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Infrastructural competition among jurisdictions
Journal
of Public
Economics
49 (1992) 241-259.
Infrastructural jurisdictions
North-Holland
competition
among
Leon Taylor* Economics
Received
Department,
Tulane
University,
May 1990, final version
New Orleuns,
LA 70118-5698, USA
received June 1991
This paper models jurisdictions that compete for an industry by building infrastructure more rapidly than their identical neighbors. Such competition can waste resources. Lucrative ‘prizes’ the tax revenues and the value of jobs that an incoming firm would provide - stimulate infrastructural spending and waste. Low initial levels of infrastructure discourage infrastructural competition. The paper urges a cautious approach toward federal subsidies of local expenditures on infrastructure.
1. Introduction
Consider an executive who must decide where to build, relocate or expand his plant. Suppose that his decision hinges upon the site’s infrastructure - for instance, its roads or utilities. Then he might resolve the matter in one of three ways. (1) He might pick one site and bargain with that jurisdiction’s officials over the facilities that he needs. Ideally, such bargaining is Coasian. (2) He might tell the officials of several jurisdictions that he will move to the first one that completes the infrastructure that he needs. This might provoke a spending race among the jurisdictions, providing him with roads and utilities quickly and cheaply. (3) He might simply pick the site that already offers the best infrastructure. In the last two scenarios, jurisdictions have reason to engage in rentseeking expenditures on their infrastructures. Consider why. In the second scenario, several jurisdictions race to complete their infrastructures, but only one will get the plant; the others must find less valuable uses for their roads and utilities. In the third scenario, jurisdictions compete to build infrastrucCorrespondence to: L. Taylor, Economics Department, Tulane University, 206 Tilton Hall, New Orleans, LA 70118-5698, USA. *I derived this paper from my dissertation, supervised by Wallace E. Oates at the University of Maryland, College Park. The paper was improved by Oates, Robert M. Schwab, seminar participants at the University of Maryland, Tulane University and Resources for the Future, and two referees. Greta Whitt provided technical help. Tulane’s economics department provided financial help. All errors are mine.
0047-2727/92/$05.00
0
1992-Elsevier
Science Publishers
B.V. All rights reserved
242
L. Tuylor,
Competition
among jurisdictions
ture today in hopes of attracting lucrative industries tomorrow, although they might not be sure of the exact nature or value of those industries. Again, the losers must find less valuable uses for their infrastructures. In sum, when jurisdictions compete for industry by building infrastructure, they might well waste resources. I do not want to overestimate this waste. If a jurisdiction builds a $100 million road in a contest that it eventually loses, it has not wasted $100 million. Some motorists will enjoy the road. I argue only that this enjoyment probably will not be worth $100 million. For if it were, and if officials of the jurisdiction were in the habit of maximizing constituents’ welfare, then they would have already built the road for that purpose. Infrastructural competition is little explored by the literature on jurisdictional contests. In fact, many analyses [e.g. Fischel (1975) Oates and Schwab (19SS)] assume two things. (1) More than one jurisdiction can attract a share of a fixed national stock of physical and divisible capital. Such ‘winner-take-some’ contests can enhance efficiency. Every locality can gain a bit of capital. Every locality can win. (2) The contest itself consumes no resources. Each jurisdiction vies by offering inducements that take effect only if the firm chooses to relocate there. Rent-seeking is not an issue. However, descriptive works on local economic development depict jurisdictional competition more starkly [Fosler (1988)]. Localities often vie for one big project, such as the supercollider or the assembly plant for the innovative Localities often commit resources to the Ford automobile, the Saturn.’ contest before the firm decides where to move. Jurisdictions might prefer to compete by building infrastructure, rather than by lowering taxes, because the former is a more powerful inducement. Infrastructure is crucial to economic development at several stages [Hirschman (1958)]. Transforming an economy based on resources into one that produces goods will require roads, ports and utilities. Transforming an economy that produces goods into one that produces information will require telecommunications [Lakshmanan (1989)]. A conservative way to gauge the allure of infrastructure is to look at the role of goods-oriented infrastructure, such as highways, in an informationoriented economy, such as the United States. In a cross-sectional study of 3,000 U.S. countries, Mills and Carlino (1989) calculate the long-run elasticities of total employment with respect to three policy instruments: local taxes per capita; the stock of local industrial development bonds; and the density of interstate highways. The tax and bond elasticities are small and ‘Apparently they hope that - through economies of agglomeration - the project will attract an expanding industry. For instance, the arrival of Motorola in Phoenix, Arizona, led to a burgeoning electronics industry there.
L. Taylor, Competition among jurisdictions
243
insignificant. The highway elasticity is large and significant. Doubling the highway density - say, by building a second highway - leads to a 54 percent increase in total employment. Infrastructural contests often focus on new-technology industries, such as measurement and control instruments, clocks and watches, and aircraft. New-technology firms often seek areas with many workers who are highly skilled. In turn, these workers are attracted by educational and cultural amenities. So jurisdictions might provide these amenities. Infrastructure used in research and development, such as university institutes, might also attract firms. In the Netherlands, Nijkamp (1987) finds evidence that proximity to R&D infrastructure is significant in explaining where new-technology firms locate. Amenities are not significant. Neither is the supply of venture capital. These results are consistent with the idea that infrastructural competition might be particularly fierce in small countries. The locational variation in wages or venture capital is too small to affect where firms locate. Infrastructural competition seems particularly vivid in the port cities of the industrialized world, such as London, Marseille, and Rotterdam. As ships have become larger, and shipping-related activities more specialized, dock activities have moved out of the central cities and onto more spacious sites. The inner cities have declined with the old docks. Since the 1960s public officials have tried to revitalize the waterfronts by building infrastructure to attract corporate headquarters, publishing, and tourism. In 1987, a quasigovernmental agency completed a light railway to the Docklands area of East London to try to attract white-collar activities [Hoyle et al. (1988)]. In sum, infrastructural competition merits economic analysis. This paper develops a dynamic model of long-run, resource-consuming competition for one project under uncertainty.2 Jurisdictions compete for the project by developing their roads, utilities, facilities, schools, and colleges. The faster a jurisdiction spends money, the faster it develops its infrastructure. This boosts its chances of winning the industry. Each jurisdiction seeks the rate of spending that will maximize its expected gain from competing. We want a sense of how much waste can occur and under what conditions. So this paper analyzes a case that seems particularly likely to engender waste: identical jurisdictions that can spend as much as they want to build infrastructure that (for the losers) has no alternative value.3 ‘They are based on Kamien and Schwartz’s (1982) models of innovation in the private sector. 30ne justification of a limitless budget is that the federal government often has subsidized infrastructural competition, particularly in site development and in construction of wastewater treatment plants. However, one can introduce this budget constraint into the problem, where Y(t) gives tax revenues: Y’(t)=rY(t)-X(t)
/ Y(O)= YO.
One can show that, even under
this constraint,
the jurisdiction
will undertake
the infrastructural
244
L. Taylor,
Competition
among jurisdictions
Section 2 develops the model. Section 3 presents comparative statics. If a lucrative industry is at stake, then jurisdictions will spend more, at any given moment, to try to attract it. But if the jurisdictions begin with poor infrastructures, they will spend less, at any given moment, on improving them. Section 4 calculates the amount of waste that occurs. When competition is quite keen, the waste exceeds the value of the industry for which jurisdictions are vying. Section 5 presents conclusions and reflections.
2. The model A total of IZ identical jurisdictions plan to compete for an industry by building infrastructure. At time t, a jurisdiction’s level of infrastructure is Z(t). The initial level is Z(0) =Z, 20. The infrastructure does not depreciate: dZ/d t 2 0. The jurisdiction will win the industry if it is the first to develop its infrastructure to level A. The jurisdiction plans to achieve this by time 7: when it plans to end the project. The jurisdiction’s problem is to pick a construction plan, Z(t), and a completion date, z that will maximize its expected payoff from the project. The jurisdiction will spend money on its infrastructure at a rate X(t). Z(t) is the capital stock; X(t) is the flow of investment dollars. X(t) is the proxy for all the inputs that the jurisdiction buys, such as labor and machines. The infrastructural production function has the general form Z(t)=f[X(t)],
df/dtzO.
For sharp results, I will specify the functional form of (1). If a jurisdiction wants to build its infrastructure faster, it will have to spend money on it faster. A sensible cost function is X(t) = Q(t)” 1Q(t) =dZ/dt.
