The paper specifies the model and analyzes the optimal conditions, which allow us to conclude, for example, that the non null subsidies/taxes policy is optimal in some cases

An Optimal Urban Planning Tool: Subsidies and Regulations.

Dr. Francisco Martínez Associate Professor

Felipe Aguila

Industrial Engineering (c) and MSc (c) Department of Civil Engineering

University of Chile POBox 228-3 Santiago, Chile Tel: 56-2-6784380, Fax:56-2-6718788

Email: [email protected] [email protected]

Abstract

The urban planning discipline has zoning regulations and locations subsidies as tools for inducing an intuitive notion of citizens welfare. The aim of this paper is to develop an economic method that identifies rigorously the optimal combination of subsidies and regulations in a city given its welfare function. The method defines an urban optimal planning problem whose objective is to obtain the maximum aggregated social benefit across consumers and suppliers in the real estate market. All agents are described by their stochastic behavior and a logit based sub-model defines the static equilibrium under constrains that represent regulations. Behavior is assumed to be affected by externalities and economies of agglomeration and of production scale and scope. The paper specifies the model and analyzes the optimal conditions, which allow us to conclude, for example, that the non null subsidies/taxes policy is optimal in some cases. A prototype application generate some preliminary results that confirm our theoretical findings and provides a first algorithm to attain solutions.

Topic Area: Urban Planning, Urban Economics.

Key Words: Optimal Planning, Land Use, Subsidies, Regulations.

1 Introduction.

There is a common view among urban planners that regulations allows citizens to achieve a higher social benefit. However, there is few theoretical studies that sustain such assertion, and assuming that is true there is no methodology to identify an optimal policy. Our hypothesis is that the existence of location externalities among consumers agents, like neighborhood and agglomeration economies, may justify introducing regulatory schemes combining zone regulations and subsidies/taxes (Martínez, 200). Additionally, assuming that the real estate market behaves as an auction (Alonso, 1964) induces a strategic behavior of suppliers in order to obtain the maximum profit.

Fujita (1989) analyzes the residential sub-market for the circular city using Herbet-Stevens (1960)’s model, including transport costs and residential agglomeration economies in firms production. Rossi-Hansberg (2004) extent this case to the analysis not only for residential but for firms and workers (residential) locations regarding to their preferences under production externalities and agglomeration economies. In both contexts efficient regulations and subsidies are identified.

In this paper we study an heterogeneous city with fixed transport costs and affected by location externalities. We define an optimal planning problem whose solution identifies a set of regulations and subsidies that maximize social benefits in an competitive auction market that attain the Walrasian demand-supply equilibrium. The market equilibrium is calculated assuming a stochastic behavior of bidders and real state suppliers, using the random bidding and supply model RB&SM developed by Martínez and Henríquez (2003). The social benefit is calculated by measuring the aggregate equivalent income variation of consumers (weighted by income cluster) and the profit variation of suppliers (weighted by real estate type and zone). The model is specified and the analytical optimal conditions are derived, a prototype application is developed from which we obtain a set of preliminary empirical results. One of these results if the evidence that the optimal social subsidy policy is not always the null subsidy, even when consumers’ benefits are equally valued by the society.

Section 2 defines the optimal planning problem, and section 3 describes the equilibrium model. In section 4 we specify the social function for a market equilibrium consitent with the RB&SM model. In section 5 we develop the model that we call the Random Bidding and Supply Optimal Planning Model (RB&SMOPM) and in section 6 we discus the interesting case of no location externalities and fixed supply. Finally, section 7 presents some results based on a prototype application, followed by some conclusions and proposals for further research.

