UNIVERSITY OF ILLINOIS LIBRARY AT URBANA-CHAMPAIGN BOOKSTACKS Digitized by the Internet Archive in 2011 with funding from University of Illinois Urbana-Champaign http://www.archive.org/details/noteonsufficient799brue FACULTY WORKING PAPER NO. 799 A Note on Sufficient Conditions for Negative Exponential Population Densities Jan K. Brueckner College of Commerce and Business Administration Bureau of Economic and Business Research University of Illinois, Urbana-Cham.paign P^Ocii -71 FACULTY WORKING PAPER NO. 799 College of Commerce and Business Administration University of Illinois at Urbana-Champaign September 1981 A Note on Sufficient Conditions for Negative Exponential Population Densities Jan K. Brueckner, Associate Professor Department of Economics Abstract This note shows that one of the conditions commonly thought to be sufficient for urban population densities to follow the negative expo- nential function is incorrect. It is shown that the correct condition on housing demand is that the income-compensated price elasticity equals -1. Many investigators erroneously believe that a unitary uncompensated elasticity is required. A Note on Sufficient Conditions for Negative Exponential Population Densities by Jan K. Brueckner 1. Introduction The main testable prediction of the Mills-Muth model of urban spatial structure is that population density declines as distance to the urban center increases. The repeated confirmation of this predic- tion in empirical studies has led to a widespread consensus on the va- lidity of the basic urban model, a rare achievement in applied economics. The most common specification of the population density function is negative exponential: density D is related to distance to the urban ^YX center x by the function D = D„e , where y > and D_ is population density at x = . Although the negative exponential is a convenient functional form, most empirical investigators are aware that it is appro- priate only under strong restrictions on the housing production tech- nology, consumer tastes, and the nature of commuting costs. Moreover, it is common for researchers to cite Mills [3] for the list of conditions which justify the function's use (see Kau and Lee [2] and McDonald and Bowman [4]). Mills' conditions are 1) a Cobb-Douglas housing production function; 2) commuting costs which are linear in distance; 3) a unitary price elasticity of housing demand. The purpose of this note is to point out the last of these conditions is incorrect: the appropriate restric- tion on housing demand is that the income compensated price elasticity (not the regular price elasticity) is unitary. While this restriction is correctly stated by Muth [7, Ch. 4], the failure of empirical re- searchers to grasp the difference between the elasticity restrictions -2- (and the incorrectness of Mills' version) is no doubt due to the lack of clarity of Muth's argument. In the next section of this note, analysis which is equivalent to Muth's is clearly presented, and the error leading to Mills' incorrect restriction is discussed. 2. Analysis The basic urban model assumes that the city is inhabited by indi- viduals with uniform incomes and identical tastes over consumption of housing services q and a numeraire non-housing good c. Commuting cost as a function of radial distance to the urban center is t(x), so that disposable income at x is y - t(x), where y is income at the center. Housing services are produced with capital N and land I according to the function H(N,£), which is homogeneous of degree one (housing ser- vices are best viewed as the services derived from floor space, with H the production function for floor space) . The rental price per unit of housing services (per square foot of floor space) is p, which will de- pend on X, while the rental prices of capital and land are i and r respectively, with i constant over x but r free to vary. Population density may be computed by noting that the ratio of square feet of housing per acre of land to square feet of housing per dwelling equals dwellings per acre of land. Thus the ratio (H/!i)/q equals dwellings per acre of land and is proportional to population density when households are of uniform size. Now the appropriate- ness of the negative exponential density function depends on form of the x-derivative of the natural logarithm of density. If this derivative is constant over x, then densities will follow the negative exponential function. Letting * denote natural log, the derivative of interest is -3- 1 dP _ dP* d[(H/£)/q]* _ d(H/0* dq* P dx " dx dx dx dx ^ ' The following analysis will indicate when (1) will indeed be constant over X. The first step is to calculate d(H/£)*/dx making use of the housing producer's first-order conditions. Housing producer profit is pK(N,t) - iN - r£, and the necessary conditions for a maximum are pK = i and pH^ = r (subscripts denote partial derivatives). These equations together iir.ply H (N, S.)/H^ (N, 2.) = r/i, and the zero-degree homogeneity of K and K^ allows this equation to be rewritten as K_(N/ll,l)/lL (N/£,l) = r/i. The last equation determines the capital-land ratio N/ £ solely as a function of the factor price ratio r/i (this, of course, is a reflection of the fact that constant returns functions are homothetic) . Now the dependence of H/£ on x follows from the dependence of N/£ on r/i and the dependence of r on x. To see this, note first that d(H/£)'^- ^ J. dH(N/£,l) dx K dx = |H,(N/£,l)^i^ ^1^^/^'^^ d(N/£)^ H dx ' ^^ where the first equality uses K(N,£)/£ = H(N/£,1). Now since H = i/p and N/£ is a function only of r/i, (2) may be written iN d(N/£)* d(r/i)* ^ dr* , , pH d(r/i)* dx ^N^N£ dx ' ^^ where u H iN/pK is capital's factor share in housing production and a is the elasticity of substitution between capital and land in housing -4- (note dl*/dx = and d(N/£)*/d(r/i)* = [d(N/li)*/d(H2/lL|_)*][d(H2/H^)*/d(r/i)*] = a^^ • 1). While (3) shows how d(H/Jl)*/dx is related to dr*/dx, one further step shows the connection between dr*/dx and dp*/dx. Total differentiation of the zero-profit identity pH - rA - iN = (profits are identically zero by constant re- turns) yields using the producer's first-order conditions. Rearrangement of C^) yields dx rZ dx Mj^ dx ' ^^ where \i = r£/pH is land's share in housing production. Combining (3) and (5) then gives djE/D* ^ %'^m dp* dx ^z ^^ ' - ^^"^ Computation of dp*/dx makes use of the first-order conditions for the consum.er optimization problem. The problem is to maximize the utility function u(c,q) subject to c + pq = y - t(x) by choice of c, q, and X. Substitution yields the maximand u(y - t - pq,q) and the first- order conditions u^/u^ = p and 4^+q-^=0. (7) dx ^ dx Rearrangement of (7) then yields d£* ^ -dt/dx ^ Q (g^ dx m ' -5- where m H pq is expenditure on housing. Now in order for the identical urban residents to live voluntarily at different locations within the city, each individual must be loca- tionally indifferent. That is, every location must be optimal, implying that condition (7) must hold at all urban values of x. Locational in- difference will not obtain, of course, unless utilities are uniform across x. It is easy to see, however, that satisfaction of (7) at all X guarantees x- invariant utilities. This follows from totally differ- entiating u(y - t - pq,q), which yields -u (dt/dx + qdp/dx) + (u„ - pu^)dq/dx, a quantity which equals zero at all x when (7) and u /u = p hold ever>a;here. In essence, the spatial variation in p reconciles urban residents to differences in communting costs; the decline of p with x (see (8)) cancels the utility-decreasing effect of longer commutes and leaves consumers locationally indifferent. The final step in deriving dD*/dx is the computation of dq*/dx, a task which is immediate in light of the preceding discussion. Since the decline with x in the price per square foot of housing keeps con- sumer utilities constant in the face of higher commuting costs, it is clear that the change in q caused by x-induced changes in p and dispos- able income follows from movement along an income-compensated (constant utility) demand curve. Therefore ial = 1 d£ ^ 1 22.1 . d£^ ^ d£l .g. dx q dx q 3p | u=constant dx dx ' where " is the income-compensated elasticity of demand. Substituting (6) and (9) into (1) then yields -6- dD* / ^N°NJl / dp* dx ^nV ' \ dt/dx ,,-, where the second equality uses (8) . Since all terms in (10) are posi- tive except for n. (10) is unambiguously negative; population density declines with x. The intuition for this result is easily stated: lower housing prices at greater distances lead to larger dwellings (dq*/dx = ndp*/dx > 0) while lower land rents at greater distances make lower capital-land ratios optimal, and thus lead to fewer square feet of housing per acre (d(H/J.)* = u a dr*/dx < 0). Together, these effects yield fewer dwellings per acre and lower population densities at greater distances. It is now possible to consider the central question of this note: when will (10) equal a constant, leading to a negative exponential den- sity function? A sufficient condition for this outcome is clearly that 2 each element in (10) is itself independent of x. In other words, y , p , , n, dt/dx, and m all should be invariant with x (note that the factor shares and elasticities will not in general be indepen- dent of x) . Now if the housing production function is Cobb-Douglas (H(N,S-) E N £ ), then a = 1, y = a, and y = 1-a. Moreover, if _--. t(x) = (ft + 6x, then dt/dx =5. If in addition the income compensated demand elasticity r, is constant and if movement along the income- compensated housing demand curve yields constant expenditures m, then the sufficient conditions for constancy of (10) will be met. Clearly, n = -1 leads to satisfaction of both these requirements since a unitary price elasticity yields expenditures which are independent of price and -7- thus constant over x. In summary, sufficient conditions for the validity of the negative exponential density function are 1) a Cobb-Douglas housing production function; 2) commuting costs which are linear in distance; 3) a unitary income-compensated price elasticity of housing demand . Although Mills [6] imposes specific functional forms at the begin- ning rather than at the end of his derivation, much of his analysis parallels the above. He implicitly calculates d(H/£)*/dx as in (6), and condition (7) enters importantly in his argument. Mills' derivation differs from the above, however, in the computation of dq*/dx. First, he assumes that the demand function for housing has the constant elas- ticity form q = AI p , where I is income and p as before is price. While such a demand function is perfectly acceptable. Mills' error is to substitute gross income y instead of disposable income y - t(x) in place of I. The resulting demand relationship q = Ay p ignores the crucial fact that consumer purchasing power declines with x! Calcula- tion of dq*/dx based on this inappropriate relationship yields dq*/dx = (3q*/9p*) (dp*/dx) = 6dp*/dx. Recalling (6) and (8), substitution in (1) then gives dP* ^ [ ^N°N£. _ I dp* dx ^ ^c ' "^^ /^N°N£ Jdt/dx U, I m e^i^^. (11) Note in (11) that the uncompensated price elasticity 6 incorrectly takes the place of the compensated elasticity r (compare (10) and (11)), Now since the above demand relationship implies m = Ay p , a unitary -8- price elasticity (6 = -1) implies m = Ay . Imposing in addition the assumptions of a Cobb-Douglas production function (which yields \i^o /Ur. = a/(l-a); see above) and linear commuting costs (implying dt/dx =6), it follows from (11) that dD*/dx = (- —^ + 1)6/Ay^ = -6/(l-a)Ay®, a constant. This discussion shows how misspecification of the demand relation- ship by use of an improper income variable leads to the erroneous con- clusion that a unitary uncompensated price elasticity, together with the other restrictions, yields a constant dD*/dx, To show explicitly that this elasticity assumption will not yield the desired result when the proper substitutions are made, note that if q = A(y - t(x)) p , then dq* ^ e_ dt dp* dx y-t dx dx where e^^ = m/(y-t) is the expenditure share for housing (the second inequality uses dt/dx = -mdp*/dx) . Imposing a Cobb-Douglas production function and linear commuting costs and assuming 6 = -1, it follows after substituting (12), (6), and (8) into (1) that dp* ^ _,_a. _ S_ dx ~ 1-a ^q m = -(l^ - eA(y-t(x))®"^) -, (13) ^"'^ A(y-t(x))^ a quantity which is clearly not constant over x (note m = A(y-t) when £ = -1). The non-constancy of (13) is, of course, simply a reflection of the fact that a unitary uncompensated price elasticity generally im- plies a non-unitary compensated elasticity, which from above must yield -9- a non-constant dD*/dx. Indeed, it is easv to show that Qt - 1, which q appears in the position of ri in (10) , is precisely the compensated price 3 elasticity for the given demand function when 3 = -1. The obvious restriction 9 > (housing is a normal good) implies that the compen- sated elasticity 9e - 1 exceeds -1, accounting for the non-constancy q of dD*/dx in (13). A wealth of empirical evidence on the parameters of housing demand is now in existence (see Mayo [3] for a recent survey). An important question is whether this evidence justifies the use of the negative exponential density function in empirical studies (in other words, is the evidence consistent with n = -1?) Unfortunately, the answer to this question is negative: nearly all studies show that the uncompen- sated price elasticity of housing demand exceeds -1, implying that the compensated elasticity is substantially greater than -1. Thus, empir- ical evidence on housing demand suggests that any negative exponential 4 population density regression will involve specification error. As a final observation, it is interesting to note that the pre- ceding discussion necessitates reinterpretation of the results of one important empirical study of population densities. Recognizing the possibility of specification error, Kau and Lee [2] used an ingenious application of the Box-Cox estimation procedure to test for the ap- propriateness of the negative exponential function rather than imposing it a priori . However, since they invoked Mills' assumptions, Kau and Lee interpreted a divergence from the negative exponential form as evi- dence of a non-unitary uncompensated price elasticity of housing demand. Their empirical results implied £ > -1 for half the sample cities, with -10- B = -1 holding for the remainder, conclusions which appear consistent with the independent evidence of price inelasticity. As should be clear from the above discussion, however, the correct inference to be drawn from Kau and Lee's results concerns the compensated price elasticity of housing demand. Correctly interpreted, their results show that this elasticity frequently exceeds -1, a conclusion which need not imply overall price inelasticity. -11- Footnotes For a recent study estimating such a function, see Glickman and Oguri [1]; for earlier studies, see Mills [5] and Muth [7]. 2 Although this is obviously not a necessary condition for the constancy of (10) , it is very hard to imagine a situation in which the elements of (10) are non-constant over x while (10) itself is x-invariant, 3 Using the Slutsky equation ia. = lal - q ^ 3p 3p|u=constant 31' it follows that q 3p q q 31 q 4 The erroneous justification for the negative exponential func- tion (S = -1) is, of course, less strongly contradicted by the demand evidence. -12- Ref erences [1] Glickman, Norman J. and Yukio Oguri. "Modeling the Urban Land tlarket: The Case of Japan," Journal of Urban Economics , 6 (1978), 505-525. [2] Kau, James B. and Cheng F. Lee. "Functional Form, Density Gradient and Price Elasticity of Demand for Housing," Urban Studies , 13 (1976), 193-200. 13] Mayo, Stephen K. "Theory and Estimation in the Economics of Housing Demand," Journal of Urban Economics . 10 (1981), 95-116. [4J McDonald, John F. and H. Woods Bowman. "Some Tests of Alternative Urban Population Density Functions," Journal of Urban Economics , 3 (1976), 242-252. 15] Mills, Edwin S. "Urban Density Functions," Urban Studies , 7 (1970), 5-20. [6] . Urban Economics . Glenview, Illinois: Scott Foresman, 1972. [7] Muth, Richard F. Cities and Housing . Chicago: University of Chicago Press, 1969. M/D/360 HECKMAN BINDERY IMC.