Here, a> 1. If as 1, then the jurisdiction might as well build its infrastructure instantly. It is more reasonable to think that marginal cost rises as the jurisdiction tries to build its infrastructure faster. Here are three reasons.4 project in a wide variety of cases. The assumption that the infrastructure has no Generally, infrastructure might have alternative ‘adaptable’ if it can serve other purposes with highway which residents can use as well as firms. down and reconstitute it into capital for other laboratory for the general chemistry classes. 41 have drawn upon Scherer (1967).
alternative use best describes sports facilities. uses if it is adaptable or malleable. It is only minor changes, for example an urban Infrastructure is ‘malleable’ if you can break it purposes. You can pillage a biotechnological
L. Taylor, Competition among jurisdictions
245
(1) Participants have less time to expand their information set. Building a capital project, such as a highway, is heuristic. What employees learn at stage I can cut costs at stage II. Speeding up construction leaves employees with less time to gather and assess information. For instance, if the jurisdiction is in a hurry to build a bridge, it might skip the prototype, despite the eventuality of higher construction costs. (2) Embarking on several plans at once will save time but incur costs. The state will simultaneously study several possible routes for the new highway, since it will not have time to launch a second study should the first route fail to pan out. Duplication and waste seem inevitable. (3) The marginal products of purchased inputs decline for conventional reasons. To build the highway faster, the state can employ more people at any given time. More laborers will line the ditches; more surveyors will crowd around the quadrant. The marginal product of labor will diminish. So will the marginal product of money spent on labor. The model’s basic results hold for any value of a greater than 1. However, for concreteness, let a = 2. Invert (2) to obtain a production function: Q(t) =X(t)+. The boundary
conditions
Z(O)= zo,
are
Z(T)=A.
Some scenarios will let Z, vary from one jurisdiction to another. The jurisdiction that wins the project will gain R dollars. This represents the present value of net real personal earnings and tax revenues, discounted to T.’ R reflects the possibility that some earnings come from added work effort at the expense of leisure. 6 Furthermore, the jurisdiction knows R. In that sense, the analysis corresponds to the second scenario of the introduction. The discount rate is r. Suppose that the jurisdiction could be certain of winning the project. Then the value of its efforts would be
5R connotes that the jurisdiction knows for sure the project’s value. In reality, jurisdictions often build industrial parks and roads without knowing precisely the type or value of industry that would be attracted. This source of uncertainty is vital. But I ignore it to focus on the jurisdiction’s uncertainty over who will win the contest. Uncertainty over the winner characterizes most infrastructural contests, even those where the jurisdictions know the value of the industry at stake. Modeling this type of uncertainty gets top priority. Nevertheless, extensions of the model should consider other types of uncertainty. 6Assume that only current residents provide the additional work offered by the courted industry. The relocation of the industry causes no labor immigration.
246
L. Taylor,
Comperition
among jurisdictions
e -“R-;e-r’X(t)dt.
(5)
0
In truth, the jurisdiction cannot be certain of winning the industry. A rival might develop its infrastructure to level A first. So before the jurisdiction decides how much to spend on the race, it must estimate its chances of winning. I assume that the jurisdiction is well informed about the prize and the nature of the race. In fact, it knows the probability distribution that governs the race. I will now characterize that distribution. The probability that the jurisdiction will lose in the next moment, given that it has not already lost, is a non-negative constant, h. F(t) is the cumulative probability that the jurisdiction will lose the race by time t. Then F’(t)/[ 1 -F(t)]
= h.
.(6)
A hazard rate, h, expresses the keenness of the race. The greater is h, the greater the chances that a rival will beat the jurisdiction in the next moment.7 I will later discuss the determinant of h. The probability that the jurisdiction will lose the race right off the bat is zero: F(0) = 0. With this boundary condition, integrate (6) to recover F(t): F(t)= 1 -emh*. Subject
(7)
to (3) and (4), the jurisdiction
emrTR[l -F(T),-je-“X(t)ll-F(t)ldt.
will maximize
(8)
0
The expected gain of competing is the present value of the prize, eprTR, times the cumulative probability that the jurisdiction will win, 1 -F(T). The
‘1 assume a constant hazard rate because it simplifies the analysis and because I find no compelling empirical reason to assume a nonconstant rate. However, the model can handle a more general probability distribution, the Weibull function. Let F(r)=l-exp(-bH(s)ds), H(t) = h(t), u(t)=wtw-‘.
As a special case, I have set w= I to obtain the exponential distribution. I use this function because, without more information about infrastructural competition, it seems reasonable to pick a probability distribution that well describes a variety of social processes. The reader might ask why the model must use a specific probability distribution. There are two reasons: it sharpens the analysis, and it paves the way for simulation and empirical studies.
241
L. Taylor, Competition among jurisdictions
expected cost of competing at time t is the present value of the amount spent that the race has not yet on the infrastructure, e -“X(t), times the probability ended, 1 - F(t). The keenness of the competition depends on the number of entrants, n: h = h(n),
dh/dn > 0.
(9)
In the absence of barriers, rivals will enter the race until the expected gain of doing so goes to zero. Let n* denote the number of jurisdictions in a competitive equilibrium. The jurisdiction will use h*= h(n*) in preparing its construction plans. Via the calculus of variations, the solution yields’ Z(t)=c[e”‘-l]/u+Zo,
05tsT.
(10)
Here, u= h+r, and c is a constant to be determined. In addition to the Euler equation, the optimal spending the transversality condition, which yields:
path must satisfy
X(T)=R[F,(T)/(l-F(T))+r].
(11)
Last-minute spending relates directly to the reward and to the probability that the jurisdiction will lose at time T, given that it is still in the race. The optimal completion date is T = -In
{ 1 - AO(v/R)*}/v,
(12)
where A, = A-Z,. A, is all the infrastructure that the jurisdiction build to win the industry. Call A, the ‘infrastructural deficit’. The optimal rate of spending is X(t) = [I@ - AOut]'u e’“‘. To interpret (13), return for positive numbers,
to (12). Since the natural
R>(A,)2(r+h).
must
(13) log function
is defined
only
(14)
When (14) does not hold for some t in [0, T], then X(t)=O. The inequality in (14) suggests that the jurisdiction might balk at competing if the required
‘The appendix
derives eq. (10) and others
248
L. Taylor, Competition among jurisdictions
effort is high; if the rivalry is stiff; or if the opportunity cost of spending rather than saving is high. These considerations might help explain why poor jurisdictions, such as Kentucky and West Virginia, invest little in their roads and colleges, despite their need for economic development: they would have to undertake a huge effort to attract industry. There is also a hint that - since h depends on the number of potential rivals - infrastructural competition might occur more often among states than localities. Sufficient conditions. Let G[Q(t), t] be the integrand in (8). A choice of [X(t), T] that satisfies (12) and (13) is locally optimal, since Go, 2 0 for all t.9 As the jurisdiction tries to build more hastily, the drain on its coffers grows faster. So its marginal expected profit from the project does not rise.
3. Comparative
statics
3.1. Infrastructural
deficit
Many firms make location decisions in two stages. First, they pick a region; then they pick a jurisdiction in that region. But the level of infrastructure often varies by region. In the Southeast of the United States, the infrastructural deficit seems large. How would regional differences affect spending on infrastructure? How much will Southeastern states spend on schools and roads to attract industry? The answer is not obvious. A jurisdiction with a large deficit might compete half-heartedly to minimize the costs of losing the race: dT/dA, >O. On the other hand, a jurisdiction with a small deficit might also ‘take it easy’, because it does not have far to go to complete the race. Let us resolve the matter. Differentiate (12) and (13): (15) dX/dA,
= -2v*[R*-
A,u*]ue2”‘<0.
The bigger the deficit, the less the jurisdiction will spend at infrastructure - and the less inclined to pursue a crash program. These results suggest that some regional economies will backward. Suppose that the jurisdictions within a region are suppose that one region (the ‘North’) has more infrastructure region (the ‘South’). Northern jurisdictions will continually region’s average level of infrastructure by competing avidly with Southern jurisdictions will rarely compete with one another, so 'G
PQ=
-2e-"[l-F(t)]~o,
OZ;tST.
(16) any time on long remain alike. Also than another add to the one another. they will add
249
L. Taylor, Competition among jurisdictions
little to what they North to the South.
have.
