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The possibility of positive rent gradients reconsidered
J0UlW.U
OF URBAN
ECONOMICS
9, 165- 172 (1981)
The Possibility
of Positive Rent Gradients Reconsidered HELEN TAUCIEN’
Department of Economics, Uniuersi~ of Norih Carolina, Chapel HifI, North CaroIina 27514 Received March 19, 1979; revised December 27, 1979 Despite empirical evidence to the contrary most residential location models are consistent only with a negative rent gradient. Richardson has suggested including average neighborhood plot size as an argument of the utility function; a positive rent gradient is not then a priori inconsistent with equilibrium. However, Richardson does not define the concept of equilibrium, and the purpose of this paper is to suggest a reasonable definition and to show that the rent gradient is negative if the utility function is strictly quasi-concave and if individual and average neighborhood plot sire are substitutes.
The one common property of residential location models is the downward-sloping equilibrium rent function. It is understandable then that empirical evidence to the contrary has spurred interest in developing a theoretical model that would permit a positive rent gradient. Richardson addressesthis question in a recent article of this journal [2]. His purpose is to develop a model in which the rent distance function may be upward sloping even if all individuals are identical and land differs only in its distance from the city center. The critical assumption of Richardson’s model is that average neighborhood plot size (the reciprocal of neighborhood population density) enters the utility function directly. The neighborhood population density is used as a surrogate variable describing local environmental amenities. Since population density is closely related to many of the features of a neighborhood, this approach is an appealing method of introducing local externalities. In this model a positive rent gradient is not a priori inconsistent with equilibrium; lower land rents and transportation costs nearer the city center may be offset by the higher population density there. The definition of equilibrium used in the paper is not given, although with average neighborhood plot size as an argument of the utility function, some thought must be given to the appropriate concept. In particular the variables which individuals take as given in their consumption and location choices must be specified. The two alternative assumptions would *I would l&e to thank Ann Witte, George Tauchen, and an anonymous referee for their comments and suggestions. 165 ooW1190/81/020165-08302.00/0 Copyrgllt 0 1981 by Academic Pms, All righu of repmduction in my form -cd.
Inc.
I66
HELENTAUCHEN
seem to be (1) that average neighborhood plot size is a traded good, or (2) that individuals take the population distribution, like prices, as given. In either case an equilibrium price system would be defined as having the usual property that the quantity demanded of all traded goods is no greater than the quantity supplied at these prices. In addition, the individual plot sizes demanded in a neighborhood must be consistent with average neighborhood plot sizes. For the first of the above assumptions this property implies that the average of individual plot sizes demanded equals the average neighborhood plot size demanded. For the second assumption this property implies that the population distribution resulting from individual utility maximization equals the distribution that individuals take as given. It can be shown that for either definition of equilibrium sufficient conditions for a downward-sloping rent function are (QC) that the utility function is strictly quasi-concave and (S) that individual and average neighborhood plot size are substitute goods. These conditions are not equivalent to Richardson’s condition that the marginal utility of neighborhood plot size be greater than the marginal utility of individual plot size. The resolution of this difference appears to lie in the definition of equilibrium. Richardson’s failure to give a definition appears to have resulted in erroneous conditions for a locational equilibrium and hence to erroneous conditions for a positive rent gradient. In the first section of this paper the location model is described and the properties of a Pareto optimal allocation are derived. It is shown that the government must levy a location tax or operate an explicit market in average neighborhood plot size in order to support the Pareto optimal allocation as an equilibrium relative to a price system. The properties of the no tax market equilibrium considered by Richardson are then described. Conditions (QC) and (S) are shown to be sufficient for a negative rent gradient in either the Pareto optimal or second best case. In the last section a numerical counterexample in which Richardson’s condition is met and yet the equilibrium rent gradient is negative is presented. PARETO OPTIMAL ALLOCATION Each of the fl identical residents of the city travels daily to the city center. The yearly round trip cost for a person living r miles from the city center is t(r) dollars. Utility is a function of a Hicksian good (c), individual plot size (s), and average neighborhood plot size (a). The Hicksian good is elastically supplied at a price of $1 per_ unit; the opportunity cost of land is its value in agricultural use of P dollars a square mile per year. Individual income is y dollars per year. A Pareto optimal allocation in which each individual has equal utility can be determined as the optimum of the Rawlsian social welfare function.
