Exercises

197

where E is the expectations operator.

where r is the interest rate and W is its total wealth, which may be stochastic.

(1) Let us first suppose that W is nonstochastic and equal to Wo > 0. Characterize the utility maximizing choice of ci and c₂.

(2) Compute the intertemporal elasticity of substitution.

(3)

choice of ci and compute the coefficient of relative risk aversion. Provide conditions under which the coefficient of relative risk aversion is the same as the intertemporal elasticity of substitution. Explain why the two differ and interpret the conditions under which they are the same.

Exercise 5.3. Prove Theorem 5.2.

Exercise 5.4. Prove a version of Theorem 5.3 when there is also production.

Exercise 5.5. * Generalize Theorem 5.3 to an economy with a continuum of commodities.

Exercise 5.6. (1) Derive the utility maximizing demands for consumers in Example

5.1 and show that the resulting indirect utility function for each consumer is given by (5.5).

(2) Show that maximization of (5.6) leads to the indirect utility function corresponding to the representative household.

Repeat the same com­putations and verify that the resulting indirect utility function is homogeneous of degree 0 in p and y, but does not satisfy the Gorman form. Show, however, that a monotonic transformation of the indirect utility function satisfies the Gorman form.

Exercise 5.7. Construct a continuous-time version of the model with finite lives and random death. In particular suppose that an individual faces a constant (Poisson) flow rate of death equal to ν > 0 and has a true discount factor equal to ρ. Show that this individual will behave as if he is infinitely lived with an effective discount factor of ρ + ν.

Exercise 5.8. (1) Will dynastic preferences as those discussed in Section 5.2 lead to

infinite-horizon maximization if the instantaneous utility function of future genera­tions are different (u_t (∙) potentially different for each generation t)?

(2) How would the results be different if an individual cares about the continuation utility of his offspring with discount factor β, but also cares about the continuation utility of the offspring of his offspring with a smaller discount factor δ?

Exercise 5.9. Prove Theorem 5.8.

Exercise 5.10. Consider the sequential trading model discussed above and suppose now that households can trade bonds at time t that deliver one unit of good 0 at time t'. Denote the price of such bonds by q_tt∣.

(1) Rewrite the budget constraint of household h at time t, (5.22), including these bonds.

(2) Prove an equivalent of Theorem 5.8 in this environment with the extended set of bonds.

Exercise 5.11. Consider a two-period economy consisting of two types of households. Na households have the utility function

where c^f( and c) denotes the consumption of household h into two periods. The remaining Nb households have the utility function

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with â_â < βa. The two groups have, respectively, incomes yA and óâ at date 1, and can save this to the second date at some exogenously given gross interest rate R > 0.

Exercise 5.12. Consider an economy consisting of N households each with utility function

with β ∈ (0,1), wheredenotes the consumption of household h at time t. The economy starts with an endowment of Y units of the final good and has access to no production technology. This endowment can be saved without depreciating or gaining interest rate between periods.