Throughout this problem set, we will look at exchange economies with two goodsand two agents. Let X = R², let u denote agent i's utility, and let wie X denoteagent i's endowment.1. Suppose u¹(x¹) = min{ri, 2} and wi = (4,8) for both agents i.(a) Argue that every Pareto optimal allocation has r≥r for both agentsi.(b) Argue that every allocation z with r≥r for both agents i is Paretooptimal.(c) Draw an Edgeworth box, with a picture depicting every Pareto-optimalallocation. In this picture, also draw the endowment allocation, and draweach agent's indifference curve through the endowment.(d) Argue that, in any competitive equilibrium, the price of good 2 must bezero.(e) Find all competitive equilibria.

Pareto Optimality and Pareto Efficiency are concepts used in welfare economics to evaluate the…

1. Suppose u²(x¹) = min{x1, x₂} and w² = (4,8) for both agents i. (a) Argue that every Pareto optimal allocation has r≥r for both agents i. (b) Argue that every allocation är with x ≥ xi for both agents i is Pareto optimal. (c) Draw an Edgeworth box, with a picture depicting every Pareto-optimal allocation. In this picture, also draw the endowment allocation, and draw each agent's indifference curve through the endowment. (d) Argue that, in any competitive equilibrium, the price of good 2 must be zero. (e) Find all competitive equilibria.

1. Suppose u²(x¹) = min{x1, x₂} and w² = (4,8) for both agents i. (a) Argue that every Pareto optimal allocation has r≥r for both agents i. (b) Argue that every allocation är with x ≥ xi for both agents i is Pareto optimal. (c) Draw an Edgeworth box, with a picture depicting every Pareto-optimal allocation. In this picture, also draw the endowment allocation, and draw each agent's indifference curve through the endowment. (d) Argue that, in any competitive equilibrium, the price of good 2 must be zero. (e) Find all competitive equilibria.

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Question

Throughout this problem set, we will look at exchange economies with two goods
and two agents. Let X = R², let u denote agent i's utility, and let wie X denote
agent i's endowment.
1. Suppose u¹(x¹) = min{ri, 2} and wi = (4,8) for both agents i.
(a) Argue that every Pareto optimal allocation has r≥r for both agents
i.
(b) Argue that every allocation z with r≥r for both agents i is Pareto
optimal.
(c) Draw an Edgeworth box, with a picture depicting every Pareto-optimal
allocation. In this picture, also draw the endowment allocation, and draw
each agent's indifference curve through the endowment.
(d) Argue that, in any competitive equilibrium, the price of good 2 must be
zero.
(e) Find all competitive equilibria.

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Step 3: Determine both agents to be Pareto Optimal

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2(b)There are three people and two goods in an economy. The utility functions and the initial bundles are given below: U = X11X 12 U2 = 2x 21+ X 22 U3 = X 31 X 3z+X 32 W; = (0,8) W;= (2,2) W3= (8,0) Suppose it is suggested that the three traders move to the allocation x given by x= (1,2), x2= (8, 4), X3= (1,4) C. Show that x is not in the core.
7. In an Edgeworth box for two consumers, A and B, with endowments of commodities Xa and Ya are A's endowments, Xb and Yb are B's, and Xo = Xa + Xb, Yo = Ya + Yb), the competitive equilibrium allocation of the two commodities represents a mutual tangency of both consumers' indifference curves with each other and with a common budget line. (a) The conditions require that A and B consume at a point on their budget lines where their indifference curves have the same marginal rate of substitution, and equilibrium for the market as a whole requires that the sum of the individuals' demands for each commodity must equal the totals (Xo and Yo) available. Mathematically express the point of equilibrium, that is, express the above statement with equations in relating each endowments, equilibrium points (Xa, Xb, Ya, Yb), and totals ((Xo, Yo). (b) Suppose MRSa = Ya/Xa, MRSB = Yb/Xb, Xa = 10, Ya, = 100, Xb = 50, Yb = 20, and let good Y be the numeraire (Py =1). Verify that the competitive…
6. In an Edgeworth box for two consumers, A and B, with endowments of commodities Xa and d Ya are A's endowments, Xb and Yb are B's, and Xo = Xa + Xb, Yo = Ya + Yb), the competitive equilibrium allocation of the two commodities represents a mutual tangency of both consumers' indifference curves with each other and with a common budget line. (b) Suppose MRSa = Ya/Xa, MRSb = Yb/Xb, Xa = 10, Ya, = 100, Xb= 50, Yb = 20, and let good Y be the numeraire (Py = 1). Verify that the competitive equilibrium price is Px = 2.
Find all Pareto optimal allocations, the egalitarian allocation and the competitive equilibrium with equal incomes for the following problems. 1 Ann and Bob need to divide 4 items. Their preferences are as follows: Y W Ann 25 25 25 25 Bob 10 20 30 40
case 1,2,3
9. Consider an Edgeworth box economy with two consumers, whose utility func- tions and endowments are u'(x},x}) = (x}Xx})} (5,5) In the following, use the normalization p2 = 1. (a) Find the competitive equilibrium price. (b) State the first fundamental theorem of welfare and verify that it holds in this economy. (e) Consider the allocation ĩ = (x',) = (2,3), (8, 7). Show whether this allo- cation can supported as an equilibrium with transfers. (d) State the second fundamental theorem of welfare, and briefly discuss whether the result in part (c) conform with or violate this theorem. 3.