some spatial equilibria in facility investment under uncertain demand

EIGHTH PACIFIC CONGRESS OF THE REGIONAL SCIENCE ASSOCIATION

SOME SPATIAL EQUILIBRIA IN FACILITY INVESTMENT UNDER UNCERTAIN DEMAND

John Roy Commonwealth Scientific and Industrial Research Organization Division of Building Research Highett, Victoria, Australia Btirje Johansson and Giorgio Leonardi Regional and Urban Development Group International Institute for Applied Systems Analysis Laxenburg, Austria

ABSTRACT This paper describes the spatial distribution of customer demand, supply of customer services and facility investment, as the outcome of a three-level game-like interaction between customers (e.g., shoppers), suppliers (e.g., retailers) and developers (e.g., landlords). Treating both the suppliers and developers at each centre as competitor s , oligopolistic equilibria of the Nash-Stackelberg type are developed, and conditions for their existence and uniqueness established. Uncertain customer demand is specified in probabilistic terms, representing the suppliers' perception of expected customer behaviour.

1, INTRODUCTION Much of the work on facility location in the operations research field has

been confined to cases where the objectives are rational. In this paper, we are investigating the decision problem of facility operators (suppliers) and facility developers (landlords) who recognize a degree of uncertainty in the choice of alternative facilities by customers or users. The users are assumed to make selections according to trade-offs between accessibility criteria, prices and the intrinsic advantages of the facilities themselves, exemplifying a 'user-attracting' system as defined by Leonardi (1981).

The analysis focuses on location of, and investments in, private facilities, demand for floorspace by profit-motivated operators, and customers' demand for services supplied by facility operators. Thus, we identify different objectives for each of these three categories of agents and examine corresponding supply/ demand equilibria. Previous work on this problem is marked (among others) by Lakshmanan and Hansen (1965), Harris and Wilson (1978), Leonardi (1978), Rijk and Vorst (1983), Roy and Johansson (1984) and Roy and Lesse (1983).

From the viewpoint of a planning authority, different criteria apply for the location of public facilities (Leonardi, 1981) as compared with private facilities. In the latter case, the authorities must contemplate the competition between developers (owners) as well as between the operators of the facilities. Our equilibrium analysis uses a Stackelberg leader-follower chain between developers and operators, as well as between operators and customers (Intriligator, 1971). Special emphasis is devoted to expanding current information theory estimation procedures to consider how customers trade off price of goods, quantity of goods purchased, shopping travel costs and quality attributes of the centres and goods.

Perceived and Estimated Behaviour of Customers In the model framework presented, customers are demanding services from

the facility operators whom we call suppliers; the latter are demanding floorspace

216 PAPERS OF THE REGIONAL SCIENCE ASSOCIATION, VOL. 56, 1985

which is supplied by developers. The customers make their decisions without contemplating the effects their behaviour may have on the decision-making of the suppliers and developers. The behaviour of customers is estimated by means of a facility choice model based on information theory, which includes budget constraints for shopping travel plus purchases. The estimation procedure is described in Section 4.

A Two-Level Oligopoly Structure The model is specified in such a way that the suppliers in each centre,

treated as independent agents, have two decision variables; the size of rented floorspace and a price level for the goods and/or services they supply. We assume that in each centre the supplier maximizes profits, in non-cooperative Cournot competition with suppliers at all other centres (Intriligator, 1971). Whilst the suppliers take the decisions of the developers as given, they anticipate the mode of reaction of the customers, implying that each supplier is a 'leader' with respect to the customer demand pool.

Two decision variables are assigned to each developer: the size of available floorspace and the average rent level. The developer maximizes annualized profits at each centre, in non-cooperative Cournot competition with developers at all other centres. The developers anticipate the mode of reaction of the suppliers, implying that each developer is a leader and each corresponding supplier a follower at this level.

In practice, the above structure represents a hierarchy in which perceived customer behaviour is embedded in the profit function of each supplier, and the resulting behaviour of the latter embedded in the objective of each developer. 1 For this multilevel game, we examine the existence of Nash equilibria, contingent on prevailing perceptions (Johansson, 1978), attempting to distinguish between the rules of the game and how it is played (Shubik, 1959). Some partially related work appears in Sherali et al. (1983) and concurrently in Sherali (1984), where several leader firms compete for shares of a common follower demand pool.

