Econ 0A Midterm Th 29 September 2004. You have approximately hour 20 minutes to answer the questions in the midterm. I will collect the exams at 2.30 sharp. Show your work, good luck! Problem. Utility maximization. (57 points) In class, we have considered the case of maximization of utility with Cobb-Douglas utility function with goods x (baseball) y (cupcakes) as arguments. We have also considered the case of intertemporal maximization of utility with consumption of goods today, c 0, inthefuture,c. This problem asks you to combine the two elements reconsider some of the properties we had found in the two separate cases. Consider a consumer with utility function u(x 0,,x,y )(x 0 ) α ( ) β + +δ (x ) α (y ) β () where x t is the quantity of baseball x consumed at time t 0,. Similarly, y t is the quantity of cupcakes y consumed at time t 0,. For reasons that we will see below, we will assume α + β<, so do not substitute β α. As usual, α>0 β>0 also hold.. What is the interpretation of δ in the utility function? Interpret in particular the special cases δ 0 δ. (6 points) 2. Explain the procedure to determine if u(x 0,,x,y ) is concave in (x 0,,x,y ). Youshouldnotdo the computations. (4 points) 3. We now consider the budget constraint. Both in period 0 in period the price of music x is p x the price of cupcakes y is p y. We denote income at time 0 by M 0 income at time as M. Consumers can borrow across periods 0 at interest rate r. Show that this implies that the (intertemporal) budget constraint is: (6 points) p x x 0 + p y + +r (p xx + p y y ) M 0 + +r M. (2) 4. The consumer maximizes utility () with respect to the four consumption variables (x 0,,x,y ), subject to budget constraint (2). Assume that the budget constraint is satisfied with equality (it is) write the Lagrangean function. Derive the set of first order conditions. (5 points). 5. Show that the first order conditions with respect to (x 0,,x,y ) lead to the familiar conditions α (3) β x 0 p y α y. (4) β x p y Give an economic interpretation of these conditions. Are you surprised that the same conditions that hold in the one-period Cobb-Douglas utility maximization also hold in the 2-period one? (5 points) 6. [Harder] If you can, show that the first order conditions imply y (α+β) x x 0 (α+β) +r +δ (5) (You may want to leave this for last you can continue answering the rest of Problem without solving this point) (0 points)
7. Keep assuming α + β < consider equation (5). How does relative consumption in period compared to period 0 (y / x /x 0 )varyasr increases? Explain intuitively why an increase in the interest rate increase consumption in the future. Similarly, how does relative consumption in period compared to period 0 (y / x /x 0 )varyasδ increases? Provide intuition. (6 points) 8. What does equation (5) imply about y / for r δ? Provide intuition on why at the optimum the agent smooths consumption of goods x y across periods. (6 points) Extra credit: Why is it crucial for this answer that α + β<? 9. For r δ, solve for the optimal value of x 0. Hint: use the budget constraint equation (3). (5 points) 0. Use the expression for x 0 that you obtained in point 9. Differentiate x 0 with respect to M 0, that is, compute x 0/ M 0. Is music x a normal good (for all levels of price income)? (4 points) Solution to Problem.. Parameter δ has the interpretation of psychological discount rate. It indicates how much the individual discounts future utility as of the present. The higher δ, the more impatient the individual is, the less she value the future. Special case δ 0: in this case the individual gives equal weight to consumption of baseball (or cupcakes) in the present or in the future. Special case δ : the individual is myopic, does not give any weight to future utility. 2. To determine whether the utility function u is concave in (x 0,,x,y ) we should set up the matrix: x 0, x 0 x 0, x x 0, x 0, y H x, x 0 x, x x, x, y, x 0, x,, y y, x 0 y, x y, y, y The condition for concavity is that the first subdeterminant be negative or equal to zero, the second be positive (or equal to zero), the third be negative (or equal to zero), the fourth (the whole matrix) be positive (or equal to zero). Importantly, these conditions have to hold for all (x 0,,x,y ) in R + 4. 3. In period 0 the consumer can spend income M 0 on good x 0 or on good. The savings in period 0 s 0 M 0 (p x x 0 + p y ) canbesavedforperiodearninterestrater. Notice that s 0 can be negative, that is, the individual may wish to borrow from period. In period 2 the individual can spend M +( + r) s 0 M +( + r)[m 0 (p x x 0 + p y )]. The budget constraint in period therefore can be written as p x x + p y y M +(+r)[m 0 (p x x 0 + p y )]. By rearranging terms, we obtain ( + r)(p x x 0 + p y )+(p x x + p y y ) M +(+r) M 0, which is equivalent to (2) (just divide by ( + r)). 4. The consumer maximizes max (x 0) α ( ) β + (x 0,,x,y ) +δ (x ) α (y ) β s.t. p x x 0 + p y + +r (p xx + p y y ) M 0 + +r M This is equivalent to maximizing the Lagrangean L(x 0,,x,y,λ) (x 0 ) α ( ) β + +δ (x ) α (y ) β λ p x x 0 + p y + +r (p xx + p y y ) M 0 + +r M. 2
