Conference 9Intermediate Microeconomics - Fall 201813/11/2018Problem 1: Public Good ProvisionTwo flatmates are considering whether to buy a coffee machine that costs c = $50. Theyeach have private valuation vi, and report ˆvi, i = 1, 2. To decide whether they will buy thecoffee machine, flatmates use an incentive compatible mechanism M that outputs decisionxM(ˆv1, ˆv2) ∈ {0, 1} and implements some transfers tM(ˆv) =tM1(ˆv), tM2(ˆv), with no deficits.1. What would be the conditions for the mechanism to be:- Incentive compatible?Incentive compatibility means we want agents to reveal their true valuations. That isfor every i = 1, 2, and every report ˆv−i, it is better to report vithan any other ˆvi:vixMvi, ˆv−i− tMi(vi, ˆv−i) ≥ vixMˆvi, ˆv−i− tMi(ˆvi, ˆv−i)- Without deficit?The no deficit constraint is tM1(ˆv) + tM2(ˆv) ≥ cxM(ˆv)- Budget balanced?The no budget balanced constraint is tM1(ˆv) + tM2(ˆv) = cxM(ˆv)- Efficient?A mechanism is efficient if it maximizes total utility, that is if the coffee machine isbought when the flatmates’ sum of valuations is more than$50.. The condition forefficiency is therefore:v1+ v2≥ c ⇒ xM(ˆv1, ˆv2) = 12. Assume flatmate 1 has private valuation v1= 25 and flatmate 2 has v2= 30. If flatmateswere to use the VCG mechanism to decide whether to buy the coffee machine, what wouldbe the outcome? Would it be efficient? Would it run a deficit?It is a dominant strategy to report truthfully with the VCG mechanism, therefore flatmatesreport ˆv1= 25 and ˆv2= 30. Based on these truthful reports, the VCG mechanism imple-ments the efficient choice xMˆv1, ˆv2= 1, since total utility is 55 with the coffee machineand 0 without the coffee machine.Flatmates transfers are:- Flatmate 1: 30 − 30 = 0- Flatmate 2: 25 − 25 = 01Therefore the VCG mechanism runs a deficit because flatmates do not put money on thetable to buy their coffee machine.We are now looking for a mechanism that runs no deficit and incentivizes flatmates to revealtheir true valuations.3. Consider the following mechanism: the coffee machine is bought as soon as ˆv1+ ˆv2= c, andflatmates make a transfer t1= t2= $25 each if xMˆv1, ˆv2= 1. What does the decision rulelook like on a graph? Will flatmates report truthfully? Does this mechanism run a deficit?The provision region is a line:0 10 20 30 40 50 60 700204060Provision regionˆv2t1v1v2The mechanism is not incentive compatible: suppose flatmate 2 reports ˆv2at$15 as drawnon the graph.- If v1≥ 25 then flatmate 1 is better off reporting ˆv1= $35, paying$25 and gettingpositive payoff, whereas if he reports anything different from$35 the good will not beprovided- If v1< 25 then flatmate 1 truthfully reports and gets utility 0 (the good is notprovisonned), instead of negative utility if the good is provisonned.The mechanism is budget balanced as flatmates pay each$25 as soon as xMˆv1, ˆv2= 1.4. Consider another mechanism: the coffee machine is bought as soon as ˆv1+ ˆv2≥ c, andflatmates make a transfer t1= t2= $25 each if xMˆv1, ˆv2= 1. What does the decision rulelook like on a graph? Will flatmates report truthfully? Does this mechanism run a deficit?Is it efficient?The provision region is now a large area:20 10 20 30 40 50 60 700204060Provision regiont2t1v1v2As the previous decision rule, it is not incentive compatible but it is budget balanced. It isefficient in the sense that xM= 1 as soon as both flatmates’ valuations are above$50.5. Now consider the mechanism in which flatmate 1 makes payment t1= $35 and flatmate2 make payment t1= $15 if when a provision is made, and xMˆv1, ˆv2= 1 if ˆv1> t1andˆv2> t2. What does the decision rule look like on a graph? Will flatmates report truthfully?Does this mechanism run a deficit? Is it efficient?The decision region is now a rectangle:0 10 20 30 4050 60 700204060Provision regiont2t1v1v2This mechanism is incentive compatible: from flatmate 1’s point of view (same reasonningapplies for flatmate 2), either- If v1≥ 35, then flatmate 1 is better off reporting his true value than reporting ˆv1< 35:if flatmate 2 reports below 15, the good is not provisionned no matter what flatmate1 does, and if flatmate 2 reports above 15, flatmate 1 prefers reporting v1to get thegood provisionned.- If v1< 35, then flatmate 1 is better off reporting v1so that the good is not provisionned,in which case his payoff is 0, rather than v1− 35 < 0.3The mechanism does not run a deficit since flatmates pay 35 + 15 = 50 if xM= 1.It can only be second-best efficient since the good is not provisonned every time v1+v2≥ 50.Problem 2: Trade in large marketsAssume there are 4 buyers and 4 sellers for a single type of good. Each seller has one unit ofthis good to sell. Buyers’ and sellers’ good valuations are the following: v1B= 10, v2B= 10,v3B= 8, v4B= 5 and v1S= 12, v2S= 10, v3S= 9, v4S= 4.1. How many trades take place if the market designer arbitrarily sets the price to (a) 12, (b)10, (c) 8 or (d) 4?(a) No buyer is willing to pay 12, so no trade takes place(b) 2 buyers are ready to buy the good for 10, and 3 sellers are ready to sell it for 10, so2 exchanges take place.(c) 3 buyers are ready to buy the good for 8, and 2 sellers are ready to sell it for 8, so 2exchanges take place.(d) All buyers are ready to buy the good for 4, and 1 seller is ready to sell it for 4, so 1exchanges take place.2. Place sellers and buyers values on a graph. What prices result from the adpated secondprice auction described in the slides for large markets? How many trades are realized?0 2 4 6 8051015v1Bv2Bv3Bv4Bv1Sv2Sv3Sv4SBuyers’ and sellers’ valuationsOnly one trade is realized: by buyer 1 and seller 1. Buyer 1 pays 10 and seller 1 pays 9,such that buyer 1 makes a transfer of 1 to buying seller.3. Illustrate why this mechanism is incentive compatible by looking at buyers and sellers’possible deviations from revealing their true valuations.Let us take the buyers’ side (the reasonning is similar for sellers).Buyer 1: if buyer 1 reveals ˆv1> v1, he still gets to trade at the same price, hence he isindifferent. If he reveals ˆv1< v1, then either ˆv1> ˆv2and he still gets to trade at thesame price, or ˆv1< ˆv2and he misses the trade (buyer 2 gets to do the trade then)4Buyer 2: if buyer 2 reveals ˆv2> v2, then either ˆv1> ˆv2and he still does not get totrade, or ˆv1< ˆv2and he gets to trade at price ˆv1= v1= 10 but