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Guvenen, Fatih. "Macroeconomics with Heterogeneity: A Practical Guide." Economic Quarterly. Federal Reserve Bank of Richmond. 2011. HighBeam Research. 23 Jul. 2018 <https://www.highbeam.com>.
Guvenen, Fatih. "Macroeconomics with Heterogeneity: A Practical Guide." Economic Quarterly. 2011. HighBeam Research. (July 23, 2018). https://www.highbeam.com/doc/1G1-295172567.html
Guvenen, Fatih. "Macroeconomics with Heterogeneity: A Practical Guide." Economic Quarterly. Federal Reserve Bank of Richmond. 2011. Retrieved July 23, 2018 from HighBeam Research: https://www.highbeam.com/doc/1G1-295172567.html
What is the origin of inequality among men and is it authorized by natural law?
--Academy of Dijon, 1754 (Theme for essay competition)
The quest for the origins of inequality has kept philosophers and scientists occupied for centuries. A central question of interest--also highlighted in Academy of Dijon's solicitation for its essay competition (1)--is whether inequality is determined solely through a natural process or through the interaction of innate differences with man-made institutions and policies. And, if it is the latter, what is the precise relationship between these origins and socioeconomic policies?
While many interesting ideas and hypotheses have been put forward over time, the main impediment to progress came from the difficulty of scientifically testing these hypotheses, which would allow researchers to refine ideas that were deemed promising and discard those that were not. Economists, who grapple with the same questions today, have three important advantages that can allow us to make progress. First, modern quantitative economics provides a wide set of powerful tools, which allow researchers to build "laboratories" in which various hypotheses regarding the origins and consequences of inequality can be studied. Second, the widespread availability of rich micro-data sources--from cross-sectional surveys to panel data sets from administrative records that contain millions of observations--provides fresh input into these laboratories. Third, thanks to Moore's law, the cost of computation has fallen radically in the past decades, making it feasible to numerically solve, simulate, and estimate complex models with rich heterogeneity on a typical desktop workstation available to most economists.
There are two broad sets of economic questions for which economists might want to model heterogeneity. First, and most obviously, these models allow us to study cross-sectional, or distributional, phenomena. The U.S. economy today provides ample motivation for studying distributional issues, with the top 1 percent of households owning almost half of all stocks and one-third of all net worth in the United States, and wage inequality having risen virtually without interruption for the last 40 years. Not surprisingly, many questions of current policy debate are inherently about their distributional consequences. For example, heated disagreements about major budget issues--such as reforming Medicare, Medicaid, and the Social Security system--often revolve around the redistributional effects of such changes. Similarly, a crucial aspect of the current debate on taxation is about "who should pay what?" Answering these questions would begin with a sound understanding of the fundamental determinants of different types of inequality.
A second set of questions for which heterogeneity could matter involves aggregate phenomena. This second use of heterogeneousagent models is less obvious than the first, because various aggregation theorems as well as numerical results (e.g., Rios-Rull [1996] and Krusell and Smith [1998]) have established that certain types of heterogeneity do not change (many) implications relative to a representative-agent model. (2)
To understand this result and its ramifications, in Section 1, I start by reviewing some key theoretical results on aggregation (Rubinstein 1974; Constantinides 1982). Our interest in these theorems comes from a practical concern: Basically, a subset of the conditions required by these theorems are often satisfied in heterogeneous-agent models, making the aggregate implications of such models closely mimic those from a representative-agent economy. For example, an important theorem proved by Constantinides (1982) establishes the existence of a representative agent if markets are complete. (3) This central role of complete markets turned the spotlight since the late 1980s onto its testable implications for perfect risk sharing (or "full insurance"). As I review in Section 2, these implications have been tested by an extensive literature using data sets from all around the world--from developed countries such as the United States to village economies in India, Thailand, Uganda, and so on. While this literature delivered a clear statistical rejection, it also revealed a surprising amount of "partial" insurance, in the sense that individual consumption growth (or, more generally, marginal utility growth) does not seem to respond to many seemingly large shocks, such as long spells of unemployment, strikes, and involuntary moves (Cochrane [1991] and Townsend [1994], among others).
