = 4 3 tɛ3 fn. Suppose n 2ɛ. Then, noting that v f n t 2, we have m 2f n v m ti 2 fi n m 2f = v f n v f ti 2 fi n t2 1 2 f. with eq. iff m = f n Thus, we obtain the resut. Remember that 2ɛ is a potentia maxima ength of the interva a singe pubic faciity can attract. Thus, n < 2ɛ correspons to the case where the number of pubic faciities is reativey sma so that not a consumers access the pubic faciity no matter how it is ocate. The emma impies that the pubic faciity shou be ispose as foows: i faciities attract as many con- 8

9 sumers as possibe; ii each faciity attracts the same number of consumers; iii each faciity attracts consumers in the neighborhoo of it. 3.2 Optima number of pubic faciities In this subsection, we consier the optima number of pubic faciities given optima ocation. By Proposition 1, the probem is reuce to maximizing W n V n C n with respect to the number n of pubic faciities, where V n max { ; ˆ Σ is NE, =n} V ; 4 3 tɛ3 fn if n < 2ɛ = v f n t2 1 f otherwise. 2 For convenience, we negect the integer probem of the number of pubic faciities. Let V n 2ɛ im n 2ɛ V n n an Vn 2ɛ + im n 2ɛ + V n n. Note that Vn 2ɛ Vn 2ɛ +. Then, we have the foowing resut: Proposition 2. The number n of pubic faciities is optima ony if V n n = C n n hos. Moreover, if C n 2ɛ V n 2ɛ or Vn 2ɛ + C n 2ɛ, the converse is aso true. There are some remarks on the proposition: i note first that the equation is a variant of Samueson 1954 s conition where the pubic faciity shou be provie up to the eve at which its margina socia vauation uner the optima ocation is equa to its margina cost. ii if C n 2ɛ V n 2ɛ or Vn 2ɛ + C n 2ɛ ho, the optima number is uniquey etermine. 9

10 iii even if Vn 2ɛ < C n 2ɛ < V n 2ɛ + hos, the number of the soutions to the equation is at most two so that we can easiy fin out the optima number. Given t, v, f,, et n t, v, f, be the optima number of faciities. Then, for any θ {t, f, }, we have n t, v, f, θ = W nθ n t, v, f, W nn n t, v, f, Vnθ = n t, v, f, Vnn n t, v, f, Cnn n t, v, f, everywhere except at n t, v, f, = 2ɛ. In aition, if a tota measure of consumers keeps constant, say, f = F, the effect of the city ength to the number is n t, v, F, = V nf n t, v, F, F + V 2 n n t, v, F, Vnn n t, v, f, Cnn n t, v, f, everywhere except at n t, v, F, = 2ɛ. The resut of comparative statics is summarize in Tabe 1. Noting that the number 2ɛ of pubic faciities is the bounary of whether a consumers access the pubic faciity uner the optima ocation or not, C n 2ɛ < V n 2ɛ correspons to the case where t an are reativey ow an pubic faciities shou be provie sufficienty to attract a consumers. Instea, Vn 2ɛ + < C n 2ɛ correspons to the case where t an are reativey high an pubic faciities shou be provie sufficienty to attract a consumers. The resut of t, an with f constant, is summarize in next coroary: Coroary 1. As for the optima number of pubic faciities, the foowings are true: Tabe 1: The resut of comparative statics n t n f n n f=f C n 2ɛ < V n 2ɛ t, : sma Vn 2ɛ < C n 2ɛ < V n 2ɛ + + or + + or 0 + or Vn 2ɛ + < C n 2ɛ t, : big

11 i the effect of t to the optima number of pubic faciities is positive if t is sufficienty sma, an the effect is negative if t is sufficienty arge; ii if f keeps constant, the effect of to the optima number of pubic faciities is positive if is sufficienty sma, an the effect is negative if is sufficienty arge. Moreover, in the case of pubic faciities without congestion, iii as t increases, the optima number of pubic faciities increases at first but finay ecreases; iv if f keeps constant, as increases, the optima number of pubic faciities increases at first but finay ecreases. Coroary 1 roughy states that the optima number of pubic faciities is reativey sma when t is sufficienty ow or high, an when is sufficienty ow or high if f keeps constant. Why? Intuitivey, this reason is as foows. If t with f constant is sufficienty ow, a few pubic faciities are sufficient for each consumer to use the pubic faciity with sufficienty ow transport cost. Thus, margina socia vauation by an aitiona pubic faciity is ow if these pubic faciities have been provie, because a consumers have areay accesse the pubic faciity with sufficienty ow cost. If t with f constant is sufficienty high, margina socia vauation by an aitiona pubic faciity is originay ow, because consumers continue to bear high cost to access the pubic faciity even if one pubic faciity is ae. In summary, if t with f constant is sufficienty ow or high, margina socia vauation by an aitiona pubic faciity is ow, which impies that the optima number of pubic faciities is reativey sma. 4 Concusion In this paper, we consier the optima number an ocation probem of pubic faciities without obigation of use. There remains severa probems: i in this paper, we confine our attention ony to an uniform istribution of consumers to simpify the anaysis. However, of course, their istribution is not uniform in actuaity so that an anaysis with more genera istribution of consumers is require; ii in ong run, ocations of pubic faciities are consiere to be reate to 11