POSITIVE RENT GRADIENT RECONSIDERED
167
The resource constraint for this problem is
c+t)ndr-rR2P=0, where
Ris the city radius and n(r)
is the population density function,
n(r) = 2mr/s(r).
(2)
The population constraint is
%=I Rndr. 0
(3)
The neighborhood for an individual at r is only the land r miles from the city center. Average neighborhood plot size at r is therefore equal to individual plot size at r, or u(r) = s(r). (4) The optimum of the Rawlsian social we&are function is the maximum of social welfare, c, subject to U(c, S,a) 2 U for all r and constraints (l)-(4). The optimum can be characterized as the saddle point of the Lagrangian u+ A,(r)(U(c,s,u)
- 6) - A,(c + t)n + X2(1)(2?rT - ns) - h,n + h4(r)(u - s).
The Euler conditions for c, S, a, and n are
~owu:=h,n*,Xo(r)u,’ - A,(+*=A,(r),Ao(r)U; = -A&), Xl(P + t) - h,(r)s*
- A, = 0;
(5)
R*)
and the transversality condition is A,( = h ,E Eliminating X,(r) and h4(r) from the first three Euler conditions gives
(q + q)/u: =A,(r)/&.
(6)
Note that given the resource constraints, all individuals will have the same utility level at an optimum, or
u(c*,s*,u+) = 3 for all r.
(7)
168
HELEN TAUCHEN
Consider now a market economy. Individuals take the distribution of neighborhood plot size as given; all other goods including land are purchased at market prices. An individual chooses his consumption bundle and location in order to maximize his utility subject to the budget constraint. The city limits are at the distance where residential land rent equals the agricultural opportunity cost. Residential rents in excessof the agricultural value and any taxes collected are divided equally among the population. An equilibrium price system has the property that the quantity demanded of all goods is no greater than the quantity supplied or that (l)-(3) are satisfied. In addition, the plot size chosen by individuals at r must equal the given neighborhood plot size at r, or (4) must hold. It will be argued that the rent function p(r) = U-J’/UT and tax T(r) = a*U,* /U: on residents of r will support the Pareto optimal allocation as a market equilibrium. Note that this tax is a location tax rather than a tax on land consumption. An ad valorem per unit tax on land would not suffice. The Pareto optimal allocation of course satisfies (l)-(4). In order to complete the argument it must also be shown that an individual at r will choose the bundle (c*(r), s*( r)) and that n*(r) is the equilibrium population density. The budget constraint for an individual at r is c + p(r)s + t(r) + T(r) = jJ + II = -h,, where II is the individuals’s share of tax and The bunrent collections and II = { Jt Tn* dr + (EPSON*dr - pR’F}/g dle (c*(r), s*(r)) satisfies the constraint. If the utility function is strictly quasi-concave, it is the only bundle on the indifference surface I?* satisfying the budget constraint. If the individual could choose a bundle within the budget constraint having utility greater than c*, it would be possible to define an allocation satisfying (l)-(4) that yields utility greater than 3. This of course contradicts the allocation being the maximum of the Rawlsian social welfare function, The Pareto optimal population density n*(r) is therefore consistent with individual utility maximization. The derivative of the equilibrium rent function is obtained by differentiating (4)-(7) with respect to r and is
e(r) = -tZ;/s*(Z;
+ Z,‘),
where zi =
-
{4f”11
-
2qiuli
+
Gi}
-
{ &q#,,
-
q&1
-
q2’%3
+
v,3>
and qi = G/U,. If the utility function is strictly quasi-concave, Z, plus Z, and the first terms of Z, and of Z3 are positive. The second terms of Z, and of Z3 are
169
POSITIVE RENT GRADIENT RECONSIDERED
positive if s and a are substitutes according to the following definition: Goods i and n + 1 are substitutes (S) if &vi(p,jg,+l)/ilx,+l is negative where x(p, js,,, i) minimizesp-x subject to U(x,Z”+,) = v.2 Good n + 1 is not a traded good and corresponds to average neighborhood plot size. To say that s -and a are substitutes implies then that if average neighborhood plot size increased (no change in land prices), the compensated demand for individual land area would fall. In order that utility remain constant when a increases, the compensated demand for at least one of the two goods, land and the Hicksian good, must of course fall. Assumptions (QC) and (S) thus imply that the rent gradient is negative. The derivatives of c and s are i = -i
+ u:)/s*(z;
+ 2;)
and
3 = iu:/s*(z;
+ zg.
