A dynam ic m odel of quality com petition with endogenous prices - - PowerPoint PPT Presentation

A dynam ic m odel

• f 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

• f

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

• ver

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

• ptimal 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

• f

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

• ptimally provided with simultaneous decision making, whereas

sequential quality and price decisions imply an under-provision

• f 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 zi ∈ {0,1}, is given by

 Under the full-market coverage, the market demand for firm i is  Product quality changes over time, due to investment by firms (I(t))

and depreciation δ>0:



Cost in each point of time:

 where c>0, γ>0 and β>0. (Constant marginal cost of production, and

increasing and strictly convex costs of quality investments) Ux,zi  v  kqi − |x − zi| − pi,

xi

D  1 2  kqi−qj  2

−

pi−pj 2 dqit dt

:



qi t  Iit − q it

Cxi

D,Ii,qi  cxi D  1 2 I i 2  qi 2

The model: Basics

 Objective of firm i is: Max  The instantaneous profit of firm i is given by  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

• ptimal 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). it  pit − cxi

Dqit,qjt,pit,pjt −  2 Iit2 −  2 qit2





ite−tdt

The Open-loop solution

 Problem of player i:  Let μi(t) and μj(t) be the current value co-state variables associated with

the two state equations. The current-value Hamiltonian is:

 The solution has to meet the following conditions:

(a) ∂Hi/∂Ii=0, (b) ∂Hi/∂pi=0, along with the adjoint conditions and the transversality condition. Specifically:

Iit, p it

Maximise 



ite −tdt,

subject to



qi t  Iit − qit,



qj t  Ijt − qjt, qi0  qi0  0, qj0  qj0  0.

Hi  pi − c

1 2  kqi−qj 2

−

pi−pj 2

−

 2 Ii 2 −  2 qi 2 − F  iIi − qi  jIj − qj

The Open-loop solution

limt→ e−titqit  0

 FOCs:  Adj eqs:  Transv:  SOC: satisfied if the Hamiltonian is concave in the control and state

variables (Léonard and Van Long, 1992). This is the case.

 Steady state: , :  From which…



qi 0

0  OL    qOL − kpOL − c 2 , IOL  qOL,



i 0

i  Ii, 1 2  kqi − qj 2 − pi − pj 2  pi − c 2 ,



i  i    qi − pi − ck 2 ,



j    j  pi − ck 2 ,



qi  Ii − qi,

The Open-loop solution

 Steady state allocation: qi=qj=qOL , and pi=pj=pOL ,  The steady state is locally stable in the saddle sense; around the steady

state the system is:

 (Jacobian matrix with negative determinant and positive trace).  p turns out to be constant over time.

pOL  c  

qOL 

k 2



I t



q t     1

 

− It qt  − k

2

The Open-loop solution

 The price equates the sum of marginal production and transportation

cost.

 Transportation cost (τ) is a parameter inversely related to the degree of

• competition. (Result

analogous to the Nash equilibrium of an equivalent static model).

 Steady-state investment and quality are decreasing in the marginal cost

• f quality (β) and investment (γ),

 The higher the depreciation rate of quality (δ), the lower the steady-

state quality; the effect of δ on investment can be non-monotonic and depends on the exact parameter configuration.

 Most

striking result: the independence between marginal transportation costs and steady-state quality. When the firm s use open-loop decision rules, steady-state quality does not depend on the degree of com petition in the m arket (as captured by τ).

The Open-loop solution

The explanation:

 All else equal, stronger competition increases the elasticity of (firm-

specific) demand w.r.t. both price and quality  this leads to lower prices.

 This has two counteracting effects on quality provision:

(1) a positive direct effect (larger quality elasticity); (2) an indirect negative effect (lower price reduces the incentive to increase quality).

 In standard static spatial competition models, these two effects

exactly cancel each other; hence, competition intensity does not affect equilibrium quality provision (Ma and Burgess (1993) for the case of Hotelling competition; Gravelle (1999) for the Salop)

 This ‘neutrality’ result carries over to a dynamic setting, as long as

the firms use open-loop decision rules.

The Closed-loop solution

 Under the closed-loop solution concepts, each firm can observe (and

therefore react to) the quality stock of the competing firm not only in the starting time, but also in any subsequent point in time.

 We present the feedback closed-loop solution, where the players –

at each point in time– make decisions by taking into account the current value of states (which summarizes the entire past history of the game).

 The feedback closed-loop solution is strongly time-consistent, but

considerably more complicated to calculate.