2 The optimization model

The model is designed to identify the optimal urban planning set, including regulations (R) and subsidies (Y) for city defined by the total population –households or firms- in each cluster, Hh, and accessibility indexes; both exogenous to the model. Let us define Y∈R and R∈R⁺. The urban optimal planning problem (UOPP) is:

( )

{

(

^{, ,}

)

^{, ( , )}

}

x ,

R Y EQ x x R Y BSN max

R R R

Y ₊ ∈

∈ (1)

where x is the vector of variables of the location model that represents an equilibrium (EQ) associated to the regulation set (Y,R). BSN is a net social benefit function defined by the planner from political wills. Note that a solution of this problem with at least one element Yk ≠0 implies that the usual no-subsidy equilibrium is sub-optimal. This problem automatically assures that the optimal plan will be in equilibrium, therefore no enforcement is required in additions to the optimal planning tools Y* and R*.

2.1 The equilibrium sub-model

The main complexity of the UOPP is that the solution must be an equilibrium point, which requires an efficient model to calculate equilibrium with subsidies and regulations.

In this paper we use the random bidding and supply model (RB&SM) recently developed by Martínez and Henríquez (2003), which calculates the urban equilibrium assuming that location options are discrete differentiable goods trade in an auction market to the best bidder. Consumers are households and firms segmented by clusters, with intra-cluster idiosyncratic behavior variability, whom bid for location options according to their willingness to pay. Suppliers maximize their stochastic profit subject to zone regulations by offering to the market the optimal set of real estate options in each zone. The model takes (R,Y) as an input and calculates the equilibrium location of agents, real estate development, suppliers’ profit and rents.

The operative RB&SM has an algorithm that converges to a unique solution, then EQ(Y,R) is a singleton. In this model the consumers’ behavior is described by their willingness to pay or bid function (Alonso, 1964) defined as the inverse of the indirect utility function conditional on the location choice (Rosen, 1974; Solow, 1973). Consumers’ taste variability is represented by stochastic bid functions, distributed Gumbel, independent and identical, which follows Ellikson’s (1981) argument that within a cluster the only relevant bid for the auction process is highest within the cluster, hence the extreme value distribution, like the Gumbel, is appropriate. Additionally, by assuming the utility function quasi-linear, the following bid function is derived:

( )h hvi( )vi hvi hvi

hviY b U b z Y

B ( )= _h + + +ε

(2)

with zvi the vector of relevant characteristics of a site with building v located in zone i, including accessibility indices and neighborhood quality indices; Uh the utility level obtained in location vi if paying Bhvi, with bh the correspondent money value of utility; bhvi(zvi)is the consumer’s value of the location amenities; Yhvi is the subsidy/tax; εhvi is the stochastic term. Firms have an equivalent bid function, assumed to have the same form derived from their maximum profit problem. In this case the most relevant attributes of a site are attractiveness of clients, access to providers and agglomeration economies. In this paper, however, we develop the model for the residential urban market, although it may be extended for firms with some extra research.

If the population of agents is segmented into clusters identified by sub-index h, with Hhmembers, then the probability that an member of cluster h is the best bidder in the auction of a site vi,

defined as _



 



 ≥

= gvi

hvi g vi

h P B maxB

P , is given by the following multinomial logit (MNL) expression:



 



 

 − +

= _h µ _{hvi vi} ηµ

vi

h H B r

P exp (3)

with rvi the expected rent defined by the expected maximum bid and given by (µ ) _µ^η

µ +



 



= ∑

h

hvi h

vi H B

r 1ln exp

(4) where µ is the scaling parameter of the Gumbel distribution of bids (inversely proportional to the standard deviation of bids) and η≈0.577 is the Euler´s constant.