If (QC) holds, E is negative and i is positive. Assumptions (QC) and (S) are not sufficient to sign T. If average neighborhood plot size were treated as a traded good its equilibrium price would be q(r) = Uj+/ Uf. No location tax or tax on land consumption is then required to support an equilibrium. The derivative of q is - iz,*/s*(z,* + z;>. By the above arguments 4 is negative if (QC) and (S) hold. Note that@ + 4 is -t/s. The rent plus externality rent therefore declines with r and behaves precisely as rent in the standard case. SECOND BEST ALLOCATION A Pareto optimal allocation can be supported as an equilibrium relative to a price system only if the government levies a location tax or operates an explicit market in average neighborhood plot size. Consider then the market solution in absence of these policies. Assuming that individuals take the population density as given, the budget constraint for a resident of r is c + P(r)s + t(r) = 7 + II where P(r) is the price of land at r. The 21n the above definition good n + 1 is not viewed as being a traded good. The cornpensated demand for good i depends on the quantity, not the price, of good n + 1. In the usual case all goods are traded, and the compensated demand function depends on all n + 1 prices. Goods i and n + 1 are substitutes (S’) if the compensated demand for good i increases with P.+,. The definitions (S) and (S’) are equivalent in the following sense. Suppose the usual compensated demand for good n + 1 at prices (P,P,+,) is X,+l. Then goods i and n + 1 are substitutes at (P,Zn+,) according to (S) if they are substitutes at (P, P,+,) according to (S).
170
HELEN TAUCHEN
Lagrangian for the individual utility maximization problem is U + h(y + II - c - Ps - t). To maximize utility subject to, the budget_constraint a resident of r will choose a bundle (5,s”) with U, = X and U, /P = A. A market equilibrium must satisfy these equalities and (l)-(4). In addition, the utility level must be the same for all r, and residential land rent at the city limits must equal P. The derivative of the equilibrium rent gradient is obtained by differentiating the above equilibrium conditions and is i(r) = - &!2 /(t& + r?,). If conditions (QC) and (S) are met, & is positive, and @is therefore negative. Note that with the usual assumptions U, is zero and @ is unambiguously negative. Richardson considers the second best allocation in his paper. The problem lies with the first order condition for the individual’s location decision. From the Lagrangian above it can be seen that this condition is BU,/A = & + i. Richardson instead has the term ui on the left side of the equation where x is defined to be U, /X. By this definition x = PU, / U,. He then implicitly assumesthat U,/U, is constant throughout the city and concludes that i = FU, / u2. From his first order condition and this conclusion it follows that P = - iU, /(U, - U,)s or that $ is positive if and only if U, is greater than U,. NUMERICAL
SOLUTION
Consider a Pareto optimal and a second best allocation having the same utility level. More resources are required of course to support the second best allocation. One of the few qualitative comparisons can be made without specifying the functional form of the utility function is that if (QC) and (S) hold, land consumption at the edge of the city is greater in the Pareto optimal than in the second best city. The relative size of the cities is indeterminant. The model can be solved numerically for a Cobb-Douglas utility function, U = c’-“‘-BsaaB. Th’ is f unction has properties (QC) and (S), and the equilibrium rent function is downward sloping for both a Pareto TABLE 1 Pareto Optimal and Second Best Allocations Compared
0.25 0.025 0.0025 --~___~~
5%Difference in resource4required
% Difference in city radius
23.319 0.336 0.015
63.852 19.880 2.070
POSITIVE
RENT GRADIENT TABLE
Equilibrium
2
rent and tax gradients’ Pareto optimal allocation
Miles 5.00 10.00 15.00 20.00 R’ = 22.59 30.00 35.00 40.00 R’ - 43.80
171
RBCONSIDERED
Rent/acre
Externality rent/acre
$61.33 50.38 40.93 32.82 29.15 20.20 15.43 11.53 9.09
$613.30 503.84 409.29 328.31 29 1.47 202.02 154.30 115.34 90.91
Second best dl~tiOll LOCdOIl
tax $2849.39 2699.99 2549.99 2399.99 2307.22 2099.92 1949.99 1799.99 1685.97
Rent/acre $203.53 168.61 138.24 112.05 100.00 -
“Iv = 400,ooo; t = $0.40; p = s 12,ooo; F = s64,ooo