 Notice that the firm's instantaneous objective function is  Which –faced with the linear dynamic constraint– gives rise to a

linear-quadratic problem.

pi − c

1 2  kq i−q j 2

−

p i−p j 2

−

 2 Ii 2 −  2 qi 2

The Closed-loop solution

 The value-function is  Define the solution to be found as: Ii=φi(qi,qj) and Ij=φj(qi,qj) .  The value function has to satisfy:  Max of the right-hand-side w.r.t. Ii yields  And similarly for Ij.  Max of the right-hand-side w.r.t. pi yields  And similarly for pj.

Viqi,qj  0  1qi  2qj  3/2qi

2  4/2qj 2  5qiqj

Ii  iqi,qj 

13q i5q j 

pi  iqi,qj  c   

kq i−q j 3

Viqi,qj  max pi − c

1 2  kq i−q j 2

−

p i−p j 2

−

 2 Ii 2 −  2 qi 2

Vq i

i qi,qjIi − qi  Vq j i qi,qjIj − qj

.

The Closed-loop solution

 From

• ne can see that at each point in time, there is a positive

relationship between the quality stock and the price charged by each firm.

 All else equal: higher quality  higher demand,  less price-elastic

demand  this increases the profit-maximizing price.

 An increase in the competitor’s quality level has the opposite effect.  Since the two firms optimal pricing rules are symmetric, it follows

that

 i.e., at each point in time, the firm with higher quality charges a

higher price.

pit : i

CLqit,qjt  c    kq it−q jt 3

pit − pjt 

2kq it−q jt 3

The Closed-loop solution

 After substituting into the HJB eq., we obtain that the following

has to hold:

 We have to find the alpha coefficients (to satisfy the above

relation as an identity).

 Six solutions are possible, but only one is consistent with global

stability (α₃<0; α₃+α₅<0; α₃-α₅<0).

 (Details in Appendix) Viqi,qj   

kq i−q j 3 1 2  kq i−q j 6

− 1

2 1  3qi  5qj2 −  2 qi 2

1  3qi  5qj

13q i5q j 

− qi 2  4qj  5qi

13q j5q i 

− qj

The Closed-loop solution

 After having find the alpha coefficients, The optimal quality

investment rule for firm i is: where:

Iit : CLqit,qjt 

1  1  3qi  5qj

1 

k 3−3  0

3  s −

 54

4 y − 2gy  5y − 2g  0 5  − 1

2 4 27

y − g − yy − 2g  0

y : 6s2  , s :   1

2 

g :

k2 

The Closed-loop solution

The key property: the negative sign of α₅,

  quality investments are intertem poral strategic substitutes

(Jun and Vives, 2004);

 this means that the control of each player responds negatively

to the state of the other player: the higher the quality stock of a given firm, the lower the optimal investment level of the competing firm. The intuition (related to the interaction between price and quality investment choices):

 All else equal, an increase in the quality stock of firm j leads to reduced

demand for firm i  Firm i will therefore optimally reduce its price. However, this  a lower price-cost margin for firm i,  a reduction in the marginal profit gain of attracting more demand by increasing quality. Firm i will therefore respond by reducing its quality investments.

The Closed-loop solution

STEADY STATE UNDER THE FEEDBACK CLOSED-LOOP SOLUTION

 In steady state, where qi=qj , equilibrium prices are given by

pCL=c+τ (The same as under open-loop and static game).

 Steady-state quality is  How does steady-state quality under feedback rules depend on the

degree of competition (inversely measured by τ)?

 Since α₃+α₅ does not depend on τ, it is easy to see that

qCL 

k 3−3−35 .

∂q CL ∂



 ∂q CL ∂3  ∂3 ∂g − ∂g ∂  0

The Closed-loop solution

When firms adopt feedback (closed-loop) decision rules, steady-state quality is increasing in the degree of competition.

 Two effects at work: (1) For given prices, lower transportation costs

make demand more quality elastic, which increases the profit-gain of quality investments. (2) On the other hand, lower transportation costs also make demand more price elastic, leading to lower prices, which in turn dampens incentives for quality investments.

 In contrast to the open-loop case, where these two effects exactly cancel

each other in steady state, the first (direct) effect dominates the second (indirect) effect under dynamic competition with feedback rules, 

 Positive

relationship between competition intensity and quality provision in steady state.

 However, Steady-state quality is lower in the closed-loop solution

than in the open-loop solution.

Comparison: Closed-loop vs. Open-loop

Steady-state quality is lower in the closed-loop solution than in the

• pen-loop solution.

 Higher competition intensity leads to higher steady-state quality levels

in the closed-loop solution, but quality provision is nevertheless always lower than in the open-loop setting. Intuition: is related to how current quality investments affect future price competition.

 Suppose that, at time t, firm i has a higher quality (i.e., qi(t)>qj(t)). The

• ptimal pricing rule dictates that j should ‘compensate’ for the lower

quality stock, by setting a lower price. Thus, higher quality investments by one firm today will trigger stronger price competition from the other firm in the future, which dampens the incentives for quality investments.

