THE UNIVERSITY OF MARYLANDCOLLEGE PARK MARYLANDDaniel R. Vincent ECONOMICS 414 March 1, 2001Office: Tydings 4128bPhone: 301-405-3485Midterm Exam (Closed Book)Please show the reasoning by which you reached your answer.1.. There are two firms in a market. Firm A selects its price first. Then Firm B selects its priceafter observing the price selected by Firm A. Let pA and pB be the levels they choose. Thequantity sold by firm A is given by, QA =50-2pA +pB and the quantity sold by firm B is QB =50-2pB + pA . The cost of production for each firm is zero. You should assume firms choose pricesfrom the set of real numbers (so that you can use calculus.)i) (5 points) What characterizes a typical subgame for Firm B?Firm B moves after observing Firm A’s action (that is, choice of price). Thus, the full descriptionof the relevant environment for Firm B is the price, pA selected by Firm A in stage 1.ii) (10 points) Characterize Firm B’s best response function.For a given price, pA Firm B’s profits are given byAB (pB;pA)=(50-2pB +pA)pBSetting the derivative of this with respect to pB equal to zero yields0=(50-2pB +pA)-2pBorpB=(50 +pA)/4as the best response for Firm B. iii) (10 points) Assume that the firms play a subgame perfect (or “Rollback”) equilibrium. Forevery pA that Firm A selects, what does it know will be selected by Firm B?Firm A knows that whatever Firm B “promises” it will best be expected to select its bestresponse given the prices chosen by A. iv) (20 points) Compute the optimal price choice of Firm A.If it knew pB Firm A’s profits are given byAA (pA,pB)=(50-2pA +pB)pAIt does not “know” pB, but it expects pB=(50 +pA)/4. Plug this into the equation to getAA (pA,pB(pA))=(50-2pA +(50 +pA)/4)pA =(62.5-1.75pA)pA Setting the derivative of this with respect to pB equal to zero yields0=(62.5-3.5pA)orpA=17.86as the best response for Firm A. This means that Firm B’s price is 16.96. Firm B profits are (50-2*16.96+17.86)*16.96=(33.94)*16.96=575.5. Firm A profits are (50-2*17.86+16.96)*17.86=(31.24)*17.86=555.92. You are the manager of a delivery firm. Among the population of drivers from which you hireyour employees you know that the whole population has a probability of an accident in a givenyear of 5%. You also know that in the same population there are mostly good drivers but that10% are bad drivers. Finally, you know that the probability that a bad driver has an accident in agiven year is 10%.i) (20 points) A driver you have hired in the past year has an accident. What probability do younow assign to that driver being a bad driver?Let the unconditional probability of an accident be p(A)=.05. The unconditional probability of abad driver is p(B)=.1. The probability of an accident conditional on being a bad driver isp(A*B)=.1. By Bayes’ rule we have p(B*A)= p(A*B)p(B)/p(A)=.1*.1/.05=.2 so the probability ofbeing a bad driver is now 20%.ii) (20 points) Suppose now that the unconditional probability of an accident when it is rainingrises to 10%. Suppose that the probability that a bad driver has an accident when it is raining is10%. Now it happens that A driver you have hired in the past year has an accident but it wasraining. What probability do you now assign to that driver being a bad driver?Now let AR represent an accident in the rain. Modifying the above we get the unconditionalprobability of an accident in the rain is p(Ar)=.1. The unconditional probability of a bad driveris still p(B)=.1. The probability of an accident in the rain conditional on being a bad driver isstill p(AR*B)=.1. By Bayes’ rule we have p(B*AR)= p(AR*B)p(B)/p(AR)=.1*.1/.1=.1 so theprobability of being a bad driver remains at 10%.iii) What is the intuition for the differences in your answers above?The difference here is that once it is raining, bad drivers are no more likely to have an accidentthan good drivers. Therefore, we do not get any distinguishing information when the accidenthappens when it is