The Stackelberg Sequential Move Model - Assignment Example

due The Stackelberg sequential move model: Part a: Assume 2 firms and not entry from other firms. The firms face aconstant marginal cost (Cj(Qj) = cqj for c > 0, and c1 = c2 = c for each firm. Also assume a linear inverse demand function, P(Q) = a – bQ, for a = demand intercept, and b > 0 being the slope. Firm 1 (the leader chooses irreversible quantity. Then firm 2 chooses quantity given the choice of Firm 1. Derive the Subgame Perfect Equilibrium using backward induction. Assume firm 1, opts to produce quantity q1 Firm 2 will produce q2 = Q – q1 where Q is the total quantity demanded And thus q2 = f (q1) Assuming that firm 2’s objective is to maximize profit: it problem can be stated as Max. π2 = P.q2 – TC Π2 = [a- b. (q1 + q2)]. q2 – cq2 = (a-c)q2 – bq1q2 –bq22 At maximum profit = (a- c) - bq1 – 2bq2 Expressing q2 = f (q1) q2* = Firm1, profit maximization problem can be expressed as: Π1 = [a- b. (q1 + q2)]. Q1 – cq1 = (a-c)q1 – bq1q2 –bq12 Substituting for equilibrium q2 = (a-c) q1 – bq1* –bq12 = At maximum profit = 0 q2* = Total quantity supplied Q = = Part b: Compare the output of the Stackelberg to the Cournot duopoly output. (* of course you will have know, or look up the Cournot model that we dealt with in the first section of the course (Chs. 6 & 7 of Shy or the notes). In a cournot game both firms act simultaneously. Firm 1’s problem; Π1 = [a- b. (q1 + q2)]. q1 – cq1 = (a-c)q1 – bq1q2 –bq12 At maximum profit = (a- c) - bq1 – 2bq2 Expressing q1 = f (q2) q1* = Since the two firms have identical costs, then firm 2’s reaction function would be : q2* = Substituting q2 into equation 1 q1* = q1* = = q2* By being the first mover, firm 1 is advantaged in the stackelberg’s equilibrium and thus q1s > q1c while firm 2, the follower produces a lower quality q2s< q2c Total quantity supplied; Qc = = Qs > Qc Part c: Derive the market price, Ps, under the Stackelberg case and compare with the Cournot solution for price. P = a – bQ For stackelberg model Ps = = Pc = = The cournot model solution is higher than the stackelberg’s solution price. Part d: What are the profits of the two firms and how do they compare with the profits for the Cournot model of oligopoly? Derive and compare. Π = (P – c)qi Stackelberg π1 = = π2 = = Cournot π1 = = π2 = = Firm 1 is better of as leader, since π1s>π1c, while firm two earns a higher profit in the simultaneous game. For firm 2, π2s MC, in an effort to maximize profit. Firm 2 will have an incentive to lower the price such that p1>p2>MC, to capture a larger market share. Rationally, consumers will shift to the lower price, p2 and firm 1 will make zero profits. The assumption is that, consumers are well informed and the products are homogeneous. Firm 1will thus cut its price making it lower than P2. The game continues until both firms charge a price equal to MC, and make zero economic profits. At this price no firm would either raise prices; leads to zero sales, or lower prices; at p < mc the firm is experiencing losses. 5) a) Below is a table showing the payoffs for price strategies that each of 2 firms can make. These profits are derived from the fact that the demand function in the particular market = Q = 130 – P, where Q = demand, and P = price with the demand intercept being = 130. Constant marginal cost is equal to 10. Derive the Nash equilibriums, if any exists, and tell us whether this is a Monopoly, Stackelberg, Cournot, Bertrand or monopolistically competitive market situation. Provide your explanation. Firm 2 P = 70 P= 50 P= 10 Firm 1 P = 70 (1800, 1800) (0, 3200) (0, 0) P= 50 (3200, 0) (1600, 1600) (0, 0) P = 10 (0, 0) (0, 0) (0, 0) Suppose firm 1 moves first, it will choose P= 70(highest price), firm 2 chooses P= 50, to maximize profits. Firm 1 realizes that its profits have been reduced to zero, and hence reduces its price p=50. The results remain unchanged regardless of the first mover. A Nash equilibrium is attained at p1=p2=50 and the payoffs (1600, 1600). Note that the first mover might also decide to cut prices to 10, where p= mc. To maintain a proportionate share of the market, the follower chooses price equal to ten. Another Nash equilibrium can be attained at p1=p2=10 and payoffs of (0, 0). However, this option is irrational. The Nash equilibrium gives a monopolistically, competitive situation as equilibrium is attained at a p>mc. The second equilibrium is the Bertrand situation, where P=MC and economic profits are zero. b) What strategy decisions should be made by the 2 firms? Explain given your derivation and explanation of the above payoff matrix. Each firm aims at maximizing profits, to avoid unnecessary price cuts, (any price lowers than 50); the companies could consider differentiating their products hence acquiring some monopoly power. 6) What is product differentiation? Under what market types should product differentiation take place? Explain. The process of making a product appears different from its competing products. Product differentiation can be attained by rebranding, changing quality and components or strategic marketing. Product differentiation is mostly effective in a monopolistic competition market structure. In this structure, there are several sellers and buyers, but the products offered are not direct substitutes and hence customers choose the best product based on their preferences. 