optimal and second best city. Whether the Pareto optimal and second best allocations differ significantly depends on the size of the parameters a and /I. For a = 0.025 and values of j3 ranging from 0.0025 to 0.25, the percentage differences in the value of required resources for a Pareto optimal and second best allocation are reported in Table 1. The parameters of the model were specified as a = 0.025, p = $64,000 per square mile per year, N = 400,000 and u = $12,000. The value of 0.025 for a is consistent with a family spending one fourth its income on housing and one tenth that cost being land. At a ten percent interest rate $64,000 is the interest on one square mile of land valued at $1000 per acre. The transportation cost function was assumed to be linear with the daily cost of a round trip being 2tr and t = $0.40. For each a and p the percentage difference in required resource decreaseswith population and transportation cost and increases with income and agricultural land rent. However, the order of magnitude of these percentage differences does not Iary with a fifty to onEhundred percent change in the parameters F, v, P or t. For instance, if N were doubled to 800,000, the column in Table 1 showing the percentage difference in required resources would be 19.2%,0.285% and 0.014%. The ratio of the marginal utility of average neighborhood plot size to the marginal utility of individual plot size is p,/aa. Recall that at an equilibrium s and a are equal. Thus if a = 0.025 and p = 0.25, Richardson’s condition for a positive equilibrium rent gradient holds. The rent function and location tax for both the Pareto optimal and second best allocations are shown in Table 2. Note that the rent gradient is not positive.
172
HELEN TAUCHEN
REFERENCES 1. E. S. Mills, “Studies in the Structure of the Urban Economy,” John Hopkins Press, Baltimore (1972). 2. H. W. Richardson, On the possibility of positive rent gradients, J. Urban Econ., 4, 60-68 (1977). 3. J. G. Riley, Optimal residential density and road transportation, J. Urban Econ., 1, 280-286 (1974).
OF URBAN
ECONOMICS
9, 165- 172 (1981)
The Possibility
of Positive Rent Gradients Reconsidered HELEN TAUCIEN’
Department of Economics, Uniuersi~ of Norih Carolina, Chapel HifI, North CaroIina 27514 Received March 19, 1979; revised December 27, 1979 Despite empirical evidence to the contrary most residential location models are consistent only with a negative rent gradient. Richardson has suggested including average neighborhood plot size as an argument of the utility function; a positive rent gradient is not then a priori inconsistent with equilibrium. However, Richardson does not define the concept of equilibrium, and the purpose of this paper is to suggest a reasonable definition and to show that the rent gradient is negative if the utility function is strictly quasi-concave and if individual and average neighborhood plot sire are substitutes.
The one common property of residential location models is the downward-sloping equilibrium rent function. It is understandable then that empirical evidence to the contrary has spurred interest in developing a theoretical model that would permit a positive rent gradient. Richardson addressesthis question in a recent article of this journal [2]. His purpose is to develop a model in which the rent distance function may be upward sloping even if all individuals are identical and land differs only in its distance from the city center. The critical assumption of Richardson’s model is that average neighborhood plot size (the reciprocal of neighborhood population density) enters the utility function directly. The neighborhood population density is used as a surrogate variable describing local environmental amenities. Since population density is closely related to many of the features of a neighborhood, this approach is an appealing method of introducing local externalities. In this model a positive rent gradient is not a priori inconsistent with equilibrium; lower land rents and transportation costs nearer the city center may be offset by the higher population density there. The definition of equilibrium used in the paper is not given, although with average neighborhood plot size as an argument of the utility function, some thought must be given to the appropriate concept. In particular the variables which individuals take as given in their consumption and location choices must be specified. The two alternative assumptions would *I would l&e to thank Ann Witte, George Tauchen, and an anonymous referee for their comments and suggestions. 165 ooW1190/81/020165-08302.00/0 Copyrgllt 0 1981 by Academic Pms, All righu of repmduction in my form -cd.
Inc.
