Strategic Location and Manager Delegation with Concentrated Consumer Demand

1 Introduction

The intra-firm organization of big firms is usually characterized by a separation of ownership and management. In managerial economics, this owner-manager relation is treated as a strategic delegation problem where the managers’ decisions are incentivized by the design of their compensation schemes. An extensive literature has emerged demonstrating that it is not the optimal strategy of owners to give their managers incentives to maximize firm profits. Instead, compensation contracts are offered that usually consist of a linear combination of profits and sales or revenues (see, for example, Vickers 1985; Fershtman and Judd 1987, 2006; Sklivas 1987). Empirical evidence for such contracts is provided, e.g. by Murphy (1985), Jensen and Murphy (1990) and Conyon (1997). The basic result of these models is that in the case of quantity competition, managers are incentivized to aggressively increase production whereas they are incentivized to increase prices in the case of price competition thereby relaxing the intensity of competition.

In a broader framework, these strategic delegation decisions of owners interact with other strategic decisions such as demand-enhancing or cost-reducing investment, or the location of firms in geographical or product space. In industrial economics, an overwhelmingly large literature on the location topic in isolation exists. A rather restrictive assumption of earlier location models is that consumers or their product preferences are uniformly located along a linear Hotelling (1929) line. In such a setting, duopolistic firms locate at the binding corners of this line (D’Aspremont, Gabszewicz, and Thisse 1979) or even outside if there is no spatial restriction (Lambertini 1994; Tabuchi and Thisse 1995).^[1] The assumption of a uniform distribution is chosen to keep the models simple. However, as pointed out by Neven (1986) and Tabuchi and Thisse (1995) among others, increasing demand densities towards the center seem more appropriate. It is well-known from urban economics that the geographical distribution of consumers is concentrated around the central business district. Similarly, marketing studies show that consumer preferences are clustered around some fashionable features in the product space.

Anderson, Goeree, and Ramer (1997) have shown that the duopolistic location models can even be solved explicitly for wide classes of non-uniform demand distributions.^[2] As a remarkable result, it has turned out that log-concave and symmetric demand densities with a peak in the center lead to a subgame perfect location equilibrium with an extraordinarily pleasant property: The mere information about the demand density in the center is sufficient for the determination of the equilibrium solution. Information about the density’s curvature in the center, for example, is not necessary. The decisive result of this class of models is that higher density in the center induces firms to move closer together.

Various attempts to analyze manager delegation and location as interrelated strategic variables of firm owners have been made by Tan (2002); Matsumura and Matsushima (2012); Liang, Tseng, and Wang (2011); Heywood and Wang (2014). Of particular interest for our study are the models of Barcena-Ruiz and Casado-Izaga (2005) and Wang and Buccella (2020) where owners strategically decide on the location of their firms and then on the design of the manager-compensation schemes, before managers operationally decide on the product prices. Such a separation of long-run and short-run decisions can often be observed in real markets. To keep the analysis simple, these models assume a uniform distribution of demand. The basic results are that delegation of price decisions to managers leads to greater differentiation in geographical or product space, higher prices and higher firm profits. This paper shows that these results are not robust but crucially depend on the shape of the demand distribution.

In order to study the mutual relations between the strategic delegation and location effects in more detail, we extend the framework used by Barcena-Ruiz and Casado-Izaga (2005) and Wang and Buccella (2020) by allowing for more general demand distributions. Our contribution is twofold: First, we demonstrate that our generalized version of the model can even be solved explicitly for non-uniform demand densities that are symmetric around their peak in the center. It turns out that, in addition to the density in the center, the function’s curvature at that point plays now a crucial role. This will make it advantageous to define and distinguish between “broad-peaked” and “narrow-peaked” demand distributions, depending on the mass around the center.

Second, we show that the result derived by Barcena-Ruiz and Casado-Izaga (2005) as well as Wang and Buccella (2020) that manager delegation leads to more distant locations, higher prices and profits, continues to hold for broad-peaked demand distributions. However, for narrow-peaked distributions, we derive the qualitatively novel results that delegation leads to less distant locations of the firms and in extreme cases even to lower prices and profits.

