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Traditionally an object of interest in microeconomics, models with privately informed agents have recently been used to study numerous topics in macroeconomics. (1) Characterization of Pareto-optimal allocations is an essential step in these studies, because the structure of optimal institutions of macroeconomic interest depends on the structure of optimal allocations. In models with privately informed agents, however, characterization of optimal allocations is a complicated problem, relative to models in which all relevant information is publicly available, especially in dynamic settings with heterogenous agents, which are of particular interest in macroeconomics.
The objective of this article is to characterize Pareto-optimal allocations in a simple macroeconomic environment with private information and heterogenous agents. We focus on the impact of private information on the implications of Pareto optimality. To this end, we consider two economies that are identical in all respects other than the presence of private information. In each economy, we fully characterize the set of all Pareto-optimal allocations. By comparing the structure of the sets of optimal allocations obtained in these two cases, we isolate the effect private information has on the implications of Pareto optimality.
The economic environment we consider is, on the one hand, rich enough to have features of interest in a macroeconomic analysis, and, on the other hand, simple enough to admit elementary, closed-form characterization of Pareto-optimal allocations, both with and without private information. The model we use is a stylized, two-period version of the Lucas (1978) pure capital income economy that is extended, however, to incorporate a simple form of agent heterogeneity. We assume that the population is heterogenous in its preference for early versus late consumption. In particular, we assume that a known fraction of agents are impatient, i.e., have a strong preference for consumption in the first time period, relative to the rest of the population. In the economy with private information, individual impatience is not observable to anyone but the agent. A detailed description of the environment is provided in Section 1.
In our analysis, we exploit the connection between Pareto-optimal allocations and solutions to so-called social planning problems, in which a (stand-in) social planner maximizes a weighted average of the individual utility levels of the two types of agents. These planning problems are defined and solved for both the public information economy and the private information economy in Section 2. The solutions obtained constitute all Pareto-optimal allocations in the two economies.
In the third section, we compare the Pareto optima of the two economies along two dimensions. First, we examine their welfare properties by comparing the utility levels provided to agents in the cross-section of Pareto-optimal allocations. The range of individual utility levels supported by Pareto optima in the private information economy turns out to be much smaller than that of the public information economy. In this sense, private information imposes limits to redistribution that can be attained in this economic environment. Then, we compare the structures of optimal intertemporal distortions, which are often called intertemporal wedges, across the Pareto optima of the two economies. With public information, all Pareto-optimal allocations are free of intertemporal wedges. In the economy with private information, we find Pareto-optimal allocations characterized by a positive intertemporal wedge, and others characterized by a negative intertemporal wedge. We close Section 3 with a short discussion of the implications of wedges for the consistency of Pareto-optimal allocations with market equilibrium outcomes, which are studied in many macroeconomic applications. Section 4 draws a brief conclusion.
1. TWO MODEL ECONOMIES
We consider two parameterized model economies. The two economies have the same preferences and technology. They differ, however, with respect to the amount of public information.
The following features are common to both economies. Each economy is populated by a unit mass of agents who live for two periods, t = 1,2. There is a single consumption good in each period, [c.sub.1], and agents' preferences over consumption pairs ([c.sub.1], [c.sub.2]) are represented by the utility function
[theta]u([c.sub.1]) + [beta]u([c.sub.2]),
where [beta] is a common-to-all discount factor, and [theta] is an agent-specific preference parameter. Agents are heterogenous in their relative preference for consumption at date 1. We assume a two-point support for the population distribution of the impatience parameter [theta]. Agents, therefore, can be of two types. A fraction [[mu].sub.H] of the agents are impatient with a strong preference for consuming in period 1. Denote by H the value of the parameter [theta] representing preferences of the impatient agents. A fraction [[mu].sub.L] = 1 - [[mu].sub.H] are agents of the patient type. Their value of the impatience parameter [theta], denoted by L, satisfies L < [H.sup.2].
In the economies we consider, the production side is represented by a so-called Lucas tree. We assume that the economy is endowed with a fixed amount of productive capital stock--the tree. (3) Each period, the capital stock produces an amount Y of the consumption good--the fruit of the tree. The consumption good is perishable--it cannot be stored from period 1 to 2. The size of the capital stock, i.e., the tree, is fixed: the capital stock does not depreciate nor can it be accumulated.
In our discussion, we will focus attention on a particular set of values for the preference and technology parameters. This will allow for explicit analytical solutions to the optimal taxation problem studied in this article. In particular, we will take
u(*) = log(*), [beta] = [1/2], H = [5/2], L = [1/2], [[mu].sub.H] = [[mu].sub.L] = [1/2], Y = 1. (1)
Roughly, the model period is thought of as being 25 years. The value of the discount factor [beta] of 1/2 corresponds to the annualized discount factor of about 0.973. The fractions of the two patience types are equal, and preferences are logarithmic. With H/L = 5, we consider a significant dispersion of the impatience parameter in the population. The per-period product of the capital stock is normalized to one.
The two economies we consider differ with respect to the scope of public knowledge of each agent's individual impatience parameter. In the first economy we consider, each agent's preference type is public information, i.e., it is known to the agent and everyone else. In the second economy, each agent's individual impatience is known only to himself.
2. PARETO-EFFICIENT ALLOCATIONS
An allocation in this environment is a description of how the total output (i.e., the economy's capital income Y) is distributed among the agents each period. We consider only type-identical allocations, in which all agents of the same type receive the same treatment. An allocation, therefore, consists of four positive numbers, c = ([c.sub.1H], [c.sub.1L], [c.sub.2H], [c.sub.2L]), where [c.sub.t[theta]] denotes the amount of the consumption good in period t assigned to each agent of type [theta].
In this section, we describe the efficient allocations. We use the standard notion of Pareto efficiency applied to type-identical allocations. We say that an allocation c is Pareto-dominated by an allocation [^.c] if all types of agents are at least as well off at [^.c] as they are at c and some are strictly better off. In our model, allocation c is Pareto-dominated by an allocation [^.c] if
[theta]u([[^.c].sub.1[theta]]) + [beta]u([[^.c].sub.2[theta]])[greater than or equal to][theta]u([c.sub.1[theta]]) + [beta]u([c.sub.2[theta]])
for both [theta] = H, L, and if
[theta]u([[^.c].sub.1[theta]]) + [beta]u([[^.c].sub.2[theta]]) > [theta]u([c.sub.1[theta]]) + [beta]u([c.sub.2[theta]])
for at least one [theta]. An allocation c is Pare to-efficient in a given class of allocations if c belongs to this class and is not Pareto-dominated by any allocation [^.c] in this class of allocations.
Pareto Optima in the Public Types Economy
In our economy with public preference types, resource feasibility is the sole constraint on the class of allocations that can be attained. …
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