The suppliers’ behavior is assumed to produce the number of different sites (Svi) that maximizes profit subject to supply a total of S sites in the city and to comply with zone regulations. The profit is defined as the difference between rents rvi and production costs cvi, with cost assumed potentially affected by economies of scale and scope by:

vi S

g a S c

wj wj wj vi vi vi

wj ∀

+

=

∑

^{ξ χ}

)

( (5)

Additionally, profits are assumed stochastic, independent and identically distributed Gumbel, with leads to an optimum supply probability with the MNL form. Then the optimal number of supply sites is:

[ ]

( )

[ ]

( )

∑

^{−− −−}

=

wj

j wj wj

i vi vi

vi r c

c S r

S λ γ

γ λ

exp

exp (6)

with λ the scale parameter of the Gumbel distribution of profits. Optimum supply given by (6) is constrained to comply with a set of linear regulations in each zone, defined as

∑

^≤

v

k i vi k

viS R

a ,

which is the role of the set of lagrangian multipliers (γ_i^k,∀ik). Because at each location there is only one relevant binding regulation we have that ⁼

∑

k k i

i γ

γ . These multipliers can be obtaining by replacing (6) in the binding constraint obtaining:

( )

( )

( )

( )

_^

 − −

=



 



 − −

=

∑

∑

wz

z wz wz v

vi vi vi

i i

c r

c r R a

S

γ λ λ

π

π λ λ

γ

exp 1ln

with ˆ

exp ˆ

1ln _*

*

(7)

where π) is interpreted as the expected profit constraint by regulations and vectors R* and a*

are the parameters of the binding regulation (of course if none regulation is active in zone i then γi=0).

Using equation (3) and (6) the number of agents allocated at each location type (vi) is

vi h vi

hvi S P

N = _/ , which is the result of simulating an auction process with stochastic bids at each offered site.

The equilibrium condition in this model is the usual one in urban economics, i.e. every agent is located somewhere, or all agents are located. This expressed by _h

vi

vi h

viP H

S =

∑ , which can be

solved for the b_h terms of bids functions to obtain:



















 

 



 + − +

−

= ∑

vi

vi hvi hvi vi

h S b Y r

b µ

µ η µln exp

1 (8)

The RB&SM is described by two sets of MNL probability expressions, (3) and (6), and two sets of logsum functions, (7) and (8), that represent linear constraints. The complexity of this equation system is not only associated with its application to large problems, defined by the large number of agents clusters (h) and location options (vi), but also because of its non-liner nature. They are associated with agents iterations in their decision process, due to location externalities and agglomeration economies described by the attribute set, zvi=zvi(P·/·i,S·i), which implies that Bhvi=Bh(zvi) and rvi=rvi(zvi). Moreover, economies of scale and scope in building production cvi=cvi(S··) only add other non linear effects. Thus, the model is specified in a system of four interdependent fixed point problems, conditional on subsidies and regulations, resumed as:

( )

( )

( )

(

^{P S b Y}

)

^hvi

f b

R Y b S P f

Y b S P f S

Y b S P f P

h

i i i

vi

vi i i vi

h

∀

=

=

=

=

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

;

; , ,

,

; , , ,

; , , ,

; , ,

4 3 2

1

γ γ

γ (9)

The solution of this equation system yields the equilibrium location distribution of agents and the land and buildings development associated to such equilibrium. Economic variables, like bids, rents and profits are obtained as relative values only because the equilibrium is invariant to translation of these variables. A simple method to have absolute values for bids and rents is to calculate a constant value across the city, b, referring the model rents to an exogenous rent rex. This reference is usually taken an the rent of agricultural land, but refer the average rent to an exogenous constant, that is ^{= −}

∑

vi vi vi

ex S r

r S

b 1 ^*

where r^* is the rent defined in (4).

Notice that the equilibrium solution defined by (9) yields the lagrangian multipliers of each regulation, which represent their respective shadow prices and measure the social cost of the regulation. This information constitutes a valuable data for assessing economically each regulation.

In this paper, the application of the RB&SM model will consider the following linear form for the bid function term associated with the valuation of two types of attributes:

( )

h vh h i

w wi wi h gw

h vi wi g h i i

hvi P S P S I S X d d

b ^{•• , •} =α ∑ ^/ +β ∑ +σ +ω ⁽¹⁰⁾

where the first term represents a neighborhood quality attribute, based on residents income (Ih), and the second term represents built environment attributes; the third and forth term are valuation of dwelling characteristics and zonal characteristics like accessibility attributes, respectively.