It is well-known from managerial economics that in the case of price competition, strategic compensation schemes incentivize managers to charge higher prices, thereby increasing the firm profits. However, the consequences of delegating price decisions to managers on the owners’ location decisions are rather complex. A firm’s relocation induces a strategic price effect and a demand effect. Decisive is whether the additional manager-induced price effect dominates the demand effect or vice versa. In the case of broad-peaked demand distributions, the manager-induced price increase dominates such that owners decide to locate their firms further away from each other. However, in the case of narrow-peaked demand distributions, the demand effect dominates, thereby inducing owners to locate their firms closer to each other. Therefore, the delegation of price decisions to managers can either increase or reduce the differentiation in geographical or product space.

The remainder of the paper is organized as follows: Section 2 describes the basic setting of the model. Section 3 solves the three-stage game where owners decide on the location and the design of the manager-compensation schemes before managers decide on the prices. Section 4 contrasts this solution for the case of separation with the one of fully owner-controlled firms to derive the impact of manager delegation. Section 5 compares the scenarios for a general class of symmetric beta distributions on the Hotelling line with demand concentration around the center. An analysis of specific demand densities with an infinite length of support is provided in Section 6. We consider the normal distribution as an example of a broad-peaked demand concentration and the logistic distribution as an example of a narrow-peaked demand concentration. Section 7 concludes the paper.

2 The Setting of the Model

The interdependence between intra-firm manager delegation and inter-firm competition has been discussed intensively in the managerial economics literature. The corresponding models deal with two-stage duopoly games where the owners design observable and irreversible manager-compensation contracts^[3] in the first stage and the managers decide on prices or production quantities in the second stage. According to Vickers (1985), Sklivas (1987) and Fershtman and Judd (1987, 2006), we consider take-it-or-leave-it compensation contracts, consisting of a fixed salary and a performance-dependent compensation part that linearly depends on firm profits and revenues. Total payments are given by

where A _i denotes the fixed salary for the manager of firm i and B _i  > 0 denotes a weight parameter that, in combination with A _i , guarantees that the total payment for each manager is equal to the exogenously given market-specific reservation payment Ψ ̄ . The managers’ objective functions

represent the performance-dependent payments consisting of the weighted sums of the performance measures profit Π _i and revenue R _i  = p _i q _i , i.e. sales q _i multiplied by prices p _i . The weights attached to these components are decisive for the intensity of competition.^[4] Contract parameters α _i  > 1 indicate compensation schemes which induce managers to increase prices, thereby relaxing the intensity of competition.

Concerning the location scenario, we follow Anderson, Goeree, and Ramer (1997) and Lambertini (1994) by assuming that consumers of mass one are distributed on a one-dimensional Hotelling line where each consumer buys one unit of the supplied goods. The demand density is assumed to be log-concave.^[5] With mill prices and transportation costs, quadratically increasing in the distance between a consumer and a seller, the location of a consumer being indifferent between buying from firm 1 at location x ₁ or from firm 2 at location x ₂ is determined via

so that the indifference point is located at

The owners’ (gross) profit functions read as^[6]

where q 1 = F ( x ̃ ) is the mass of consumers buying from firm 1, q 2 = 1 − F ( x ̃ ) is the mass of consumers buying from firm 2 and c denotes the firms’ marginal production costs which are quantity-invariant and of equal size. The performance-dependent managers’ objective functions can then be rewritten as ψ _i = (p _i  − α _i c)q _i , i = 1, 2, i.e.,

As it is well-known from industrial economics, location decisions are not restricted to the geographical space. The same framework can be used to study horizontal product differentiation. In this case, the geographical location of consumers has to be reinterpreted in terms of their preferences for the product’s attributes, and the transportation costs must be reinterpreted in terms of a loss of utility when consumers are not able to purchase the products with their most preferred attributes. In this modified interpretation of the location game, the firm owners choose the horizontal (brand) positioning of their products.

3 The Game with Manager Delegation

We are interested in the subgame perfect equilibrium of the three-stage game, where the owners strategically decide on the locations x _i in stage 1 and design their managers’ compensation contracts, characterized by the weight parameters α _i , in stage 2 before managers charge prices p _i in stage 3. It proves useful to first solve the game for a general distribution F(x) of consumer demand and thereafter to provide a toolkit for appropriate specifications of the demand distributions.