2. MULTI-CENTRE NASH EQUILIBRIA FOR SUPPLIERS We shall study a system with M customers who visit supplier centres in

which goods and/or services are supplied from facilities located in the centres, indexed by j. The customers' zone of origin is indexed by i and their income group by k. In order to describe the customer behaviour we must introduce the suppliers' decision variables:

Wj = amount of floorspace utilized by suppliers in centre j yj = (average) price of a typical basket of goods selected in centre j by

members of lowest income group k

In addition, we introduce an estimated factor fj = measure of local attractiveness quality, reflecting convenience and

comfort in centre j (see Section 4)

Customer Behaviour The proportion of customers in origin i and belonging to income group k

is oik. The following average quantities and (estimated) parameters are used to describe the behaviour of customers:

t To some extent, this is a hierarchy in which each adjustment process is embedded in a relatively slower process of change.

ROY ET AL.: EQUILIBRIA IN FACILITY INVESTMENT 217

cijk, tog --- average travel cost and time respectively between zone i and centre j for customers of category k

ak -- ratio of 'quantity' of the commodities purchased by income group k compared with that of the poorest group

ffk = parameter describing commodity price and travel cost sensitivity of customers in income group k

= parameter describing customers' travel time sensitivity

Since we are studying basic purchasing behaviour by shoppers, a price elasticity of demand of zero is assumed, implying a constant average 'quantity' of goods being purchased by each household in an income group (see Section 4 for possible refinements). In essence, we advocate a travel cost (time) vs. price tradeoff, rather than the traditional quantity vs. price tradeoff. For given price levels yj, the average shopping purchase plus travel budget bk per trip in each income group k is expressed as

(~, o,k) bk = ~ P~jk (ak Yk + Cijk) (1) i ij

where the behaviour of customers is described by probabilities P0~ showing the likelihood of customers being of type k from origin i and visiting centre j. Defining the interaction effect, f~jk, in terms of the variables and parameters introduced above as

f~jk = exp {- ~kkakyj -- ffk C0k -- /~ tok} (2)

the choice probabilities are determined in the form (see Section 4)

P,jk = o,k ~ I~fj~k (3) 3"

where a is an estimated parameter reflecting how the scale of a centre affects the overall destination attractiveness for customers. The probabilities in Equation (3) are also assumed to reflect the suppliers' perceptions of customer behaviour.

Existence and Character of Nash Equilibrium The decision problem of suppliers is conceived as a competition among

centres, anticipating expected customer reactions. We assume that the supplier profits at each centre j are maximized contingent on the decisions in all other centres. Let the profit function of centre j be

rj Wj = M ~ ak P0k (Yj -- Y~ -- (rj + wj)Wj (4) ik

where wj are costs proportional to the floorspace and y] costs proportional to the sales volume, and where ~rj and rj are profit and rent per unit floorspace, respectively. M, w~ and y~ enter as exogenous parameters, W i and yj are decision variables, and rj is assumed to be fixed by developers. The Stackelberg assumption implies that the customer decisions P~jk are embedded as functions of Wj and y~ via Equations (2) and (3) in the supplier's decision functions of Equations (4), implying that suppliers have perfect perceptions of expected customer reactions to their quantity and price decisions.

Proposition 1: Identify each centre as a decision-making unit selecting Wj from a closed interval on the real line. Let for each centre the objective

218 . PAPERS OF THE REGIONAL SCIENCE ASSOCIATION, VOL. 56, 1985

function be given by Equation (4) and let a < 1. Then for given rent levels and prices yj > y;, there exists a unique Nash equilibrium { l~j).

Proof: A unique Nash equilibrium exists if (i) the decision sets are compact and convex, (ii) the profit function is continuous in all variables, and (iii) strictly concave in Wj (see, for example, Berge 1957). Properties (i) and (ii) are obviously satisfied. Property (iii) can be demonstrated with the help of the first and second order derivatives:

a(~ jw j ) : ,~ w ; 1 ~ T~ - (rj + wj) a ~

�9 api+k aPijk using O W i ffi Wj (1 - puJoik)

where Tj = ~ ak Pijk (1 - p~jJo~k)] and Yj : M (.vj - yj*). ik

a~('~JwJ) = - ~ w ; ~ ~ ( T j - ~ U~) (5) a~ where Uj = ~ ak p;jk ( 1 - p,j~/o,k) (1 - 2 PuJO~k). As Uj is less than Tj, it follows

i k

from Equation (5) that the second order derivative is always negative for a -< 1, and the solution obtains for

: a Yj ~/(rj + w~) (6)

where superscript ^ refers to optimum values or expressions as functions of optimum values. The situation described in Proposition 1 is illustrated by Figure 1 (Case a).