4 Measuring Social Benefits from the RB&SM equilibrium.

In this paper we consider a Benthamite’s type of social benefit measure, aggregating consumers’

and producers’ benefits. The quasi-linear utility function makes the common consumer benefits’

measures, compensating and equivalent income variations, identical.

In the RB&SM model benefits are stochastic variables because bids and rents are. Hanneman and Kanninen (1998) suggest that adequate measure of benefits in the stochastic case is the expected value and the median of the surplus. For operational reason me use the expected value, which is given in our case, with linear random terms, directly by the deterministic term of behavior functions (bids and profits); moreover, linear aggregations of benefits retain this property.

For a individual agent belonging to cluster h, the equivalent variation (VE) associated with two equilibrium points E⁰ and E¹, is obtained solving the following equation for the indirect utility function conditional in the location: ^Vhvi

(

^{P,Ih+Yhvi0 −rvi0+VEhvi01,z0vi}

) (

^{=Vhvi P,Ih+Yhvi1 −rvi1,z1vi}

)

. Inverting in the second component and solving for VE yields:

(

^{b b}

)

^h

b b

VE_h⁰¹= _h⁰ + ⁰− _h¹+ ¹ ∀ (11)

This result is identical to the following difference on bid functions )

, ( ) ,

( ^{0 0 0 1}

01

h vi h h vi

h z U B z U

B

VE = − , that is, the willingness to pay for the change in the equilibrium utility levels associated to the equilibrium points compared, but holding the location constant.. We notice that if we have no externalities and homogeneous subsidy policy, Yhvi=Y ∀hvi, equation (11) becomes equal to Y, a constant variation in income across consumers. This is because average rents are referred to an external rent, so rents absolute values remain unchanged by homogeneous subsidies, hence, subsidies are direct benefits to consumers. Conversely, for non homogeneous subsidies there will be a distribution of benefits over demand and supply.

Aggregating these benefits across consumers, weighting them by social distribution parameters ωh

(exogenous), yields the following total consumers’ benefits:

[ ]

∑

^{+ − −}

=

h

h h

h

hH b b b b

BSC⁰¹ ω ^{0 0 1 1} (12)

We have not introduced yet mobility cost. Firms have an equivalent benefit measure associated to their willingness to pay functions.

Suppliers benefits is measured as the aggregate change in profit, weighted by social distribution parameters τvi (also exogenous):

( ) ( )

[ ]

∑ ^{− − − −}

=

vi

vi i vi vi i vi vi

viS r S r c

BSP⁰¹ τ ^{1 1} γ^{1 0 0} γ^{0 01} (13)

where c_vi⁰¹ are cost variation discounting real estate stock from total supply. For our analysis of the model we will simply assume c_vi⁰¹ =S¹_vic¹_vi −S_vi⁰c_vi⁰.

Add consumers’ and suppliers’ benefits, discount the cost of subsidies ∑

hvi hvi hviY

N and potential administrative costs for changing regulations C^R, and refer the benefit to a situation with constant benefit BSN(E⁰)=K⁰, to obtain:

(

^{b b}

)

^S

[

^{r b c}

]

^{N Y C K0}

H

BSN ^R

hvi

hvi hvi vi

i vi vi

vi vi h

h h

h + + + − − − +

−

= ∑^ω ∑^{τ * γ -}∑ ⁽¹⁴⁾

with r^* and bh obtained directly from the model output before making the correction to absolute values b.

Notice that BSN remains invariant under policies of homogeneous subsidies (where Yhvi=Y

∀hvi), if social values are such that H H

h h

h =

∑^ω . This implies that the UOPP have no a unique solution in that case, which justify the introduction of a subsidy budget constraint G (in monetary units), i.e. N Y G

hvi hvi

hvi =

∑ ^.

