Exercise 3: Consider two households, A and B. They must decide whether to purchase a bench that they would jointly own. The price of the bench is 40. The value of the bench to each household is privately known. What is known to all is that They decide to meet and report what the bench is worth to them. They also agree to the following rule. Let a [0,1]. A reports $0 i. ii. Household A values the bench 80 with probability p E[0,1] and 0 with probability (1 - p). Household B values the bench 120 with probability q E[0,1] and 0 with probability (1-q). iii. A reports $80 B reports $0 Bench is not purchased Bench is purchased A pays $40 B reports $120 Bench is purchased B pays $40 Bench is purchased A pays $40a B pays $40(1-a) Explain, in a few lines and using words as opposed to a mathematical proof, whether the variables p or q should be high or low to facilitate honest reports of a high valuation. Assume that p = q = 0.8. Is the bench ever ordered? And, if so, how much does each household pay for it? (Prove all answers.) Let p = q = 0.8, and assume now that households A and B amend the rules in such a way that they now agree to the following: