ECON 1200: MidtermThursday July 9, 2015DUE: Tuesday July 14, 2015Q1[25pts]Consider the following strategic-form game G1 px, yq where x, y P R are parameters of thegame.Figure 1: Strategic-form game G1 px, yqRow: TBColumn:LR4x, 34, 22, yx, 71. For what values of x does the Row player have a strictly dominated strategy? Provideboth the set of possible values of x and the strategy that is dominated.2. For what values of y does the Column player have a weakly dominated strategy?Provide both the values of y and the strategy that is dominated.3. For values of px, yq is pB, Rq a pure strategy Nash Equilibrium. For what values ofpx, yq is pT, Lq a pure strategy Nash Equilibrium?4. Now assume that x 5 and y 2. Find a totally mixed Nash Equilibrium, (i.e.,pp, 1 ´ pq for Row, pq, 1 ´ qq for Column, such that p, q P p0, 1q.)Q2[25pts]Two rms are price-competing as in the standard Bertrand model. Each faces the marketdemand function Dppq 100 ´ p. Firm 1 has constant marginal cost c1 40, while rm2 has the constant marginal cost c2 50. If either of the rms has the lower price, itcaptures the entire market, and when they charge exactly the same price, they split themarket evenly, as in the standard model.1. Draw the prot function 1 pp1 , p2 q when p2 60. (Mark sure to label all points ofinterest on all graphs: maximum values, axis crossings, points of discontinuity, axeslabels)2. Draw the prot function 2 pp1 , p2 q when p1 60.3. Write out the best-response correspondences B1 pp2 q and B2 pp1 q as functions of p2 andp1 respectively.4. Is there an equilibrium for this game as dened?5. Suppose S1 S2 t0.00, 0.01, 0.02, . . . , 1000.00u.

That is, instead of any real numberwe force prices in whole cents.(a) Why must there now be an equilibrium, regardless of the exact prot functions?(b) Provide a Nash Equilibria in pure strategies for this game.(c) Calculate the prots for each rm under any one of these equilibria, accurate tothe nearest cent.1Q3[25pts]Recall in class, that we showed that the general solution for the N rm Cournot model wasfor each rm to producepa´cqN .Now we will investigate more closely the three rm case.Each rm i P t1, 2, 3u simultaneously chooses a quantity to produce qi . The market priceis determined by an inverse demand which depends on the total quantity produced by allthree rms Q q1 ` q2 ` q3 , where the price per unit is given by P pQq max ta ´ Q, 0u. Each rm has the same marginal cost c of producing every unit of the good, such thata ° c ° 0. So the total prot is given by:$&pa ´ q ´ q ´ q q ¨ q ´ c ¨ q123iii pq1 , q2 , q3 q%´c ¨ qiif q1 ` q2 ` q3 § aif q1 ` q2 ` q3 ° a1. Write out the best-response correspondence B1 pq2 , q3 q for every possible combinationof q2 and q32. Show that the symmetric Nash Equilibrium quantity for each rm is given by q1q2 q3pa´cq4 .(I.e., show that this is a mutual best response)3. Give an expression for each rms equilibrium prots i pq1 , q2 , q3 q as a function of aand c at this equilibrium.4. Provide any symmetric point q1 q2 q3 such that each rms prot i pˆ1 , q2 , q3 q °ˆˆˆq ˆ ˆi pq1 , q2 , q3 q.Q4[25pts]Consider the following game. There are two players. Player 1 has 10 dollars, and proposesany split of the ten dollars between himself and player 2 (i.e., A1 r0, 10s). Then, afterthe proposal, player 2 can accept of reject the split. The payos given by (a = accept, r =reject)u1 px, aq 10 ´ xu2 px, aq xu1 px, rq 0u2 px, rq 01. How many subgames are there in this game? Describe the set of all subgames.2. Verify that the following strategies constitute a Nash Equilibrium: p6; a if x 6, r if x6q.3. Consider another NE strategy prole that achieves the same outcome (i.e., x 6 andaccept).4. In a subgame perfect NE, what must P2s action be for any x ° 0?5. Show that in a subgame perfect NE, P2 cannot reject (with any positive probability)even after x 0. (Hint: assume this was the case and nd a contradiction).6. What is the unique subgame perfect NE of this game?2