Then, the social net benefits in equation (14) have the final form:

(

^{b b}

)

^S

[

^{r b c}

]

^{G C K0}

H

BSN ^R

vi

i vi vi

vi vi h

h h

h + + + − − − +

−

= ∑^ω ∑^{τ * γ - (15)}

5 The optimal planning model RB&SOPM.

The Random Bidding and Supply Optimal Planning Model (RB&SOPM) is designed and analyzed in this section. The model finds a set (Y,R)∈ΣxΘ that induce an equilibrium defined by the RB&SM which maximizes the BSN defined in (15).

The subsidy set ^{Σ =}

{

Y^∈Mhxvi

( )

R /Yhvi ^∈

[

ahvi,bhvi

]

^⊂ℜ

}

is compact for any real value of ahvi and bhvi exogenously determined. The set Θ is the set of linear constraints with the form

ik R S

a _i^k

v vi k

vi ≤ ∀

∑ ^{, with avik} parameters associated to the calculation of densities, square meters, etc. Then ^Θ=

{

^{R∈ℜixk /Rik ≥rk ∈ℜ}

}

^{, where rk} are the set of lower bounds for regulations such that an equilibrium is feasible.

The optimization problem is then expressed as:

[ ]

G Y N

ik R S a

k r R

hvi b

a Y a s

S P

b b BSN

hvi hvi hvi

k i v

vi k vi

k k i

hvi hvi hvi R Y_hvi ^k_i

=

∀

≤

∀

≥

∀

∈

∑

∑

•

•

•

•

•

•

•

, .

.

) , , , , (

max, γ

(16)

where (b.,P./..,γ.,S..) are the solution of (9) and b is the correction on bh we need to get a absolute values of equilibrium utilities. Notice that the set of regulations constraints is redundant since γi variables assure that equilibrium fulfils regulations (they have been included only for exposition purposes).

The RB&SOPM has a solution if exits a RB&SM equilibrium contained in Σ, because this space is compact and total supply S is exogenously fixed to be equal the total number of consumers.

This means that there exist a set of r^k bounds which beyond them γi=0 ∀i. Additionally, all equilibrium variables are continuous functions of subsidies Yhvi, hence, BSN is also continuous in subsidies. Moreover, all equilibrium variables are differentiable in Yhvi, with the exception of γi

variables, which makes the BSN function not generally differentiable, except when r^k minimum bounds are large enough, so γi=0 ∀i. In this special case, the first order conditions may be calculated and the problem accepts an analytical solution, while for the case with binding constraints we need a specifically designed algorithm. Accordingly, we proceed analyzing the first order conditions of the non-regulated case.

Notice that the global optimal solution of the UOPP (16) will occur on a non-regulated situation (or when r^k are sufficiently large enough); the optimal solution on a restricted domain is a sub- optimal point.

5.1 First order conditions for the non-regulated RB&SOPM.

The Lagrange’s function of problem (16) is

( ) ( ) ( )

^



 



 −

+

− +

− +

= ^BSN

∑

^{L a Y}

∑

^{L Y b L}

∑

^{N Y G}

Y

hvi

hvi hvi hvi

hvi hvi hvi hvi

hvi hvi hvi hvi

2

l 1 (17)

where C^R has been omitted assuming that changes in regulations are null of fixed and all other constant are omitted too. Differentiating respect to each component Ygwj , and equalizing to zero, we get:

gwj

1 0

2 − = ∀

+









 +

∂

⋅ ∂

∂ −

∂

∑

^{gwj gwj gwj}

hvi

hvi gwj

hvi gwj

L L N Y Y

L N Y

BSN (18)

( )

^{0 gwj}

1gwjagwj−Ygwj = ∀

L (18a)

( )

^{0 gwj}

2_gwjY_gwj−b_gwj = ∀

L (18b)

[

^,

]

^∀gwj

∈ gwj gwj

gwj a b

Y (18c)

gwj 0 , ²

1_gwj L_gwj≥ ∀

L (18d)

G Y N

hvi

hvi

hvi =

∑

^(18e)

