OVERCONFIDENCE 1197 The next theorem shows that,for each c,there exists a unique pair (k',B)that solves equations (19)and (20).The smooth pasting con- ditions are sufficient to determine the function gand the "trading point" k'. THEOREM 1.For each trading cost c>0,there exists a unique k'that solves (20).If c 0,then k'=0.If c>0,k">c(r+A). The next theorem establishes that the function g described by equa- tion (18),with B,and k given by (19)and(20),solves (13).The proof consists of two parts.First,we show that (14)and (15)hold and that g'is bounded.We then use a standard argument to show that in fact g solves equation (13)(see,e.g.,Kobila [1993]or Scheinkman and Zar- iphopoulou [2001]for similar arguments). THEOREM 2.The function q constructed above is an equilibrium op- tion value function.The optimal policy consists of exercising immedi- ately if g">k';otherwise wait until the first time in which g>k'. It is a consequence of theorem 2 that the process g will have values in (-o,k").The value k'acts as a barrier,and when g"reaches k',a trade occurs,the owner's group switches,and the process is restarted at -k".The function q(g")is the difference between the current owner's demand price and his fundamental valuation and can be legitimately called a bubble.When a trade occurs,this difference is 1h(-k) b≡g(-k)= (21) r+入h()+h(-k*) Using equation (21),we can write the value of the resale option as b h(x) for x<k h(-k) 9(x)= (22) x。 、h(-x)-c for x≥ r+入h(-k) B.Duration betiween Trades We let w(x,k,r)E"[e,with T(x,k)infis:g,>given g=xsk.The term w(x,k,r)is the discount factor applied to cash flows received the first time the difference in beliefs reaches the level of k given that the current difference in beliefs is x.Standard arguments (e.g.,Karlin and Taylor 1981,p.243)show that w is a nonnegative and strictly monotone solution to 20wx一px0=rw,w(k,h,）=1. Reproduced with permission of the copyright owner.Further reproduction prohibited without permission.Reproduced with permission of the copyright owner. Further reproduction prohibited without permission