DynamicPriceCompetition.doc

industrial economics iiprof. davide vannoni handout 2dynamic price competition and tacit collusion 1. informal introduction2. the basic model3. applications application 1: tacit collusion and concentration, application 2: tacit collusion and information lags, application 3: multimarket contact (bernheim and whinston, 1990, rand), application 4: fluctuating demand (rotemberg and saloner, 1986, aer; green and porter, 1984, econometrica).good reviews of this literature:ivaldi, jullien, rey, seabright, tirole (2003) “the economics of tacit collusion”, final report for dg competition, european commission. feuerstein s. (2005) “collusion in industrial economics a survey”, journal of industry, competition and trade, pp. 163.198.1. informal introductionin real world, firms are likely to interact repeatedly. in general, firms do not enter an industry, play their simultaneous one shot game in prices and then disappear.if this is the case, in choosing their current price firms have to take into account not only the impact on current profits but also on future profits.for instance if a firm decides to undercut the current price it has to compare the increase in current profits (short-run gain) with the reduction in future profits (long-run loss) which might occur if the undercutting strategy makes a price war more likely. in turn, the possibility of a price war might be a sufficient threat to deter the temptation to cut prices. hence, oligopolists might be able to sustain prices above the competitive level in a purely noncooperative manner (tacit collusion).2. the basic model (repeated game or supergame)model set up (duopoly case)firms replicate t+1 (t can be finite or infinite) times the basic bertrand game.remainder of the basic bertrand gametwo firms (1,2) produce identical goods with equal and constant marginal costs . in addition each firm supplies the demand it faces in each period. in each period consumers buy from the firm with the lowest price . if the same price is charged by both firms we simply assume that each firm supplies half of the market demand at the common price. firms choose their prices simultaneously and noncooperatively in each period.price choices at time t are allowed to depend on the history of previous prices. therefore the price strategy, may depend on the history where:each firm maximises the present discounted value of its profits, that is:where is the discount factor. note that close to 1 implies that future profits are not heavily discounted. in this sense, we can interpret the discount factor as a measure of firms patience.equilibrium conceptprice strategies are required to form a perfect equilibrium. for any price history , firm is price strategy forms a perfect equilibrium if from time t on it maximises the present discounted values of profits given firm js price strategy from time t on.case 1: finite horizon by using backward induction, it is immediate to prove that the only perfect equilibrium of the repeated game is the (t+1) repetition of the solution of the once shot bertrand game.conclusion: with a finite horizon the dynamic element does not provide any additional hint compared to the one shot game.caveat: in some circumstances (craziness, -irrationality, multiple equilibria), even with a finite horizon, collusion can emerge.case 2: infinite horizon the bertrand equilibrium repeated infinitely is a perfect equilibrium of the infinite horizon price game.this can be easily understood by noting that if firm i chooses a price equal to marginal cost in each period t (regardless of the history of the game up to t), firm j can do no better than to charge the competitive price itself.however there are also other price strategies which form a perfect equilibrium (embarrassment of riches).for instance lets consider the following (symmetric) price strategy, commonly known as trigger strategy:firm i plays the monopoly price at time 0.it also plays the monopoly price at time t if both firms have played the monopoly price in every period before t, otherwise it plays the competitive price forever.this symmetric trigger strategy forms a perfect equilibrium if the long-run loss from undercutting the monopoly price is greater than the corresponding short run gain: in a price competition setting (bertrand)which obviously impliesand thereforeconclusion: provided the discount factor is sufficiently high, this model is consistent with both firms setting the monopoly price in a non-cooperative way.additional remarkthere are many other equilibria in this game. any price between the competitive price and the monopoly price can be in fact sustained as an equilibrium price if the discount factor is greater than . in fact, for any given price in the relevant range the following condition must hold:which again implies:given this embarrassment of riches is there any “focal equilibrium” firms are likely to coordinate on?in the literature the following assumption is often made. in a symmetric game the focal equilibrium is symmetric and must be pareto optimal from the viewpoint of the two firms.general formulationthe symmetric trigger strategy forms a perfect equilibrium if the long-run loss (difference between profits in a collusive agreement and non cooperative profits) from undercutting the monopoly price is greater than the corresponding short run gain (difference between profits from deviation and profits in a collusive agreement):which obviously impliesand thereforebertrand: ; ; . thus: cournot: ; ; . thus: in a duopoly collusion is easier with price competition!application 1 : tacit collusion is harder to sustain with a larger number of firmsthis can be easily understood by computing the short-run gain and the long-run loss from undercutting the monopoly price in an industry with n firms competing la bertand.the short-run gain from deviating is:whereas the long-run loss is:tacit collusion is sustainable only if the long-run loss is greater than the short run gain. this turns out to be the case only iftherefore an increase in the number of firms in the industry requires a