A dynam ic m odel of quality com petition with endogenous prices by Roberto CELLINI(University of Catania) Luigi SICILIANI (University of York) Odd Rune STRAUME (University of Minho) NIPE working paper 0 8 / 20 15 cellini@unict.it Brescia Workshop, October 2015
Motivations In many industries, quality is a highly important aspect of the goods or services offered. Quality, along with price, are relevant choice variables for firms and for market competition. However, since a firm's incentive for attracting more demand by providing higher quality is positively related to the price of the product offered, price and quality decisions tend to interact in a way that m akes the effect of com petition on quality am biguous: Theoretically, a higher degree of competition has two counteracting effects on quality provision: (i) more competition increases the incentives to provide quality for given prices, but (ii) more competition also reduces the price-cost margin, which in turn reduces the incentives for quality provision.
Motivations The effect of competition upon quality provision is also a question of great interest for policy makers (particularly in sectors like health care, long-term care, education, child care…). In these industries, prices tend to be regulated in some countries and unregulated in others. We revisit the question of how competition affects quality in a dynamic context, with quality as a stock, the quality stock can be increased over time through investment. The relationship competition - quality is closely related to the question of whether an unregulated market will produce a socially optimal quality provision.
Aims To study the relationship between competition, price and quality provision. We use a differential-game approach to derive the equilibrium price and quality provision. Price and investment (for quality increase) are the choice variables. We compare the benchmark open-loop solution against the feedback closed-loop solution. (Feedback closed-loop behavior rule implies ‘true’ strategic dynamic interaction over time). We find that: …
Aims We find that: … Steady-state quality in the open-loop solution is at the socially optimal level and independent of competition intensity. (The degree of competition intensity, as measured by a reduction in transportation costs along the Hotelling line.) In contrast, steady-state quality in the closed-loop solution is (i) increasing in the degree of competition between firms, (ii) lower than in the open-loop solution, (iii) lower than the socially optimal level. Thus, our analysis identifies dynamic strategic interactions between competing firms as an independent source of inefficiency in quality provision.
Structure of the (remainder of) presentation (Motivation) Related literature The basics of the model The open-loop solution The closed-loop solution Comparison Welfare analysis Concluding remarks
Related literature (1) Optimal quality level: the firm perspective vs. the social perspective Spence (1975) on monopoly: a monopolist will provide a quality level that is higher (lower) than the socially optimal level if the marginal valuation (WTP) of quality is higher (lower) for the marginal than for the average consumer. Ma and Burges (1993) in Oligopoly (one-shot game): quality is optimally provided with simultaneous decision making, whereas sequential quality and price decisions imply an under-provision of quality in equilibrium
Related literature (2) Quality level provision over time Piga (1998, 2000) in dynamic oligopoly: firms set price and advertising levels. (Advertising similar to quality = a tool to increase the perceived product quality). However, in Piga’s models, Advertising has a public good component that increases market size. (In contrast, quality investments have a purely business-stealing effect in our model). In Piga, the ranking of desirability of the outcomes depend on the information rule adopted (open-loop vs. feedback).
Related literature (2) Quality level provision over time (cont.) Cellini et al. (2008) focus on persuasive advertising and compare the outcomes of price and quantity competition. Conclusion: Price competition entails more advertising. Investment in R&D, affecting the production cost or product characteristics --with some parallels to investment in product quality-- are studied by Hinloopen (2000, 2003) and Cellini and Lambertini (2005, 2009), among others. Conclusions: Intensity in R&D, and the incentive towards cooperative behaviour, depend on the form of market competition (price vs. quantity competition) and the information structure, with a variety of possible outcomes. In general, more intense competition arises when the firms' choice variable is price (rather than quantity).
Related literature (2) Quality level provision over time (cont.) Brekke et al. (2010, JHE ) provide a model where oligopolistic firms set qualities in the presence of regulated prices . The degree of competition (captured by travel cost of consumers) has different effect on quality provision, depending on cost structure and the information rule: (Closed-loop favors collusive behavior, provided that quality levels are strategic complements lower quality level in equilibrium) Brekke et al (2012, JEMS ): extension to sluggish beliefs abut quality Siciliani et al. (2013, JEDC ): extension to motivated providers Cellini and Lamantia (2015, JEE ): extension to MQS In all these models, prices are regulated (exogenous for competing firms)
Motivation NOVELTIES OF OUR PRESENT MODEL: (1) We make the price endogenous, while most available models generally consider prices as regulated when quality is the choice variable (see, e.g., Brekke et al., 2010; Siciliani et al., 2013; Cellini and Lamantia, 2015). (2) We take a differential-game approach, which allows us to highlight how price and quality choices interact when firms make their decisions in a dynamic framework. Our model highlights the effect of current quality on rivals' future price decisions, which is shown to play a crucial role in firms' decision making.
The model: Basics A market with two firms located at either end of the unit line S =[0,1]. A uniform distribution of consumers, with total mass normalized to 1. Assuming unit demand, the utility of a consumer who is located at x ∈ S and buys from firm i , located at z i ∈ {0,1}, is given by U x , z i v kq i − | x − z i | − p i , Under the full-market coverage, the market demand for firm i is D k q i − q j p i − p j 2 − 1 x i 2 2 Product quality changes over time, due to investment by firms ( I(t) ) and depreciation δ >0: dq i t : q i t I i t − q i t dt Cost in each point of time: D 2 q i C x i D , I i , q i cx i 2 I i 2 1 where c >0, γ >0 and β >0. (Constant marginal cost of production, and increasing and strictly convex costs of quality investments)
The model: Basics Objective of firm i is: Max i t e − t dt 0 The instantaneous profit of firm i is given by 2 I i t 2 − i t p i t − c x i D q i t , q j t , p i t , p j t − 2 q i t 2 To solve the model, we consider two different solution concepts, which correspond to two different sets of information used by players when setting the optimal plan: The open loop – players know the initial state of the world, and set the optimal plan at the beginning, than stick to it forever. The solution is such that the plan of control variable only depends on time and initial condition The closed loop – players observe the evolution of the state(s); the control variable(s) depend on the current state (only on the current state under the Markovian assumption, or feedback rule).
The Open-loop solution Maximise i t e − t dt , Problem of player i : I i t , p i t 0 q i t I i t − q i t , subject to q j t I j t − q j t , q i 0 q i 0 0, q j 0 q j 0 0. Let μ i (t) and μ j (t) be the current value co-state variables associated with the two state equations. The current-value Hamiltonian is: 2 − 2 q i 2 − F i I i − q i j I j − q j k q i − q j p i − p j H i p i − c 2 − − 1 2 I i 2 2 The solution has to meet the following conditions: (a) ∂ H i / ∂ I i =0, (b) ∂ H i / ∂ p i =0, along with the adjoint conditions and the transversality condition. Specifically:
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