This yields the following five first order conditions: f.o.c. x 0 : α (x 0 ) α ( ) β λp x 0 f.o.c. : β (x 0 ) α ( ) β λp y 0 f.o.c. x : ³α β (x ) α (y ) +δ f.o.c. y : +δ f.o.c. λ : 5. First order conditions 2 lead to ³β (x ) α (y ) β λ λ +r p x 0 +r p y 0 p x x 0 + p y + +r (p xx + p y y ) α (x 0 ) α ( ) β λp x β (x 0 ) α ( ) β λp y M 0 + +r M 0 Taking the ratio of these two equalities, we obtain α. (6) β x 0 p y Similarly, the ratio of first order conditions 3 4 lead to α y. (7) β x p y These conditions indicates the Marginal Rate of Substitution between goods x y in period 0 equate the ratio of the prices, similarly so for goods x y in period. The condition MSRratio of prices that denotes the tangency of indifference curve budget set, therefore, extends to the 2-period case. 6. First order conditions 3 lead to α (x 0 ) α ( ) β λp x ³α β (x ) α (y ) +δ λ +r p x. Taking the ratio of these two equalities, we obtain α β x0 y0 +r x y +δ. (8) Note now that (6) (7) can be combined to yield x 0. y x We label R y0 y x0 x the ratio of consumption in period 0. Then (8) implies (α+β) (α+β) R α+β y x +r +δ 7. As r increases, the right-h side of the equality increases. This implies that the left-h side must increase as well. Given α + β<, this implies that y / x /x 0 must increase. Intuitively, as the interest rate increases, it becomes more expensive to consume in period 0 more advantageous to consume in the later period, period. The opposite effect applies when impatience increases. The right-h side decreases, y / x /x 0 must decrease as well. Intuitively, if individuals become more impatient they consume more in the present. x 0 3
8. For r δ, we obtain y y 0 y x x 0 x, that is, the consumer consumes the same quantity of baseball (cupcakes) in each of the periods. The reason for this is that the utility function is concave in the amount of consumption in each period. This implies that marginal utility is decreasing in the amount of consumption in each period. The consumer prefers to consume equal amounts rather than consuming a lot one period very little the next. 9. Under the assumption r δ, we know that y y 0 y x x 0 x. Further, equation (3) implies p y y (β/α) p x x. We substitute into the budget constraint to get or or p x x +(β/α) p x x + +r (p xx +(β/α) p x x )M 0 + +r M p x x α + β α + M 0 + +r +r M x +r α M 0 + +r M. (9) 2+r α + β p x 0. The derivative of (9) with respect to M 0 is x 0 +r M 0 2+r α α + β The good is a normal good since increases in income M 0 induce an increase in the quantity consumed x 0 for all levels of p x r. p x. 4
Problem 2. (8 points). Consider the maximization problem max f (x, p) x where both x p are real numbers (that is, not vectors). This leads to the first order condition f 0 x (x,p)0. Use the implicit function theorem to give an expression for dx /dp. Argue that, as long as x is a maximum, the sign of dx /dp is given by fx,p 00 (x,p). (8 points) 2. Consider a consumer with preferences over two goods, music x cupcakes y, with x 0 y 0. The consumer is addicted to cupcakes y is loss averse to consuming less than Y > 0 units of cupcakes. Here are the preferences of the consumer. A bundle (x, y) is weakly preferred (%) toa bundle (x 0, ) if only if the following is true: (y Y )+x ( Y )+x 0 for y Y Y 2(y Y )+x ( Y )+x 0 for y<y Y (y Y )+x 2( Y )+x 0 for y Y <Y 2(y Y )+x 2( Y )+x 0 for y<y <Y Can you provide a utility function that represents these preferences? Plot some of the indifference curves. Why could these indifference curves capture an aspect of addiction? (0 points) Solution to Problem 2.. Using the implicit function theorem we have dx dp (f 0 x (x,p)) p (f 0 x (x,p)) x f x,p 00 (x,p) fx,x 00 (x,p). Since in a point of maximum the second order condition guarantees fx,p 00 (x,p) < 0, we can conclude that the sign of dx /dp is equal to the sign of the numerator, that is, to the sign of fx,p 00 (x,p). 2. A utility function that represents the preferences in the problem is u (x, y) x +(y Y ) if y Y x +2(y Y ) if y<y See the Figure for indifference curves. The consumer has the same preferences for x y as long as y is at least as large as Y. That is, she is willing to trade one unit of music for one unit of cupcake. If y falls below the minimum level Y, though, the preferences of the consumer change. For y below Y, the consumer wants two units of cupcake in exchange for one unit of music. Intuitively, this captures the existence of a reference point with respect to which the consumer is loss averse: the relative preference for good y increases when y falls below the reference point. This may capture a feature of addiction insofar as falling below the addictive level Y makes the consumer desire y more. 5