For a > 1, we consider the following proposition (illustrated by Case b in Figure 1):

Case a Case b

^ (rj+wj)Wj

Rj

wj wj FIGURE 1. Illustration of Proposition 1 (Case a)

and Proposition 2 (Case b) k j = M z p,jk ak (yj - ~)

RJA pjRj


Proposition 2: Let the assumptions in Proposition 1 be retained, and consider the profit function ~j Wj = M ~, Pijk ak (yj - y~ (1 - o j), in which land

lords charge rents proportional to the income of each centre, and where oj represents this share taken for rents. With this function there exists a Nash equilibrium.

Outline of a Proof: We shall not show that the equilibrium is unique. Hence, instead of strict concavity (as in Proposition 1) we shall only require quasi- concavity of each profit function (Berge 1957). ~j Wj is obviously positive and monotonically increasing, with the first order derivative a I4/~j a (1 - #j) Yj T~ > 0.

The function has an inflection point from Equation (5) when Tj - a ~ = 0. To the left of this point it is positive, increasing and convex. To the right it is increasing and concave. Evidently, like puk(Wj), it is quasi-concave. From Equation (5) it is seen that the chosen floorspace may fall within the concave region for cases with a small number of centres.

In order to establish the existence of a Nash equilibrium for the price decisions, we need a compact (closed and bounded) decision space for each zone. As a lower bound we can select y~ > 3~. We can also prevent yj from getting arbitrarily large, since pj. -, 0 as yj -, m at a faster rate than that of yj itself, where the destination probability pj = ~ Pok.

ik

Proposition 3: Identify each zone as a decision-making unit selecting yj from a compact set, and let the profit function be rj Wj in Equation (4). Then for given sizes V/j and given rents there exist an n-tuple (for n zones) {pj} which, under elastic demand conditions, constitutes a Nash equilibrium such that

A

39j = Y; + L / T j (7)

where ~j = ak/3~j~ and Tj = a~ #k Pu~ (1 - t3oJoik) using OpjOy i

= - ak ffk Pok (1 - Pijkl/Oik).

Proof: The profit function is continuous and the decision set compact. Then it remains to show that the profit function is quasi-concave. We shall do this by examining the first and second order derivatives with respect to yj. Let them be denoted by ff~ and 1T~ respectively.

[ ~ a~ ~kPok(l --Pok,/Oik)]} (8)

= M { S j - ( y j - Y3 Tj}

IIj" - - M ~ {a~ ffk P0k (1 -- Pok/O,k) ik

[2 - a• (yj - y~ (1 - 2 PodO,k)]} (9) --- - M { 2 T~- (yj -y) U~j}

U~ = [ ~ a~, 4,~ pok (1-- p,jO,k) (l -- 2 Pa~/Oi~)]'-I For ( y j - Y3 small, IIj where


> 0 and ITj < 0. Treating ~j = Sj as the 'quantity normalized' demand at j, and realizing t__hat the price elasticity of demand 8j defined as cj = (O~j/Oyj)yj/-~j yields ej = -yj T/Sj, it is seen that a price equilibrium exists as given in Equation (7) for the case where demand is elastic, i.e. l ejl > Yj/(Yj - Y'J) at yj = pj.

Corollary 1: By inserting pj into Equation (9), we can see that usually ~. < 0 at the point yj = pj. This is because the inner bracket in Equation (9) becomes {2 - (1 - 2 puJo~k)[(~ ~kk)/(~ ~ ~bk Pok (1 -po#otk))] [ ( ~ a~ Po'k)/ak]} at the

ik ik optimum, where the value of the variable term will usually be in the neigh- bourhood of unity for cases with a reasonably large number ofcentres and a not too extreme distribution of budgets over the income groups k. In addition, as the higher income groups (with higher values of a~) will tend to be less sensitive to price (have lower values of ~k~) than lower income groups, the (ak ~k) values will automatically tend to even out, further ensuring negativity of ~ at yj. = pj. Thus, under the above restrictions, the ~rj Wj relation will be strictly concave within a generous band about the optimum, and a 'stable' Nash price equilibrium will exist.