7) What is the meaning of a “Network Good”? Explain A network good is a good whose value and importance increases with the number of people accessing it. For instance, a mobile phone application such as WhatsApp, can be termed as network good, in the sense that, it becomes more effective to derive maximum utility from the good, if almost everyone in the community is using it. 8) Network goods sometimes indicate different operating or receiving “standards”, such as HD-DVD, or Blu-Ray, or operation on a standard  versus a standard β. Given the “standards game” illustrated below, derive the Nash equilibriums, if any compatibility exists, and briefly explain the strategies. Firm B Standard strategy Standard  Standard β Firm A Standard  (a, b) (c, d) Standard β (d, c) (b, a) Assume that a>c and b>d. If firm A chooses standard  and implements it, firm B will opt for the standard α, and opt for β if firm A chooses β. The reverse is also true. The Nash equilibriums will thus be (α, α) and (β, β)… hence the two standards are not comparable. 9) What is the Hotelling linear location model solution and how do we get to the solution if we assume prices are fixed exogenously, free entry into the market, and fixed costs. Derive, explain and tell us the impact of such a solution. Hottellings linear model is concept in economics that embraces on minimizing product differentiation. By assuming that consumers are linearly located, the model alludes, in a case of identical products, profits can maximize by increasing transport costs. For a firm to produce some output q, it has to incur a set up cost, c. Consider a situation where new firms enter chronologically, assuming that incumbent firms cannot afford the moving costs involved. New firms can only enter if the expected profit exceeds cost i.e. π > c and hence entrance rational only when v≤ π that is l /2 ≤ π Wher v and l are the lowest peripheral length and lowest interior interval that can invite entering firms respectively. Hence the firms maximum reach L =2f. Assume that there are L customers then the maximum number of firms, N = L/w = L/2f In accordance with the above, the equilibrium number of firms is directly influenced by L, but inversely related to the expected earnings. 10) a) Define what is meant by a “Nash Equilibrium”. This refers to a situation where the players are contented with the payoffs and lack the incentive to change their strategy. Nash equilibrium might not be the best strategy, but deviating from the equilibrium leads to a loss in payoffs. b) Alex and Tyler are choosing software. The utilities of each software to Alex and Tyler are given in the payoff matrix given below. Derive the Nash equilibriums if any such equilibriums exist in this case. What do the Nash equilibriums mean if they exist at all? Tyler Strategy Apple Microsoft Alex Apple (11, 11) (3, 3) Microsoft (3, 3) (10, 10) Suppose Alex chooses Apple, Tyler will also choose apple, if Alex chooses Microsoft, Tyler chooses Microsoft too, to reap the higher payoffs. On the other hand, if Tyler was the first mover, Alex would choose apple and Microsoft if Tyler chooses apple and Microsoft respectively. Hence the Nash equilibrium would be Tyler Strategy Apple Microsoft Alex Apple (11, 11) (3, 3) Microsoft (3, 3) (10, 10) Nash equilibrium exists if the two players use the same type. The implication is that a higher utility level is attained with the two consuming the same product. c) There is a “Mixed Strategy” equilibrium in the case above which gives a (7/15) probability to Apple software and an (8/15) probability to Microsoft software. What do these mixed strategies mean in this case relative to the pure Nash equilibriums you chose in part b) above? Explain. Depending on the first mover’s choice, Alex and Tyler might end up acquiring lower payoffs, of (10, 10). Assigning probability to the option provides a new equilibrium. That is: Given the probabilities, the resultant payoff following first movers choice is is as illustrated below. Strategy Apple Microsoft Alex Apple (11, 11) (3, 3) 11* 7/15 + 3 *8/15 = 11 Microsoft (3, 3) (10, 10) 3* 7/15 + 10* 8/15 = 6.73 11* 7/15 + 3 *8/15 = 11 3* 7/15 + 10*8/15 = 6.73 d) Is it possible for the market to “lock in” the wrong product in this case? Explain given the Nash equilibrium conditions you derived above and the mixed strategy conditions in this network good example. In the Nash equilibrium, the choices for unlike products i.e (apple, Microsoft) and Microsoft, apple) are locked out as they are not pareto efficient. (Apple, apple) happens to be the best combination using the two models hence there is no chance of locking it out. Though, (Microsoft, Microsoft) could be locked in, despite producing lower payoffs. Work cited Dwivedi, D N. Microeconomics: Theory and Applications. New Delhi: Pearson Education, 2006. Print Read More