Industries
seeking
infrastructure
will
prefer
the
3.2. Reward Many jurisdictions go after very large, very visible projects. strategy affect infrastructural spending? Compute the statics: dT/dR=
-{A,~/2(o/R)~R~}{l/u[l
How does that
-A&/R)+}
(17)
dX/dR = CR* - A,&] R - *Ue’“’ > 0. The bigger the reward, complete its infrastructure
(18)
the harder the jurisdiction more quickly.
will compete,
trying
to
3.3. The discount rate and the degree of competition The discount and hazard rates express certain opportunity costs of infrastructural spending. The discount rate reminds us that the jurisdiction could have invested its money in other pursuits (which I have not explicitly modeled). The hazard rate reminds us that the jurisdiction might be throwing away its money. How might these parameters affect the jurisdiction’s spending plan? It is tempting to assume that a rise in r or h must cause the jurisdiction to spend less on infrastructure, since it seems to signal higher opportunity costs of spending. However, matters are not so simple. A rise in h might dissuade the jurisdiction from attempting the race - or it might prod the jurisdiction to pick up its pace. The jurisdiction is like a runner at the starting line who glances around him and finds that the field is unexpectedly strong. He might pull up his starting blocks and walk away from the race; or he might dig in his heels. Much depends on the pain and the prize that he expects from the race. To say more than this, we must undertake the comparative statics of u. In this matter, modern control methods yield cleaner results than the calculus of variations. Form the Hamiltonian H[Z(t),
X(t), Tp;r,h,t]
= -(e-“X(t)[l s.t. Z(0) =
-F(t)]
+pX(t)*
(19)
z,,
Z(T)=A. Z(t) is the state variable.
X(r) is its control.
The optimality
condition
is
250
L. Taylor, Competition
dH/dX= The multiplier
among jurisdictions
-e-“[l-F(t)]+p/[2X(t)*]=O, equation
- dH/dZ
(20)
is
= 0 = dp/dt.
(21)
Here, p is the expected value of the discounted marginal cost of building the infrastructure at time t. The jurisdiction will equate this marginal cost at each moment. This will minimize the expected expense of building the infrastructure to level A over the selected planning period. The state equation is dH/dp = X(t)+ = dZ/dt = Q(t). Solving
(20) for X(t) yields: X(t)*=pe”/[2(1
From
(22)
-F(t))].
(21), p/2 is a constant
- call it k. Manipulating
Z(t) = k/v e”* + d,
(23) (23) yields: (24)
where d is another constant. Eq. (24) is equivalent to (10). Since the Hamiltonian and the calculus of variations yield equivalent solutions to the problem, we might borrow from both approaches to analyze the effects of u. Solve explicitly for p by equating (23) to an expression for Q(t) derived from (13): p = 2&[R’
- ,Q+],
dP/dv = [R+ - 2A,vt]/vb.
(25) (26)
If the reward, R, is large enough, then d,u/dv>O: a rise in the probability of losing a lucrative race will spur the jurisdiction to spend more. If the infrastructural deficit, A,, is large enough, then dp/dv ~0: a rise in the probability of losing an already-bleak race will cause the jurisdiction to hedge its bets. In particular, when A, approaches its upper bound of (R/u)*, then dp/dva
- (Rv);.
(27)
When the jurisdiction has a huge deficit, then a small rise in the probability of losing might well trigger a large cut in planned expenditures.
L. Taylor,
Competition
251
among jurisdictions
3.4. An interpretation One can tell a story that is consistent with these diverse elements. Suppose that West Virginia vies with other states that have large infrastructural deficits. If the reward is large, then West Virginia might compete wholeheartedly. Otherwise, it will spend little on its infrastructure, and it will not hurry to completion. Broadly, these results also hold as West Virginia’s probability of losing rises (dp/dv > 0 when R is large, but dp/dv O).”
4. Does jurisdictional
competition
waste resources?
It is in the firm’s interest to goad jurisdictions to race one another in developing sites. But is it in society’s interest? Consider the amount of waste generated by the contest. Summed over all competitors, spending is
S(n,T,R,Ao,v)=njem”X(t)dt 0
=[n(Ri-A,vi)2v/(2v-r)][e(2”-“TDefine ‘waste’ as the aggregate value jurisdictions expended on the contest:
of the
11. resources
(28) that
the
~(~,A,,7;R,n,r,h)={[(n-l)(R~-A,v~)~v]/(2v-r)){e’~“-“~-l).
losing
(29)
When, if ever, does aggregate spending exceed the reward, which (by hypothesis) is also the social value of the project? More formally: Under what conditions does the following expression hold? (30) One can show that (30) holds if n +co or T-co. If the competition industry is keen enough or long enough, then it will produce a net loss.
for
Industrial policy From (29) we can minimize waste by setting n= 1. The social planner would pick the jurisdiction to receive the project and would specifiy the “This
derivative
comes from (11). using (15).
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L. Taylor, Competition
among jurisdictions
completion date for the infrastructure. The other contenders could save their money. This justification for an industrial policy is naive if the social planner responds to political pressure. The jurisdictions would shift resources from producing infrastructure to producing political pressure. Waste would occur in a different garb - lobbying, advertising, etc. Whether we choose competition or industrial policy depends on which process generates less waste. More important, an unlimited industrial policy seems undesirable - and a limited one seems unworkable. We do not want the social planner to direct all locational decisions. For instance, many locational decisions hinge on labor costs. We would prefer that markets drive those decisions by generating information in the form of wages. We want the planner to direct just the locational decisions that depend on infrastructure. But how can we separate these decisions from the others?
5. Conclusions The jurisdiction’s rate of infrastructural spending depends on the reward that it expects from the courted industry. A large reward might induce many jurisdictions to build up their infrastructures, although the industry can use only one of them. The amount of resulting waste depends on how many jurisdictions compete and on how long they compete. If the contest is very long, or if it involves oery many contestants, then it can produce a net social loss. However, the model analyzes a case that seems likely to produce social waste - the case of identical jurisdictions that compete by building infrastructure with no alternative value. Certain more realistic cases might produce less waste. Suppose that a jurisdiction knows that it has less infrastructure than its rivals. Then it knows that its chances of winning are lower, and it is less likely to compete. Infrastructural competition among jurisdictions depicts the big tradeoff between equality and efficiency. When all regions start with the same infrastructure - start on a level playing field - then competition might waste substantial resources. But when regions differ in their initial infrastructures might increase when the field favors some players - then the contest inequality. Recall that dX/dA,, ~0: the weak regions drop out, and the spoils go to the strong. On the other hand, this outcome reduces waste: at least the weak regions are not throwing their money away. Such issues matter only if infrastructural competition is important. However, Peterson (1984) concludes that net investment in structures by states and localities in the United States began falling precipitously in 1970. It approximated zero in the early 1980s and triggered popular fears of an ‘infrastructural crisis’. So why might a competitive model of infrastructural spending pertain to current policy?
253
L. Taylor, Competition among jurisdictions
One reason: infrastructural spending does not always show up on the jurisdiction’s books. The appropriations committees in the U.S. House and Senate dole out $1 billion a year to localities in pork-barrel projects notably high-technology centers for universities and Economic Development Administration grants - that enable localities to compete for economic activity. Typically, these grants are not subjected to merit review. Instead, they result from log-rolling in conferences between leaders of the two committees [Morgan (1989)]. Log-rolling can represent a Pareto improvement for the log-rollers. But the instruments of log-rolling (such as university centers) can waste national resources if they are built without a compelling need. Let us reconsider infrastructural projects that are funded mainly by the localities that would use them. A steady decline in net aggregate investment implies that competitive spending is not increasing relative to capital depreciation. I speculate that losers of infrastructural contests eventually turn to tax competition. They can offer lower taxes than other jurisdictions since they do not have to recoup infrastructural costs. They will let their excess bridges and schools fall apart. Empirical work might focus fruitfully on whether jurisdictions build infrastructure to accommodate recent growth or to lure new growth. If competitive spending is large, then we might beware of proposals to subsidize it. If accommodative spending is large, and if infrastructural decay is due more to a region’s poverty than to the misfortunes of competition, then we might view subsidies more sympathetically.
Appendix Equation paper.