Using Equation (7) to eliminate the price term Yj from Equation (6), one obtains the following implicit equilibrium demand function for quantity of floorspace by a retailer at j at equilibrium prices, in terms of any rent level rj set by the developers

A

= Ma L ~/[Tj (rj + wj)]. (10)

As P~k is still itself a function of j~j, an iterative solution of Equations (7) and (10) is necessary to solve for l~j and 33j simultaneously.

Monopoly and Collapse of the Spatial Structure The oligopolistic setting in the preceding subsection turns out to be essential

for preserving the multi-centred structure of solutions. If a single decision-maker controls all suppliers, the model generates a monopoly solution which utilizes only one centre.

Assume that a monopoly has the objective to maximize the_sum of profits emanating from all centres, subject to spatial constraints Wj _-< Zj, where Zj is the floorspace supplied at j. The corresponding Lagrange function is

L = Z [R~ - (r~ + w~) W~] + Z x~(Z - w~) (11) J )

where Rj = Y j ( ~ a~pok). The standard opt imum conditions are

oL { dl, Vj = ( M a l W i ) Ziqk ak Pijk [(Yj -- Y ~ l n

- (y , - y 3 p,/,~] } - (rj + wj + xj) _-< 0

(12a)

OL - - Wj = 0 ( 1 2 b ) ow~

where the conditional probability Pq/ik is defined as piqJo~k and n is the number of retail centres.


Proposition 4: Let there be one decision-maker who maximizes the total profits over all centres as specified in Equation (11). Moreover, let rent plus wage levels (rj + wj) > 0 and prices y: > ~ be given. Then, for non-identical centres, the maximum is obtained by selecting only one centre.

Outline of a Proof: Observe that, for at least one centre, the income

(~akp~jk(yj-y~)attheactualpricemargin(y:-3~) will be less than the

income(~akp~jk[~ (yq- yq)p~/jk])attheaveragepricemargin[~ (yq- .] \ ik

y'q) pq/~k I for each (ik) combination making the summation term in Equation

(12a) negative. Since rj + wj + ~: > 0, this implies according to Equation (12b) that ~ = 0 is the only feasible solution. Having observed that ~ = 0, we can apply the same argument for still another centre, and continue to eliminate centres till only one is left.

Remark 1: The result in Proposition 4 may be prevented if we introduce congestion effects or a simple density constraint of the following type pj M/ W: _-< d for each j.

The statement of Remark 1 follows directly from inspection of the associated Lagrangian, which can yield positive solutions for ~ when the new multiplier is relatively large, that is, when the density constraint is strongly active. Remark 1 also reflects the fact that a customer density relation is lacking in the oligopoly situation. However, in that case, the non-cooperative setting is enough to preserve the multi-centred structure.

3. DECISION PROBLEM OF DEVELOPERS In this Section, the floorspace demand function [Equation (10)] in terms of

rents is directly embedded into the developer income term for the short-run rent setting problem. For long-term capacity expansion decisions, Equation (10) is inverted into a rent function in terms of floorspace capacity, ensuring that demand will equal supply. For both of these Stackelberg assumptions, the current derivations assume that developers have perfect perceptions of expected supplier reactions. First we_ consider the short term problem for which the available floorspace, ~ -- Z:, is fixed, so that the rent level is the only decision variable. In the second step we allow for investments in new floorspace and associated infrastructure.