12 the an market see Fujita 1986 an Sakashita 1987, an the number is aso thought to be reate. Thus, an anaysis which incue the infuence of an market is aso important. Appenix In this appenix, we prove the statements in Exampes 1 an 2. To prove them, the foowing properties, which are intuitivey cear, are usefu: Lemma 3. Take any = x j n j=1 where x 1 < < x n an any Nash equiibrium ˆ Σ. Then, the foowings are true: i u x x, ˆ, u x x, ˆ x 0; x ii x < x ; iii u xj = u xj+1 for any j {1,, n 1}; iv x1 0 u x1 = 0, xn u x n = 0 an xj+1 xj u xj = 0 for any j {1,, n 1}. Proof. Omitte. Property i states that if consumers at bounaries of those using a faciity use the faciity, they can obtain non-negative payoff. Property ii states that a measure of consumers using a faciity is positive. Properties iii an iv state that the utiity of a consumer at a right-han bounary of those using a faciity an that of one at a eft-han bounary of those using the right next faciity are zero if they are not in the same ocation. Proof of Exampe 1 Proof. Take any Nash equiibrium ˆ Σ. We first show that x1 = x 1 + ɛ. Suppose x 1 + ɛ < x1. If consumer x 1 ɛ uses faciity x 1, her payoff is u x 1 ɛ x 1, ˆ x 1 ɛ = u x 1+ɛ x 1, ˆ x 1+ɛ > u x1 x 1, ˆ 0, x1 12

13 which impies that x1 x 1 ɛ an m x1 > 2ɛf. However, if consumer x1 uses faciity x 1, her payoff is u x1 x 1, ˆ x1 = v m x1 x 1, ˆ t x1 2 x x1 1 < v 2fɛ tɛ 2 = 0, which is a contraiction. Suppose x1 < x 1 + ɛ. If x 1 ɛ < x1, we have m x1 < 2ɛf. eviation of consumer j x 1 ɛ, x1 from ˆ j = to x 1. Then, we have Consier a u j x 1, ˆ j = v mx1 x1, ˆ j t j x1 2 > v 2fɛ tɛ 2 = 0 = u j, ˆ j = u j, which is a contraiction. Thus, we must have x1 x 1 ɛ. Then, we have u x1 u x1 x 1, ˆ x1 > u x1 x 1, ˆ 0. x1 Therefore, x2 = x1 < x 1 + ɛ x 2 ɛ. Moreover, 0 u x2 x 2, ˆ = v m x2 x2 x 2, ˆ t x2 x2 2 x 2 < v mx2 tɛ 2, which impies that m x2 < 2fɛ. Therefore, x2 < x 1 + 3ɛ x 2 + ɛ. Then, we have u x2 u x2 x 2, ˆ x2 > u x2 x 2, ˆ 0. x2 Therefore, x3 = x2 < x 2 + ɛ x 3 ɛ. Moreover, 0 u x3 x 3, ˆ = v m x3 x3 x 3, ˆ t x3 x3 2 x 3 < v mx3 tɛ 2, which impies that m x3 < 2fɛ. Therefore, x3 < x 2 + 3ɛ x 3 + ɛ. Iterating simiar argument yies xn < x n ɛ an xn < x n + ɛ. However, consier a eviation of 13

