1. Suppose there are two consumers, A and B. The utility functions of each consumer are given by: UA(X,Y) = X*Y UB(X,Y) = X*Y3 Therefore: • For consumer A: MUX = Y; MUY = X • For consumer B: MUX = Y3; MUY = 3XY2 The initial endowments are: A: X = 10; Y = 6 B: X = 14; Y = 19 show all work a) Suppose the price PY = 1. Calculate the price of X, PX that will lead to a competitive equilibrium. b) How much of each good does each consumer demand in equilibrium? Consumer A’s Demand for X: Consumer A’s Demand for Y: Consumer B’s demand for X: Consumer B’s demand for Y: c)What is the marginal rate of substitution for consumer A at the competitive equilibrium?

Consider an economy with 2 goods and 2 agents. The Örst agent has the utilityfunction, u (x1; x2) = ln x1 + 2 ln x2, and the other one has u (y1; y2) = 2 ln y1 + ln y2.The aggregate endowments of the 2 goods are given by (50; 100). Suppose there is asocial planner who cares about agents equally.(a) Set up the plannerís problem b) Calculate the first-best outcome (i.e., the social plannerís solution).

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. Answer all parts (a) (c) of this question. (a) Consider an agent whose preferences over any couple (x1, x2), where 2₁ ER+ and x2 € R+, e.g., apples and oranges, is such that she prefers the bundle that is closest to having the same number of apples and oranges. Write a utility function u: R² → R+ which represent these preferences. A politician remarks "Our recent increases in the wage rates of teachers has been a total suc- cess! The shortage of teachers has been reduced drastically. Another, similar wage increase should eliminate this shortage entirely" (b) Explain and illustrate in a diagram what is meant by "income effects" and "substitution effects" of a wage rate change. (c) Explain and illustrate how you would model the labour supply decision of a potential teacher. Do you agree that the wage increase will increase the labour supply in this case? Carefully outline the assumptions underlying your argument.

2. General Equilibrium Consider a market with two goods, x and z, and two consumers, A and B. The utility functions for consumers A and B are as follows UA ÜB 2 1 == XBZB and the initial endowments for each consumer are @ eA = (4,2) еB = (2,6) where consumer B is endowed with 2 unit of good x and 6 units of good z, respectively. a) Draw the Edgeworth Box (Don't worry about the shape of the utility curves. Just pick a general shape that we have used before). b) Derive the contract curve.

Exercise 4. In the lectures we have been working the the CRRA utility function, u : R+ which we defined, for any n≥0, as → R, Prove that u(x): -{ 21--1 1-n if ŋE [0,1) U (1,∞) In a if n = 1. 2--1 lim == Inz. →1 1-7 (7) Hint: Use L'Hôpital's Rule. Exercise 5. Consider the following two-period optimization problem we studied in class: max u (x1)+ẞu (x2), s.t. x1+px2=y, (1.22)ER² (8) where u is given by (7), x; represents consumption of good i = {1,2}, ẞ € [0, 1] is the discount factor, p = R++ is the relative price of good 2 in terms of good 1, and y = R++ is income expressed in terms of good 1. For the following two questions, consider an arbitrary n > 0. 1. Find the bundle (1, 2) that solves (8). 2. Notice that the solution (1, 2) depends on the parameters (n,3,p,y), which the con- sumer takes as given. Derive the expressions for ǝr/Op and ǝx/Əp.

***Please provide an answer and explanation for BOTH parts (A & B).  B builds on and cannot be completed without A.

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3.