Subsidy design for facility location under price-sensitive demands.

ABSTRACT

In this paper we propose a model for solving a subsidy design problem from a regional planner's standpoint to optimally influence the private firm's decisions on pricing as well as the location of uncapacitated facilities. The problem arises when there are conflicting objectives between a private firm and a regional planner. The primary contribution is to develop a new algorithm for coordinating the decisions of two parties with different preferences on the outcomes of facility location. Study of the classic location-pricing problem has focused mainly on optimizing the facility location and selling prices in a centralized setting. We extend the classic location-pricing problem to a decentralized setting, in which a social planner wants to influence a firm's location and pricing decisions by offering subsidization. Designing such an agreement or incentive to coordinate parties with different positions is also known as the contract design problem under the Principle-Agent framework. We show that this optimal subsidization agreement will always improve the coordination efficiency as compared with the situation when subsidization is absent.

1. INTRODUCTION

In the context of location modeling (see Daskin 1995 for an overview), most models consider two levels of decisions, viz., the firm's decisions and the consumer's decisions. The model considered in this paper comprises three, rather than two, levels of decisions. In addition to aforementioned decisions made by a firm and customers, there are also decisions made by a regional planner who manages the space where the firm and customers are located. Whereas decision makers on low levels will act as followers in the sense that they take the decisions from higher levels as given, decision makers on higher levels will, of course, have to consider the reactions from the lower levels.

In our model, the customers occupy the lowest level, as they set no policies themselves, but only react to the firm's pricing policies by deciding the quantity that they purchase. The firm belongs to the next higher level, as some variables can be set according to the firm's objective. Typically in location models, these variables are location and price. While firms can locate and price their goods at will, they must take the consumers' reaction into consideration. Finally, a regional planner belongs to the highest level, as the planner does not take directions from agents at the lower levels but must consider the reactions of the firm to its policies. The main decision variables at the planner's disposal are tax incentives and subsidies, which they can employ to directly or indirectly influence the decisions made by the agents on lower levels. Because a private firm's decisions on pricing and facility locations often have an impact on social well-being, public policy makers may have an interest in asserting influence, via appropriate incentive plans, such as subsidization or taxation, to affect the private firm's decisions. This is true especially when the regional planner has objectives that are different from the ones of the private firm. In this study, we will assume that a regional taxation incentive is expressed by the subsidies offered. Specifically, we assume that the corporate tax rate is exogenous and the subsidy offered from the regional planner to the firm is tax-free. This avoids the inclusion of additional variables in the model to keep the illustration simple. The relaxation of this assumption is straightforward. Our model incorporates the subsidizing mechanism that allows coordination between the social planner and the private firm. When considering the public-sector objectives, Wagner and Falkson (1975) first proposed a joint price-location model to maximize net social benefits. Erlenkotter (1977) examined a "quasi-public" setting that desires to maximize net social benefits subject to a constraint ensuring sufficient revenues to cover the firm's costs.

A broad range of literature in location theory has incorporated the price-sensitive demand of individual markets into the modeling to consider the impact of selling-price on profitability. These studies provide an excellent vehicle for examining how location problems interact with marketing issues to optimize decision-making for the private firm. In the context of economic theory, Greenhut et al. (1987) studied the impact of different pricing policies on the firm's location decisions. In location theory, Hanjoul et al (1990) proposed a unified framework that covers the different price policies and developed a general price-searching algorithm to solve a variety of the pricing-location problems. The optimization technique developed in our model is rooted in location theory. Nevertheless, the analysis in this paper is closely related to the logics in the Principle-Agent (PA) framework born in the economics field.

2. THE MODELS

In keeping with the setting in most location-pricing problems (references), we also assume that a firm produces a single commodity under monopolistic conditions and supplies n markets. The space considered in this model is a graph G = (N, A), where N = {1,2, ..., n} represent the set of demand points and A ={i, j} represents the set of arcs. Consider now the processes that govern the decisions made by the three levels of decision makers outlined in the Introduction. Each individual customer on the lowest level is assumed to decide whether or not to buy one unit of the commodity at the price offered by the firm. Since customer behavior and wealth level vary individually, based on the price offered in market i, some of the customers are willing to buy the commodity while others may decide not to buy. Given a price [pi], with each market i is associated a demand function [D.sub.i]([pi]) that symbolizes the aggregated amount of the commodity desired by those customers in market i. Further, [D.sub.i]([pi]) is assumed to be continuous and decreasing in [pi]. This approximation is appropriate when the number of customer is sufficiently large at each market.