 Under feedback rule, each firm has a strategic incentive to reduce its

quality investments in order to dampen future price competition.

Comparison: Closed-loop vs. Open-loop

 There is it a striking parallel with the difference between

simultaneous and sequential decisions in a one-shot version of the game:

 As shown by Ma and Burgess (1993), equilibrium quality is lower

when quality and price decisions are made sequentially rather than simultaneously.

 The reason is precisely the strategic incentive to lower quality in

• rder to dampen price competition when quality decisions are made

before prices are set.

 This suggests that, in the case at hand, simultaneous-move and

sequential-move games in a static setting provide results which are reasonable parallels of the open-loop and the closed-loop solutions, respectively, in a dynamic setting.

The social welfare perspective

 Social welfare is the present value of the sum of aggregate consumer

surplus and profits accruing over the infinite time horizon.

 Since total demand is fixed, this is equivalent to aggregate gross

consumer utility minus the total costs of production, transportation and quality provision.

 Formally:

Iit, Ijt, xi

Dt

Maximise W  





xi

Dt

v − x  kqitdx  

xi

Dt

1

v − 1 − x  kqjtdx −c −

 2 Iit2 −  2 qit2 −  2 Ijt2 −  2 qjt2

e−tdt, subject to



qi t  Iit − qit,



qj t  Ijt − qjt, qi0  qi0  0, qj0  qj0  0,

Comparison: Closed-loop vs. Open-loop

 The function to be maximized simplifies to:  (A simpe otpimal control problem; no strategic interaction).  FOCs, adjoint eqs and transversality conditions are as usual.  Under the symmetry assumption, the steady state quality is:

(The same as in open-loop)

 Compared with the first-best optimal level, quality is optimally

provided under the OL solution and is under-provided under the FB-CL solution.

 (Price is immaterial to the solution; it only affects the redistribution

• f surplus between consumers and firms)

W  



v − c − 

2  kqitx i Dt  kqjt1 − x i Dt

−

 2 Iit2 −  2 qit2 −  2 Ijt2 −  2 qjt2

e−tdt

q ∗ 

k 2

Comparison: Closed-loop vs. Open-loop

Intuitions:

 The welfare-optimal quality provision in the open-loop solution is

partly explained by the linearity of the demand system, which implies that consumers' marginal and average valuations of quality are identical.

 As demonstrated by Spence (1975) in a monopoly setting, whether

quality is over- or under-provided depends on the difference between marginal and average willingness-to-pay for quality.

 However, dynamic strategic interaction (with FB decision rules)

creates an inefficiency that leads to under-provision of quality in the FB solution.

Comparison: Closed-loop vs. Open-loop

Moreover,

 The welfare properties of the OL and FB-CL solutions mimic the

welfare properties of the Hotelling model with price and quality competition in a 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 (Ma and Burgess, 1993). [No surprise]

 However, a relevant difference occurs. Indeed, the quality in FB-CL

dynamic game depends on transportation cost, while equilibrium quality in the sequential-move game does not. Put differently, the ‘neutrality’ result obtained by static games, according to which lower transportation cost (and hence fiercer competition) has no effect on quality, no longer holds in a dynamic model.

Concluding remarks

 We have taken a differential game approach to study price

competition and choice concerning quality in a model of oligopoly.

 We

have considered different assumptions concerning the information set used by firms over time (OL; FB-CL rule).

 The

properties

• f

the equilibria generated under these two assumptions are studied and compared with the conclusions provided by static models of price and quality competition (e.g., Ma & Burgess, 1993).

 Dynamic strategic interaction between competing firms creates an

additional inefficiency that leads to under-provision of quality.

 The steady-state quality in the closed-loop solution is increasing in

the degree

• f

competition, as measured by a reduction in transportation costs. This is in contrast to the outcome from an equivalent static game and from the open-loop solution, where the equilibrium quality does not depend on the degree of competition.

Concluding remarks

 At the same time, relevant and non-trivial effects have been

uncovered, both from a positive and a normative point of view.

 Strategic dynamic substitutability between quality and price leads to

under-provision of quality.

 Since quality is generally non-verifiable (hard to regulate), and since

the under-provision result is caused by dynamic interaction between price and quality choices, our analysis suggests a potential role for price regulation as an instrument that can be used to avoid an inefficient outcome with respect to quality provision.

 Such a policy intervention might be unnecessary if firms instead are

committed to long-term plans regarding quality investments (in which case the relevant solution concept is open-loop).

 The consideration of time, hence, simply represents an additional

source of evaluation in the never-ending debate about the necessity and desirability of public intervention in market economies.

Thanks!

Roberto

(also on behalf of Luigi and OddRune!)

cellini@unict.it http://www.robertocellini.it