A.I.
numbers
without
the prefix
A refer to equations
in the main
Deriving (10)
max e ~‘TR[l-F(7)1-~e~“CQ(1)~2CI-F(t)~dt. 0 (Z(O.T) Let G[Z,(t), t] denote the Euler equation
the integrand
of (A.l).
64.1)
The optimal
Z(t)
G, = dG,/dt. Since Z(t) is not an independent argument in G, the left-hand is zero. The right-hand side is computed from (A.l):
must
satisfy
(A.2) side of (A.2)
L. Taylor, Competition
254
d{ -2eP”Q(t)[1 Integrate
among jurisdictions
-F(t)]}/dt=O.
(A.3)
both sides: -2e-*‘Q(t)[l Q(t)=
-F(t)]
=cl,
-cIer’/2[1-F(t)]=c,e*‘/[l-F(t)],
(A.4)
c2 = -c,/2. Integrate
both sides of (A.4):
(A.9
Z(t) + cj = 1 c1 e*‘/[ 1 - F(t)] dt. Substitute
(7) into (AS): Z(t)+c,
=jc2e”/emh’dt,
Z(t)+c,=S~,e”‘~“dt, v=r+h,
Z(t) + c3 = j c2 e”’ dt,
(-4.6)
Z(t) + cj = c2 e”‘/v + c.+, cg=cq-CJ.
Z(t)=~~e”*/v+c,,
Solve for the constant
c5 by using Z(0) =Z,:
Z(O)=Z,=c,/u+c,, (A.7) cg = z, - c2 Jv. Simplify (A.7), suppressing Z(t)=c[e”‘-
the subscript
in cz:
l]/u+Z,.
(10)
A.2. Deriving (11) In addition to the Euler equation, the transversality condition G-Q*dG/dQ+H,=O,
at 7;
the optimal
spending
path must satisfy
(A8
L. Taylor, Competition
where H(T) =e-‘TR[l
-F(T)].
dG/dQ=2e-“[Q(T)][l
among jurisdictions
Let us compute
the components
255
of (A.8):
-F(T)],
-Q*dG/dQ=2e-“[Q(T)]2[1-F(T)],
(A.9)
H,= -re-‘TR[l-F(T)]-Ft(T)e-‘TR, H,= -e-rTRIFI(T)+r(l-F(T))]. Substitute
(A. 10)
(A.9) and (A.lO) into (A.8) and simply:
F,(T)/(l-F(T))+r=h+r=v, ce “T = (Ru)‘.
(A.1 1)
To solve for 7: we must also solve for c. Solving for two unknowns two independent equations. We have one in (A.ll). For the second, (A.7) with the boundary condition Z(T) = A: Z(T) = c[eVT - I]/0 + Z, = A.
requires combine
(A.12)
Solve (A.12) for c: c = [A -Z,]
u/[euT - l] = A,u/[euT - 11,
(A.13)
A,-A-Z,.
From (A. 11): e “T = (Ro)+l/c, e “‘Substitute
1 = (Ru)+/c - 1 = [(Ru)+ - cl/c.
(A.14) into (A.13) and solve again for c:
(A.14)
256
L. Taylor, Competition among jurisdictions
c = A,u/[euT-
l] = &v/([Rv)f
1 = A,v/[(Rv)f
- c],
-cl/c,
(A.15) (Ru)’ -c = A& c = (RLp - A,v. To solve for T, substitute
(A.15) into (A.7) and take logs:
ce “T =(Ru)“, [(Ru)* - A,v) evT =(Ru)*, e “T= (Ro)+/[(Ru)+ - A,u] = { [(Ru)+ - Aou]/(Ru)+} - I,
(12) evT = { 1 - A(u/R)*}
‘,
UT= -ln{l-A,(u/R)*}, T = -In
{ 1- Ao(u/R)f}/u.
A.3. Deriving (13) With T in hand, use (4):
we need only solve for X(t).
X(t)=Q(t)*=[~e”‘]*=c*e*“~. Substitute
Differentiate
(10) again
and
(A.16)
(A.15) into (A.16):
(13) X(t) = [R* - AOu~]2ue2”t.
A.4.
Deriving (24)
Rewrite p/2 as a constant X(t)*= Substitute
ke”/(l
-F(t)).
(22) into (A.17):
k: (A.17)
L. Taylor, Competition
among jurisdictions
257
Q(t) = k e”/( 1 -F(t)). Since F(t)=
1 -emhI,
Q(t) = k e”‘, Integrate
(A.18)
v=r+h.
(A. 19)
both sides of (A.19): Z(t) = k/v cut + d.
A.5. Deriving
Summed
(24)
(28)
over all competitors,
spending
is
S(n,T,R,A,,v)=nie-“X(t)dt 0
=nv[Rf-A,vt]2~e(2”~r)‘dt. 0
To solve the right-hand
integral,
substitute
u =(2v-
r)t:
(Zu-r)T
e(2”Pr”dt=
1/(2v-r)
s
e”du=[1/(2v-r)]
[e(2”-r)T-
11,
0
S(n, 7; R, A,, v) = [n(R” - A,v~)~v/(~v-
A.6.
Deriving
Under
results from
what conditions
[eT](2”-r)>
Using (12):
reprT
11.
(28)
(30)
does the following
[n(R~-Aov~)?v/(2v-r)][e’2”-r’Te(2umr)T>
r)] [e(2”-r)T-
R(2v - r)]/[nu(R+
[eCrTR(2v -r)]/[nv(R*
expression
hold?
l] >eprTR, - Aouf)‘]
+ 1,
- AOv*)‘] + 1.
(30)
L.
258
Taylor,
Competition
among jurisdictions
> [eerTR(2u - r)]/[nu(R* - &I~)~] + 1, {exp( -In >
{ 1 -Ao(u/R)f})}~~2V~r~~“~
[e~‘TR(2u-r)]/[nu(Rt-A,uf)2]
+ 1,
{l/( 1 - A,,(u/R)~)~[(~“~‘)~“~ > [e -‘TR(2u-r)/[nu(R”-A,uf)2] {R+/(R+ -&~+)}[(~“-r)‘“l>
[e
-‘TR(2u - r)/[nu(R+ - &I*)~]
+ 1, + 1, (A.21)
1 >[e~‘T(2u-r)/nu]R’i2”[(Rf-A,uf)]~-’i”)+[(Rf-A,ut)/R3]2-““, 1 >[e-‘T(2u-r)/nu][R~-A,uf)/R~]‘~“”)+[(R~-A,~~)/R~]2-““, 1 >[Rf-A,u~)/R~]‘-““‘[e~‘T(2u-r)/nu+((R~-A,uf)/Ri)2], [(R+--,4,g+)/R+]““-
[(R*-Aou*)/Rf12
>e-‘T(2u-r)/nu.
Since u = r + h > r, 2u - r > 0. So the right-hand side is non-negative. It goes to zero if n-+cc or T+co. In those cases, (A.21) certainly holds, since the lefthand side is positive. l1 Furthermore, by working backward, we can show that if (A.21) holds, so does (30). “Recall that 0 <(RI’* - A,u”~)/R”~< 1. So, on the left-hand side of (A.21), the left-hand is greater than the right-hand term if and only if r/u < 2. Since u = r + h, this holds.
term
Fischel, W., 1975, Fiscal and environmental considerations in the location of firms in suburban communities, in: E. Mills and W. Oates, eds., Fiscal zoning and land use controls (D.C. Heath, Lexington, MA) 119-l 74. Fosler, R.S., ed., 1988, The new economic role of American states: Strategies in a competitive world economy (Oxford University Press, New York). Hirschman, A.O., 1958, The strategy of economic development (Yale University Press, New Haven, CT). Hovle, B.S., D.A. Pinder and M.S. Husain, 1988, Revitalising the waterfront: International dimensions of dockland development (Bellhaven Press, London). Kamien. M. and N. Schwartz, 1982, Market structure and innovation (Cambridge University Press, Cambridge). Lakshmanan, T.R., 1989, Infrastructure and economic transformation, in: A.E. Andersson et al., eds., Advances in spatial theory and economics (North-Holland, Amsterdam) 241-261. Mills, E.S. and G. Carlino, 1989, Dynamics of country growth, in: A.E. Andersson et al., eds., Advances in spatial theory and economics (North-Holland, Amsterdam) 195-205. Morgan, D., 1989, Select few in Congress decide how the money will be spent, Washington Post, 30 May, Al.