Using equation (15),

[ ] _∑

∑

∑

_^











∂

− ∂

∂ + ∂

∂ + ∂

−

∂ + + ∂













∂ + ∂

∂

− ∂

∂ =

∂

vi gwj

vi gwj gwj vi vi vi vi

vi vi

gwj vi vi

h gwj gwj

h h h

gej Y

c Y

b Y

S r c

b Y r

S Y

b Y

H b Y

BSN _* ^*

τ τ

ω (19)

In the equation system (18) each component gwj of the optimal subsidy is dependent on the other components through the second term in (18). Another observation is that, from (18a and b), one of he langrange multipliers should be zero unless the interval (18c) is a singleton the subsidy is defined by the subsidy constraints. Finally, the lagrange multiplier L is equal to unity by construction, because if we increase the budget constraint parameter G we decrease BSN in the same amount.

5.2 Analysis of optimal conditions: equilibrium derivatives.

To analyze the gradient of BSN, we need to calculate the derivatives of the equilibrium fixed points system:

( )

(

^{Y b r}

)

^h

S b

vi

vi hvi hvi vi

h =−_µ^{1 ln}

∑

^{exp µ} + − ∀ ⁽²⁰⁾

( )

(

^{b Y b r}

)

^hvi

H

P_hvi = _hexp µ _h + _hvi + _hvi − _vi ∀ (20a)

( )

( )

( )

(

^{rr cc}

)

^vi

S S

wj

wj wj

vi vi

vi ∀

−

= −

∑

^{expexpλλ (20b)}

considering the set of complementary equations:

∑

−

=

vi vi vi

ext S r

r S

b 1

(20c)

( )

(

^{b b Y}

)

^vi

H r

h

hvi hvi h h

vi = _µ^1ln

∑

^{exp µ} + + ∀ ^(20d)

(

^{P S}

)

^{P S I S X d d hvi}

b _{h vh h i}

w

wi wi h gw

h vi wi g h i i

hvi ^{••, •} =^α

∑

^/ +^β

∑

+^σ +^ω ∀ ^(20e)

vi S

g a c

wj wj wj vi vi vi

wj ∀

+

=

∑

^{ξ χ (20f)}

hvi P

S

N_hvi = _{vi hvi} ∀ (20g)

Differentiating them respect to each subsidy component yields:

Y hgwj P S

Y r Y

N b H

Y b

vi gwj

vi vi h

vi gwj

hvi vi gwj gwj hvi hvi h

gwj

h ∀













∂ + ∂













∂

− ∂

∂ +

− ∂

∂ =

∂ ¹

∑

^δ _µ¹

∑

⁽²¹⁾

hvigwj Y

r Y

b Y

P b Y

P

gwj hvi vi

gwj gwj hvi gwj

h vi h gwj

vi

h ∀













∂

− ∂

∂ + + ∂

∂

= ∂

∂

∂ µ δ (21a)

vigwj Y

c Y

P r Y

c Y

S r Y

s

sq gwj

sq gwj

sq sq gwj

vi gwj

vi vi gwj

vi ∀

























∂

− ∂

∂

− ∂

∂

− ∂

∂

= ∂

∂

∂ ^λ

∑

^(21b)













∂ + ∂

∂

− ∂

∂ =

∂

∑ ∑

vi gwj

vi vi vi

vi gwj

vi

gwj Y

S r Y r

S S

Y

b 1

(21c) vigwj

Y b Y

P b Y

r

h

hvi gwj gwj hvi gwj

h vi h gwj

vi  ∀









 +

∂ + ∂

∂

= ∂

∂

∂

∑

^{δ (21d)}

hvigwj Y X

I P Y P P Y P

P Y

b

s

si gwj

si h

ps

h gwj

si si p si gwj

si p h

gwj

hvi ∀

∂ + ∂













∂ + ∂

∂

= ∂

∂

∂ ^α

∑

^β

∑

^(21e)

vigwj Y

S S Y g

c

sq gwj

sq sq

sq sq vi gwj

vi sq ∀

∂

= ∂

∂

∂

∑

^{ξ χ χ −1 (21f)}

hvigwj Y

S P Y P

S Y

N

gwj vi h vi vi h gwj

vi gwj

hvi ∀

∂ + ∂

∂

= ∂

∂

∂ (21g)