In the third stage of the game, risk-neutral managers maximize their objective functions (3) with respect to the prices p _i . From the first-order conditions, we obtain the price equations

With anticipation of these price decisions, the owners’ profit functions (2) read

and the indifference Equation (1) can be rewritten as

Totally differentiating the implicit function (6) gives

In the second stage of the game, risk-neutral firm owners maximize their profit functions (5) with respect to the weight parameters α _i of the manager-compensation contracts, using the differentials (7). From the first-order conditions, we derive the weight parameters

Inserted into (4), we obtain the prices

and, inserted into (6), the indifference equation

With anticipation of the managers’ price-setting behavior (9), the owners’ reduced-form profit functions (5) read as

In the first stage of the game, firm owners maximize these profit functions with respect to the locations x _i . This gives the first-order conditions

where (10) can be used to derive the differentials d x ̃ / d x i .

For demand densities, symmetrically shaped around the center x ̃ , where F ( x ̃ ) = 1 / 2 and f ′ ( x ̃ ) = 0 , these first-order conditions read as^[7]

and lead to the equilibrium distance

and, hence, to the locations

where the index S denotes the considered case of separation. After observing these locations, firm owners design the manager-compensation contracts with the weight parameters

which give the managers incentives to increase prices, thereby relaxing the intensity of competition.

Finally, given these strategic owner decisions, managers charge the prices

and firm owners realize the (gross) profits

For the class of uniform distributions with location-invariant densities f ( x ) = f ̄ , i.e. f″(x) = 0 ∀ x, the solution simplifies to x 1 S = x ̃ − 5 / ( 4 f ̄ ) and x 2 S = x ̃ + 5 / ( 4 f ̄ ) for the locations, α S = 1 + 5 / ( 2 c f ̄ 2 ) for the contract parameters, p S = c + 5 / f ̄ 2 for the prices and Π S = 5 / ( 2 f ̄ 2 ) for the profits. The numerical solution x ₁ = −3/4, x ₂ = 7/4, p = c + 5, Π = 5/2, derived by Barcena-Ruiz and Casado-Izaga (2005) and Wang and Buccella (2020), proves to be a special case when the demand center is located at x ̃ = 1 / 2 and the distribution is uniform on the unit interval ( f ̄ = 1 ) .

For non-uniform distributions with f ″ ( x ̃ ) < 0 , the mere information of the density in the demand center proves to be no longer sufficient for the calculation of the subgame perfect equilibrium of the model. In addition, the curvature of the density function in the center is crucial. Compared to the uniform distribution in previous models, an increasing curvature | f ″ ( x ̃ ) | leads to (i) locations closer to the center, (ii) compensation contracts with weight parameters giving the managers declining incentives for relaxing price competition, (iii) lower prices, and lower profits. In the limit case of total consumer concentration in the center, the extreme solution of Bertrand competition results, regardless of whether price decisions are delegated or not.

We are now ready to analyze how manager delegation influences the location and price decisions of the firms. To derive this impact, we have to compare our solution for the case of separation to the one for the benchmark case of fully owner-controlled firms.

4 The Influence of Manager Delegation

Let us consider the benchmark solution for the case of fully owner-controlled firms where the owners themselves decide on locations and prices. This solution (index NS) can be simply derived by setting α ₁ = α ₂ = 1 and neglecting the manager payment ψ ̄ . We obtain the subgame perfect locations

the prices

and the firm profits

These are the solutions derived by Neven (1986), Tabuchi and Thisse (1995) and Anderson, Goeree, and Ramer (1997) and used for the toolkit provided by Meagher, Teo, and Wang (2008).

A comparison of the solution Equations (11) for the case of manager delegation and (14) for the case of fully owner-controlled firms shows that locations are more distant in the former case if

This is obviously the case in the models of Barcena-Ruiz and Casado-Izaga (2005) and Wang and Buccella (2020) where f ″ ( x ̃ ) = 0 and f ( x ̃ ) = 1 . The greater distance is supported by the strategic incentive for managers to increase prices. This is of advantage for the owners since the demand effect without consumer concentration is low. However, if this inequality is not satisfied, manager delegation leads to closer locations of the firms. The reason is that otherwise the owners must fear a sharp drop in the mass of demand around the center as a consequence of the incentive for managers to increase prices.

