Spatial Duopoly Research Articles

Location, Location, Quality:The Fixed Differentiation Principle

AbstractThis work revisits the classic spatial duopoly location-then-prices competition with quadratic transportation costs, in the presence of vertical differentiation. We show that contrary to the substitutability between horizontal and vertical differentiation that has appeared thus far in the literature, higher-quality asymmetry does not affect locational differentiation in equilibrium. While the better-quality seller indeed approaches the city center, the worse-quality seller is simultaneously driven further away from the city, and ultimately out of the market when quality differences become large enough. Interestingly, although vertical differentiation weakens competition, total welfare increases – even when the better-quality seller ultimately monopolizes the market.

Impact of the New International Land–Sea Transport Corridor on Port Competition between Neighboring Countries Based on a Spatial Duopoly Model

The development of international land–sea transport corridors has provided more convenient access to the sea for inland areas and promoted the improvement of transportation efficiency, environmental improvement, and the strengthening of international cooperation. However, the construction of international land–sea transport corridors has also intensified competition among the ports, which has extended from the local and regional to the national and even international levels. This paper explores the impact of international land–sea transport corridors on oligopolistic port competition between neighboring countries using the Hotelling model. By setting up the utility of the shipper’s port selection, the equilibrium price, market share, and profit of duopoly ports in neighboring countries are analyzed under different conditions of cross-border land transportation and maritime transportation. It is found that the high cross-border transportation cost of the international land–sea transport corridor is not conducive to increasing the market share of the overseas oligopolistic ports in the region. If the maritime transportation cost of overseas oligopoly ports is too high compared with domestic oligopoly ports, it will offset the land transport advantages brought by international land–sea transport corridors. The findings in this paper could provide support for strategic decision making in port markets and cross-border transport corridor development.

Optimal patent licensing—Two or three‐part tariff

AbstractWe look into technology transfer by an insider patentee in a spatial duopoly model under three types of licensing contracts—(i) two‐part tariff with fixed fee and per‐unit royalty, (ii) two‐part tariff with fixed fee and ad‐valorem royalty and (iii) general three‐part tariff with fixed fee, per‐unit and ad‐valorem royalties. Under two‐part tariff contracts, the licenser is better off with the per‐unit royalty contract but the general contract does better than the other contracts. In contrast to the existing literature, all three licensing contracts may make the consumers worse‐off compared to no licensing, with the lowest consumer surplus achieved under the general licensing contract. Welfare under the general licensing contract is equal to the welfare under two‐part tariff with ad‐valorem royalty and it is higher than the welfare under no licensing but lower than the welfare under two‐part tariff with per‐unit royalty. Hence, the general three‐part licensing contract is privately optimal but not socially optimal. Similar conclusions hold also under a nonspatial linear demand model with differentiated products.

Transportation gas emissions with online retailing: a spatial model

ABSTRACT We analyse the impact of online retailing on transportation gas emissions in a spatial duopoly with upstream suppliers. We consider: (1) pure offline competition with two physical retailers; and (2) online/offline competition, where an online retailer competes against an offline retailer. When the upstream suppliers are located in the city centre, transportation gas emissions are lower under online/offline competition. However, when the upstream suppliers are at the endpoints, online retailing might increase transportation gas emissions. This happens when: (1) in the online/offline competition case, the physical retailer does not locate at the endpoint; (2) the transportation costs of the suppliers’ commercial trucks are sufficiently larger than those of the consumers’ private cars; and (3) the competitiveness of the offline retailer is high enough.

Demand uncertainty, product differentiation, and entry timing under spatial competition

We investigate entry timing and product position decisions with market size uncertainty under continuous-time Hotelling type spatial duopoly competition. We show how entry timing and product position interact under uncertainty. The expected result under real options holds: the leader’s entry threshold monotonically increases with volatility, which means that the leader abandons earlier period monopoly profits, which are not so time discounted. The significant findings are as follows: First, from the expected result, the larger the abandoned monopoly profits in earlier periods through volatility, the closer to the edge the leader’s product position. Second, this property of the leader’s decision regarding its product position makes the follower’s entry threshold non-monotonic with a positive correlation to volatility, in contrast to previous studies in the real options literature. The above two results have an important implication: the leader values the abandoned monopoly profits in earlier periods through volatility more than the highly increased monopoly profits in later periods through delayed entry by the follower.