This raises the more practical question of "how far are we from the complete markets benchmark?" To answer this question, researchers have recently turned to directly measuring the degree of partial insurance, defined for our purposes as the degree of consumption smoothing over and above what an individual can achieve on her own via "self-insurance" in a permanent income model (i.e., using a single risk-free asset for borrowing and saving). Although this literature is quite new--and so a definitive answer is still not on hand--it is likely to remain an active area of research in the coming years.
The empirical rejection of the complete markets hypothesis launched an enormous literature on incomplete markets models starting in the early 1990s, which I discuss in Section 3. Starting with Imrohoroglu (1989), Huggett (1993), and Aiyagari (1994), this literature has been addressing issues from a very broad spectrum, covering diverse topics such as the equity premium and other puzzles in finance; important lifecycle choices, such as education, marriage/divorce, housing purchases, fertility choice, etc.; aggregate and distributional effects of a variety of policies ranging from capital and labor income taxation to the overhaul of Social Security, reforming the health care system, among many others. An especially important set of applications concerns trends in wealth, consumption, and earnings inequality. These are discussed in Section 4.
A critical prerequisite for these analyses is the disentangling of "ex ante heterogeneity" from "risk/uncertainty" (also called ex post heterogeneity)--two sides of the same coin, with potentially very different implications for policy and welfare. But this is a challenging task, because inequality often arises from a mixture of heterogeneity and idiosyncratic risk, making the two difficult to disentangle. It requires researchers to carefully combine cross-sectional information with sufficiently long time-series data for analysis. The state-of-the-art methods used in this field increasingly blend the set of tools developed and used by quantitative macroeconomists with those used by structural econometricians. Despite the application of these sophisticated tools, there remains significant uncertainty in the profession regarding the magnitudes of idiosyncratic risks as well as whether or not these risks have increased since the 1970s.
The Imrohoroglu-Huggett-Aiyagari framework sidestepped a difficult issue raised by the lack of aggregation--that aggregates, including prices, depend on the entire wealth distribution. This was accomplished by abstracting from aggregate shocks, which allowed them to focus on stationary equilibria in which prices (the interest rate and the average wage) were simply some constants to be solved for in equilibrium. A far more challenging problem with incomplete markets arises in the presence of aggregate shocks, in which case equilibrium prices become functions of the entire wealth distribution, which varies with the aggregate state. Individuals need to know these equilibrium functions so that they can forecast how prices will evolve in the future as the aggregate state evolves in a stochastic manner. Because the wealth distribution is an infinite-dimensional object, an exact solution is typically not feasible. Krusell and Smith (1998) proposed a solution whereby one approximates the wealth distribution with a finite number of its moments (inspired by the idea that a given probability distribution can be represented by its moment-generating function). In a remarkable finding, they showed that the first moment (the mean) of the wealth distribution was all individuals needed to track in this economy for predicting all future prices. This result--generally known as "approximate aggregation"--is a double-edged sword. On the one hand, it makes feasible the solution of a wide range of interesting models with incomplete markets and aggregate shocks. On the other hand, it suggests that ex post heterogeneity does not often generate aggregate implications much different from a representative-agent model. So, the hope that some aggregate phenomena that were puzzling in representative-agent models could be explained in an incomplete markets framework is weakened with this result. While this is an important finding, there are many examples where heterogeneity does affect aggregates in a significant way. I discuss a variety of such examples in Section 6.
Finally, I turn to computation and calibration. First, in Section 5, I discuss some details of the Krusell-Smith method. A number of potential pitfalls are discussed and alternative checks of accuracy are studied. Second, an important practical issue that arises with calibrating/estimating large and complex quantitative models is the following. The objective function that we minimize often has lots of jaggedness, small jumps, and/or deep ridges because of a variety of reasons that have to do with approximations, interpolations, binding constraints, etc. Thus, local optimization methods are typically of little help on their own, because they very often get stuck in some local minima. In Section 7, I describe a global optimization algorithm that is simple yet powerful and is fully parallelizable without requiring any knowledge of MPI, OpenMP, and so on. It works on any number of computers that are connected to the Internet and have access to a synchronization service like DropBox. I provide a discussion of ways to customize this algorithm with different options to experiment.
