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Question 3 Consider an economy with two goods, a private good and a public good subject to congestion. One unit of private good can be transformed into a unit of public good : if z units of private good are used then y = 2, where y is the quantity of public good produced. There are / agents with initial resources (w);2, in private good who can use the public good with varying intensity. Let q' denote the intensity of use of agent i. Think of y as a number which summarizes the characteristics of a freeway (width, length, road quality) and of q' as the miles that agent i drives on the freeway. When agent i uses the public good with intensity q', it costs the agent c(q' ) of private good (think of c(q') as the cost of gas). The utility of an agent for the public good depends not only on y but also on the congestion Q = _i, q' of the public good. That is, the agents' preferences depend on Y = f(y, (), which we can call the quality of public good, where f is increasing in y and decreasing in Q. The preferences of agent i are thus represented by a function u'(a', q', Y) where r' is the consumption of private good, q' the intensity of use of the public good, Y is the quality of the public good and u' is an increasing function. All functions considered in this problem are continuously differentiable. (a) Write a maximum problem whose solutions are the Pareto optimal allocations of this economy. Be careful to write correctly the feasibility constraints. (b) Write the first-order conditions for an interior solution (r' > 0,y > 0, q' > 0 for all i ) of this maximum problem. (c) By eliminating the multipliers in (b), show that (i) the optimality of the provision y* of the public good requires that a condition, akin to the Samuelson condition, must be satisfied, involving all the agents' valuations Sy / 2: of an additional unit of quality of the public good and 9%, all evaluated at the Pareto optimum; (ii) for each i = 1. ..., I, the optimality of the intensity of use q"* of the public good implies a condition involving the agent's valuation as / gp of intensity of use, the marginal cost c'(q'), all agents' valuations of / 297 . j = 1,.... I, of the quality of the public good and go, all evaluated at the Pareto optimum. (d) To check that your FOCs are right, derive the formulae in (c) by marginal reasoning, evalu- ating the marginal cost of, and the agents' propensity to pay for, a marginal increase in the Page 3 of 6 amount y produced, and the agents' marginal willingness to pay or need to be compensated for a marginal change in the intensity q' of agent i's use of the public good. (e) Consider an equilibrium ((F', q')_1, 5. Q, Y, 7) such that: (i) the government taxes the agents' endowments at the rate 7 to finance the provision g = 7 21, w' of the public good; (ii) agents are small and take the level of congestion @ as independent of their own intensity of use. Show that, no matter how F is chosen, the equilibrium is not Pareto optimal. Explain why, and suggest a way of improving on the equilibrium.Question 4 We were all pretty relieved that Ali from our first prelim got such a simple consumption problem. Yet, we also realized that putting him into the Cobb-Douglas straightjacket missed some features of the story. Since the overall theme of the second prelim should be "We can do better!", let's also improve Ali's consumption model. As before, write Ta, It, Is 2 0 for the amounts of his consumption of almonds, toothpicks, and gifts, respectively. We assume for simplicity that these goods are infinitesimally divisible. Let I = (Ta, It, I,). Instead of assuming a utility function of the Cobb-Douglas form, we assume now that his utility function is given by u(Ia, It, Ig) = aln(Ta - ba) + 0In(rt - b;) + (1 -a -0) In(1, - by) with a, 0, b; > 0, a + 0 0 for i E {a, t, g}. As before, his income or wealth is denoted by w > 0. Finally, we denote by Pa; Pt, P, > 0 the prices of almonds, toothpicks, and gifts, respectively, and let p = (Pa, Pt, Pg). a. Use the Kuhn-Tucker approach to derive step-by-step the Walrasian demand func- tion r(p, w). Verify also second-order conditions. b. Verify that the demand function is homogenous of degree zero and satisfies Walras' Law. c. Provide a verbal interpretation of this demand system. (It will be helpful to consider the demand system in its "expenditure form" by multiplying both sides of each demand equation by its respective price.) d. You would expect that the more almonds Ali eats, the more they get stuck in his teeth and the more toothpicks he purchases. In light of such considerations, does it make sense to assume Ali has the utility function above? (Consider changes in the demand for almonds and toothpicks caused by changes in the price of almonds and changes in the parameters b, and o, respectively.) e. Consider now a utility function given by i(In, It, Ty) = (1. - ba)" (It - b;)(1, - by)]-a-e with a, 0, b; > 0, a + 0 0 for all i E {a, t, g}. How is this utility function related to the one given at the beginning of Question 4? f. Remember that when Professor Schipper interviewed Ali about how exactly he arrives at his optimal consumption bundle, Ali expressed ignorance about maxi- mizing utility subject to his budget constraint. Instead, he seemed to minimize his expenditure on consumption such that he reaches a certain level of utility. A smart undergraduate student walked by and claimed that this is clear evidence against the assumption of utility maximization in economics. Since Professor Schipper Page 5 of 6 likes Linear Expenditure Systems as much as Cobb-Douglas utility functions, he conveniently sent the student to you so that you can show him how expenditure minimization works. Again, use the Kuhn-Tucker approach to derive the Hicksian demand function but use the utility function in part e instead. Simplification: Let's not write our fingers to the bone. Assume from now on (for all parts g to () that Ali got rid of his girlfriend. Sure you must be sad about it but there is clearly a tradeoff between having a girlfriend and completing successfully and on time a prelim exam. Most important to Ali: No more gifts! Thus, we can consider now the case of two goods, almonds and toothpicks, only. Set 0 = 1 - a to economize on parameters. g. Derive the expenditure function. h. Show that the expenditure function is homogeneous of degree 1 in prices, strictly increasing in a as well as nondecreasing and concave in the price of each good taken separately. i. Derive Ali's indirect utility function using the expenditure function just derived. j. Verify that Ali satisfies Roy's identity with respect to almonds. k. Verify the (own price) Slutsky equation for the example of almonds