“Gammaville”: An optimal town

JOURNAL

OF ECONOMIC

THEORY

6, 471-482 (1973)

“Gammaville”:

An Optimal

Town*

JOHN G. RILEY Boston

College,

Chestnut

Hill,

Massachusetts

02167

Received June 20, 1972

In the first issueof this journal, Beckmann [l] discussedthe distribution of urban rent and residential density. Treating the commuting population as price takers in all markets, including the market for land, he attempted to solve explicitly for a competitive equilibrium. To do this he made some specific assumptions about the form of each commuter’s utility function (Cobb-Douglas), and also about the distribution of income (Pareto distribution). That Beckmann’s solution was faulty has been shown by Delson [2]. However, the latter author did not succeedin correcting the errors in the original discussion. Since then, Montesano [5] has fully demonstrated the complexity of the problem and obtained complete solutions for some special cases. In the following analysis we focus again upon this very simple urban model. However, instead of seeking an equilibrium for some given income distribution, we ask whether there is a “best” distribution of income. Although there are significant differences in specification between the model discussedhere and the original “optimal town” of Mirrlees [4], much of the analysis parallels closely his work. Our approach is to maximize a welfare function of the utilities of N identical individuals located around a central businessdistrict (CBD) of given radius 7. All transportation to work is by automobile, and the simplifying assumption is made that the daily commuting time and the resource costs of transportation (wear and tear on the car, gas, etc.) are monotonic functions of distance from the CBD. Each individual is assumedto have a utility function of four variables: consumer goods “c”, the area on which he lives “a”, the time he spends at home (leisure) “I& ,” and his distance from the center of the CBD “r”. Later on in the analysis, we will make the further simplifications that * This paper forms part of my Ph.D. dissertation submitted to the Massachusetts Institute of Technology in June, 1972. I am indebted to R. M. Solow, P. A. Diamond, R. Findlay, and a referee for their valuable comments and suggestions. Copyright All rights

0 1973 by Academic Press, Inc. of reproduction in any form reserved.

471

472

RILEY

transport costsare linear functions of distance, and that the utility function is a Cobb-Douglas function. Many researchers in the field of residential location have included distance in the utility function primarily as the disutility of travel time. However, it is important to note that this is not the casehere. We assume that there is an equal disutility to an hour of work and an hour of travel time.l Therefore, the cost of transportation is implicitly accounted for by the inclusion of leisure in the utility function. The distance component is then just a residual. If people enjoy the excitement of the city, then the marginal utility of distance will be negative. On the other hand, if people prefer to get away from the noise, pollution, and high crime rate often associatedwith the CBD, then the marginal utility of distance is positive. We must also make a more specific assumption about the social welfare function. Guided by the ethical principle that everyone should have an equal opportunity, and constrained by the necessity of mathematical simplicity, we seek to maximize the product of each person’s utility2

w = fi ui, i=l

or, taking the logarithmic transformation W = In w = f

In V.

i=l

Now consider all individuals living in a band of radius r, width A. It is easy to show that any allocation in which two people living in the same band receive different bundles of goods is sub-optimal. Then we can write the three choice variables 8, ai, Aai as functions of distance alone. Furthermore, since every person living in the band (r, A) must have the sameutility, we can write W = Zn(r) lnU(c(r), a(r), As(r), r), where n(r) is the number of people in the rth band. 1 The results are essentially unchanged if individuals have constant marginal rates of substitution of work for leisure, and of travel for leisure which exceed unity (Riley [6], p. 53). 2 This social welfare function was also adopted recently by R. C. Fair in his paper, “The Optimal Distribution of Income,” Quarrerly Journal ofEcconomics, Vol. 85 (1971), p. 565.

AN

OPTIMAL

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As a final simplification, it is assumed that a given fixed proportion (1 - k) of the land in each band is reserved for transportation. Clearly this is not completely satisfactory, and analysis of the problems of road congestion and the possibility of different road technologies would yield a superior solution. Indeed, there have been several attempts to resolve these issues (e.g., Mills [3], Solow [8], and Riley [7]). However, results can be gained only at the cost of significant increases in complexity. Proceeding then with the simplification, the area available for residential use is Zknrd, and it follows that 2kmA w(r) = ___ a(r)

.