Impacts of variable interest rates on the market areas of a spatial duopoly in supply chains operating on the finite horizon

The paper aims to present a theoretical study of inventory allocation in two retailer warehouses that form a spatial duopoly on the finite horizon, where the shortage of goods, and not only prices, determines their market areas, depending on the customer's travel decisions to minimise not only the cost of purchasing the items but also the travel expenses under uncertainties of availability of the products. The customers' decision on which store to visit first is influenced by their estimation of regarding shortages of goods. The demand size required by arrived customers can be one or more units of goods. Delays in the required amount and all flows in the production–inventory–logistic system have varying parameters, simulated by the Network Simulation Method (NSM). The impact of delay in supply chain, which provides the products and consequently creates the shortages on market areas, is studied at varying interest rates and the shortened, unknown length of the time horizon. The annuity stream approach of evaluation of ordering policies is applied in areas of spatial duopoly where the optimal ordering policy depends on the interaction between the prices and the shortages of goods in the studied duopolies. Customer travelling problem (CTP) is defined, which determines the market area for allocated inventories. In our study, various distributions of customers' quantity demand at varying shortages and delays of the provision are supposed, presenting their impact on the market area. NSM helps in describing how to study the impact of varying interest rates and stochastic length of the time horizon in the case of shortages of goods as a consequence of delays in supply system activities and how to improve the estimates of NPV and profit in the case of low and varying interest rates.

Equilibrium existence in the Hotelling model with convex production costs

We revisit the spatial duopoly model à la Hotelling (1929), to show that, provided the parameter scaling marginal cost is sufficiently high, quadratic production costs guarantee equilibrium existence in presence of linear transportation costs and a uniform distribution, with minimum product differentiation and no undercutting. We also discuss the conditions under which partial coverage arises. Finally, we extend the duopoly game to generalise the condition for the existence of the pure-strategy equilibrium to a class of convex production cost functions.

Stackelberg-Nash Equilibrium in the Linear City Model

In the paper [1], the spatial duopoly of firms under Stackelberg competition in which one of the firms is the leader, both by the volume of goods supply and location, was studied. In this paper, we consider the case of leadership only by the volume of goods supply. The Stackelberg-Nash equilibrium in the quantity and spatial strategies of firms are found. In the course of stability analysis of this equilibrium, it is proved that the transport tariff is a bifurcation parameter for firms. It is established that the change of the central agglomeration strategy into the differentiation strategy occurs at the point of transcritical bifurcation. The conditions for full coverage of the markets for both strategies are determined. As is demonstrated below, the Stackelberg information asymmetry leads to the asymmetry of equilibrium locations of firms.

Impact of rail transport services on port competition based on a spatial duopoly model

Rail transport plays an important role in port-based freight flows. In the context of port competition, the change of inland transport modes would directly influence pricing strategies and profits of ports. Therefore, this paper provides a method to analyze how the introduction of rail transport services affect port competition. Firstly, a two-stage game model is constructed based on a Hotelling model, which consists of two ports and one rail operator. Then, this paper demonstrates how the existence of pure strategy equilibrium prices of ports depends on the value of marginal profits of rail transport services. After comparing the results of different situations, our analysis shows there is a negative effect on ports' service prices and profits when the rail transport services are introduced in the port. As shown in the numerical example, the profits of ports would be reduced by about 10% (when rail transport services are only provided to one port) and 20% (provided to both ports) respectively. From the perspective of government, there exists an incentive to support the development of rail transport due to the promotion of social welfare. In addition, the rail operator is also likely to expand rail services in order to improve profits. Finally, the effect of economies of scale of rail transport is extendedly analyzed and three policies are proposed to weaken the loss of ports’ profits, including implementing refunding scheme, setting price floor on port service price and providing government subsidy. The findings in this paper could provide support for the decision making towards comprehensive transportation management in coastal zone.

Demand Uncertainty, Product Differentiation, and Entry Timing Under Spatial Competition

We investigate the decision of entry timing and product positions under market size uncertainty with Brownian motion for continuous-time spatial duopoly competition. We show the following results. First, the leader is more likely to increase the degree of product differentiation as volatility increases and to decrease it as volatility decreases. Second, the entry threshold of the follower non-monotonically increases in volatility, which is in contrast to previous studies in the real options literature. Finally, a more significant entry cost asymmetry between firms places the leader closer to the edge in a preemptive-entry equilibrium and closer to the center in a sequential-entry equilibrium.

Mixed spatial duopoly, consumers’ distribution and efficiency

We solve a mixed spatial duopoly with a generic log-concave consumers’ distribution. We show that the sub-game perfect equilibrium in prices and locations exists and is generally inefficient, so that the efficiency in the standard uniform distribution case stands out as an exception. Notable examples show that the inefficiency may increase as the distribution becomes more concentrated.