L. Taylor, Competition among jurisdictions
259
Nijkamp, P., 1987, New technology and regional development, in: T. Vasko, ed., The long-wave debate (Springer-Verlag, Berlin) 274284. Oates, W. and R. Schwab, 1988, Economic competition among jurisdictions: Efficiency enhancing or distortion inducing?, Journal of Public Economics 35, 333-354. Peterson, G., 1984, Financing the nation’s infrastructure requirements, in: R. Hanson, ed., Perspectives on urban infrastructure (National Academy Press, Washington, D.C.) llG-142. Scherer, F.M., 1967, Research and development resources allocation under rivalry, Quarterly Journal of Economics 81. 359-394.
of Public
Economics
49 (1992) 241-259.
Infrastructural jurisdictions
North-Holland
competition
among
Leon Taylor* Economics
Received
Department,
Tulane
University,
May 1990, final version
New Orleuns,
LA 70118-5698, USA
received June 1991
This paper models jurisdictions that compete for an industry by building infrastructure more rapidly than their identical neighbors. Such competition can waste resources. Lucrative ‘prizes’ the tax revenues and the value of jobs that an incoming firm would provide - stimulate infrastructural spending and waste. Low initial levels of infrastructure discourage infrastructural competition. The paper urges a cautious approach toward federal subsidies of local expenditures on infrastructure.
1. Introduction
Consider an executive who must decide where to build, relocate or expand his plant. Suppose that his decision hinges upon the site’s infrastructure - for instance, its roads or utilities. Then he might resolve the matter in one of three ways. (1) He might pick one site and bargain with that jurisdiction’s officials over the facilities that he needs. Ideally, such bargaining is Coasian. (2) He might tell the officials of several jurisdictions that he will move to the first one that completes the infrastructure that he needs. This might provoke a spending race among the jurisdictions, providing him with roads and utilities quickly and cheaply. (3) He might simply pick the site that already offers the best infrastructure. In the last two scenarios, jurisdictions have reason to engage in rentseeking expenditures on their infrastructures. Consider why. In the second scenario, several jurisdictions race to complete their infrastructures, but only one will get the plant; the others must find less valuable uses for their roads and utilities. In the third scenario, jurisdictions compete to build infrastrucCorrespondence to: L. Taylor, Economics Department, Tulane University, 206 Tilton Hall, New Orleans, LA 70118-5698, USA. *I derived this paper from my dissertation, supervised by Wallace E. Oates at the University of Maryland, College Park. The paper was improved by Oates, Robert M. Schwab, seminar participants at the University of Maryland, Tulane University and Resources for the Future, and two referees. Greta Whitt provided technical help. Tulane’s economics department provided financial help. All errors are mine.
0047-2727/92/$05.00
0
1992-Elsevier
Science Publishers
B.V. All rights reserved
242
L. Tuylor,
Competition
among jurisdictions
ture today in hopes of attracting lucrative industries tomorrow, although they might not be sure of the exact nature or value of those industries. Again, the losers must find less valuable uses for their infrastructures. In sum, when jurisdictions compete for industry by building infrastructure, they might well waste resources. I do not want to overestimate this waste. If a jurisdiction builds a $100 million road in a contest that it eventually loses, it has not wasted $100 million. Some motorists will enjoy the road. I argue only that this enjoyment probably will not be worth $100 million. For if it were, and if officials of the jurisdiction were in the habit of maximizing constituents’ welfare, then they would have already built the road for that purpose. Infrastructural competition is little explored by the literature on jurisdictional contests. In fact, many analyses [e.g. Fischel (1975) Oates and Schwab (19SS)] assume two things. (1) More than one jurisdiction can attract a share of a fixed national stock of physical and divisible capital. Such ‘winner-take-some’ contests can enhance efficiency. Every locality can gain a bit of capital. Every locality can win. (2) The contest itself consumes no resources. Each jurisdiction vies by offering inducements that take effect only if the firm chooses to relocate there. Rent-seeking is not an issue. However, descriptive works on local economic development depict jurisdictional competition more starkly [Fosler (1988)]. Localities often vie for one big project, such as the supercollider or the assembly plant for the innovative Localities often commit resources to the Ford automobile, the Saturn.’ contest before the firm decides where to move. Jurisdictions might prefer to compete by building infrastructure, rather than by lowering taxes, because the former is a more powerful inducement. Infrastructure is crucial to economic development at several stages [Hirschman (1958)]. Transforming an economy based on resources into one that produces goods will require roads, ports and utilities. Transforming an economy that produces goods into one that produces information will require telecommunications [Lakshmanan (1989)]. A conservative way to gauge the allure of infrastructure is to look at the role of goods-oriented infrastructure, such as highways, in an informationoriented economy, such as the United States. In a cross-sectional study of 3,000 U.S. countries, Mills and Carlino (1989) calculate the long-run elasticities of total employment with respect to three policy instruments: local taxes per capita; the stock of local industrial development bonds; and the density of interstate highways. The tax and bond elasticities are small and ‘Apparently they hope that - through economies of agglomeration - the project will attract an expanding industry. For instance, the arrival of Motorola in Phoenix, Arizona, led to a burgeoning electronics industry there.
L. Taylor, Competition among jurisdictions
243
insignificant. The highway elasticity is large and significant. Doubling the highway density - say, by building a second highway - leads to a 54 percent increase in total employment. Infrastructural contests often focus on new-technology industries, such as measurement and control instruments, clocks and watches, and aircraft. New-technology firms often seek areas with many workers who are highly skilled. In turn, these workers are attracted by educational and cultural amenities. So jurisdictions might provide these amenities. Infrastructure used in research and development, such as university institutes, might also attract firms. In the Netherlands, Nijkamp (1987) finds evidence that proximity to R&D infrastructure is significant in explaining where new-technology firms locate. Amenities are not significant. Neither is the supply of venture capital. These results are consistent with the idea that infrastructural competition might be particularly fierce in small countries. The locational variation in wages or venture capital is too small to affect where firms locate. Infrastructural competition seems particularly vivid in the port cities of the industrialized world, such as London, Marseille, and Rotterdam. As ships have become larger, and shipping-related activities more specialized, dock activities have moved out of the central cities and onto more spacious sites. The inner cities have declined with the old docks. Since the 1960s public officials have tried to revitalize the waterfronts by building infrastructure to attract corporate headquarters, publishing, and tourism. In 1987, a quasigovernmental agency completed a light railway to the Docklands area of East London to try to attract white-collar activities [Hoyle et al. (1988)]. In sum, infrastructural competition merits economic analysis. This paper develops a dynamic model of long-run, resource-consuming competition for one project under uncertainty.2 Jurisdictions compete for the project by developing their roads, utilities, facilities, schools, and colleges. The faster a jurisdiction spends money, the faster it develops its infrastructure. This boosts its chances of winning the industry. Each jurisdiction seeks the rate of spending that will maximize its expected gain from competing. We want a sense of how much waste can occur and under what conditions. So this paper analyzes a case that seems particularly likely to engender waste: identical jurisdictions that can spend as much as they want to build infrastructure that (for the losers) has no alternative value.3 ‘They are based on Kamien and Schwartz’s (1982) models of innovation in the private sector. 30ne justification of a limitless budget is that the federal government often has subsidized infrastructural competition, particularly in site development and in construction of wastewater treatment plants. However, one can introduce this budget constraint into the problem, where Y(t) gives tax revenues: Y’(t)=rY(t)-X(t)
/ Y(O)= YO.
One can show that, even under
this constraint,
the jurisdiction
will undertake
the infrastructural
244
L. Taylor,
Competition
among jurisdictions
Section 2 develops the model. Section 3 presents comparative statics. If a lucrative industry is at stake, then jurisdictions will spend more, at any given moment, to try to attract it. But if the jurisdictions begin with poor infrastructures, they will spend less, at any given moment, on improving them. Section 4 calculates the amount of waste that occurs. When competition is quite keen, the waste exceeds the value of the industry for which jurisdictions are vying. Section 5 presents conclusions and reflections.
