Player two accepts or rejects. If player 2 accepts, they split the pie. Otherwise, player 2 gets r 2 and player 1 gets r 1, where r 1 +r 2 < 1.

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1 The Coase Theem A popular idea in economics is called, the Coase theem, and it goes like this: Suppose all the parties to a negotiation are completely infmed and there are no costs to coming to an agreement. Then we should expect them to choose the most efficient outcome. F example, suppose a railway goes by a cn field and tends to throw sparks, which set the crops on fire. A sparkguard costs $0 dollars. The profits of the railway are $00, and the profits of the farm are $X. When is the sparkguard installed? Does this depend on the legal property rights of the agents? Bargaining In bargaining games, we are generally trying to model agents haggling back-and-fth over how to split the value surplus from trade. They typically have the following ingredients: There s a pie /value/surplus of fixed value. We usually nmalize it to, so the players are trying to get the largest share they can. The players can choose any split, (s, s). There s an extensive fm that describes who makes offers and counteroffers when, and under what circumstances bargaining breaks down and the players walk away from the trading opptunity. Iftrade breaksdown, the playersgettheir outside option payoffs, r i, which represent what the player can get on his own. We usually assume that r + r 2 < in two-players bargaining games, so that trade would be efficient if it can be arranged. The Dictat Game Player one proposes a split of the pie, (t, t), with t going to player. Player two accepts rejects. If player 2 accepts, they split the pie. Otherwise, player 2 gets r 2 and player gets r, where r +r 2 <. What s the subgame perfect Nash equilibrium, especially when r = r 2 = 0? Is this a good prediction? We solve f the SPNE through backwards induction:

2 In the last subgame, player 2 is deciding between accept and reject. He should accept if t r 2 r 2 t. He should reject otherwise. If player wants player 2 to accept, player must solve maxt t subject to r 2 t. Since player wants to maximize t, he should pick t = r 2, since that is the largest value he can select where player 2 will still accept. Then the SPNE is Player offers t = r 2 Player 2 accepts if t r 2, and rejects otherwise Then we expect to see the players actually choose Player offers t = r 2 Player 2 accepts But if r = r 2 = 0, this means that t = and t nothing. = 0, so player 2 gets Discounting Suppose an agent gets a payoff u f each date t = 0,,2,3,...,T. Then the discounted payoff when an agent has discount fact δ is given by the sum S = u+δu+δ 2 u+...+δ T u where δ <. If the interest rate is R, then a dollar today is wth and investing it gives (+R) dollars tomrow, so we can ask, What is the discount fact that makes you indifferent between a dollar today and receiving +R dollars tomrow? = δ(+r) δ = +R So if R =.05, which is a reasonable investment, the discount fact would be δ =.05 =.95 2

3 But suppose that rather than accept t = r 2, the second player makes a counteroffer. To make it interesting, we imagine that there are costs to delay, so that the players payoffs after a counteroffer are discounted by δ, giving us an extensive fm: In particular, if r = r 2 = 0, what is the SPNE? We use backwards induction: In the final subgame, player is deciding between Accept and Reject. Player should accept if δt 2 δr Player should reject otherwise. Then player 2 is trying to maximize t 2 r max t 2 δ( t 2 ) subject to t 2 r. Since player 2 s payoff is decreasing in t 2, he should minimize it, choosing the lowest t 2 possible, t 2 = r. Thatmeans thepayoffsfollowinga rejectionbyplayer2willbe (δr,δ( r )) Now player 2 is deciding between accepting and getting t, rejecting and getting δ( r ). Player 2 should accept if Player 2 should reject otherwise. t δ( r ) δ( r ) t 3