The crucial importance of the ratio | f ″ ( x ̃ ) | / f ( x ̃ ) 3 leads us to define the demand distribution as “broad-peaked” if the inequality (16) is fulfilled, and as “narrow-peaked” if not. “Broad-peaked” distributions are characterized by a weak curvature in the center so that a high proportion of mass of demand is concentrated around the center. “Narrow-peaked” distributions are characterized by a strong curvature in the center so that only a small proportion of mass is concentrated around the center.

For demand distributions where | f ″ ( x ̃ ) | = 6.4 f ( x ̃ ) 3 holds (unchanged location distance), we obtain from (12)

This implies that managers are still incentivized to increase the prices. Therefore, even if manager delegation does not change the location decisions of owners, these incentives lead managers to set higher prices than owner themselves would, i.e.,

However, a comparison of the price Equations (13) And (15) shows that the managers set prices higher than the owners themselves would only if

With stronger curvatures, the price-increasing incentive effect via the compensation contract is dominated by the necessity to reduce prices as a consequence of the strong demand effect caused by the high demand concentration around the center. Finally, if | f ″ ( x ̃ ) | = 10 f ( x ̃ ) 3 , the delegation effect diminishes (α ^S  = 1), firms locate back to back in the center ( x S = x ̃ ) and prices fall to the unit cost (p ^S  = c) as it is the case in fierce Bertrand competition.

Summarizing, demand concentration can reverse conventional wisdom about the impact of manager delegation: it is possible that owners move their firms closer together and that managers set lower prices leading to lower profits for the owners. Therefore, it seems important to consider some usual specifications of demand densities with a peak in the center. Our first candidate is the beta distribution.

5 Demand Concentration Around the Center of the Hotelling Line

An obvious generalization of the uniform distribution on the Hotelling line is the beta distribution with a demand peak in the center x = 1/2. The class of symmetric beta distributions reads as^[8]

and has mean 1/2 and variance 1/[4(2n + 3)]. The parameter n is an inverse measure of dispersion. The limit case of n = 0 captures the uniform density f(x) = 1 as used in the models of Barcena-Ruiz and Casado-Izaga (2005) and Wang and Buccella (2020). Positive values of n generate density distributions with a peak in the demand center. Figure 1 illustrates the special beta function with n = 2, i.e. f(x) = 30x ²(1 − x)².

Figure 1:

Density of the beta distribution with n = 2.

The beta distribution is characterized by

and

Substitution into (11) gives the subgame perfect equilibrium locations

The subgame perfect equilibrium contract parameters (12) are

leading to the prices

and firm profits

Without separation of decisions, according to (14) firms locate at

implying in (15) the prices

and the firm profits

Since the inequality (16) is fulfilled for all parameter values n ≥ 0, we classify the symmetric beta distribution as “broad-peaked”. As a consequence, locations are more distant in the case of separation. Both, the larger distance and the manager incentives relax the intensity of price competition, leading to higher prices and firm profits. As can be seen in Figure 2, the distance between the locations resulting under separation (S) and no separation (NS) is largest in the extreme case of a uniform distribution (n = 0). As soon as there is demand concentration around the demand center (n > 0), the firms move towards the hub and the impact of manager delegation declines. A negligible dispersion leads to the Bertrand solution, regardless of whether managers are involved or not.

Figure 2:

Location distance with beta-distributed consumer demand.

6 Dropping the Borders of the Hotelling Line

Especially in the case of horizontal product differentiation, modern location models prefer to drop the borders of the Hotelling line. Instead, they assume demand distributions on the support −∞ < x < ∞ with a peak in the center x = 0. Prominent examples are the normal and the logistic distribution (see, e.g. Meagher, Teo, and Wang 2008).

