Concordia University - Econ 302 Winter 2023 Assignment 4 . The due date is Thursday, April 6th. . The total is 58 points. 1. (10 points) Bargaining for a house. Skyler wants to buy a house which she values at V. If she finds a price p, which is satisfactory for her and the seller, the seller receives p and she gets V - p. If they cannot agree, Skyler receives zero the seller receives a value Pout. You can interpret pout as the price another buyer is ready to pay for the house. . Assume that when one is indifferent between accepting and rejecting, one accepts an offer. . Assume both Skyler and the seller know Skyler's valuation of the house and the value Pout . . We are looking for a subgame perfect Nash equilibrium. . There is no discounting. a) (4 points) If Skyler makes an offer and the seller cannot make a counter offer: i) What is the offer? ii) What is the minimum value of V (as function of Pout) such that Skyler wants to make the offer (instead of walking away)? b) (4 points) What if the seller is the one making the offer to Skyler (which Skyler can then accept or reject, without counter offer). i) What will the seller offer Skyler? ii) What is the minimum value of V (as function of Pout ) such that the seller wants to make the offer (and not take the outside option Pout)? c) (2 points) In this game, which side has the advantage, the one who makes or receives the offer? 2. (48 points) Two consumers, two goods Consider an exchange economy with two consumers, A and B, and two goods, 1 and 2. . Consumer A's initial endowment is WA = (wh.wh) = (6,9) and consumer B's endowment is ws = (WB, wg) = (6,3). . Consumer A's consumption is FA = (7),$2) and consumer B's consumption is TB = (TB. FB). . Prices are p1, P2. They will be determined endogenously. Both consumers take prices as given. . Consumer A's utility is uA (r, x2) = (xA) ra. Consumer B's utility is us (13, 2) = a) (4 points) Write conditions on feasable allocations of r'4, r; and ra, r? given the initial endowments. (Assume consumers consume all available resources.consumers and the initial endowment (note it w). Make it sufficiently large as many other things will be drawn. And of course Always label your axes B 3 of 3 c) (6 points) Find the contract curve: all the Pareto-optimal allocations. Use the condi- tions on feasable allocations computed in a) to substitute .3, T; and express the curve as a% as function of a'4. Draw it on your graph. To make sure it is drawn accurately, compute zz on the curve at z) = 0,71 = 4,x1 = r, x) = 8,1) = 12. d) (6 points) Find consumer A's demands as function of his initial endowment and the prices: IA (P1, P2, WA, WA), D'A (pi, pz, WA,WA). Substitute the numerical values of the initial endowment WA, WA. e) (6 points) What are consumer A's demands (r), 27) if the relative prices are B = 1, 21 = 2, 21 = 3? Use them to represent on your graph consumer A's price-offer curve. f) (4 points) Find the formula for consumer A's price-offer curve. Remember, it will be an equation of a', as function of ', (and the parameters). (Hint: use P to substitute r. in the demand for 3.) g) (4 points) The market equilibrium can be found at the intersection of A's price offer curve and the contract curve. Use your answers in c) and f) to find the market equilibrium (74, 17). Note it X on your graph. h) (4 points) What is the slope of A's budget constraint passing through the market equilibrium? Draw the budget constraint on your graph. i) (6 points) Find consumer B's demands T'; (p1, pz, w;. w;), 2; (PI. Pz.wh,wh) and draw his price offer curve as well. j) (4 points) Draw (approximate) indifference curves of A and B passing through the market equilibrium.the offer (instead of walking away)? b) (4 points) What if the seller is the one making the offer to Skyler (which Skyler can then accept or reject, without counter offer). i) What will the seller offer Skyler? ii) What is the minimum value of V (as function of Pout) such that the seller wants to make the offer (and not take the outside option Pout)? c) (2 points) In this game, which side has the advantage, the one who makes or receives the offer? 2. (48 points) Two consumers, two goods Consider an exchange economy with two consumers, A and B, and two goods, 1 and 2. . Consumer A's initial endowment is WA = (wh.wh) = (6,9) and consumer B's endowment is ws = (wg, wh) = (6,3). . Consumer A's consumption is FA = (TA, ') and consumer B's consumption is TB = (TB. 13). . Prices are p1, P2. They will be determined endogenously. Both consumers take prices as given. . Consumer A's utility is uA (rA, rA) = (xA) ra. Consumer B's utility is up (13, 13) = a) (4 points) Write conditions on feasable allocations of TA, I; and ra, r given the initial endowments. (Assume consumers consume all available resources. b) (4 points) Draw an Edgeworth box representing the total resources available to both consumers and the initial endowment (note it w). Make it sufficiently large as many other things will be drawn. And of course Always label your axes c) (6 points) Find the contract curve: all the Pareto-optimal allocations. Use the condi- tions on feasable allocations computed in a) to substitute , T; and express the curve as a, as function of 4. Draw it on your graph. To make sure it is drawn accurately, compute r2 on the curve at rA = 0,24 = 4,x4 = r, r) = 8,21 = 12. d) (6 points) Find consumer A's demands as function of his initial endowment and the prices: T'A (P1, P2, WA,WA), T'A (PI, pz, WA, WA). Substitute the numerical values of the initial endowment WA, WA. e) (6 points) What are consumer A's demands (r), r2) if the relative prices are B = 1, 21 = 2, 21 = 3? Use them to represent on your graph consumer A's price-offer curve. f) (4 points) Find the formula for consumer A's price-offer curve. Remember, it will be an equation of a'7 as function of r, (and the parameters). (Hint: use # to substitute rA in the demand for 13-) g) (4 points) The market equilibrium can be found at the intersection of A's price offer curve and the contract curve. Use your answers in c) and f) to find the market equilibrium (r4, xx). Note it X on your graph. h) (4 points) What is the slope of A's budget constraint passing through the market equilibrium? Draw the budget constraint on your graph. i) (6 points) Find consumer B's demands .'; (p1, p2, ".W3), 13 (P1, Pz, Wh, wg) and draw his price offer curve as well