2. The model A total of IZ identical jurisdictions plan to compete for an industry by building infrastructure. At time t, a jurisdiction’s level of infrastructure is Z(t). The initial level is Z(0) =Z, 20. The infrastructure does not depreciate: dZ/d t 2 0. The jurisdiction will win the industry if it is the first to develop its infrastructure to level A. The jurisdiction plans to achieve this by time 7: when it plans to end the project. The jurisdiction’s problem is to pick a construction plan, Z(t), and a completion date, z that will maximize its expected payoff from the project. The jurisdiction will spend money on its infrastructure at a rate X(t). Z(t) is the capital stock; X(t) is the flow of investment dollars. X(t) is the proxy for all the inputs that the jurisdiction buys, such as labor and machines. The infrastructural production function has the general form Z(t)=f[X(t)],
df/dtzO.
For sharp results, I will specify the functional form of (1). If a jurisdiction wants to build its infrastructure faster, it will have to spend money on it faster. A sensible cost function is X(t) = Q(t)” 1Q(t) =dZ/dt.
Here, a> 1. If as 1, then the jurisdiction might as well build its infrastructure instantly. It is more reasonable to think that marginal cost rises as the jurisdiction tries to build its infrastructure faster. Here are three reasons.4 project in a wide variety of cases. The assumption that the infrastructure has no Generally, infrastructure might have alternative ‘adaptable’ if it can serve other purposes with highway which residents can use as well as firms. down and reconstitute it into capital for other laboratory for the general chemistry classes. 41 have drawn upon Scherer (1967).
alternative use best describes sports facilities. uses if it is adaptable or malleable. It is only minor changes, for example an urban Infrastructure is ‘malleable’ if you can break it purposes. You can pillage a biotechnological
L. Taylor, Competition among jurisdictions
245
(1) Participants have less time to expand their information set. Building a capital project, such as a highway, is heuristic. What employees learn at stage I can cut costs at stage II. Speeding up construction leaves employees with less time to gather and assess information. For instance, if the jurisdiction is in a hurry to build a bridge, it might skip the prototype, despite the eventuality of higher construction costs. (2) Embarking on several plans at once will save time but incur costs. The state will simultaneously study several possible routes for the new highway, since it will not have time to launch a second study should the first route fail to pan out. Duplication and waste seem inevitable. (3) The marginal products of purchased inputs decline for conventional reasons. To build the highway faster, the state can employ more people at any given time. More laborers will line the ditches; more surveyors will crowd around the quadrant. The marginal product of labor will diminish. So will the marginal product of money spent on labor. The model’s basic results hold for any value of a greater than 1. However, for concreteness, let a = 2. Invert (2) to obtain a production function: Q(t) =X(t)+. The boundary
conditions
Z(O)= zo,
are
Z(T)=A.
Some scenarios will let Z, vary from one jurisdiction to another. The jurisdiction that wins the project will gain R dollars. This represents the present value of net real personal earnings and tax revenues, discounted to T.’ R reflects the possibility that some earnings come from added work effort at the expense of leisure. 6 Furthermore, the jurisdiction knows R. In that sense, the analysis corresponds to the second scenario of the introduction. The discount rate is r. Suppose that the jurisdiction could be certain of winning the project. Then the value of its efforts would be
5R connotes that the jurisdiction knows for sure the project’s value. In reality, jurisdictions often build industrial parks and roads without knowing precisely the type or value of industry that would be attracted. This source of uncertainty is vital. But I ignore it to focus on the jurisdiction’s uncertainty over who will win the contest. Uncertainty over the winner characterizes most infrastructural contests, even those where the jurisdictions know the value of the industry at stake. Modeling this type of uncertainty gets top priority. Nevertheless, extensions of the model should consider other types of uncertainty. 6Assume that only current residents provide the additional work offered by the courted industry. The relocation of the industry causes no labor immigration.
246
L. Taylor,
Comperition
among jurisdictions
e -“R-;e-r’X(t)dt.
(5)
0
In truth, the jurisdiction cannot be certain of winning the industry. A rival might develop its infrastructure to level A first. So before the jurisdiction decides how much to spend on the race, it must estimate its chances of winning. I assume that the jurisdiction is well informed about the prize and the nature of the race. In fact, it knows the probability distribution that governs the race. I will now characterize that distribution. The probability that the jurisdiction will lose in the next moment, given that it has not already lost, is a non-negative constant, h. F(t) is the cumulative probability that the jurisdiction will lose the race by time t. Then F’(t)/[ 1 -F(t)]
= h.
.(6)
A hazard rate, h, expresses the keenness of the race. The greater is h, the greater the chances that a rival will beat the jurisdiction in the next moment.7 I will later discuss the determinant of h. The probability that the jurisdiction will lose the race right off the bat is zero: F(0) = 0. With this boundary condition, integrate (6) to recover F(t): F(t)= 1 -emh*. Subject
(7)
to (3) and (4), the jurisdiction
emrTR[l -F(T),-je-“X(t)ll-F(t)ldt.
will maximize
(8)
0
The expected gain of competing is the present value of the prize, eprTR, times the cumulative probability that the jurisdiction will win, 1 -F(T). The
‘1 assume a constant hazard rate because it simplifies the analysis and because I find no compelling empirical reason to assume a nonconstant rate. However, the model can handle a more general probability distribution, the Weibull function. Let F(r)=l-exp(-bH(s)ds), H(t) = h(t), u(t)=wtw-‘.
As a special case, I have set w= I to obtain the exponential distribution. I use this function because, without more information about infrastructural competition, it seems reasonable to pick a probability distribution that well describes a variety of social processes. The reader might ask why the model must use a specific probability distribution. There are two reasons: it sharpens the analysis, and it paves the way for simulation and empirical studies.
241
L. Taylor, Competition among jurisdictions
expected cost of competing at time t is the present value of the amount spent that the race has not yet on the infrastructure, e -“X(t), times the probability ended, 1 - F(t). The keenness of the competition depends on the number of entrants, n: h = h(n),
dh/dn > 0.
(9)
In the absence of barriers, rivals will enter the race until the expected gain of doing so goes to zero. Let n* denote the number of jurisdictions in a competitive equilibrium. The jurisdiction will use h*= h(n*) in preparing its construction plans. Via the calculus of variations, the solution yields’ Z(t)=c[e”‘-l]/u+Zo,
05tsT.
(10)
Here, u= h+r, and c is a constant to be determined. In addition to the Euler equation, the optimal spending the transversality condition, which yields:
path must satisfy
X(T)=R[F,(T)/(l-F(T))+r].
(11)
Last-minute spending relates directly to the reward and to the probability that the jurisdiction will lose at time T, given that it is still in the race. The optimal completion date is T = -In
{ 1 - AO(v/R)*}/v,
(12)
where A, = A-Z,. A, is all the infrastructure that the jurisdiction build to win the industry. Call A, the ‘infrastructural deficit’. The optimal rate of spending is X(t) = [I@ - AOut]'u e’“‘. To interpret (13), return for positive numbers,
to (12). Since the natural
R>(A,)2(r+h).
must
(13) log function
is defined
only
(14)
When (14) does not hold for some t in [0, T], then X(t)=O. The inequality in (14) suggests that the jurisdiction might balk at competing if the required
‘The appendix
derives eq. (10) and others
248
L. Taylor, Competition among jurisdictions
effort is high; if the rivalry is stiff; or if the opportunity cost of spending rather than saving is high. These considerations might help explain why poor jurisdictions, such as Kentucky and West Virginia, invest little in their roads and colleges, despite their need for economic development: they would have to undertake a huge effort to attract industry. There is also a hint that - since h depends on the number of potential rivals - infrastructural competition might occur more often among states than localities. Sufficient conditions. Let G[Q(t), t] be the integrand in (8). A choice of [X(t), T] that satisfies (12) and (13) is locally optimal, since Go, 2 0 for all t.9 As the jurisdiction tries to build more hastily, the drain on its coffers grows faster. So its marginal expected profit from the project does not rise.
3. Comparative
statics
3.1. Infrastructural
deficit
Many firms make location decisions in two stages. First, they pick a region; then they pick a jurisdiction in that region. But the level of infrastructure often varies by region. In the Southeast of the United States, the infrastructural deficit seems large. How would regional differences affect spending on infrastructure? How much will Southeastern states spend on schools and roads to attract industry? The answer is not obvious. A jurisdiction with a large deficit might compete half-heartedly to minimize the costs of losing the race: dT/dA, >O. On the other hand, a jurisdiction with a small deficit might also ‘take it easy’, because it does not have far to go to complete the race. Let us resolve the matter. Differentiate (12) and (13): (15) dX/dA,
= -2v*[R*-
A,u*]ue2”‘<0.
