References and Literature

This chapter covered a lot of ground and in most cases, many details were omitted for brevity. Most readers will be familiar with much of the material in this chapter. Mas-Colell, Winston and Green (1995) have an excellent discussion of issues of aggregation and what types of models admit representative households.

The best reference for existence of competitive equilibrium and the welfare theorems with a finite number of consumers and a finite number of commodities is still Debreu’s (1959) Theory of Value. This short book introduces all of the mathematical tools necessary for general equilibrium theory and gives a very clean exposition. Equally lucid and more modern are the treatments of the same topics in Mas-Colell, Winston and Green (1995) and Bewley (2006). The reader may also wish to consult Mas-Colell, Winston and Green (1995, Chapter 16) for a full proof of the Second Welfare Theorem with a finite number of commodities (which was only sketched in Theorem 5.7 above). Both of these books also have an excellent discussion of the necessary restrictions on preferences so that they can be represented by utility functions. Mas-Colell, Winston and Green (1995) also has an excellent discussion of expected utility theory of von Neumann and Morgenstern, which we have touched upon.

Neither of these two references discuss infinite-dimensional economies. The seminal refer­ence for infinite dimensional welfare theorems is Debreu (1954). Stokey, Lucas and Prescott (1989, Chapter 15) presents existence and welfare theorems for economies with a finite number of consumers and countably infinite number of commodities. The mathematical prerequisites for their treatment are greater than what has been assumed here, but their treatment is both thorough and straightforward to follow once the reader makes the investment in the neces­sary mathematical techniques. The most accessible reference for the Hahn-Banach Theorem, which is necessary for a proof of Theorem 5.7 in infinite-dimensional spaces are Kolmogorov and Fomin (1970), Kreyszig (1978) and Luenberger (1969). The latter is also an excellent source for all the mathematical techniques used in Stokey, Lucas and Prescott (1989) and also contains much material useful for appreciating continuous time optimization. Finally, a version of Theorem 5.6 is presented in Bewley (2006), which contains an excellent discussion of overlapping generations models.

5.13. Exercises

(1) Consider the following optimization problem

Although you do not need to, you may assume that G is continuous and convex, and u is continuous and concave.

Prove that any solutionto this problem is time consistent.

(2) Now Consider the optimization problem

the problem

Exercise 5.2.

where E is the expectations operator. The budget constraint of the household is 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 c∙₂.

(2) Compute the intertemporal elasticity of substitution.

(3) Now suppose that W is distributed over the supportwith some distribution

function G(W), whereCharacterize the utility maximizing

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 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.

(3) Now suppose thatRepeat the same com­

putations and verify that the resulting indirect utility function is homogeneous of degree 0 in p and y, but does not satisfy 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 (i.e., 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 individuals 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, (5.18), 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

with P â < β a. Each group, respectively, has income yA and óâ at date 1, and can save this to the second date at some exogenously given gross interest rate R.

Exercise 5.12. Consider an economy consisting of N households each with utility function at time t = 0 given by

withdenotes the consumption of household i 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.

(1) What are the Arrow-Debreu commodities in this economy?

(2) Characterize the set of Pareto optimal allocations of this economy.

(3) Does Theorem 5.7 apply to this economy?

(4) Now consider an allocation of Y units to the households,such that

Given this allocation, find the unique competitive equilibrium price vector and the corresponding consumption allocations.

(5) Are all competitive equilibria Pareto optimal?

(6) Now derive a redistribution scheme for decentralizing the entire set of Pareto optimal allocations?

(2) Suppose that the production structure is given by a neoclassical production function, where the production vector at time t is only a function of inputs at time t and capital stock chosen at time t — 1, and that higher capital so contributes to greater

Exercise 5.14. *

(3) Show that this modified version of Theorem 5.7 covers the economy in case 1 of this exercise.