I66
HELENTAUCHEN
seem to be (1) that average neighborhood plot size is a traded good, or (2) that individuals take the population distribution, like prices, as given. In either case an equilibrium price system would be defined as having the usual property that the quantity demanded of all traded goods is no greater than the quantity supplied at these prices. In addition, the individual plot sizes demanded in a neighborhood must be consistent with average neighborhood plot sizes. For the first of the above assumptions this property implies that the average of individual plot sizes demanded equals the average neighborhood plot size demanded. For the second assumption this property implies that the population distribution resulting from individual utility maximization equals the distribution that individuals take as given. It can be shown that for either definition of equilibrium sufficient conditions for a downward-sloping rent function are (QC) that the utility function is strictly quasi-concave and (S) that individual and average neighborhood plot size are substitute goods. These conditions are not equivalent to Richardson’s condition that the marginal utility of neighborhood plot size be greater than the marginal utility of individual plot size. The resolution of this difference appears to lie in the definition of equilibrium. Richardson’s failure to give a definition appears to have resulted in erroneous conditions for a locational equilibrium and hence to erroneous conditions for a positive rent gradient. In the first section of this paper the location model is described and the properties of a Pareto optimal allocation are derived. It is shown that the government must levy a location tax or operate an explicit market in average neighborhood plot size in order to support the Pareto optimal allocation as an equilibrium relative to a price system. The properties of the no tax market equilibrium considered by Richardson are then described. Conditions (QC) and (S) are shown to be sufficient for a negative rent gradient in either the Pareto optimal or second best case. In the last section a numerical counterexample in which Richardson’s condition is met and yet the equilibrium rent gradient is negative is presented. PARETO OPTIMAL ALLOCATION Each of the fl identical residents of the city travels daily to the city center. The yearly round trip cost for a person living r miles from the city center is t(r) dollars. Utility is a function of a Hicksian good (c), individual plot size (s), and average neighborhood plot size (a). The Hicksian good is elastically supplied at a price of $1 per_ unit; the opportunity cost of land is its value in agricultural use of P dollars a square mile per year. Individual income is y dollars per year. A Pareto optimal allocation in which each individual has equal utility can be determined as the optimum of the Rawlsian social welfare function.
POSITIVE RENT GRADIENT RECONSIDERED
167
The resource constraint for this problem is
c+t)ndr-rR2P=0, where
Ris the city radius and n(r)
is the population density function,
n(r) = 2mr/s(r).
(2)
The population constraint is
%=I Rndr. 0
(3)
The neighborhood for an individual at r is only the land r miles from the city center. Average neighborhood plot size at r is therefore equal to individual plot size at r, or u(r) = s(r). (4) The optimum of the Rawlsian social we&are function is the maximum of social welfare, c, subject to U(c, S,a) 2 U for all r and constraints (l)-(4). The optimum can be characterized as the saddle point of the Lagrangian u+ A,(r)(U(c,s,u)
- 6) - A,(c + t)n + X2(1)(2?rT - ns) - h,n + h4(r)(u - s).
The Euler conditions for c, S, a, and n are
~owu:=h,n*,Xo(r)u,’ - A,(+*=A,(r),Ao(r)U; = -A&), Xl(P + t) - h,(r)s*
- A, = 0;
(5)
R*)
and the transversality condition is A,( = h ,E Eliminating X,(r) and h4(r) from the first three Euler conditions gives
(q + q)/u: =A,(r)/&.
(6)
Note that given the resource constraints, all individuals will have the same utility level at an optimum, or
u(c*,s*,u+) = 3 for all r.
(7)
168
HELEN TAUCHEN
Consider now a market economy. Individuals take the distribution of neighborhood plot size as given; all other goods including land are purchased at market prices. An individual chooses his consumption bundle and location in order to maximize his utility subject to the budget constraint. The city limits are at the distance where residential land rent equals the agricultural opportunity cost. Residential rents in excessof the agricultural value and any taxes collected are divided equally among the population. An equilibrium price system has the property that the quantity demanded of all goods is no greater than the quantity supplied or that (l)-(3) are satisfied. In addition, the plot size chosen by individuals at r must equal the given neighborhood plot size at r, or (4) must hold. It will be argued that the rent function p(r) = U-J’/UT and tax T(r) = a*U,* /U: on residents of r will support the Pareto optimal allocation as a market equilibrium. Note that this tax is a location tax rather than a tax on land consumption. An ad valorem per unit tax on land would not suffice. The Pareto optimal allocation of course satisfies (l)-(4). In order to complete the argument it must also be shown that an individual at r will choose the bundle (c*(r), s*( r)) and that n*(r) is the equilibrium population density. The budget constraint for an individual at r is c + p(r)s + t(r) + T(r) = jJ + II = -h,, where II is the individuals’s share of tax and The bunrent collections and II = { Jt Tn* dr + (EPSON*dr - pR’F}/g dle (c*(r), s*(r)) satisfies the constraint. If the utility function is strictly quasi-concave, it is the only bundle on the indifference surface I?* satisfying the budget constraint. If the individual could choose a bundle within the budget constraint having utility greater than c*, it would be possible to define an allocation satisfying (l)-(4) that yields utility greater than 3. This of course contradicts the allocation being the maximum of the Rawlsian social welfare function, The Pareto optimal population density n*(r) is therefore consistent with individual utility maximization. The derivative of the equilibrium rent function is obtained by differentiating (4)-(7) with respect to r and is
e(r) = -tZ;/s*(Z;
+ Z,‘),
where zi =
-
{4f”11
-
2qiuli
+
Gi}
-
{ &q#,,
-
q&1
-
q2’%3
+
v,3>
and qi = G/U,. If the utility function is strictly quasi-concave, Z, plus Z, and the first terms of Z, and of Z3 are positive. The second terms of Z, and of Z3 are
169
POSITIVE RENT GRADIENT RECONSIDERED
positive if s and a are substitutes according to the following definition: Goods i and n + 1 are substitutes (S) if &vi(p,jg,+l)/ilx,+l is negative where x(p, js,,, i) minimizesp-x subject to U(x,Z”+,) = v.2 Good n + 1 is not a traded good and corresponds to average neighborhood plot size. To say that s -and a are substitutes implies then that if average neighborhood plot size increased (no change in land prices), the compensated demand for individual land area would fall. In order that utility remain constant when a increases, the compensated demand for at least one of the two goods, land and the Hicksian good, must of course fall. Assumptions (QC) and (S) thus imply that the rent gradient is negative. The derivatives of c and s are i = -i
+ u:)/s*(z;
+ 2;)
and
3 = iu:/s*(z;
+ zg.
If (QC) holds, E is negative and i is positive. Assumptions (QC) and (S) are not sufficient to sign T. If average neighborhood plot size were treated as a traded good its equilibrium price would be q(r) = Uj+/ Uf. No location tax or tax on land consumption is then required to support an equilibrium. The derivative of q is - iz,*/s*(z,* + z;>. By the above arguments 4 is negative if (QC) and (S) hold. Note that@ + 4 is -t/s. The rent plus externality rent therefore declines with r and behaves precisely as rent in the standard case. SECOND BEST ALLOCATION A Pareto optimal allocation can be supported as an equilibrium relative to a price system only if the government levies a location tax or operates an explicit market in average neighborhood plot size. Consider then the market solution in absence of these policies. Assuming that individuals take the population density as given, the budget constraint for a resident of r is c + P(r)s + t(r) = 7 + II where P(r) is the price of land at r. The 21n the above definition good n + 1 is not viewed as being a traded good. The cornpensated demand for good i depends on the quantity, not the price, of good n + 1. In the usual case all goods are traded, and the compensated demand function depends on all n + 1 prices. Goods i and n + 1 are substitutes (S’) if the compensated demand for good i increases with P.+,. The definitions (S) and (S’) are equivalent in the following sense. Suppose the usual compensated demand for good n + 1 at prices (P,P,+,) is X,+l. Then goods i and n + 1 are substitutes at (P,Zn+,) according to (S) if they are substitutes at (P, P,+,) according to (S).
