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Introduction to Modern Economic Growth Exercise 5.3 Prove Theorem 5.2 Exercise 5.4 Prove 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 ¢ σ−1 P ¡ i i σ − ξ Repeat the same x (3) Now suppose that U i (xi1 , , xiN ) = N j j j=1 computations and verify that the resulting indirect utility function is homogeneous of degree 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 Is this sufficient to ensure that the economy admits a representative household? 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 > 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 (i.e., ut (·) 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 individuals can trade bonds at time t that deliver one unit of good at time t0 Denote the price of such bonds by qt,t0 252