1. AGGREGATION
Even in a simple static model with no uncertainty we need a way to deal with consumer heterogeneity. Adding dynamics and risk into this environment makes things more complex and requires a different set of conditions to be imposed. In this section, I will review some key theoretical results on various forms of aggregation. I begin with a very simple framework and build up to a fully dynamic model with idiosyncratic (i.e., individual-specific) risk and discuss what types of aggregation results one can hope to get and under what conditions.
Our interest in aggregation is not mainly for theoretical reasons. As we shall see, some of the conditions required for aggregation are satisfied (sometimes inadvertently!) by commonly used heterogeneous-agent frameworks, making them behave very much like a representative-agent model. Although this often makes the model easier to solve numerically, at the same time it can make its implications "boring"--i.e., too similar to a representative-agent model. Thus, learning about the assumptions underlying the aggregation theorems can allow model builders to choose the features of their models carefully so as to avoid such outcomes.
A Static Economy
Consider a finite set I (with cardinality I) of consumers who differ in their preferences (over l types of goods) and wealth in a static environment. Consider a particular good and let xi (p, wi) denote the demand function of consumer i for this good, given prices p [euro] R' and wealth [w.sub.i] Let ([w.sub.1], [w.sub.2], ..., [w.sub.1]) be the vector of wealth levels for all I consumers. "Aggregate demand" in this economy can be written as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
As seen here, the aggregate demand function x depends on the entire wealth distribution, which is a formidable object to deal with. The key question then is, when can we write x(p, [w.sub.1], [w.sub.2], ...,[w.sub.n]) [equivalent to] x(p, [SIGMA] [w.sub.1])? For the wealth distribution to not matter, we need aggregate demand to not change for any redistribution of wealth that keeps aggregate wealth constant ([SIGMA] d[w.sub.i] = 0). Taking the total derivative of x, and setting it to zero yields
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
for all possible redistributions. This will only be true if
([delta][x.sub.i] (p, [w.sub.i]/[delta][w.sub.i])) = ([delta][x.sub.j] (p, [w.sub.j]/[delta][w.sub.j])) [for all]i, j [euro] I.
Thus, the key condition for aggregation is that individuals have the same marginal propensity to consume (MPC) out of wealth (or linear Engel curves). In one of the earliest works on aggregation, Gorman (1961) formalized this idea via restrictions on consumers' indirect utility function, which delivers the required linearity in Engel curves.
Theorem 1 (Gorman 1961) Consider an economy with N < [infinity] commodities and a set I of consumers. Suppose that the preferences of each consumer i [euro] 1 can be represented by an indirect utility function (4) of the form
[v.sub.i] (p, [w.sub.i]) = [a.sub.i] (p) + b (p) [w.sub.i],
and that each household i [euro] I has a positive demand for each commodity, then these preferences can be aggregated and represented by those of a representative household, with indirect utility
v (p, w) = a (p) + b (p) w,
where a(p) = [[summation over (term)].sub.i] [a.sub.i] (p)and w = [[summation over (term)].sub.i] [w.sub.i] is aggregate income.
As we shall see later, the importance of linear Engel curves (or constant MPCs) for aggregation is a key insight that carries over to much more general models, all the way up to the infinite-horizon incomplete markets model with aggregate shocks studied in Krusell and Smith (1998).
A Dynamic Economy (No Idiosyncratic Risk)
Rubinstein (1974) extends Gorman's result to a dynamic economy where individuals consume out of wealth (no income stream). Linear Engel curves are again central in this context.