Short run selection of rent level In the short term, having sunk capital into the infrastructure, the developers

manipulate rent levels to maximize their operating profits; bearing in mind the expected adjustments of the suppliers. Their constrained operating profit function g: is given as

gj = rj l ~ j - o j Z : + h j ( Z j - 1~i) (13)

where oj is the operating cost per unit of supplied floorspace and )~j the multiplier on the capacit_y constraint. As Equation (10) represents an implicit demand function for W: in terms of rents, we cannot differentiate its right-hand side directly, but can rewrite Equation (10) as


A

?= (rj+ wk ~ 5 - M - L L - - 0. Then, the total differential d ~/drj , given as - (OF/Ors)/(OF/O l~j), can be shown to yield

A A A

d fJ~j/drj = - f.V s T~/{(rj + wj) (Tj + a Uj) (14)

A

where U: = ~ a 2 ffk Pijk (1 - /~ljk/O,k) (1 - 2 l~ijkl/Oik)o ik

Upon setting 0 gJOrj = ~ + (5 - Xj) d VVj/drj to zero and removing the M term in Equation (14) via Equation (10), a solution exists for the profit maximizing rents fj (under the conditions in Corollary 1) as

A & A ^ . A

a A

which, when substituted into Equation (10), yield ~ as zx /x A

= M o~ 2 [Tj (L ~ + ~ ) - Us L L]/(ws ~ ) (16)

with no active capacity constraints. This may be solved simultaneously with pj from Proposition 3 and iteratively for all centres j.

In determining the nature of the equilibrium, the second derivative O2~g./ 0~, given here as (2 d l'PJdr i + (rj - Xj) d: gVJd ~), is examined. Defining Dj, the denominator of Equation (14) after substitution of Equation (10), as

A A A

~, = (rj + w,) {(T~ + <~ ~) - o, ~ (L ~ + ~ ) / g L}

the second derivative of ~ becomes A A a

ce ~td 4 = ~- Tj { (~ + o, ~.) + 0 ~;/0 r~}/~ z~

noting that the derivative of any term such as Ts with respect to rj is given as

(OTJOf, Vj)(d VVJd r~). Upon extraction from Equations (14) and (15) that (g. - _ _ A

~i) = DJTj, the second derivative 02gJ0~ at the optimum rj = ~ comes out as A A A

o 2 g j o ~,~:,~ = - ~ . {2 T~ - [(T~ + ~ ~ ) + o ~ j 0 ~]}/~.

Under the conditions of Corollary 1, the second derivative at the optimum can be approximated by -Wj {2 - [2 - a]}/[(1 - a)(r~ + wj)] which is negative for a < 1, implying a quasi-concave result. Now, it can be seen that at r~ = 0, 0gJ 0r~ equals a positive value ( ~ - k~ d VVJd r~) and 02gJOd is strongly negative. As shown in Figure 2, OgJOr~ -- 0 at rj = ~. and 02gJ0d is less strongly negative. As 5 increases further, OgJOrj turns negative, and because the positive ( r jd ~ W~/ d ~) contribution to 02g~/0~ only increases with r~ according to the ratio r~/(r: + wj) and from Equation (14) ~ < w~ (i.e. ~ ~ w~ (1 - a)/a), then r~ can increase considerably beyond ~ before the point 0:gJ0~ = 0 is reached.

Remark 2: Under the conditions of Corollary 1, it can be shown that ~- at 02gJO~ = 0 approximates 2 ~, thus guaranteeing a large concave region


about the optimum (see Figure 2), and ensuring strictly concave behaviour about the Nash equilibrium.

Floorspace Supply Decisions in Centres If the optimal combination of floorspace demands ~ and developer rents

Pr in the previous Section initially yielded some positive excess demands, there is clearly the scope for some developers to increase their constructed floorspace Z r, say by Zj to ~ = (Z r + Zj). In order to determine the rent level ~ which ensures that the optimal floorspace demand ~ by the retailers from Equation (10) equals totaldeveloper supply ~ , Equation (10) is inverted and ~ substituted for Wj to give r r as a function of ~ , yielding

, x

r; = M L / ( Z 3 - wj.

Letting rj and oj refer to annual rent income and operating costs respectively, and F~ the unit annualized investment costs when providing Z r units of new floorspace, the developer at each centre j can be considered to choose Z r to maximize the surplus gj, given as

gr = (6 - or) ~ - Fj Zj. (17)

Defining R r = (r~ - or) ZTj as the net operating surplus, we may conclude the following:

Remark 3: For a < 1, Rj is concave under the same physical conditions as expressed under Remark 2.

gj

I Point of

/r,. i n

r

FIGURE 2. Developer Income for Short-Run Rent Setting


Proof."