4 Then player is trying to solve max t t subject to δ( r ) t. Since player s payoff is increasing in t, he should pick the highest t possible without causing a rejection, t = δ( r ) Then the SPNE is Player offers t = δ( r ) Player 2 accepts if t δ( r ) and rejects otherwise. If a rejection occurs, player 2 offers t 2 = r. Player accepts the offer if t 2 r and rejects otherwise. Then we expect to see player make an offer t = δ( r ) in the first round, player 2 accepts, and the game ends. If r = r 2 = 0, then t = δ, and t 2 = δ. So the players DO split some of the profits, but not very equally. Or... Great practice! If we set r = r 2 = 0, a pattern begins to emerge: The game ends in the first round, with the first player offering t = t 2 = δ 4

5 t 3 = δ( δ) = δ +δ2 t 4 = δ( δ( δ)) = δ +δ 2 δ 3... t T = δ +δ2 δ 3 +δ 4...±δ T = Σ T t=0 ( δ)t But these are somewhat confusing because of the... terms. Geometric Sums But how do we compute a sum like S = y + xy + x 2 y x T y? First multiple S by x: xs = xy +x 2 y +x 3 y +...+x T+ y Then Or S = y +xy +x 2 y +x 3 y +...+x T y xs = xy x 2 y x 3 y... x T y - x T+ y S( x) = y - x T+ y S(T) = xt+ x y Note that S( ) = x y By the way, I deliberately switched from δ and u to x and y because later on, x will be δ, not δ, and y will be one, so we get S(T) = ( δ)t+ ( δ) and S( ) = But we can always use this fmula f geometric sums as long as < x <, so there s nothing special about the discounted utility framewk. Then using the geometric summation fmula on yields t T = δ +δ2 δ 3 +δ 4...±δ T = Σ T t=0 ( δ)t t T = ΣT t=0 ( δ)t = ( δ)t If we take the limit as we allow the bargaining to go on indefinitely, T, and t = 5

6 and the players payoffs are, δ So as δ, we get lim δ t = lim δ = 2 so the players split the value equally. Likewise, if δ = e R, so that interest compounds continuously at rate R over a sht period of time of length, lim 0 t = lim 0 +e R = 2 So if the players become very patient, the time between offers goes to zero, the players split the pie equally: Bargaining power here comes from agenda control and the ability to waste resources credibly. If these are removed by allowing many offers and counter-offers, the time between offers goes to zero, then the players split the pie equally in the SPNE. What s missing from this model of bargaining that is probably imptant in real applications? Bargaining with Investment In many situations, opptunities to bargain and the inability to commit lead to inefficient outcomes, even when property rights are clearly defined. The student decides on an amount of education to get, e 0, which costs c 2 e2 The firm makes the student a wage offer, w 0 The student can accept the job, receiving a payoff of w c 2 e2, reject, and open his own business that makes profits be. If the student accepts the job, the firm makes profits ae, where a > b, and otherwise the firm makes nothing. Sketch an extensive fm f this game and solve f the subgame perfect Nash equilibrium. What is the efficient level of education? Is the student over- under-educated in the SPNE relative to the efficient outcome? If the firm could tie its hands and commit to making a wage offer w = a (a/c), would the problem be resolved? 6

7 In the last subgame, the student accepts as long as and the student rejects if w < be. w c 2 e2 be c 2 e2 w be At the offer stage, the firm chooses the offer w to maximize max w ae w subject to the constraint that the student accept the offer, w be. Since the firm s profits are decreasing in w, they want to pick the lowest w possible, w = be. At the education stage, the student chooses e to maximize Maximizing over e gives a condition So the SPNE is The student chooses e = b/c The firm offers a wage w(e) = be maxbe c e 2 e2 b ce = 0 e = b c The student accepts the offer if w(e) be, but rejects otherwise What we expect to see the players actually do is The student chooses e = b/c The firm offers a wage w = b 2 /c The student accepts the offer The efficient outcome is characterized by So the efficient level of education is Since a > b, max (w c e 2 e2 )+(ae w) = maxae c 2 e2 e o = a c e o = a c > b c = e So the student in the SPNE gets an inefficiently low level of education. e 7