It will, therefore, be convenient to express the social welfare function as an integral. i.e., W= For expositional

2krr

y In U dr. s 4-1

ease we introduce

the density of population

,J Then

and W = 2kn The total population following condition:

s

rfln

is N. Therefore,

2kz- J’ rfdr

U dr. welfare

= R.

(1) is constrained

by the

(2)

To close the model, we must make some assumptions about the production sector. Since the day is assumed to be divided between work h, , commuting time h,(r) and leisure 17, we must have h, + h,(r) + 17, = 24, (3) with 17, , 17, 2 0. By assuming that the marginal disutility of less leisure increases without bound as leisure time falls, we ensure that h, is positive everywhere in the optimal town. However, there is no equally straightforward way of ensuring that h, is positive. Production of consumables “c” and transportation goods “s” is

474

RILEY

assumed to be constrained linearly by the labor supply. Choosing the units for “c” and “s” as that amount which can be produced by one hour of labor, we have 2kn j crf dr + 2krr 1 srf dr < 2kn J‘ h,rfdr

total consumption

+ total production

of transport goods ,< total labor supply.

We can now place a stronger restriction on hours worked, for it will not be efficient for a man to use up more transport goods than he can make, i.e., all 11, > 0. h, > s(r), Furthermore, one can at least visualize the possibility of a solution in which those living beyond some radius do not work at all. These “lucky” people can then enjoy every minute of the day “doing their thing” without having to travel into the CBD. Although this solution is of some mathematical interest, it seems likely to be of only marginal economic significance, and will be ignored for the remainder of this paper. We are, therefore, assuming that the boundary conditions for the outer limits of the city will be such as to ensure that the total time required to commute from the outskirts is a significant, but small, fraction of the day. A more detailed discussion of this problem is given in Riley [6], Chapter 4. The problem we are left with, then, is the maximization of (l), subject to the two integral constraints (2) and (4), and the time constraint (3). Substituting for h, from (3), this is a standard problem in the calculus of variations. Tntroducing two multipliers h and p for the constraints (2) and (4), we can form a Lagrangian L = 2ka J rf[ln U - A - ~(c + s(r) + h,(r) + h, - 24)] dr.

Since there are no first derivatives of the three variables c(r), a(r), and the first order (Euler) conditions reduce to

h,(r) in the Lagrangian,

p ] =o

L.=2kmf[+-

[In U - X - q

L, = -2kmf2

zzz0 L,, = 2kmf

[+

- p ] =o.

(5)

- p(c + s(r) + h,(r) + h, - W]

AN

OPTIMAL

TOWN

475

Before discussing the particular solution in the Cobb-Douglas case, we turn to the issue of decentralization. Can we find a rent and income profile that supports the optimal allocation ? We have assumed that the production constraint is linear. Therefore, if we make the price of “other consumer goods” the numeraire, an individual commuter’s budget constraint will be c i q(r)

+ s(r) < h, + Jf,

(8)

where q(r) is the rent at distance r, M is non-wage income, and h, + h,(r) i- h, = 24. Assuming no corner solutions, first order conditions for individual maximizations are

We, therefore, seek to derive these conditions equations. From (5) and (7) it follows immediately that

Furthermore,

differentiating

from

the three Euler

(6) with respect to r, we obtain

UP --adr U

d - ull ( u

1

=

p(s’

f

hn’),

i.e., U -Y-a-In U,

d dr

AU = 2 ( U 1

U 2-a-h U,

d dr

u, = g (s’ + hs’), ( U, 1 n

(s’ + hz’),

i.e.,

since UC/U is constant. Writing 7-U U”

p = U,jU, , this reduces to P‘ “p=---

s’ + h,’ n ’

i.e., I/ Q= P

UT ap’+s’+h,’

UC =T’

(9)

RILEY

476

These are the remaining necessary conditions for local maximization, if the rent profile q(r) is given by q(r) = p(r)

== [+I*.

To complete the decentralization, we must give each individual a subsidy M*

= c*(r)

+ a*(r)

[+I*

+ s(r) + h,(r) + h,*(r)

- 24.

