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Problem 1 Consider an economy in which there are two goods, 1 and 2, whose prices are p, > 0 and p2 > 0, respectively. The two goods can only be consumed in non-negative amounts x, and X2, respectively. A consumer has preferences over R? which are represented by the utility function 1:R - R. (x1 12) " 1(x1, X2) := (x1 + 2)X2. The consumer's income is / > 0. (a) Formulate the consumer's utility maximization problem, find the first-order conditions for utility maximization, and find the Marshallian demand functions' x1(p1, p2, /) and x2(p1, p2. I) for goods 1 and 2, respectively. (Note: Use the Lagrangian method. Assume that the budget constraint holds with equality and that the solution is interior (i.e. x1 > 0 and x2 > 0), thus disregarding the non-negativity constraints on x, and x2.) Check that the second order conditions are satisfied.(f) Verify that the Slutsky equation for good I with respect to its own price holds in this case. To do this, first write the the Slutsky equation for a general utility function : R} - R, (x1, X2) # 1(x1, x2). (Note: Assume that the associated Marshallian demand function is differentiable with respect to prices and income and that the compensated demand function is differentiable with respect to prices.) Then use your derivations in (a), (b) and (c) to substitute for the various terms in this expression and show that this equality holds for the current utility function.(g) The motivation that is commonly given for studying income and substitution effects is that it helps understand the possibility of a Giffen good. Using the Slutsky equation, explain the connection between Giffen goods and inferior goods. In particular, which of the following two statements is necessarily true and which one is not always so? Why? (i) A Giffen good is an inferior good; (ii) An inferior good is a Giffen good.Problem 2 In most of the utility maximization problems we encounter in this course, the solution is interior. This means that a strictly positive quantity of each good is demanded at the utility maximizing bundle (x),....x*) (ie. x/ > 0 for i = 1,....n.) However, this is not always the case, as the following problem shows. 13 Consider again an economy in which there are two goods, 1 and 2, whose prices are p1 > 0 and p2 > 0, respectively. The two goods can only be consumed in non-negative amounts x, and x2, respectively. A consumer has preferences over R? which are represented by the utility function 1: R - R. (x1, *2) 2 4(x1, 12) :3 x1+ X2. The consumer's income is / > 0. (a) Formulate the consumer's utility maximization problem, writing down explicitly all the constraints (i.e., the budget constraint and the non-negativity constraints on x, and x2).(c) Now that you have shown that the utility function is strictly increasing, you know that the budget constraint holds with equality at the utility maximizing bundle. Ignore for a moment the the non-negativity constraints on x1 and x2 and solve the utility maximization problem using the substitution method. That is: (a) using the budget constraint, express one choice variable as a function of the other choice variables and the parameters (exogenous variables) p1, p2 and I of the model; (b) plug your expression into the objective function; (c) take the first order condition and solve for X1(p1, p2, I) and x2(p1, p2. /). Skip the check of second order conditions.Problem 3 Consider once again an economy in which there are two goods, 1 and 2, whose prices are p, > 0 and p2 > 0, respectively. The two goods can only be consumed in non-negative amounts * and X2, respectively. A consumer has preferences over R? which are represented by the utility function M: R? - R. (X1, X2) - 1(x1, X2) := (ax,2+ Bx2-2)-1/2 where a and B are strictly positive real numbers. The consumer's income is / > 0. (Note: This is an example of CES utility function) (a) Formulate the consumer's utility maximization problem, find the first-order conditions for utility maximization, and find the Marshallian demand functions x1(p1, p2. /) and x2(p1. p2. I) for goods 1 and 2, respectively. (Note: Use the Lagrangian method. Assume that the budget constraint holds with equality and that the solution is interior (i.e. x, > 0 and x2 > 0), thus disregarding the non-negativity constraints on x, and x2. Do not check second order conditions.)Problem 4 As you started to see in this course, concave functions and convex sets play an important role in economic analysis. This problem will help you to gain a better grasp of these concepts. In what follows, whenever convenient for notation, we denote a generic element (x1, ....X,) of R" simply as x. For any 1 6 R and any x e R", the scalar multiplication between 1 and x, denoted by Ax, is the element of R" defined as Ax (2x1, ..., Ax,). For example, if A = 2 and x = (3.5) e R, then Ax = (6, 10). A subset S of R" is said to be convex if for all x, y e S and all 1 e [0, 1], we have Ax + (1-A)y e S. In words, the definition says that a subset S of R" is convex if for any two elements x and y belonging to S there are no elements of the straight line segment between x and y that are not elements of S. An expression of the form Ax + (1-A)y, where x, y e R" and A e [0, 1], is called a convex combination of x and y. (a) Suppose that A and B are arbitrary convex subsets of R". Show that An B is convex, while AU B need not be. (Hint: You may want to use a graphical representation of the two set operations to guide your reasoning.)(b) Consider an economy in which there are two goods, 1 and 2, whose prices are p, > 0 and p2 > 0, respectively. The two goods can only be consumed in non-negative amounts x, and x2. The budget set of a consumer with income / > 0 is defined as B(P1, P2, 1) := ((x1, x2) ERA : PIXI + P2.*2 = 1). Show that B(p1, p2, /) is a convex subset of RY
