Consider a two person two goods pure exchange economy. The goods are; xand y. The utility functions are u1(.) = x21.y1 and u2(.) = x2.y2, and theinitial endowments are e1(.) = (25, 75) and e2(.) = (25, 75). Assuming p1 = 1,compute competitive equilibria for this economy

Consider the following 3 × 3 pure exchange economy. The goods are; x, y andz. Individual utility functions are: u1 = 3x1 + 2y1 + z1; u2 = 2x2 + y2 + 3z2 andu3 = x1 + 3y1 + 2z1. The initial endowments are e1 = e2 = e3 = (1, 1, 1). Doinitial endowments(a) constitute an ‘equal division’ allocation ?(b) constitute a non-envious allocation?(c) constitute a Pareto Efficient allocation?

Consider consumers A and B with uilitiesuAxA1 ; xA2
= ln xA1 + (1 ) ln xA2uBxB1 ; xB2
= ln xB1 + (1 ) ln xB2 :Suppose the total endowment of good 1 in this economy is !1 and of good 2it is !2: Suppose !A1 ; !A2
is owned by consumer A and !B1 ; !B2
is owned byconsumer B: We have !A1 + !B1 = !1 and !A2 + !B2 = !2 by deÖnition.(a) What is the exchange economy equilibrium here.(b) What is a Pareto e¢ cient allocation? The social welfare is maximizedby allocating the goods to consumers A and B; and the allocation has torespect the resourse constraints. Formulate the search for such allocationsas an optimization program.(c) What can be said about the equilibrium prices at that allocation?

Two firms X and Y produce the samecommodity. Due to the productionconstraints, each firm is able to producepackages of 1, 3 and 5 units of the product.The cost of producing qx units for firm X is< [6 +2xq – 2qx + 5], and firm Y has theidentical cost function < [6 +2yq – 2qy + 5]for producing qy units. p is the price of oneunit for firm X. We assume that the marketis in equilibrium. The outcomes are theprofits of the firm shown in the form of amatrix A = [aij] (pay-off matrix). Write(i) a11, (ii) a22, (iii) a21, if the demandfunction D(p) is given by D(p) = 50 – p.

There are 10,000 people in the population, each of whom is willing to pay 10 for at most 1 unit of a given good (i.e. each individual is interested in buying just 1 unit of the good, not 2, and for that unit is willing to pay 10). There are 2 identical firms providing the good. Currently, both firms have a constant marginal cost of 5, and they do not bear any fixed costs.  1)     The two firms compete à la Bertrand. What is the equilibrium price and what are the firms’ profits? In case of repeated interactions (and “no game changer”), and both firms adopting a grim-trigger strategy (i.e. “I collude until you collude, if you cheat once, I will fix the minimum price possible forever”), determine the probability “p” that makes collusion sustainable.  Starting from the Bertrand equilibrium above identified, suppose now that one firm can adopt a new technology that lowers its marginal cost to 3.   2)     Is this a drastic innovation? (Just on the basis of economic reasoning, i.e. no calculus is needed), explain why yes or no. 3)     What is the equilibrium price in the market now and how much would this firm be willing to pay for this new technology? 4)     Should this innovation be prosecuted by Antitrust? Explain why yes or no.  Suppose that this new technology is available to both firms (which are always competing à la Bertrand). The cost to a firm of purchasing this technology is 10,000. Firms simultaneously decide whether to adopt the new technology or not.5)     Express the game in a normal form and determine what is (are) the Nash equilibrium (equilibria) of this game.

Suppose that in addition to the consumer described inExercise 1 we also have an another consumer, Consumer 2, whose pref-erences are given by the utility functionu2(x1, x2) = x1x2where x1 is the amount of good 1 the consumer consumes and x2 theamount of good 2. Let ωℓ2 be Consumer 2’s initial endowment of goodℓ. [This is the utility function you analysed in Homework 1.] Similarlylet u1(x1, x2) = min{x1, x2} be the utility function for Consumer 1 thatyou discussed in the previous exercise and ωℓ1 be Consumer 1’s initialendowment of good ℓ.(1) Repeat the analysis of the previous exercise for Consumer 2,finding the demand functions x12(p1, p2, ω12, ω22) andx22(p1, p2, ω12, ω22)