170
HELEN TAUCHEN
Lagrangian for the individual utility maximization problem is U + h(y + II - c - Ps - t). To maximize utility subject to, the budget_constraint a resident of r will choose a bundle (5,s”) with U, = X and U, /P = A. A market equilibrium must satisfy these equalities and (l)-(4). In addition, the utility level must be the same for all r, and residential land rent at the city limits must equal P. The derivative of the equilibrium rent gradient is obtained by differentiating the above equilibrium conditions and is i(r) = - &!2 /(t& + r?,). If conditions (QC) and (S) are met, & is positive, and @is therefore negative. Note that with the usual assumptions U, is zero and @ is unambiguously negative. Richardson considers the second best allocation in his paper. The problem lies with the first order condition for the individual’s location decision. From the Lagrangian above it can be seen that this condition is BU,/A = & + i. Richardson instead has the term ui on the left side of the equation where x is defined to be U, /X. By this definition x = PU, / U,. He then implicitly assumesthat U,/U, is constant throughout the city and concludes that i = FU, / u2. From his first order condition and this conclusion it follows that P = - iU, /(U, - U,)s or that $ is positive if and only if U, is greater than U,. NUMERICAL
SOLUTION
Consider a Pareto optimal and a second best allocation having the same utility level. More resources are required of course to support the second best allocation. One of the few qualitative comparisons can be made without specifying the functional form of the utility function is that if (QC) and (S) hold, land consumption at the edge of the city is greater in the Pareto optimal than in the second best city. The relative size of the cities is indeterminant. The model can be solved numerically for a Cobb-Douglas utility function, U = c’-“‘-BsaaB. Th’ is f unction has properties (QC) and (S), and the equilibrium rent function is downward sloping for both a Pareto TABLE 1 Pareto Optimal and Second Best Allocations Compared
0.25 0.025 0.0025 --~___~~
5%Difference in resource4required
% Difference in city radius
23.319 0.336 0.015
63.852 19.880 2.070
POSITIVE
RENT GRADIENT TABLE
Equilibrium
2
rent and tax gradients’ Pareto optimal allocation
Miles 5.00 10.00 15.00 20.00 R’ = 22.59 30.00 35.00 40.00 R’ - 43.80
171
RBCONSIDERED
Rent/acre
Externality rent/acre
$61.33 50.38 40.93 32.82 29.15 20.20 15.43 11.53 9.09
$613.30 503.84 409.29 328.31 29 1.47 202.02 154.30 115.34 90.91
Second best dl~tiOll LOCdOIl
tax $2849.39 2699.99 2549.99 2399.99 2307.22 2099.92 1949.99 1799.99 1685.97
Rent/acre $203.53 168.61 138.24 112.05 100.00 -
“Iv = 400,ooo; t = $0.40; p = s 12,ooo; F = s64,ooo
optimal and second best city. Whether the Pareto optimal and second best allocations differ significantly depends on the size of the parameters a and /I. For a = 0.025 and values of j3 ranging from 0.0025 to 0.25, the percentage differences in the value of required resources for a Pareto optimal and second best allocation are reported in Table 1. The parameters of the model were specified as a = 0.025, p = $64,000 per square mile per year, N = 400,000 and u = $12,000. The value of 0.025 for a is consistent with a family spending one fourth its income on housing and one tenth that cost being land. At a ten percent interest rate $64,000 is the interest on one square mile of land valued at $1000 per acre. The transportation cost function was assumed to be linear with the daily cost of a round trip being 2tr and t = $0.40. For each a and p the percentage difference in required resource decreaseswith population and transportation cost and increases with income and agricultural land rent. However, the order of magnitude of these percentage differences does not Iary with a fifty to onEhundred percent change in the parameters F, v, P or t. For instance, if N were doubled to 800,000, the column in Table 1 showing the percentage difference in required resources would be 19.2%,0.285% and 0.014%. The ratio of the marginal utility of average neighborhood plot size to the marginal utility of individual plot size is p,/aa. Recall that at an equilibrium s and a are equal. Thus if a = 0.025 and p = 0.25, Richardson’s condition for a positive equilibrium rent gradient holds. The rent function and location tax for both the Pareto optimal and second best allocations are shown in Table 2. Note that the rent gradient is not positive.
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HELEN TAUCHEN
REFERENCES 1. E. S. Mills, “Studies in the Structure of the Urban Economy,” John Hopkins Press, Baltimore (1972). 2. H. W. Richardson, On the possibility of positive rent gradients, J. Urban Econ., 4, 60-68 (1977). 3. J. G. Riley, Optimal residential density and road transportation, J. Urban Econ., 1, 280-286 (1974).