Note that there is an equation system for each gwj combination which is independent from other index combinations. Additionally, the each equation system is linear and higher order derivatives exist and are all linear. The system (21) has the form x=Ax+b, which has a solution if matrix (I-A) has an inverse. In particular if ||A||<1, the matrix has inverse and, moreover, a fixed point iteration for the equation converge to the unique solutions. Unfortunately the condition ||A||<1 is highly demanding and in most cases is not satisfied, then an alternative less costly methods than inverting A can be considered. One is to obtain the inverse of the three sub-matrices associated with the variable bh, Ph/vi, Svi, and iterate, although for this procedure convergence is not assured. Another one, used in this paper, is to approximate the derivatives to calculate gradients.

6. The homogenous subsidy policy

This case represents the simplest subsidy policy, where the most significant is the null subsidy case. Observe that the homogeneous subsidy consistent with (18e) is Y=G/H and this type of subsidy does not alter the equilibrium solution because it is invariant to b (see equations 20). As consequence of this invariance ∂BSN/∂Y is constant (see equations 21).

Additionally, the second term of equation (18) is null because the total number of agents is constant for all Y. Then, for this case equation (18) yields:

gwj

1 0

2 − = ∀

+

∂ −

∂

gwj gwj gwj gwj

L L Y N

BSN (22)

Here we conclude that for the homogeneous policy to be critical, very specific conditions should hold: either the first two terms are equal (see next section) or the subsidies domain is bounded and has homogeneous bounds.

A case of interest is one with no location externalities and fixed supply, because it naturally comply with the first condition for the homogeneous policy to be critical. In this case,

, 0

2 =

∂

∂

gwj hvi

Y

b vigwj

Y S Y

c

gwj vi gwj

vi = ∀

∂

= ∂

∂

∂ 0 (23)

Additionally, from equation (19) we have:

gwj h

h h vi

vi vi wj

h gwj

h vi

hvi h

h h vi

vi vi vi

h h gej

N H S S

Y N b H S S

Y H BSN











 

 



 −

− +

∂ +

∂























 

 



 −

−

−

−

∂ =

∂

∑

∑

∑ ∑ ∑ ∑

ω τ

τ

ω τ

τ ω

1

1

(24)

which, for ω_h =τ_vi =1 reduces to _gwj

gwj

Y N BSN =

∂

∂ , which proof the critical condition. However, it is worth noting that we have empirically identified cases where, although the homogeneous policy is critical, the optimal solution in non homogeneous.

7 Empirical Evidence.

We have made some simulations to explore the form of the BSN function and optimal policies.

Our main objective is to derive some preliminary conclusions from the algebraic analysis and a simple prototype application, to enlighten future research. Here we describe the most stable outputs obtained from a prototype application.

All simulations were performed using MatLab Optimal package, which approximates first and second order partial derivatives to estimate gradients and hessians. The simulations were performed for a test city defined by: three clusters of consumers (h) with homogeneous behavior and identified by their income level, two zones (i) and three dwelling types (v) ,i.e. six locations alternatives vi. Identical scale parameters were assumed for the Gumbel distribution of consumers’ and suppliers’ stochastic behaviors representing a high level of dispersion. No constant attributes, like accessibility indices, or any constant zonal attributes on bid functions, as they do not alter the analysis. The population consist of a 100 households. The subsidy budget (G) was set to zero.

The optimal test city tends to be concentrated to get the maximum possible rent for suppliers. To get that city optimal subsidies/taxes are large and different. This is an in important result because it an empirical confirmation of the existence of an optimal subsidy policy (different from the null subsidy).