(10)

What we have shown is that with the optimal rent and subsidy profiles, no commuter will do better if he makes a small move nearer the CBD or further away from it. It is natural to ask whether or not this is a global maximum for an individual. Mirrlees’ analysis of this question can easily be extended to show that this is indeed the case, if the utility function is concave for any given distance r.3 We now make use of our specific assumptions, namely, s(r) = B,r, h,(r)

= 0,r (linear transport costs),

U(c, a, h, , r) = caaBh3Yrs,

and

% 8, y > 0.

The first order conditions yield (11) p.(flr - 24),

(12)

where 6, = 0, + $2, and ‘, = p.

(13)

Then, consumption of other goods and leisure are the same for everyone in the optimal town. Furthermore U = UOeuer

uo = ULl(k PI.

(14)

Therefore, utility increasesexponentially with distance. S In the Cobb-Douglas y < 1.

case the concavity assumption will be satisfied if a + p +

AN OPTIMAL

Substituting

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for c and h, , we have a=a,r

6 uer -- 6e-F-

a0 = (+T

($7

f-lo.

(15)

These profiles are depicted in Figures I and 2. The result about utility is somewhat suprising. Even though our social welfare function is usually regarded as being quite egalitarian, it is not optimal to give the same utility to each individual. For two detailed

U

“0 r FIG. 1. The Utility Profile in the Optimal Town. a

FIG. 2. The Area Profile in the Optimal Town. 642/6/S-S

478

RILEY

discussions, the reader is referred to Mirrlees [4] and Riley [6]. But, briefly, the explanation is that there are two rather special aspects of this location problem which differentiate it from most other economic analyses. On the one hand, there is a concealed nonconvexity. Implicit in the model is the assumption that an individual can only live at one distance from the CBD. If he is given two locations he can enjoy one or the other. but not both. Second, the social value of land varies with distance from the CBD, because of the cost of transportation and the disutility (or utility) of distance itself. The primary implication of these two factors is that it is not optimal to give everyone an identical consumption bundle. As always, the first order conditions for optimality require that, in a decentralized optimal town, everyone has the same marginal utility of income. However, given the conclusion that different people have different consumption bundles, the usual implication of identical total utility levels no longer follows. To obtain the optimal income distribution, we must first find the rent profile

4 = (S,”

=

($-j(Gj

= $,

(16)

i.e., (17) Although the social value of land in its alternative (agricultural) use is not encompassed by the model, it seems reasonable to define such a value Q0,. At the boundary of the city the value of land in residential use will be just equal to its value in agricultural use Qa , and from Equation (17), we can solve for the city radius

In order to understand the reasons behind the shape of the optimal rent profile we must go back and consider the optimal area profile. It is helpful to return to the discrete formulation of the problem, and consider the effects of moving a commuter from a band of inner radius r and width d (r, d) to the adjacent band (r + d, d) without altering his consumption and leisure time. First of all there is an impact on all the people living in the two bands, because of the change in area available per person. However, in this

AN

OPTIMAL

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TOWN

special model, the gains to those remaining in the inner band are exactly compensated by the losses to those originally in the outer band, and the net impact is simply Pa’(r) A a(r>’ Second, the person moving is now further away from the CBD, so there is a direct “distance effect,” easily shown to be

Finally, there is a loss due to increased transport costs s’(r)A plus reduced labor supply h,‘(r)A. The social valuation of these costs is the multiplier II. Therefore, the total social cost of the “transportation effect” is p(s’(r)

+ h,‘(r))A

= @A.

The combined impact of the person’s move is then

and this must be zero, for the optimal allocation.” If 6 is negative, the area gradient is everywhere upward-sloping. This is also the case if 6 is positive and r > $/a, that is, the distance effect is dominated by the transportation effect. However, for small r < @/8 the opposite is true: the marginal disutility of moving closer to the CBD dominates the transport cost savings; hence, the area gradient is negative. Since the optimal rent profile is just the optimal marginal rate of substitution of consumption for area, and the latter is inversely proportional to area, rents increase for r < @/a, and decrease thereafter. We are now ready to answer our question as to the optimal distribution of income. From Equation (lo), M*(r)

= c*(r)

+ h,*(r)

+ Or -

24 + a*q*

=a+p+Y-24+&-. P *This optimality case.

condition

is just Equation (9) for the special log-Cobb-Doublas

480

RILEY

Dropping

the (*)

notation

(indicating

M(r)

= M(0)

M(o)

=

optimal

values),

we can write

+ Or

m+B+r P

(18)

-224 .

