Game Theory Problem

This problem considers a modied version of the Spence Job Market game we discussed in class. The main difference is that instead of there being two types (talented vs ordinary), there are now four types: talented youth with rich parents tgf, ordinary youth with rich parents tor, talented youth with lowincome parents {m} and ordinary youth with lowincome parents (to!)- The prior probability of the four types is: . me.) = 1r'8 and 19(th =1f8 . p) = age and pftor) = 3,38. The timing and play of the game is as in class: the youth (worker) rst chooses whether to go to an elite private college or a good state college. Two {or more) rms observe which college a potential worker went to, and then compete a la Bertrand in offering wages to the potential worker. PayoHrelated information is as follows: 0 A rm who doesn't hire gets 0. o Armwhohiresatalentedtypeand paysthemawagewgets payo'Sw. o Armwhohiresanordinarytypeand pays themawagewgets payo'Qw. o A tn. type who goes to an elite college and accepts a job with wage or gets payo w 1. o A ta type who goes to an elite college and accepts a job with wage or gets payoff w 1. o A to, type who goes to an elite college and accepts a job with wage 10 gets payoff w 2. o A to; type who goes to a elite college and accepts a job with wage or gets payoff w 3. o A tn. type who goes to state college and accepts a job with wage or gets payo w. o A ta type who goes to an state college and accepts a job with wage 10 gets payoff w. o A to, type who goes to an state college and accepts a job with wage in gets payoff is. o A to; type who goes to an state college and accepts a job with wage w gets payo' w. 1. Suppose that both of the two competing rms have some beliefs HtEHte} (i.e., a subjective prob ability distribution over the types, after seeing that they went to an elite college). Explain intu itively why an equilibrium which features these beliefs must involve both rms offering the wage is = 5 x (b[tg..|EliteJ + MEMBER\" + 2 x (b[to..|Elte) + b(tg[|EIite]) . 2. Describe the unique PBE in which Elite college attendance is positive. Explain. (a) In this equilibrium, what is the causal eect of elite college attendance on wages? (I.e., if any individual switched from a state school to an elite college, what would happen to their wage)? (b) In this equilibrium, what is the causal effect of elite college attendance on productivity? (c) In this equilibrium, what is the average wage of youth with rich parents? What is the average wage of youth with low-income parents? [d] Would you characterize the economy described in this probl- a meritocracy? Why or why not? (e) Suppose you we're \"in\" the economy of this equilibrium and you met two students at an elite college, and you learned that one of them had rich parents and the other had lowincome parents. Who should you subjectively assess is more likely to be talented? 3. Is there also a PBE with zero Elite college attendance? Explain