Consider a frictionless economy in which each individual solves an in-tertemporal consumption-savings/portfolio allocation problem. That is, every period current wealth [w.sub.t]is apportioned between current consumption [c.sub.t] and a portfolio of a risk-free and a risky security with respective (gross) returns [R.sub.t.sup.f] and [R.sub.t.sup.s]. (5) Let [a.sub.t]denote the portfolio share of the risk-free asset at time t, and [delta] denote the subjective time discount factor. Individuals solve
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
s.t. [w.sub.t] + 1 = ([w.sub.t] - [c.sub.t]) ([a.sub.t][R.sub.t.sup.f + (1 - [a.sub.t]) [R.sub.t.sup.s]])
Furthermore, assume that the period utility function, U, belongs to the hyperbolic absolute risk aversion (HARA) class, which is defined as utility functions that have linear risk tolerance: T(c) [equivalent to] -U (c)'/ U (c)" = [rho] + [gamma]c and [gamma] < 1. (6) This class encompasses three utility functions that are well-known in economics: U(c) = [([gamma] - 1).sub.-1] 9([rho] + [gamma]c).sup.1-[[gamma].sup.-1] (generalized power utility; standard constant relative risk aversion [CRRAJ form when [rho][equivalent to] 0); U (c) = -[rho] x exp(-c/[rho]) if [gamma] [equivalent to] 0 (exponential utility); and U(c) = 0.5[([rho] - c).sub.2] defined for values c < [rho] (quadratic utility).
The following theorem gives six sets of conditions under which aggregation obtains. (7)
Theorem 2 (Rubenstein 1974) Consider the following homogeneity conditions: 1. All individuals have the same resources 1110, and tastes S and U.
2. All individuals have the same S and taste parameters y 0.
3. All individuals have the same taste parameters y = 0.
4. All individuals have the same resources w0 and taste parameters p = 0 and y = 1.
5. A complete market exists and all individuals have the same taste parameter y = 0.
6. A complete market exists and all individuals have the same resources w0 and taste 8, p = 0, and y = 1.
Then, all equilibrium rates of return are determined in case (1) as if there exist only composite individuals each with resources w0 and tastes S and U; and equilibrium rates of return are determined in cases (2)-(6) as if there exist only composite individuals each with the following economic characteristics: (i) resources: [w.sub.0] = [SIGMA][w.sub.0.sup.i]/I; (ii) tastes: [sigma] = [PI] [([[sigma].sup.i]).sup.([p.sub.i]/ [SIGMA][p.sub.i]) (where [sigma] [equivalent to] 1/[sigma] - 1) or [sigma] = [SIGMA] [[sigma].sup.i]/I; and (iv) preference parameters: [rho] = [SIGMA] [[rho].sub.i]/I, and [gamma].
Several remarks are in order.
Demand Aggregation
An important corollary to this theorem is that whenever a composite consumer can be constructed, in equilibrium, rates of return are insensitive to the distribution of resources among individuals. This is because the aggregate demand functions (for both consumption and assets) depend only on total wealth and not on its distribution. Thus, we have "demand aggregation."
Aggregation and Heterogeneity in Relative Risk Aversion
Notice that all six cases that give rise to demand aggregation in the theorem require individuals to have the same curvature parameter, [gamma]. To see why this is important, note that (with HARA preferences) the optimal holdings of the risky asset are a linear function of the consumer's wealth: [K.sub.1] + [K.sub.2][w.sub.t]/[gamma], where [K.sub.1] and [k.sub.2] are some constants that depend on the properties of returns. It is easy to see that with identical slopes, ([k.sub.2]/[gamma]), it does not matter who holds the wealth. In other words, redistributing wealth between any two agents would cause changes in total demand for assets that will cancel out each other, because of linearity and same slopes. Notice also that while identical curvature is a necessary condition, it is not sufficient for demand aggregation: Each of the six cases adds more conditions on top of this identical curvature requirement. (8)
A Dynamic Economy (With Idiosyncratic Risk)
While Rubinstein's (1974) theorem delivers a strong aggregation result, it achieves this by abstracting from a key aspect of dynamic economies: uncertainty that evolves over time. Almost every interesting economy that we discuss in the coming sections will feature some kind of idiosyncratic risk that individuals face (coming from labor income fluctuations, shocks to health, shocks to housing prices and asset returns, among others). Rubinstein's (1974) theorem is silent about how the aggregate economy behaves under these scenarios.
