UC BerkeleyHaas School of BusinessEconomic Analysis for Business Decisions(EWMBA 201A)Final examAnswer Key[1] PreferencesThe marginal rate of substitution is the maximum amount of a good thata consumer is willing to give up in order to obtain one additional unit ofanother good. The marginal rate of substitution at any point is equal totheslopeoftheindifference curve.— Geoffrey’s indifference curves are “convex” (bowed inwards) so themarginal rate of substitution diminishes as we move down the indif-ference curve.— Elizabeth’s indifference curves are “concave” (bowed outwards) sothe marginal rate of substitution increases as we move down theindifference curve.Since his indifference curves are “convex” Geoffrey prefers, in terms ofconsumption bundles of classical and rap music records, averages overextremes (preference for diversification). Thus, Geoffrey’s optimal choiceis to buy both types of records, whereas Elizabeth’s optimal choice is tobuy only one type of records. This is illustrated in t he diagrams below.[2] Risk aversion, insurance and moral hazardElizabeth has a car that will be stolen with probability .1 if an antitheftdevice is not used and with probability .05 if a device is used (Elizabethlives in Oakland). The value of Elizabeth’s car is $10, 000,andthecostofthe an titheft device is $10.Suppose that Elizabeth can only insure the car for its full value, and thatthe insurance compan y assumes that she has not bought the antitheftdevice. Also assume that Elizabeth has $1, 000 in the bank. Let Elizabethutility function be =√w where w denotes Elizabeth’s wealth (the valueof the car plus money in the bank).— First note that an insurance company must be risk-neutral (otherwiseit will go out of business). In a competitive insurance market (zeroprofit), an insurance company can only charge a premium equal tothe expected loss due to the theft:premium = .1(10, 000) = 1, 000.1— Since Elizabeth can only insure the car for its full value for a premiumof $1, 000, her utility if she only purchases the theft insurance is giv enbyu(10, 000) =p10, 000 = 100.Her expected utility if she only purchases the antitheft device is givenbyE(u)=.95u(10, 000 + 990) + .05u(990)= .95p10, 990 + .05√990= 101.16(assuming that installing the antitheft device does not increase thevalue of the car). Th u s, Elizabeth should not purchase the theftinsurance.— If Elizabeth can convince the insurance company that she has pur-chased the antitheft device, the insurance company will charge herpremium = .05(10, 000) = 500which is the expected loss due to the theft. Elizabeth’s utility if shepurchases the antitheft device and the theft insurance is given byu(10, 490) =p10, 490 = 102.42.[3] Game Theory I— Since (T,L) is the a Nash equilibrium of Ga ≥ e and b ≥ d(no player has a profitable deviation) and since it is also a Nashequilibrium of G0a0≥ e0and b0≥ d0.Thus,a + a0≥ e + e0and b + b0≥ d + d0so (T,L) also an equilibrium of G00.— Since (p∗,q∗) is a completely mixed-strategy Nash equilibrium of Gq∗a +(1− q∗)c = q∗e +(1− q∗)g (1)(when player 2 assigns probability q∗to her strategy L and probabil-ity 1 − q∗to her strategy R,player1 is indifferent between playingT or B)andp∗b +(1− p∗)f = p∗d +(1− p∗)h (2)(when player 1 assigns probability p∗to her strategy T and proba-bility 1 −p∗to her strategy B,player2 is indifferent between playingL or R).2Similarly, since (p∗,q∗) is a also a completely mixed-strategy Nashequilibrium of G0q∗a0+(1− q∗)c0= q∗e0+(1− q∗)g0(3)andp∗b0+(1− p∗)f0= p∗d0+(1− p∗)h0. (4)Adding (1) and (3) and rearrangingq∗(a + a0)+(1− q∗)(c + c0)=q∗(e + e0)+(1− q∗)(g + g0) (5)and adding (2) and (4) and rearrangingp∗(b + b0)+(1− p∗)(f + f0)=p∗(d + d0)+(1− p∗)(h + h0). (6)What (5) says is that in game G00,whenplayer2 assigns probabilityq∗to her strategy B and probability 1 −q∗to her strategy S,player1 is indifferent between playing T or B. What (6) says is that ingame G00,whenplayer1 assigns probability p∗to her strategy T andprobability 1 − p∗to her strategy B,player2 is indifferent betweenplaying L or R. Hence, (p∗,q∗) also an equilibrium of G00.[4] Game Theory II— The game has no Nash equilibrium in pure strategies.— Let p be the probability that player 1 assign to strategy R.Player2will be indifferent between using her strategy L and R when player1 assigns a probability p such that her expected payoffsfromplayingL and R are the same. That is,0p +1(1− p)=1p +(1− δ)(1 − p)p∗=δ1+δHence, when player 1 assigns probability δ/(1 + δ) to her strategyR and probabilit y 1/(1 + δ) to her strategy L,player2 is indifferentbetween playing L or R or an y mixture of them.Let q be the probability that player 2 assign to strategy L.Player1will be indifferent between using her strategy L and R when player2 assigns a probability q such that her expected payoffsfromplayingL and R are the same. That is,1q +0(1− q)=0q +(1− δ)(1 − q)q∗=δ1+δHence, when player 2 assigns probability δ/(1 + δ) to her strategyL and probability 1/(1 + δ) to her strategy R,player1 is indifferentbetween playing L or R or an y mixture of them.3— When δ approaches one, the unique mixed-strategy Nash equilibriumapproaches (1/2, 1/2).Whenδ approaches zero, the unique mixed-strategy Nash equilibrium approaches a pure strategy equilibrium(L, R).[5] Cournot and StackelbergThe firms payoffs(profits) are give byπ1= Pq1− c1q1= (240 − q1− q2− c1)q1andπ2= Pq2− c2q2= (240 − q1− q2− c2)q2.— In the Cournot’s game, firm 1’s best response to any given output q2is given byq1=12(240 − q2− c1)and firm 2’s best response function is given byq2=12(240 − q1− c2).The Nash equilibrium of the Cournot’s game is the pair of outputsq∗1=240 + c2− 2c13andq∗2=240 + c1− 2c23which is the solution to the two equations above (q∗1is a best responseto q∗2and q∗2is a best response to q∗1).— In the Stackelberg’s game, firm 1’s strategy is the output q1themaximizesπ1= (240 − q1− q2− c)q1subject to q2=12(A − q1− c)Thus, firm 1 maximizesπ1= (240 − q1− (12(240 − q1− c)) − c)q1=12q1(240 − q1− c).This function is quadratic in q1that is zero when q1=0and whenq1= 240 − c. Thus its maximizer isq∗1=12(240 − c).4We conclude that Stackelberg’s duopoly game has a unique subgameperfect