14 consumer x n + ɛ from ˆ x n+ɛ = to x n. Then, we have u xn+ɛ x n, ˆ x n+ɛ > u xn x n, ˆ 0 = u xn+ɛ, ˆ x n x n+ɛ = u xn+ɛ, which is a contraiction. Thus, we concue that x1 = x 1 + ɛ. Next, we show that x1 = x 1 ɛ. Suppose x1 > x 1 ɛ. Then, we have m x1 < 2fɛ. Consier a eviation of consumer j x 1 ɛ, x1 from ˆ j = to x 1. Then, we have, u j x 1, ˆ j = v mx1 x1, ˆ j t j x1 2 > v 2fɛ tɛ 2 = 0 = u j, ˆ j = u j, which is a contraiction. Suppose x1 < x 1 ɛ. Then, we have m x1 > 2fɛ. Consier a eviation of consumer j x1, x 1 ɛ from ˆ j = x 1 to. Then, we have u j, ˆ j = 0 = v 2fɛ tɛ 2 > v m x1 x1, ˆ j t j x1 2 = u j x 1, ˆ j = u j, which is a contraiction. Thus, we concue that x1 = x 1 ɛ. Iterating simiar argument for x 2,, x n, we obtain the resut. Proof of Exampe 2 Proof. Take any Nash equiibrium ˆ Σ. We first show that x1 = x 1 +. Suppose x 1 + < x1. If consumer x 1 uses faciity x 1, her utiity is u x 1 x 1, ˆ > u x1 x x 1 1, ˆ x1 0, which impies that x1 = x 1. Note that v m x1 x1, ˆ j t j x1 2 = u j x 1, ˆ j u j x 2, ˆ j = v mx2 x2, ˆ j t j x2 2 14

15 for j D x1 x 1 +, x1 ], which impies m x2 > m x1 > f n. Thus, x2 > x 2 +. Note that v m x2 x2, ˆ j t j x2 2 = u j x 2, ˆ j u j x 3, ˆ j = v mx3 x3, ˆ j t j x3 2 for j D x2 x 2 +, x2 ], which impies m x3 > m x2 > f n. Thus, x3 > x 3 +. Iterating simiar argument yies x n + n < xn, which is a contraiction. Suppose x1 < x 1 +. Then, we have m x 1 < f n. If x 1 < x1, consier a eviation of consumer j x 1, x1 from ˆ j to x 1. Then, we have u j x 1, ˆ j = v mx1 x1, ˆ j t j x1 2 > v 2 f t 0 = u j, ˆ n j = u j, which is a contraiction. Thus, we must have x1 = x 1. Then, we have u x1 u x1 x 1, ˆ x1 > u x1 x 1, ˆ 0. x1 Therefore, x2 = x1 < x 1 + = x 2. Moreover, v m x1 x1, ˆ j t j x1 2 = u j x 1, ˆ j u j x 2, ˆ j = v mx2 x2, ˆ j t j x2 2 for j D x2 [ x2, x 2, which impies that m x2 < m x1 < f n. Therefore, x2 < x = x 2 +. Then, we have u x2 u x2 x 2, ˆ x2 > u x2 x 2, ˆ 0. x2 15

16 Therefore, x3 = x2 < x 2 + = x 3. Moreover, v m x2 x2, ˆ j t j x2 2 = u j x 2, ˆ j u j x 3, ˆ j = v mx3 x3, ˆ j t j x3 2 for j D x3 [ x3, x 3, which impies that m x3 < m x2. Therefore, x3 < x = x 3 +. Iterating simiar argument yies xn < x n xn < x n +. However, consier a eviation of consumer x n + from ˆ = to x x n + n. Then, we have an u x n+ x n, ˆ > u x n x x n + n, ˆ 0 = u x n+, ˆ = u x n x n + x n+, which is a contraiction. Thus, we concue that x1 = x 1 +. Next, we show that x1 = x 1. Suppose x1 > x 1. Then, we have m x1 < f n. Consier a eviation of consumer j x 1, x1 from ˆ j = to x 1. Then, we have u j x 1, ˆ j = v mx1 x1, ˆ j t j x1 2 > v 2 f t 0 = u j, ˆ n j = u j, which is a contraiction. Thus, we concue that x1 = x 1. Iterating simiar argument for x 2,, x n, we obtain the resut. 16

17 References [1] Beriant, M., S.K. Peng an P. Wang, 2006, Wefare anaysis of the number an ocations of oca pubic faciities, Regiona Science an Urban Economics 36, [2] Fujita, M., 1986, Optima ocation of pubic faciities: area ominance approach, Regiona Science an Urban Economics 16, [3] Hoteing, H., 1929, Stabiity in competition, Economic Journa 39, [4] Sakashita, N., 1987, Optimum ocation of pubic faciities uner the infuence of the an market, Journa of Regiona Science 27, [5] Samueson, P.A., 1954, The pure theory of pubic expeniture, Review of Economics an Statistics 36,