Let N(M) be the number of people with subsidies less than M. Since it has been assumedthat the inner radius of the residential area is i;, the minimum subsidy is M(F), and we have N(M)

= 0,

M < M(r);

N(M(R))

= m

N’(M)

= n(M)

> 0.

Since M is an increasing function of distance, the number of people living in a band (r, fl) must be N’(M)

M’(r)d.

But this is just the area of the band divided by the area per person; therefore, we must have n(M)

=

and substituting for a and M’(r) n(M)

2nkr a(r) M’(r)

’

from (15) and (18), 2vrkr

= 8a,r

-2 +& Oe B

Using (18) to eliminate r, we have, finally, n(M)

CC (n/r

M(r)

_

M,)l+ie-;(M-Mo)

< M < M(R).

This is, of course, just a truncated Gamma density function. We have, therefore, shown that social welfare is maximized if non-wage income is distributed as a Gamma distribution. As a final note, we ask what factors affect the “degree of inequality” of the optimal income distribution. Since the coefficient of variation is invariant with respect to a proportional change in non-wage income, this is a natural measure of inequality. coefficient of variation =

S.D.(M - M,) = (1 + $)li2. Mean(M - M,,)

AN OPTIMAL

TOWN

481

Therefore, if what we have called the “residual” distance factor is zero (6 zero), the degree of inequality of non-wage income is independent of the parameters of the individual commuter’s utility function. More generally, the degreeof inequality is an increasing function of the elasticity of utility with respect to distance (6). If this elasticity is positive, then the degree of inequality decreaseswith the elasticity of utility with respect to area (/3). If it is negative, the degreeof inequality increaseswith the area elasticity of utility. In order to understand the reasonsfor these results, we first note that the (S//3) term appears in c, becauseof its inclusion in the first factor of the optimal area profile -- a _UBI a*(r) = no@,p,...) r 6e 6 . Furthermore, a change in the second “parameter” ,&/,L3does not affect the coefficient of variation. We are, therefore, justified in considering the implications for a*(u) and, hence c, of a ceteris paribus change in 6; hence (S//3). Differentiating with respect to 6, we have aa*

as

1aa 1 GgO~---!C--lnr~O. a0 as B

Then, for small r, u*(r) increaseswith an increase of 6 independent of its sign, and for large r, u*(r) decreases. To achieve the redistribution of land, people must be moved away from the CBD towards the outer suburbs. But, in a decentralized world, this implies giving these people larger subsidies,which, in turn, implies a flattening of the subsidy distribution and an increase in the degree of inequality.

REFERENCES 1. M. J. BECKMANN, “On the Distribution of Urban Rent and Residential Density,” J. Econ. Theory, 1 (1969), 6tM7. 2. A. K. DELSON, “Correction on the Boundary Conditions in Beckmann’s Model of Urban Rent and Residential Density,” J. Econ. Theory, 2 (1970), 314-318. 3. E. S. MILLS AND D. M. DE FERRANTI, “Market Choices and Optimum City Size,” Am. Econ. Reu. Papers and Proceedings, 61 (1971), 346345. 4. J. A. MIRRLEES, “The Optimum Town,” Swedish Journal of Economies, 74 (1972), 114135. 5. A. MONTESANO, “A Restatement of Backmann’s Model of Urban Rent and Residential Density, ” J. Econ. Theory, 4 (1972), 329-354.

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RILEY

6. J. G. RILEY, Optimal Towns, unpublished Ph.D. Dissertation, M.I.T., June, 1972. RILEY, Optima1 Residential Density and Rood Transportation, Journal of Economics (forthcoming). 8. R. M. SOLOW, “Congestion, Density and the Use of Land in Transportation,” Swedish Journal of Economics 74 (1972), 161-173. 7. J. G. Urban