Changes in the social distribution parameters for consumers induce a differentiated optimal subsidy policy across clusters, however, the optimal test city for these parameters showed to be highly insensitive regarding the corresponding supply distribution.

Another characteristic is that the more deterministic is the consumers’ and suppliers’ behavior, i.e.

higher Gumbel’s scale parameters, the more concentrated around the average are the optimal subsidies. Hence, a less predictable behavior induces extreme subsidy values

The solutions obtained for the optimal test city when the subsidy domain tends to be relaxed (for a given G), showed that supply tends to concentrate on a reduced number of options vi, although the relative distribution of supply remains similar. In the limit, when subsidies are unconstrained (except for G), the solution concentrates supply in only one zone and building type. The main explanation for this result is that the test city does not include supply constraints, like space capacity and zone regulations. Nevertheless, this result indicates a strong tendency in the optimal city to reduce real estate differentiation. These effect seems to capture the potential of the city to make the best use of subsidies.

The test city confirms the conclusion for the fixed supply with no externalities case and homogeneous subsidies. Our empirical results yield optimal subsidies not significantly different to the null subsidy policy in this case. Conversely, in other cases with variable supply the optimal policy is to provide heterogeneous subsidies.

For a case of homogenous policies which is critical, we found non homogeneous optimum, which implies that the BSN function has more than one critical point, hence it is not concave. That requires the computation of second equilibrium derivatives in order to determine what kind of critical point is a particular critical policy: a local maximum or minimum, or an inflexion.

7 Conclusions and Next Research Steps.

We found that, in general case, the urban equilibrium is not generally efficient since the optimal policy is not always the null subsidy.

Preliminary empirical results shows that the distribution of supply is the most important factor for the market welfare, which is in line with the intuitive rule of planners that tend to improve the city by acting restraining the supply side. Less intuitive is the result that optimal equilibrium tends to show a great degree of concentration, on space and building type. This tendency, however, is restrained in the real market by the spatial capacity of zones, an effect not considered in our empirical tests but included in the model.

The BSN function considered in this paper is not concave, then it may have more than one local optimum. Since the first and higher order derivatives can be obtained analytically, it is possible to develop specific algorithm that use the property of linearity of these derivatives. Future research is require to develop such algorithms for the RB&SOPM considering the restricted supply case, with the aim of finding an optimal combination of subsidies and supply regulation policies.

References.

Alonso, W., 1964. Location and Land Use, Harvard University Press, Cambridge.

Ellickson, B., 1981. An Alternative Test of the Hedonic Theory of Housing Markets, Journal of Urban Economics 9 56-79.

Fujita, M., 1989. Urban Economic Theory, Land Use and City Size, Cambridge University Press, Cambridge.

Hanemann, W. M., Kanninen, B., 1998. The Estatistical Analisys of Discrete-Response CV Data, Nº 798 in Department of Agricultural & Resource Economics, UC Berkeley, Working Paper Series (Downloadable from http://agecon.lib.umn.edu/ucb/wp798.pdf).

Herbert, J. D., Stevens, B. H., 1960. A model of the distribution of residential activity in urban areas, Journal of Regional Science 2 21-26.

Martínez, F.J., Henríquez, R., 2003. Modelo de Equilibrio Estocástico en Localización Urbana, XII Congreso Panamericano de Ingeniería de Tránsito y Transporte, Quito, Ecuador, Novenber18-22, 2002.

Rosen, S., 1974. Hedonic Prices and Implicit Markets: Product Differentiation in Pure Competition, Journal of Political Economics 82 34-55.

Rossi-Handsberg, E., 2004. Optimal Urban Land Use Zoning, Review of Economic Dynamics 2004 7 (1) 69-106.

Solow, R.M., 1973. On Equilibrium Models of Urban Location. In Essays in Modern Economics, M. Parkin (eds), Longmans, Green, London, pp 2-16.