The bigger the deficit, the less the jurisdiction will spend at infrastructure - and the less inclined to pursue a crash program. These results suggest that some regional economies will backward. Suppose that the jurisdictions within a region are suppose that one region (the ‘North’) has more infrastructure region (the ‘South’). Northern jurisdictions will continually region’s average level of infrastructure by competing avidly with Southern jurisdictions will rarely compete with one another, so 'G
PQ=
-2e-"[l-F(t)]~o,
OZ;tST.
(16) any time on long remain alike. Also than another add to the one another. they will add
249
L. Taylor, Competition among jurisdictions
little to what they North to the South.
have.
Industries
seeking
infrastructure
will
prefer
the
3.2. Reward Many jurisdictions go after very large, very visible projects. strategy affect infrastructural spending? Compute the statics: dT/dR=
-{A,~/2(o/R)~R~}{l/u[l
How does that
-A&/R)+}
(17)
dX/dR = CR* - A,&] R - *Ue’“’ > 0. The bigger the reward, complete its infrastructure
(18)
the harder the jurisdiction more quickly.
will compete,
trying
to
3.3. The discount rate and the degree of competition The discount and hazard rates express certain opportunity costs of infrastructural spending. The discount rate reminds us that the jurisdiction could have invested its money in other pursuits (which I have not explicitly modeled). The hazard rate reminds us that the jurisdiction might be throwing away its money. How might these parameters affect the jurisdiction’s spending plan? It is tempting to assume that a rise in r or h must cause the jurisdiction to spend less on infrastructure, since it seems to signal higher opportunity costs of spending. However, matters are not so simple. A rise in h might dissuade the jurisdiction from attempting the race - or it might prod the jurisdiction to pick up its pace. The jurisdiction is like a runner at the starting line who glances around him and finds that the field is unexpectedly strong. He might pull up his starting blocks and walk away from the race; or he might dig in his heels. Much depends on the pain and the prize that he expects from the race. To say more than this, we must undertake the comparative statics of u. In this matter, modern control methods yield cleaner results than the calculus of variations. Form the Hamiltonian H[Z(t),
X(t), Tp;r,h,t]
= -(e-“X(t)[l s.t. Z(0) =
-F(t)]
+pX(t)*
(19)
z,,
Z(T)=A. Z(t) is the state variable.
X(r) is its control.
The optimality
condition
is
250
L. Taylor, Competition
dH/dX= The multiplier
among jurisdictions
-e-“[l-F(t)]+p/[2X(t)*]=O, equation
- dH/dZ
(20)
is
= 0 = dp/dt.
(21)
Here, p is the expected value of the discounted marginal cost of building the infrastructure at time t. The jurisdiction will equate this marginal cost at each moment. This will minimize the expected expense of building the infrastructure to level A over the selected planning period. The state equation is dH/dp = X(t)+ = dZ/dt = Q(t). Solving
(20) for X(t) yields: X(t)*=pe”/[2(1
From
(22)
-F(t))].
(21), p/2 is a constant
- call it k. Manipulating
Z(t) = k/v e”* + d,
(23) (23) yields: (24)
where d is another constant. Eq. (24) is equivalent to (10). Since the Hamiltonian and the calculus of variations yield equivalent solutions to the problem, we might borrow from both approaches to analyze the effects of u. Solve explicitly for p by equating (23) to an expression for Q(t) derived from (13): p = 2&[R’
- ,Q+],
dP/dv = [R+ - 2A,vt]/vb.
(25) (26)
If the reward, R, is large enough, then d,u/dv>O: a rise in the probability of losing a lucrative race will spur the jurisdiction to spend more. If the infrastructural deficit, A,, is large enough, then dp/dv ~0: a rise in the probability of losing an already-bleak race will cause the jurisdiction to hedge its bets. In particular, when A, approaches its upper bound of (R/u)*, then dp/dva
- (Rv);.
(27)
When the jurisdiction has a huge deficit, then a small rise in the probability of losing might well trigger a large cut in planned expenditures.
L. Taylor,
Competition
251
among jurisdictions
3.4. An interpretation One can tell a story that is consistent with these diverse elements. Suppose that West Virginia vies with other states that have large infrastructural deficits. If the reward is large, then West Virginia might compete wholeheartedly. Otherwise, it will spend little on its infrastructure, and it will not hurry to completion. Broadly, these results also hold as West Virginia’s probability of losing rises (dp/dv > 0 when R is large, but dp/dv
4. Does jurisdictional
competition
waste resources?
It is in the firm’s interest to goad jurisdictions to race one another in developing sites. But is it in society’s interest? Consider the amount of waste generated by the contest. Summed over all competitors, spending is
S(n,T,R,Ao,v)=njem”X(t)dt 0
=[n(Ri-A,vi)2v/(2v-r)][e(2”-“TDefine ‘waste’ as the aggregate value jurisdictions expended on the contest:
of the
11. resources
(28) that
the
~(~,A,,7;R,n,r,h)={[(n-l)(R~-A,v~)~v]/(2v-r)){e’~“-“~-l).
losing
(29)
When, if ever, does aggregate spending exceed the reward, which (by hypothesis) is also the social value of the project? More formally: Under what conditions does the following expression hold? (30) One can show that (30) holds if n +co or T-co. If the competition industry is keen enough or long enough, then it will produce a net loss.
for
Industrial policy From (29) we can minimize waste by setting n= 1. The social planner would pick the jurisdiction to receive the project and would specifiy the “This
derivative
comes from (11). using (15).
252
L. Taylor, Competition
among jurisdictions
completion date for the infrastructure. The other contenders could save their money. This justification for an industrial policy is naive if the social planner responds to political pressure. The jurisdictions would shift resources from producing infrastructure to producing political pressure. Waste would occur in a different garb - lobbying, advertising, etc. Whether we choose competition or industrial policy depends on which process generates less waste. More important, an unlimited industrial policy seems undesirable - and a limited one seems unworkable. We do not want the social planner to direct all locational decisions. For instance, many locational decisions hinge on labor costs. We would prefer that markets drive those decisions by generating information in the form of wages. We want the planner to direct just the locational decisions that depend on infrastructure. But how can we separate these decisions from the others?
5. Conclusions The jurisdiction’s rate of infrastructural spending depends on the reward that it expects from the courted industry. A large reward might induce many jurisdictions to build up their infrastructures, although the industry can use only one of them. The amount of resulting waste depends on how many jurisdictions compete and on how long they compete. If the contest is very long, or if it involves oery many contestants, then it can produce a net social loss. However, the model analyzes a case that seems likely to produce social waste - the case of identical jurisdictions that compete by building infrastructure with no alternative value. Certain more realistic cases might produce less waste. Suppose that a jurisdiction knows that it has less infrastructure than its rivals. Then it knows that its chances of winning are lower, and it is less likely to compete. Infrastructural competition among jurisdictions depicts the big tradeoff between equality and efficiency. When all regions start with the same infrastructure - start on a level playing field - then competition might waste substantial resources. But when regions differ in their initial infrastructures might increase when the field favors some players - then the contest inequality. Recall that dX/dA,, ~0: the weak regions drop out, and the spoils go to the strong. On the other hand, this outcome reduces waste: at least the weak regions are not throwing their money away. Such issues matter only if infrastructural competition is important. However, Peterson (1984) concludes that net investment in structures by states and localities in the United States began falling precipitously in 1970. It approximated zero in the early 1980s and triggered popular fears of an ‘infrastructural crisis’. So why might a competitive model of infrastructural spending pertain to current policy?
253
L. Taylor, Competition among jurisdictions
One reason: infrastructural spending does not always show up on the jurisdiction’s books. The appropriations committees in the U.S. House and Senate dole out $1 billion a year to localities in pork-barrel projects notably high-technology centers for universities and Economic Development Administration grants - that enable localities to compete for economic activity. Typically, these grants are not subjected to merit review. Instead, they result from log-rolling in conferences between leaders of the two committees [Morgan (1989)]. Log-rolling can represent a Pareto improvement for the log-rollers. But the instruments of log-rolling (such as university centers) can waste national resources if they are built without a compelling need. Let us reconsider infrastructural projects that are funded mainly by the localities that would use them. A steady decline in net aggregate investment implies that competitive spending is not increasing relative to capital depreciation. I speculate that losers of infrastructural contests eventually turn to tax competition. They can offer lower taxes than other jurisdictions since they do not have to recoup infrastructural costs. They will let their excess bridges and schools fall apart. Empirical work might focus fruitfully on whether jurisdictions build infrastructure to accommodate recent growth or to lure new growth. If competitive spending is large, then we might beware of proposals to subsidize it. If accommodative spending is large, and if infrastructural decay is due more to a region’s poverty than to the misfortunes of competition, then we might view subsidies more sympathetically.