As an appropriate approximation of the demand distribution on unrestricted support, we first consider the density function

of the normal distribution with mean 0 and variance σ ² which is characterized by f ( x ̃ = 0 ; σ ) = 1 σ 2 π and f ″ ( x ̃ = 0 , σ ) = − 1 σ 3 2 π . The equilibrium locations follow from Equations (11) as

leading to the contract parameters

prices

and profits

Without separation of decisions, according to (14) firms locate at

inducing the prices p ^NS = c + 3πσ ² < p ^S and profits Π^NS = (3πσ ²)/2 < Π ^S . Since the inequality (16) is fulfilled for all σ values, we classify the normal as a “broad-peaked” distribution. Therefore, locations are more distant in the case of separation. Additionally, α ^S > 1 gives the managers incentives to further increase prices so that profits increase.

As an important counterexample of a “narrow-peaked” demand distribution, let us finally consider the density function

of the logistic distribution with mean 0 and variance (π ²/3)s ² which is characterized by f ( x ̃ = 0 ; s ) = 1 / ( 4 s ) and f ″ ( x ̃ = 0 ; s ) = − 1 / ( 8 s 3 ) .

Figure 3 compares the shape of the logistic distribution with density f ^L (x) to the shape of the normal distribution with density f ^N (x). In order to induce an equal density f ( x ̃ = 0 ) = 1 in the center, we use the specifications s = 1/4 and σ = 1 / 2 π . As can be seen, the peak of the logistic function is more narrow-shouldered but the tails are fatter. Indeed, in contrast to the normal distribution, the logistic distribution does not fulfill the inequality condition (16), regardless of the value of the shape parameter s. Consequently, the influence of separation on the location distance is reversed.

Figure 3:

Normal and logistic distribution.

Indeed, in the case of separation, the equilibrium locations are

The contract parameter is α ^S = 1 + 16s ²/c, leading to the prices p ^S = c + 32s ² and profits Π^S = 16s ². Without separation, according to (14) firms locate at

leading to the prices p ^NS = c + 24s ² < p ^S and profits Π^NS = 12s ² < Π^S. As predicted, locations are less distant in the case of a separation of decisions. Nevertheless, since inequality (17) is violated, the price-increasing incentive effect for the managers dominates the location effect, so that prices and firm profits are still higher compared to the case of a fully owner-controlled firm.

Due to its empirical relevance, the logistic demand distribution is a good example for the qualitatively novel result of our paper. When demand concentration around the center is narrow-peaked, the demand effect dominates the price-increasing incentive effect for the managers. Therefore, in contrast to the previous literature, manager delegation can lead to less distant locations of the firms.

7 Summary and Conclusion

Ownership and management of firms often are separated in the sense that owners make strategic decisions on location in geographical or product space, whereas operational price decisions are delegated to managers. In this sequence of decisions, the designs of the performance-dependent compensation schemes are also strategic decisions of the owners. Of course, location and delegation strategies are interrelated so that location decisions influence the incentive contracts for managers and delegation decisions influence the differentiation in geographical and product space.

Previous literature on this topic proposes that a separation of decisions leads to more distant locations. This paper has shown that this result only holds with broad-peaked demand distributions for which the uniform distribution proves to be an extreme example. When the demand distribution is characterized by a narrow-peaked concentration around the center, separation leads to closer locations. For extremely curved densities in the center, it is even possible that delegation leads to lower prices and profits than it would be the case without delegation.

A technical contribution of this paper is the demonstration that the generalized model can be solved explicitly in terms of the demand density and its curvature in the center. This enabled us to solve for the subgame perfect equilibrium of the three-stage market game for a broad variety of demand distributions, including the beta, the normal, and the logistic distribution. Among others, all of them seem to be appropriate to capture relevant features of real demand concentration. While the beta and normal distributions prove to be broad-peaked, the logistic distribution turns out to be narrow-peaked. Therefore, the logistic distribution represents a prominent example of demand concentration where manager delegation leads to closer locations of firms in geographical or product space.

The model could be further extended in several directions. Other compensation contracts with market-share or relative-profit evaluation could be analyzed, even contract bargaining might be considered. The one-dimensional space could be widened to a two-dimensional area where demand is distributed on a square or a disk, for example. The deterministic setting could be extended to a stochastic one where, for example, due to fashion waves, the demand center is the realization of a random variable. An extension to more than two firms would increase the applicability of the model in many case studies. However, it is important to note that an increasing complexity usually forces one to use techniques of numerical simulations. Therefore, we are pleased that we have been able to present theoretical results which can be applied, at least in a stylized way, to the business practices of real firms.