hh tackle the following

Suppose an individual has utility function u : R2+ R where

u(x1, x2) = x1 + 4x2

Assume p1, p2 > 0 and that the consumer's income is y.

(a) Find the individual's Marshallian demand functions for goods 1 and 2.

(b) Find the individual's Hicksian demand functions for goods 1 and 2.

1(c) Derive the decomposition of the change in demand for x2 in response to a

change in p1 into an income and substitution effect. What do you notice

about the income effect? Explain!

3. Suppose a consumer's utility function is continuous, strictly increasing and strictly quasiconcave on Rn+ and that he chooses a consumption bundle x Rn+ such that p x y. As a consequence, the consumer's utility maximization problem has a unique solution. Therefore, we can use the Weak Axiom of Revealed Preference (WARP). For each one of the following constellations of choices and prices, state whether these choices satisfy WARP. Note:

p = (p1, p2), x = (x1, x2).

(a) p = (1, 3), x(p, y) = (4, 2); p = (3, 5), x(p , y ) = (3, 1)

(b) p = (1, 6), x(p, y) = (10, 5); p = (3, 5), x(p , y ) = (8, 4)

(c) p = (1, 2), x(p, y) = (3, 1); p = (2, 2), x(p , y ) = (1, 2)

(d) p = (2, 2), x(p, y) = (2, 1); p = (3, 5), x(p , y ) = (1, 2)

Interdependence across generations, fortuitous inheritance, and income distribution a) In the simple Diamond OLG model without technical progress, if not in reality, all are born with zero financial wealth, the same work ability, and the same work willingness within generations as well as across generations. Nevertheless, as long as the economy has not reached a steady state, the members of different generations get different labor incomes. Why? b) In the model, through what channel does the behavior of one generation affect the economic conditions for the next generation? We now extend the model by adding uncertainty about the time of death. We also assume that people may live three periods (childhood is ignored). But they always work only in the first two. Individual labor supply is inelastic and equals one unit of labor in each of the two periods. All people survive the two first periods of life, but there is a probability (0 1) of dying before the end of the third period. Since in period analysis events happen either at the beginning or the end of the period, we have to assume that an "early" death occurs immediately after retirement. Suppose families are single-parent families: for each parent in generation there are 1 + children and these belong to the next generation. Any financial wealth left by a person who just died is inherited equally by the 1 + children. For members of generation the probability of staying alive three periods is thus 1 . Suppose each individual "born" at time (the beginning of period )1 maximizes expected utility, = (1) +(1 + )1(2+1) + (1 )(1 + )2(3+2). c) Given the pure rate of time preference in what direction does a decrease in affect the effective rate of time preference for a middleaged person? d) Suppose that there are no life annuity markets, and that the young knows the inheritance (positive or zero) before deciding the saving in the first period of life. Assume there is a constant real interest rate, For a young belonging to generation whose parent dies at the end of period with financial wealth , the period budget constraints are 1 + = + 1 1 + 2+1 + +2 = +1 + (1 + ) 3+2 = (1 + )+2 Explain. Interpret +2 : is it saving in period + 1 or what? e) Suppose that for some unexplained reason all members of generation 1 happens to have the same financial wealth at the time of retirement. Yet, after one period an inegalitarian distribution of wealth within generations tends to arise although all individuals have the same rate of time preference. Explain in a few words why. At a certain point in time a competitive market for private life annuities arises. Then before retiring middle-aged individuals place part or all their financial wealth in life annuity contracts issued by the life insurance companies. These use the deposits to buy capital goods which are rented out to the production firms. Next period the production firms pay back a risk-free return, 1 + per unit of account invested. At the same time, the insurance companies distribute their holdings (with interest) to their surviving depositors in proportion to their initial deposits. f) Suppose that the insurance companies have no operating costs. Their aim is to maximize expected profit. Then, given free entry and exit, in equilibrium what will expected profit in the annuity industry be? g) In equilibrium, how much will each surviving depositor receive per unit of account initially deposited?

h) Suppose somebody claimed: "The middle-aged individuals will choose to hold all their financial wealth in the form of life annuities." True or false? Why? i) What will the wealth distribution within generations in the long run look like?

Consider an individual's saving problem in Blanchard's "perpetual youth" model (standard notation): max () =0 0 = Z 0 (ln )(+) s.t. 0 = ( + ) + where 0 is given, lim 0 (+) 0. a) Briefly interpret the objective function, the constraints, and the parameters. b) Derive first the first-order conditions and the transversality condition, next the Keynes-Ramsey rule. c) Derive the full solution to the problem, i.e., find the consumption function. Hint: combine the Keynes-Ramsey rule with strict equality in the intertemporal budget constraint. d) How will a rise in the interest rate level affect current consumption and saving? Comment in terms of the Slutsky effects.