higher discount rate to make tacit collusion sustainable.in the case of cournot competition: for n greater than 2, while the opposite occurs for n = 2.therefore, except for the case of duopoly, collusion is easier to sustain in a quantity competition setting (cournot)!application 2: tacit collusion is harder to sustain with longer information lagsthe idea of information lags cannot easily be accomodated in the present framework unless one is willing to make the strong assumption that each firm observes not only rivals price but also its demand and profit with a two period (instead of one period) lag.under this assumption in a duopolistic market with price competition the short-run gain from deviating is:whereas the long-run loss is:as usual tacit collusion is sustainable only if the long-run loss is greater than the short run gain. this turns out to be the case only iftherefore a longer information lag requires a higher discount rate to make tacit collusion sustainable.application 3: tacit collusion is easier to sustain with multimarket contact (berhneim and whinston, 1990, rand)let us suppose that there are two identical and independent duopolistic markets. in one market (1) information is available with a one period lag whereas in the other market (2) it is available with a two period lag.without multimarket contact tacit collusion is sustainable in market 1 but not in market 2 if the discount rate is let us now suppose that the same two firms participate in both markets. with multimarket contact tacit collusion is sustainable on both markets if the short run gain from undercutting the monopoly price simultaneously in both markets is smaller than the corresponding long-run loss, that is ifhint: the firm begins to deviate in the market with detection lag in the first period and in both markets in the second period.from the above inequality, the following condition for tacit collusion with multimarket contact can be easily derived:conclusion: if the discount rate lies between 0.64 and 0.71, collusion is sustainable in market 2 only with firms operating on both markets.irrelevance result: in order to be able to transfer a collusive potential from one market to another, there must be some asymmetries:- between markets (i.e. detection lags)- between firms- no constant returns to scaleother example: mmc and asymmetry between firmsconsider a market a in which, under collusion, firm 1 has a higher market share 1/2 (large firm).collusion is sustainable if and if that is if if = 1/2 collusion is not sustainable (asymmetry hinders collusion because pushes the small firm to deviate!)if both firms compete in a second market b, where firm 2 happens to be the large firm (with a market share 1/2), multimarket contact restores the symmetry and helps firms to collude.in such an event, for both firms, the short run gain from undercutting the monopoly price simultaneously in both markets is smaller than the corresponding long-run loss, ifthat is if application 4: evolution of demand4.1. tacit collusion could explain countercyclical markups- price wars during booms (rotemberg and saloner,1986, aer)let us assume that demand is stochastic. in each period it can be either low (1) or high (2) with equal probability. at any given price:for simplicity the demand shock is identically and independently distributed (iid) over time. therefore the state of demand today does not convey any information about the state of demand tomorrow.we look at the pair of prices such that:i) both firms charge the same price with when the state of demand is ,ii) the price configuration is sustainable in equilibrium,iii) the expected present value of future profits of each firm along the equilibrium pathis not pareto dominated by any other equilibrium payoffs.lets derive first the conditions under which the fully collusive outcome is sustainable.the long run loss from undercutting is equal towhereas the short run gain depends on the state of demand. it is when demand is low and when demand is high. since , the binding condition is:and finallynote also thatintuition: when lies between and full collusion cannot be sustained in the state of high demand, basically because the long run loss is lower compared to the case of a deterministic demand.at this point the relevant question to be answered is: what is the pair of prices such that i), ii) and iii) hold when the discount factor, lies between and ? it can be shown (and it is quite intuitive) by solving the program which maximises the firms expected payoff subject to the undercutting (incentive) constraint that the pair of prices becomes , that is monopoly price when demand is low and less than monopoly price when demand is high.this result has been interpreted by macroeconomists as evidence supporting countercyclical markups. this is not fully correct however. what rotemberg and saloner show is that the amount of sustainable collusion is lower when demand is high . it might still be the case that 4.2. cyclical demand movements and collusion (haltiwanger and harrington, 1991, rand)in rotemberg and saloner booms are not likely to continue in the future, in other words there is no correlation among demand shocks.let us assume that actual demand and profits are d(p) and (p) while future demand and profits are respectively t d(p) and t (p) with 0:if 1(1), collusion is easier to reach, while if demand is expected to decline in the future collusion is likely to break down.when the above model is adapted to account for periodical cycles of growth and decline, the main result that collusion is more likely to be sustained when demand is growing (and keeps growing for several time periods) holds.4.3. secret price cuts (green and porter, 1984, econometrica)in this model prices are not observable (in rotemberg and saloner demand was stochastic but observable)when a firm encounters a reduction of its market share it is not sure about its underlying cause (it may be due to a reduction of demand or to the secret undercutting of the rival)rigid punishment strategies such as trigger strategies are not suitable in such a situa