On the second level, the firm seeks to locate p (where p [less than or equal to] n) plants and these locations are to be chosen from a pre-selected m sites, given by M = {1,2 ... m}, m [greater than or equal to] p. Without loss of generality, we assume that M [??] N, i.e., the sites themselves are considered nodes of the graph G. We further assume that there is no capacity constraint on the demand served by each plant. Plants and sites are similarly identified by j. The cost function at site j is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where [q.sub.j] is the quantity produced by plant j, [v.sub.j] is the local constant marginal production cost, and [f.sub.j] is a fixed cost incurred if a plant were built at site j. Let [t.sub.ij] denote the unit transportation cost for supplying market i from site j. The problem of the firm is to determine the price charged to each market, the level of production at each plant, and the distribution pattern in order to maximize its profit. In addition to the allocation decision, the firm needs to decide the delivered price [[pi].sub.ij] at market i, when the commodity is shipped from plant j. We assume that [[pi].sub.ij] is decided according to the spatial discriminatory pricing policy (see e.g. Hansen and Thisse 1977, Philips 1983).

On the highest level, the corporate tax rate has been set at [lambda], which is applied to the firm's net income. If the firm agrees to locate a plant at site j, the subsidy offered is denoted by [s.sub.j]. Without loss of generality, the subsidy paid by the regional planner is assumed to be tax-free. Note that whereas 2 is a fixed parameter, [[s.sub.j]] is a vector of variables for all potential sites j = 1, ..., m.

2.1 The Firm's Decision Model

We introduce the following variables. [X.sub.ij] is set equal to 1 if market i is supplied by plant j and 0 otherwise. [Y.sub.j] is set equal to 1 if plant j is open and hence the fixed cost [f.sub.j] is incurred; [Y.sub.j] is set equal to 1 if plant j is not operating. To avoid the trivial discussion on potential ties among allocation decisions, we assume that the site with smaller index has the priority to be chosen. Under a fixed tax rate [lambda] and the subsidy vector [[s.sub.j]], the objective function of the firm equals to its after-tax profit plus the subsidy from the regional planner. As a result, the problem of optimizing the firm's pricing and allocation decisions can be expressed as follows:

(1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

s.t.

(2) [m.summation over (j=1)[X.sub.ij] = 1, [for all] i [member of] N

(3) [m.summation over (j=1)[Y.sub.j] p, [for all] j [member of] M

(4) [n.summation over (i=j)[X.sub.ij] [less than or equal to] n[Y.sub.j]; where [X.sub.ij] [less than or equal to] 0, [Y.sub.j] [member of] {0,1}

Constraints (2), and (4) are standard in any p-median problem: (2) ensures that all demand at each node gets allocated, (3) guarantees that exactly p facilities get located, and (4) makes certain that no demand node is allocated to an unopened facility. Define that price [[pi].sup.*.sub.ij] is chosen so as to maximize [D.sub.i]([pi])([pi] - [v.sub.j] - [t.sub.ij) and this may require different solution methods. We restate this result as follow;

Corollary 1: The problem (1)-(4) can be restated as follows

(5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where

(6) [[pi].sup.*.sub.ij] = arg max[[D.sub.i]([pi])([pi] - [t.sub.ij] - [v.sub.i])]

Generally, when the demand function [D.sub.i]([pi]) is concave or less convex than an equilateral hyperbola, the optimal price [[pi].sup.*.sub.ij], which is the delivered price when assuming market i is supplied by site j, can be independently obtained by standard gradient methods. For example, when using a non-negative linear function [D.sub.i]([pi])= max[[a.sub.i] + [b.sub.i][pi], 0] to represent the price-demand function in each market node i, straightforward computations show that

[[pi].sup.*.sub.ij] = ([a.sub.i] + [b.sub.i][v.sub.j] + [b.sub.i][t.sub.ij]/2[b.sub.i]

As a result, we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

3. THE REGIONAL PLANNER'S OBJECTIVE AND SUBSIDY DESIGN PROBLEM

The regional planner, often representing a politician or a legislative committee in a democratic society, determines the subsidy vector [[s.sub.j]] to optimize an objective function. We assume that the planner takes the following monetary measures into consideration: (a) Corporate tax collected from the firm. (b) Sales tax collected at each market. (c) The spinoff benefits that result from the firm's expense, such as personal consumption from workers' salary paid by the firm, as well as the fixed investment when opening plants at certain sites (d) Potential value (or damage) generate by the product sold at market i; for instance, a new appliance that can save energy or tobacco that causes higher healthcare cost.

Since the corporate tax has been discussed, we now explain the rest of the factors (b), (c) and (d). Specifically, when market i is supplied by site j, the total variable cost under the optimal price [[pi].sup.*.sub.ij] becomes

[A.sub.ij]= ([[pi].sup.*.sub.ij] - [t.sub.ij] - [v.sub.j])[D.sub.i]([[pi].sup.*.sub.ij])

[B.sub.ij]= [[pi].sup.*.sub.ij] [D.sub.i]([[pi].sup.*.sub.ij]) - [A.sub.ij]

[C.sub.ij]= [D.sub.i]([[pi].sup*.sub.ij])

Define [gamma] as the sales tax rate (i.e., the portion of revenue [A.sub.ij]+[B.sub.ij]) for all markets. It is natural that the multiplier of each site weights differently. Therefore, [[micro].sub.j] is defined as the multiplier that transfers the firm's expense (for both variable and fixed ones) in site j into the benefit in the planner's objective function. For ease of notation, we will henceforth refer to [g.sub.j] = ([[micro].sub.j] - [lambda])[f.sub.j] and summarize the planner's objective function in below. Note that the objective function (7) represents the planner's rationality in maximizing monetary benefit but may or may not reflect the fairness from the standpoint of social welfare.