This is where Constantinides (1982) comes into play: He shows that if markets are complete, under much weaker conditions (on preferences, beliefs, discount rates, etc.) one can replace heterogeneous consumers with a planner who maximizes a weighted sum of consumers' utilities. In turn, the central planner can be replaced by a composite consumer who maximizes a utility function of aggregate consumption.
To show this, consider a private ownership economy with production as in Debreu (1959), with m consumers, n firms, and l commodities. As in Debreu (1959), these commodities can be thought of as date-event labelled goods (and concave utility functions, [U.sub.i], as being defined over these goods), allowing us to map these results into an intertemporal economy with uncertainty. Consumer i is endowed with wealth ([w.sub.i1], [w.sub.i2], ..., [w.sub.il]) and shares of firms ([[theta].sub.i1],[[theta].sub.i2], ...,[[theta].sub.in]) with [[theta].sub.ij][greater than or equal to]0 and [[SIGMA].sub.m] [[theta].sub.ij] = 1. Let the vectors [C.sub.i] and [Y.sub.j] denote, respectively, individual i's consumption set and firm j's production set.
An equilibrium is an (m + n + 1)-tuple ([([c*.sub.i]).sub.i.sup.m] = 1, [([y*.sub.j]).sub.j.sup.n] = 1, [p*) such that, as usual, consumers maximize utility, firms maximize their profits, and markets clear. Under standard assumptions, an equilibrium exists and is Pareto optimal.
Optimality implies that there exist positive numbers [[lambda].sub.i], i = 1, ..., m, such that the solution to the following problem (P1),
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (P1)
s.t. [y.sub.j] [euro] [Y,sub.j], j = 1, 2, ... n;
[c.sub.i] [euro] [C.sub.i], i = 1, 2, ..., m;
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
(where h indexes commodities) is given by ([c.sub.i]) ([c*.sub.i]) and ([y.sub.i]) = ([y*.sub.j]). Let aggregate consumption be z [equivalent to] ([z.sub.1, ..., [z.sub.1]),[z.sub.h] [equivalent to] [[SIGMA].sub.i=1.sup.m][c.sub.ih]. Now, for a given z, consider the problem (P2) of efficiently allocating it across consumers:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
s.t. [c.sub.i] [euro] [C.sub.i], i = 1, 2, ..., m,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Now, given the production sets of each firm and the aggregate endowments of each commodity, consider the optimal production decision (P3):
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (P3)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Theorem 3 (Constantinides [1982, Lemma 1]) (a) The solution to (P3) is ([y.sub.j] = [y*.sub.j]) and [z.sub.h] = [[summation over (term)].sub.j=1.sup.n] [y*.sub.jh] + [w.sub.h],[[for all].sub.h].
(b) U (z) is increasing and concave in z.
(c) If [z.sub.h] = [[SIGMA][y.sub.jh]+ [W.sub.h], [[for all].sub,h], then the solution to (P2) is ([x.sub.i]) = ([x*.sub.i]).
(d) Given [[lambda].sub.i], i = 1, 2, ..., m, then if the consumers are replaced by one composite consumer with utility U (z), with endowment equal to the sum of m consumers' endowments and shares the sum of their shares, then the (1 + n + 1)-tuple ([[SIGMA].sub.i=1.sup.m][c*.sub.i], [([y*.sub.j].sub.j=1.sup.n]=1, p*) is an equilibrium.
Constantinides versus Rubinstein
Constantinides allows for much more generality than Rubinstein by relaxing two important restrictions. First, no conditions are imposed on the homogeneity of preferences, which was a crucial element in every version of Rubinstein's theorem. Second, Constantinides allows for both exogenous endowment as well as production at every date and state. In contrast, recall that, in Rubinstein's environment, individuals start life with a wealth stock and receive no further income or endowment during life. In exchange, Constantinides requires complete markets and does not get demand aggregation. Notice that the existence of a composite consumer does not imply demand aggregation, for at least two reasons. First, composite demand depends on the weights in the planner's problem and,
thus, depends on the distribution of endowments. Second, the composite consumer is defined at equilibrium prices and there is no presumption that its demand curve is identical to the aggregate demand function.