where

R'j : O RiO Z~ : M a2[T~ (Sj Q + ~ ) z x z x ^ ^

- ~ s j ~1/(~ ~) - (w: + o,). ~x

ZX ZX ZX A

+ < g [ V j - 2 U ~ J T j ] - T j [ g g + 3 g g zx

- ( L U~ + ~11~1}/(~ ~1

: ~ ak Po~ (1 - Pujoi~) [(I - 2 p~jo~k)(1 - 3 Ptjo~k) - PJogk] i t

zx

and Vj is identical, but with ak above replaced by ~ ffk. From the same conditions as imposed under Remark 2, it can be deduced

that the terms inside the { } bracket in Kj are of the order of (1 - a)/a, guaranteeing a unique maximum for a < 1.

Comparing Rj = 0 above with the solution of the short-run problem in Equation (16), it is seen that the denominator of the latter now becomes (wj + oj), implying that the maximum operating surplus occurs at a value o f ~ less than that for the short-run solution where the demand constraint I~j < Zj is not imposed (Figure 3). This will be further accentuated if the capital contribution Fj Zj is considered in Equation (17). For instance, if e is the unit annualized

i q

urplus I I I I I I

A

W ( Long~[Short~

- Run]~-Run ] Unconstrained

FIGURE 3. Developer Long-Run Income and Cost Relations


investment level for the lowest quality centre and i~ the relative investment weighting at any centre j, then neglecting scale effects, Fj is written as

-- ij e (18) /x

which brings the denominator of Equation (I 6) to (wj + oj + ire) ~ in the long- run solution, yielding

A Z~ Z~

g = M , 2 9, + - g g g l / [ (w, + o; + i, e) (19)

However, as new investment would not have been considered unless certain of the excess demands were positive in the short run, there is still a good chance that the long-run solution ~ will be greater than the existing floorspace supply z,.

In practice, the unit construction costs may not be linear as assumed in Equation (18) above. Neglecting the operating costs oj, and assuming the physical conditions exist as mentioned under Remark 2, a Nash equilibrium for the developers exists in the cases given below.

Proposition 5: Let a < 1 and regard the developers in each zone as one decision maker selecting Zj from a compact set, and let Zj >_- 0. Then there exists a constellation of floorspace decisions, {Zj}, which form a Nash equilibrium if for each j, Fj is a monotonically increasing function which satisfies one of the following conditions: (i) convex everywhere; or (ii) concave with R~ > 0, and Rj > Fj for Z~ > L'j," or (iii) concave with Zj > 0, R~ _-__ g for Z7 > Zj; or (iv) S-shaped such that Fj" > 0 in the convex segment of Fj, and Rj <

in the concave segment; or (v) concave or convex with Rj concave and peaked.

Outline of a Proof." According to Remark 3, Rj is a continuous and concave function, and Fj is continuous by assumption. Hence, & is continuous. The additional requirement on & (for an equilibrium to exist) is that g, is quasi- concave. This is satisfied if& is (I) monotonically increasing or (II) monotonically decreasing, or (III) monotonically increasing to a peak and thereafter decreasing, or (IV) & is concave. Cases (i) to (iv) are illustrated in Figure 4.

In case (i) & is the difference between a concave and a convex function. Hence, at least one of properties (I), (III) or (IV) are satisfied.

In case (ii) g, is the difference between two increasing concave functions such that property (I) is satisfied.

In case (iii) & is__monotonically increasin.g for Z 7 < Zj and monotonically decreasing for Z7 > Zj, since ~ = 0 for Zj < Zj. Hence, property (III) is satisfied.

In case (iv) property (III) is satisfied. For the convex segment of Fj we can use the result from case (i). For the subsequent segment of Fj we use the result from (iii) if R'j > 0 everywhere and from (v) if Rj is concave and peaked.

In case (v) property (III) is obviously satisfied, since & is the difference between a peaked concave and a monotonically increasing function. This completes the proof.

4. ESTIMATION AND IMPLEMENTATION

Estimation and Validation of Customer Model We estimate an aggregate customer model directly from average aggregate

data, with the inclusion of extra variance information when this can be shown


case(i)

Zj Z~ l case {iii) case{iv)

Rj Rj

case (ill

Zj Zj

FIGURE 4. Alternative Developer Investment Functions

to measurably improve the goodness of fit. At this stage, we neglect consideration of substitution or agglomeration effects between competing destinations, which result in nested recursive model forms in terms of destination clusters (Roy 1985). Nevertheless, as in Roy (1983), the quantity and quality components of destination attractiveness are accounted for in the following derivation to estimate a model for determining the probabilities Pijk of customers coming from zone L being of income group k and shopping in centre j. As for individual choice models, the selection of constraints is a 'trial-and-error' procedure, guided by goodness-of-fit comparisons.

For the estimation of the model, the maximum entropy S is determined with respect to p~jk in the form

= max - ~ p,j~ (log pjW] - I) + ~ Xi~ (o,~ - Z Po.) Pijk ijk ik j

+ B (t - ~ Pei~ t,:k) + ~k Ck (bk - ~,j P~:k (a~ yj + c,:k)) (20)

+ ~ (m - ~, p,j~ mj) + . (m2 (m) - ~ pejk m2 (mj)) ak ak

-b ~1 ( i -- ~ Pijk ij) + K (~l -- ~ Pag nj). ijk ijk

In addition to the terms defined at the start of Section 2, certain destination quality influences have also been included. Thus, 'convenience' effects are related to the amount of time rnj to park and complete the average shopping task in centre j. As this time may vary considerably for different persons at different


times, a constraint on the average variance measure m2(m) of m can also be applied in terms of the observed variances mz(mj) at each centre j. As a proxy for 'comfort" ij is taken as the average building infrastructure investment intensity in j (normalized to unity for the poorest centre), which is the only exogenous parameter directly connecting customer choice with developer decisions, as seen from Equation (18). One may also consider a binary variable n> given as unity if undercover parking exists at j and zero otherwise. The solution comes out in the form of Equation (3), where f1 = exp - (~ mj + ~ m2 (mj) + n ij + K nj).

In addition, the unknown power a, reflecting the scale effect on centre size as perceived by customers, may be simultaneously determined by minimizing the error of fit, which yields the following implicit relation for a:

log = E log J J

which only requires the observed destination probabilities ~j. Finally, the role of the budget constraints [Equation (1)] deserve special

mention. As these constraints also need to be satisfied in the forecasting time period, multipliers ~j will need to be estimated anew after the forecast values of W s and ~j have been obtained, and a new run made through the model sequence. As emphasized by Truong (1982) in his comparison of econometric demand models with individual choice models, it is often desirable to generalize Equation (1) by including non-zero elasticity effects. Thus, in the forecasting use of the model, where price changes are due to competition between centres, Equation (1) is replaced by

Pok {a;k Y1 [1 + Ek (yj - y~)/~] + cok} = ~oik b, (21) ij i

where y~ are the observed prices at the estimation time period, Ek is the price elasticity of demand for basic retailing goods by income group k, and r the observed relative quantity of goods at price Y7 purchased by shoppers from group k at j. The implications of the quadratic occurrence of yj in Equation (21) compared with its linear appearance in Equation (1) require further investigation.

5. CONCLUDING REMARKS A three-level leader-follower model has been introduced, in which at the

retailer level, represented by oligopolists at each centre j, prices and floorspace demands are set according to perceived response at the customer level. At the final level, the developers, again acting as oligopolists at each centre, make their short-run rent decisions and longer-run capacity expansion decisions depending on their perceptions of response by the retailers. Viable simultaneous equilibrium states have been derived for all categories of agents. Future developments include introducing stochastic elements into the model, building for instance on the extensions of stochastic cost-volume-profit analysis to oligopolies recently per- formed by Karnani (1983).

In the above work, we have been primarily investigating acceptable model structures for spatial oligopolistic competition in the retail sector. However, it may not be premature to attempt to implement these models.

ACKNOWLEDGMENTS The authors are grateful for the assistance of Paul Lesse on the use of

information theory concepts, and Truong Truong on various aspects of demand models.


R E F E R E N C E S

Berge, C. 1957. Theorie Generale des Jeux ~ n Personne. Memoriale des Sciences Mathematiques, Fasc 138.

Harris, B., and Wilson, A. G. 1978. Equilibrium values and dynamics of attractiveness terms in production-constrained spatial interaction models. Environment and Planning A, 10:371-388.

Intriligator, M. D. 1971. Mathematical Optimization and Economic Theory. New Jersey: Prentice Hall.

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