(7) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Corollary 2: The social planner's objective function (7) then can be restated as follows:

(8) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Having described the objective functions of both the firm and the planner, we now turn our attention to the formulation of the subsidy design problem. In order to influence the firm's decisions, the only instrument the regional planner is able to offer is a subsidy proposal [[s.sub.j]], which provides the firm with a set of site-specific incentives for allocating plants. The subsidy design problem is stated as follows:

(9) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(10) s.t. [[PI].sup.f][[s.sub.j]] [less than or equal to] [[PI].sup.F][0]

(11) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

In constraint (10), [[PI].sup.F][[s.sub.j]] is the optimal value of problem (5) and [[PI].sup.F][0] is when the subsidy vector equals to zero. Since the firm always has the option to reject the proposed agreement and select the location based on [[PI].sup.F][0], this fact dictates that the planner must provide the firm with a payment that can at least compensate the firm's loss due to selecting the location proposed by the planner. It is obvious that the equality of (10) is always held. Our model has assumed that the decision right for site allocation, along with the pricing decision that leads to [[pi].sup.*.sub.ij], belongs to the firm. As a result, constraint (11) specifies that the allocation decision is made based on the firm's best interest, which is conditioned by the subsidy vector [[s.sub.j]] proposed by the planner. Problem (9)-(11)will be denoted Subsidy Design Problem. To the best of our knowledge, no solution for this problem has been proposed so far. The method described in the next section is a binary search on the subsidy vector [[s.sub.j]]; for each vector, we then derive an upper bound and a lower bound and use the derived information to generate a new vector recursively. To do so, we have to solve two uncapacitated p-median problems in every iteration.

4. SOLVING THE SUBSIDY DESIGN PROBLEM

We now turn our attention to explaining how to search for [[s.sub.j]] that can best coordinate the decisions between the firm and the planner. At each iteration of the procedure, the bounds then can be obtained based on the results of solving two separate p-median problems (11) and (12). Based on the results, we then update the subsidy vector. It is convenient to think of this solving procedure in two stages. First, when considering the pricing decisions the firm would made, for each subsidization agreement [[s.sub.j]] proposed by the planner, we solve two optimal allocation decisions correspondent to the fir's and the planner's objective functions. Second, if the planner and the firm have a different preference on location, we modify the subsidization agreement to eliminate the gap between the upper bound and the lower bound in each iteration. The solving procedure can then be summarized as follows.

Step 1: Calculate [[PI].sup.F][0].

Step 2: Set r = 1 and initialize the subsidy vector [[s.sub.j.sup.(1)]].

Step 3: Given [[s.sub.j.sup.(r)]], solve the P-median problem (11).

Step 4: Given [[s.sub.j.sup.(r)]], solve the P-median problem (12).

Step 5: Update UB according to Proposition 5a or 5b.

Step 6: Update LB according to Proposition 6a or 6b.

Step 7: Calculate the new subsidy vector [[s.sub.j.sup.(r+1)]]

Step 8: Let r = r + 1.

Step 9: Go to Step 3 unless termination condition is met.

Finally, after completion of the exchange algorithm, the lower bound is improved by increased subsidy with the multipliers at the values computed by the exchange procedure. To this end, new subsidy vector [[s.sub.j.sup.(r + 1)]] is then generated by moving in a direction given by the step size [[DELTA].sub.i].

REFERENCES

Daskin, M.S. Network and Discrete Location, John Wiley & Sons, Inc, 1995.

Erlenkotter, D. "Facility Location with Price-Sensitive Demands: Private, Public, and Quasi-Public," Management Science, 24.4, 1977, 378-386.

Greenhut, M. L., G. Norman, C.-S. Hung The Economics of Imperfect Competition, Cambridge University Press, Cambridge (U. K.), 1987.

Hanjoul, P. and D. Peeters "A Facility Location Problem with Client's Preference Orderings," Regional Science and Urban Economics, 17, 1987, 451-473.

Hanjoul, P.,P. Hansen, D. Peeters, J.-F. Thisse "Uncapacitated Plant Location under Alternative Spacial Price Policies," Management Science, 36.1, 1987, 41-57.

Philips, L. The Economics of Price Discrimination, Cambridge University Press, Cambridge (U. K.), 1983.

Wagner, J. L. and L. M. Fallson "The Optimal Nodal Location of Public Facilities with Price-Sensitive Demand," Geographical Anal., 7, 1975, 69-83.

Author Profile:

Dr. H. Steve Peng earned his Ph.D. at York University, Canada, in 2001. Currently he is an assistant professor of operations management at California State University, Hayward.