Consider an exchange economy with two types of agents, A and B, and two goods, a and y. Preferences are given by uA(TA, YA) = XA +6 ln(yA) and ug(xB, YB) = In(rBYB). Suppose the government wants to reach the goal of rB = 5 with the competitive equilibrium by setting a redistribution r such that wa = (10, r) and wy = (0, 20 – r). Let Py = 1. Findr reaches the government's goal. Round your answer to 2 decimal points. Answer:

Consider an exchange economy with two types of agents, A and B, and two goods, x and y. Preferences are given by UA (XA, YA) = = A + 6 ln(y₁) and UB(TB, YB) = ln(TBYB). Suppose the government wants to reach the goal of XB = 5 with the competitive equilibrium by setting a redistribution r such that w₁ = (10,r) and wB = (0,20 - r). 1. Find r reaches the government's goal. Round your answer to 2 decimal points. Let Py = Answer:

a) Consider an economy with 3 agents, Mohammed (M), David (D) and Susan (S). There are two goods available, good x, and good y. The marginal rates of substitution (where good x is on the horizontal axis and good y is on the vertical axis) are given by for Mohammed,  for David and  for  Mohammed and David are both consuming twice as much of the good x than good y, while Susan is consuming equal amounts of x and y. What are the conditions for Pareto efficiency in an exchange economy? Are these consumption levels economically efficient?  Can these consumption allocations be observed in a perfectly competitive equilibrium in an exchange economy without production? Explain.   b) There are two firms in the economy. Each firm employs positive amounts of capital and labour. The technology satisfies diminishing marginal rate of technical substitution of labour for capital. Currently, A’s marginal rate of technical substitution of labour for capital is 4 while B’s marginal rate of technical…

Output √w 2. Consider a one-consumer one-firm economy in which u(x) = x₂ + √√q and f(2)=√z. (a) Derive the Pareto set of this economy. Solution. Pareto-efficient allocations are solutions to the problem ₂+√√q subject to √w- x₂ = xq, which is illustrated in Figure 3. Any solution to this problem in which , x > 0 must solve the first-order condition = f'(wx), or 2₁/x = max 0≤x≤w,xq 20 au(x*) Əxz _u(x*) Əxq By substituting = √w - x, we have that x = (¹/2)¹/³ and x = w − (1/2)³/³. Any solution to the Pareto problem with x = 0 would have to be such that ə Figure 5: No corner solution with x = 0 Əxz — *²)=0² u(x*)x=0 which, using that = √w in this case, reduces to w≤ (1/2) 8/³. In words, if the consumer's endowment of input is small, then the Pareto-efficient allocations have the firm use up all of the input to produce output, leaving the consumer with no consumption of input. This is illustrated in Figure 4. Any solution to the Pareto problem with x = 0 would have to be such that ≤ f(w-x)…

Anthony and Brandon have a utility possibility frontier that is given by the following equation: UA + 2U}? = 50. Anthony and Brandon believe that the ideal allocation is given by maximizing an appropriate social welfare function. Anthony thinks that UA = 30 and Ug = 100 is the best distribution of welfare, and presents the maximization solution to a weighted-sum-of-the-utilities social welfare function that confirms this observation. Anthony's social welfare function can be written as a) W = UA + UB b) W = UA + 10UB c) W = 10UA + UB d) W = 2UA + 10UB e) None of the above.

2(a) Suppose x is a Pareto efficient allocation in which each agent holds a positive amount of each good. Also suppose that preferences are convex, continuous and monotonic. Then show that x' is a Walrasian equilibrium for the initial endowments that are redistributed such that w, = x' for i =1,.n.

There are two goods, Computers and Wheat, and two factors of production, capital and labor. Production in both sectors satisfies constant returns to scale. Production of computers is relatively more capital-intensive. Both input and final goods markets are competitive. Goods are freely traded internationally, but factors are immobile across countries. We will focus our analysis on the Home country. In the initial equilibrium, the Home country produces both computers and wheat, but only exports wheat. In our analysis, we will consider the following sequence of time-frames: In the short run, both factors are completely specific to their initial sectors of employ- ment. (i.c. both factors are used to produce both goods, but in the short run it is impossible to relocate the factors that are currently in use for production) . In the medium run, capital is specific but labor can relocate. . In the long run, both factors are mobile across sectors. • In each run, the prices of factors adjust…

There are 40 people living in a village each having preference over apples andshirts represented by U(a,b) = a1/3s2/3, where a and s are amount of applesin pounds and shirts consumed, respectively. The price of a shirt is $10. Each villager has $300 income. Apples are supplied to the village by a farmer, whose supply function is S(pa) = 1000pa.Part a1What is the equilibrium price of apples? How many pounds of apples does eachvillager consume?Part a2What is the aggregate net consumer surplus at the equilibrium?Part a3What is the price elasticity of demand at the equilibrium?Part b1Suppose 50 more people move in to the village. What is the equilibrium pricenow? Part b2Are consumers better or worse off now? Why? If a consumer is worse off, byhow much additional income he/she needs to be compensated in order to be aswell off as in Part a.

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Consider a two-good exchange economy with two types of consumers. Type A have the utility function And an endowment of 3 units of good 1 and k units of good 2. Type B has the utility function And an endowment of 6 units of good 1 and 21 - k units of good 2. a. Find the competitive equilibrium outcome and show that the equilibrium price p* = p1/p2 of good 1 in terms of good 2 is p* = 21+k/15. b. Find the income levels (MA; MB) of both types in equilibrium as a function of k. c. Suppose that the government can make a lump-sum transfer of good 2, but it is impossible to transfer good 1. Use your answer to part b to describe the set of income distributions attainable through such transfers. Draw this in a diagram. d. Suppose that the government can affect the initial distribution of resources by varying k. Find the optimal distribution of income if (i) the SWF is W = log(MA) + log(MB) and (ii) W = MA + MB.