Conference 10Intermediate Microeconomics - Fall 201819/11/2018Problem 1: Adverse Selection and ReportingConsider a trade set up in which a company sells ice cream to its customers. Producing icecream is costly: it costs x2to produce x units of ice cream: c(x) = x2.Customers have different valuation for ice cream: type 1 consumers moderately like ice creamand have valuation v(1, x) =√x, where x is the quantity of ice cream consumed, while type 2consumers absolutely love ice cream and have valuation v(2, x) = 2√x. If consumers get no icecream, both types receive the same utility v(1, 0) = v(2, 0) = 0.1. Assume the firm has full information on customer types.i. Depending on customer types, what price does the firm choose to maximize its profits?Since the firm has full information, it can extract all surplus from consumers, therefore:p(1, x) = v(1, x) − v(1, 0) = v(1, x)p(2, x) = v(2, x) − v(2, 0) = v(2, x)ii. What quantity does the firm produce to maximize its profits?For each customer type t = 1, 2, the firm solvesmaxxp(t, x) − c(x) = maxxv(t, x) − c(x)= maxxt√x − x2FOC givest2√x− 2x = 0⇔t − 4x√x = 0⇔16x2x = t2⇔x3=t216⇔x =t423Therefore the firm sells x1=1423to type 1 customers at price1413and x2=1223to type 2 customers at price 2121312. Now assume the firm only knows customers have a34chance of being type 1 and a14chanceof being type 2. Agents choose what type they report. If the ice cream company implementsthe same program as in question 1, do customers choose to report their types truthfully?Payoffs for customers are:- Type 1 customers if they report type 1: v(1, x1) − p(1, x1) =1413−1413= 0- Type 1 customers if they report type 2: v(1, x2) − p(2, x2) =1213− 21213< 0- Type 2 customers if they report type 2: v(2, x2) − p(2, x2) = 21213− 21213= 0- Type 2 customers if they report type 1: v(2, x1)−p(1, x1) = 21413−1413=1413> 0Therefore type 1 customers report their true types, but type 2 prefer to report type 1.3. We now design a program such that customers truthfully report their types.i. What conditions on customers’ utility must we impose in order for them to truthfullyreport?Type 1 customers must prefer to report type 1 and type 2 customers must prefer toreport type 2, that is:v(1, x1) − p(1, x1) ≥ v(1, x2) − p(2, x2) (1)v(2, x2) − p(2, x2) ≥ v(2, x1) − p(1, x1) (2)ii. What conditions on customers’ utility must we impose in order for them to participatein the market and buy ice cream?Customers must get at least their reservation value:v(1, x1) − p(1, x1) ≥ v(1, 0) = 0 (3)v(2, x2) − p(2, x2) ≥ v(2, 0) = 0 (4)iii. Provide an intuition for why (3) and (2) bind (i.e. the inequality is in fact an equality)We have seen above that type 1 customers have no incentive to not report truthfullywhen the ice cream company extracts all the surplus. Therefore the firm is goingto be able to extract all surplus in this new mechanism as well, and (3) becomesv(1, x1) − p(1, x1) = 0.An intuitive explanation for why (2) binds is the following: because type 2 cus-tomers have an incentive to lie when the company extracts all the surplus, in thisnew mechanism the ice cream company is looking to make them indifferent betweenthe payoffs they get with their own optimal price and quantity and with type 1’s op-timal price and quantity, so as to remove their incentive to lie. Therefore we havev(2, x2) − p(2, x2) = v(2, x1) − p(1, x1).iv. We assume that in this mechanism the ice cream company will offer larger quantitiesto type 2 than to type 1 customers: x2≥ x1. Show that (4) and (1) are redundant.(4) can be deduced from (3) and (2):v(2, x2) − p(2, x2) − v(2, 0) ≥ v(2, x1) − p(1, x1) − v(2, 0)= 2√x1− p(1, x1) − 0≥√x1− p(1, x1) − 0≥ v(1, x1) − p(1, x1) − v(1, 0)≥ 02(1) can be deduced from (2):v(2, x2) − p(2, x2) = v(2, x1) − p(1, x1)⇔ p(2, x2) − p(1, x1) = v(2, x2) − v(2, x1)⇔ p(2, x2) − p(1, x1) = 2(√x2−√x1)≥√x2−√x1= v(1, x2) − v(1, x1)⇔ v(1, x1) − p(1, x1) ≥ p(2, x2) − v(1, x1)v. What is the ice cream company’s expected payoff it seeks to maximize?The ice cream company is looking to solve:maxx1,x212(p(1, x1) − c(x1)) +12(p(2, x2) − c(x2))vi. Deduce expressions for p(1, x1) and p(2, x2) from (3) and (2), replace in the company’sprogram and find optimal quantities and prices.(3) yields p(1, x1) = v(1, x1) and (2) yields:v(2, x2) − p(2, x2) = v(2, x1) − p(1, x1)⇔p(2, x2) = v(2, x2) − v(2, x1) + p(1, x1)⇔p(2, x2) = v(2, x2) − v(2, x1) + v(1, x1)Replacing into the company’s maximization program:maxx1,x234(p(1, x1) − c(x1)) +14(p(2, x2) − c(x2))= maxx1,x234(v(1, x1) − c(x1)) +14(v(2, x2) − v(2, x1) + v(1, x1) − c(x2))= maxx1,x234(√x1− x21) +14(2√x2− 2√x1+√x1− x22)= maxx1,x214(2√x1− 3x21+ (2√x2− x22)]The firm is maximizing on two variables x1and x2, but they appear separately in theexpression, so we can simply solve for one and then the other.FOC for x1yields:1422√x1− 6x1= 0⇔1√x1− 6x1= 0⇔1 − 6x1√x1= 0⇔36x21x1= 1⇔x1=13613=16233FOC for x2yields:1422√x2− 2x2= 0⇔1√x2− 2x2= 0⇔1 − 2x2√x2= 0⇔4x22x2= 1⇔x2=1413=1223Prices are equal to:p(1, x1) = v(1, x1)=1613p(2, x2) = v(2, x2) − v(2, x1) + v(1, x1)= 21213− 21613+1613= 21213−1613vii. How different are the results the full information benchmark’s from question 1.?Type 1 customers get a lower quantity of ice cream now (1623) than they did in withfull information (1423=11613) also for a lower price (1613against1413).Type 2 customers get the same quantity of ice cream as in the full information bench-mark (1223) at a higher price (21213−1613=1213+1213−1613≥1213).Problem 2: Adverse Selection on the Job MarketConsider a job market set up in which workers have private information about their skills ∈ {0, . . . , 3}. By default, the firm only knows that surplus is uniformely distributed in{0, . . . 3}.If a worker is hired by the firm, surplus produced is vhire(s) = s. If a worker is not hired, shehas outside option vout(s) = 0.7s.1. Assume the firm has no additional information about workers’ skills. We now detail howadverse selection plays out in this set up.- What is the maximum wage the