Appendix Equation paper.
A.I.
numbers
without
the prefix
A refer to equations
in the main
Deriving (10)
max e ~‘TR[l-F(7)1-~e~“CQ(1)~2CI-F(t)~dt. 0 (Z(O.T) Let G[Z,(t), t] denote the Euler equation
the integrand
of (A.l).
64.1)
The optimal
Z(t)
G, = dG,/dt. Since Z(t) is not an independent argument in G, the left-hand is zero. The right-hand side is computed from (A.l):
must
satisfy
(A.2) side of (A.2)
L. Taylor, Competition
254
d{ -2eP”Q(t)[1 Integrate
among jurisdictions
-F(t)]}/dt=O.
(A.3)
both sides: -2e-*‘Q(t)[l Q(t)=
-F(t)]
=cl,
-cIer’/2[1-F(t)]=c,e*‘/[l-F(t)],
(A.4)
c2 = -c,/2. Integrate
both sides of (A.4):
(A.9
Z(t) + cj = 1 c1 e*‘/[ 1 - F(t)] dt. Substitute
(7) into (AS): Z(t)+c,
=jc2e”/emh’dt,
Z(t)+c,=S~,e”‘~“dt, v=r+h,
Z(t) + c3 = j c2 e”’ dt,
(-4.6)
Z(t) + cj = c2 e”‘/v + c.+, cg=cq-CJ.
Z(t)=~~e”*/v+c,,
Solve for the constant
c5 by using Z(0) =Z,:
Z(O)=Z,=c,/u+c,, (A.7) cg = z, - c2 Jv. Simplify (A.7), suppressing Z(t)=c[e”‘-
the subscript
in cz:
l]/u+Z,.
(10)
A.2. Deriving (11) In addition to the Euler equation, the transversality condition G-Q*dG/dQ+H,=O,
at 7;
the optimal
spending
path must satisfy
(A8
L. Taylor, Competition
where H(T) =e-‘TR[l
-F(T)].
dG/dQ=2e-“[Q(T)][l
among jurisdictions
Let us compute
the components
255
of (A.8):
-F(T)],
-Q*dG/dQ=2e-“[Q(T)]2[1-F(T)],
(A.9)
H,= -re-‘TR[l-F(T)]-Ft(T)e-‘TR, H,= -e-rTRIFI(T)+r(l-F(T))]. Substitute
(A. 10)
(A.9) and (A.lO) into (A.8) and simply:
F,(T)/(l-F(T))+r=h+r=v, ce “T = (Ru)‘.
(A.1 1)
To solve for 7: we must also solve for c. Solving for two unknowns two independent equations. We have one in (A.ll). For the second, (A.7) with the boundary condition Z(T) = A: Z(T) = c[eVT - I]/0 + Z, = A.
requires combine
(A.12)
Solve (A.12) for c: c = [A -Z,]
u/[euT - l] = A,u/[euT - 11,
(A.13)
A,-A-Z,.
From (A. 11): e “T = (Ro)+l/c, e “‘Substitute
1 = (Ru)+/c - 1 = [(Ru)+ - cl/c.
(A.14) into (A.13) and solve again for c:
(A.14)
256
L. Taylor, Competition among jurisdictions
c = A,u/[euT-
l] = &v/([Rv)f
1 = A,v/[(Rv)f
- c],
-cl/c,
(A.15) (Ru)’ -c = A& c = (RLp - A,v. To solve for T, substitute
(A.15) into (A.7) and take logs:
ce “T =(Ru)“, [(Ru)* - A,v) evT =(Ru)*, e “T= (Ro)+/[(Ru)+ - A,u] = { [(Ru)+ - Aou]/(Ru)+} - I,
(12) evT = { 1 - A(u/R)*}
‘,
UT= -ln{l-A,(u/R)*}, T = -In
{ 1- Ao(u/R)f}/u.
A.3. Deriving (13) With T in hand, use (4):
we need only solve for X(t).
X(t)=Q(t)*=[~e”‘]*=c*e*“~. Substitute
Differentiate
(10) again
and
(A.16)
(A.15) into (A.16):
(13) X(t) = [R* - AOu~]2ue2”t.
A.4.
Deriving (24)
Rewrite p/2 as a constant X(t)*= Substitute
ke”/(l
-F(t)).
(22) into (A.17):
k: (A.17)
L. Taylor, Competition
among jurisdictions
257
Q(t) = k e”/( 1 -F(t)). Since F(t)=
1 -emhI,
Q(t) = k e”‘, Integrate
(A.18)
v=r+h.
(A. 19)
both sides of (A.19): Z(t) = k/v cut + d.
A.5. Deriving
Summed
(24)
(28)
over all competitors,
spending
is
S(n,T,R,A,,v)=nie-“X(t)dt 0
=nv[Rf-A,vt]2~e(2”~r)‘dt. 0
To solve the right-hand
integral,
substitute
u =(2v-
r)t:
(Zu-r)T
e(2”Pr”dt=
1/(2v-r)
s
e”du=[1/(2v-r)]
[e(2”-r)T-
11,
0
S(n, 7; R, A,, v) = [n(R” - A,v~)~v/(~v-
A.6.
Deriving
Under
results from
what conditions
[eT](2”-r)>
Using (12):
reprT
11.
(28)
(30)
does the following
[n(R~-Aov~)?v/(2v-r)][e’2”-r’Te(2umr)T>
r)] [e(2”-r)T-
R(2v - r)]/[nu(R+
[eCrTR(2v -r)]/[nv(R*
expression
hold?
l] >eprTR, - Aouf)‘]
+ 1,
- AOv*)‘] + 1.
(30)
L.
258
Taylor,
Competition
among jurisdictions
> [eerTR(2u - r)]/[nu(R* - &I~)~] + 1, {exp( -In >
{ 1 -Ao(u/R)f})}~~2V~r~~“~
[e~‘TR(2u-r)]/[nu(Rt-A,uf)2]
+ 1,
{l/( 1 - A,,(u/R)~)~[(~“~‘)~“~ > [e -‘TR(2u-r)/[nu(R”-A,uf)2] {R+/(R+ -&~+)}[(~“-r)‘“l>
[e
-‘TR(2u - r)/[nu(R+ - &I*)~]
+ 1, + 1, (A.21)
1 >[e~‘T(2u-r)/nu]R’i2”[(Rf-A,uf)]~-’i”)+[(Rf-A,ut)/R3]2-““, 1 >[e-‘T(2u-r)/nu][R~-A,uf)/R~]‘~“”)+[(R~-A,~~)/R~]2-““, 1 >[Rf-A,u~)/R~]‘-““‘[e~‘T(2u-r)/nu+((R~-A,uf)/Ri)2], [(R+--,4,g+)/R+]““-
[(R*-Aou*)/Rf12
>e-‘T(2u-r)/nu.
Since u = r + h > r, 2u - r > 0. So the right-hand side is non-negative. It goes to zero if n-+cc or T+co. In those cases, (A.21) certainly holds, since the lefthand side is positive. l1 Furthermore, by working backward, we can show that if (A.21) holds, so does (30). “Recall that 0 <(RI’* - A,u”~)/R”~< 1. So, on the left-hand side of (A.21), the left-hand is greater than the right-hand term if and only if r/u < 2. Since u = r + h, this holds.
term
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L. Taylor, Competition among jurisdictions
259
Nijkamp, P., 1987, New technology and regional development, in: T. Vasko, ed., The long-wave debate (Springer-Verlag, Berlin) 274284. Oates, W. and R. Schwab, 1988, Economic competition among jurisdictions: Efficiency enhancing or distortion inducing?, Journal of Public Economics 35, 333-354. Peterson, G., 1984, Financing the nation’s infrastructure requirements, in: R. Hanson, ed., Perspectives on urban infrastructure (National Academy Press, Washington, D.C.) llG-142. Scherer, F.M., 1967, Research and development resources allocation under rivalry, Quarterly Journal of Economics 81. 359-394.