Costly location in Hotelling duopoly

We introduce a cost of location into Hotelling׳s (1929) [9] spatial duopoly model. We derive the general conditions on the cost-of-location function under which a pure strategy price-location Nash equilibrium exists. With linear transportation cost and a suitably specified cost of location that rises toward the center of the Hotelling line, symmetric equilibrium locations are in the outer quartiles of the line, ensuring the existence of pure strategy equilibrium prices. With quadratic transportation cost and a suitably specified cost of location that falls toward the center of the line, symmetric equilibrium locations range from the center to the end of the line.

A public firm in a vertically linked price discriminating spatial duopoly

We show that, in a vertically linked duopoly where neither firm can produce all varieties demanded, spatial competition between a public and a private firm induces them to deviate from the socially optimal location. We identify specific conditions under which a change in the degree of privatization induces one firm to move toward, while the other moves away from the socially optimal location. There exists a critical level of privatization above (below) which the public and private firms will come close (drift apart) with a rise in the degree of privatization.

A public firm in a model of spatial duopoly with price discrimination

We visit the role of privatization in the location decision of firms in an industry where no firm can produce all varieties demanded. We demonstrate that the Nash equilibrium locations are socially optimal, in the presence of a publicly owned firm, notwithstanding the degree of privatization.

Spatial Advertisement Competition: Based on Game Theory

Since advertisement is an important strategy of firms to improve market share, this paper highlights duopoly advertisement under the Hotelling model. A model of advertisement under spatial duopoly is established, and corresponding effects of brand values and transportation costs are all captured. This study presents the proportion of sales revenue spending on advertisement. The condition for free-rider in advertisement investment is discussed. Under firms with the identical brand values, if firms' advertisement points to corresponding consumers, price and advertisement investment are all reduced. Therefore, advertisement is discussed under spatial competition in this work.

Price Guarantees, Consumer Search, and Hassle Costs

The paper deals with the competitive effects of price guarantees in a spatial duopoly where consumers can search for lower prices but have to incur hassle costs if they want to claim a price guarantee. It is shown that symmetric equilibria with and without price guarantees exist but price guarantees will have no effect on prices if search costs are low, hassle costs are high and the number of uninformed consumers is small. However, when both firms use price guarantees, there also exist payoff-dominant equilibria where both firms use mixed pricing strategies in the form of high-low pricing schemes, provided that the search costs are sufficiently high.

The competition of investments for endogenous transportation costs in a spatial model

This paper proposes a two-stage spatial duopoly model to explore how the firms improve the value of product by service enhancement. In the model, we assume that the higher service ability means lower transportation cost. The results show that when two private firms participate in competition, no firm tends to invest more under the profit-maximizing objection. But when one is a public firm, the private firm may participate in competition with better service and higher price relative to the public one.

Innovation Under Spatial Duopoly

Innovation is an important topic in economics. This paper highlights duopoly innovation under the Hotelling model with game theory approaches. This paper argues that market power, as measured by the cost advantage of a dominant firm over its rival, serves to enhance the incentive to innovate in a Hotelling model of spatial competition. This result implies that a firm with cost advantage will have a larger incentive than its rivals to further its cost advantage as new opportunities for innovation arise thereby implying that innovation increases concentration. This result is in contrast to the result obtained by Holmes et al. (2012) who use the Betrand model to show that "market power" lowers the incentive to innovate. We think that the inelastic demand causes this economic phenomenon.

Product differentiation decisions under ambiguous consumer demand and pessimistic expectations

This paper studies product differentiation decisions in a spatial duopoly with limited information on consumer demand. In particular, a situation is discussed in which the firms do not know the exact distribution of the random location of consumer demand and its responsiveness to price changes (measured by the scale of transport costs), but resolve the resulting ambiguity using the α-maxmin or minimax regret criteria. When the firms are sufficiently pessimistic (α is high enough), results are in contrast with the existing literature. In particular, an increase of demand location uncertainty decreases the equilibrium product differentiation, intensifying the second-stage competition in prices, although the effect is dampened by uncertainty about transport costs. Endogenizing the choice of objective function leads to the dominance of an extreme form of pessimism, which turns out to be socially-optimal.

Optimal product variety in a Hotelling model

Consider the Hotelling linear spatial duopoly with firm uncertainty over the consumer mean. As uncertainty about the mean grows relative to the dispersion of consumers, competitive locations become socially optimal. A limit result for discontinuous, log-concave densities is also established.