Thus, the usefulness of Constantinides's result hinges on (i) the degree to which markets are complete, (ii) whether we want to allow for idiosyncratic risk and heterogeneity in preferences (which are both restricted in Rubinstein's theorem), and (iii) whether or not we need demand aggregation. Below I will address these issues in more detail. We will see that, interestingly, even when markets are not complete, in certain cases, we will not only get close to a composite consumer representation, but we can also get quite close to the much stronger result of demand aggregation! An important reason for this outcome is that many heterogeneous-agent models assume identical preferences, which eliminates an important source of heterogeneity, satisfying Rubinstein's conditions for preferences. While these models do feature idiosyncratic risk, as we shall see, when the planning horizon is long such shocks can often be smoothed effectively using even a simple risk-free asset. More on this in the coming sections.
Completing Markets by Adding Financial Assets
It is useful to distinguish between "physical" assets--those in positive net supply (e.g., equity shares, capital, housing, etc.)--and "financial" assets--those in zero net supply (bonds, insurance contracts, etc.). The latter are simply some contracts written on a piece of paper that specify the conditions under which one agent transfers resources to another. In principle, it can be created with little cost. Now suppose that we live in a world with J physical assets and that there are S(> J) states of the world. In this general setting, markets are incomplete. However, if consumers have homogenous tastes, endowments, and beliefs, then markets are (effectively) complete by simply adding enough financial assets (in zero net supply). There is no loss of optimality and nothing will change by this action, because in equilibrium identical agents will not trade with each other. The bottom line is that the more "homogeneity" we are willing to assume among consumers, the less demanding the complete markets assumption becomes. This point should be kept in mind as we will return to it later.
2. EMPIRICAL EVIDENCE ON INSURANCE
Dynamic economic models with heterogeneity typically feature individual-specific uncertainty that evolves over time--coming from fluctuations in labor earnings, health status, portfolio returns, among others. Although this structure does not fit into Rubinstein's environment, it is covered by Constantinides's theorem, which requires complete markets. Thus, a key empirical question is the extent to which complete markets can serve as a useful benchmark and a good approximation to the world we live in. As we shall see in this section, the answer turns out to be more nuanced than a simple yes or no.
To explain the broad variety of evidence that has been brought to bear on this question, this section is structured in the following way. First, I begin by discussing a large empirical literature that has tested a key prediction of complete markets--that marginal utility growth is equated across individuals. This is often called "perfect" or "full" insurance, and it is soundly rejected in the data. Next, I discuss an alternative benchmark, inspired by this rejection. This is the permanent income model, in which individuals have access to only borrowing and saving--or "self-insurance." In a way, this is the other extreme end of the insurance spectrum. Finally, I discuss studies that take an intermediate view--"partial insurance"--and provide some evidence to support it. We now begin with the tests of full insurance.
Benchmark 1: Full Insurance
To develop the theoretical framework underlying the empirical analyses, start with an economy populated by agents who derive utility from consumption c, as well as some other good(s) [d.sub.t]: [U.sup.i] ([c.sub.t+a.sup.i],[d.sub.t+1.sup.i]), where i indexes individuals. These other goods can include leisure time (of husband and wife if the unit of analysis is a household), children, lagged consumption (as in habit formation models), and so on.
The key implication of perfect insurance can be derived by following two distinct approaches. The first environment assumes a social planner who pools all individuals' resources and maximizes a social welfare function that assigns a positive weight to every individual. In the second environment, allocations are determined in a competitive equilibrium of a frictionless economy where individuals are able to trade in a complete set of financial securities. Both of these frameworks make the following strong prediction for the growth rate of individuals' marginal utilities:
[[delta].sup.1] ([U.sub.c.sup.i]([c.sub.t + 1.sup.i], [d.sub.t + 1].sup.i]))/([U.sub.c.sup.i]([c.sub.t.sup.i], [d.sub.t.sup.i])) = [[LAMBDA].sub.t + 1]/[[LAMBDA].sub.t]
where [U.sub.c]. denotes the marginal utility of consumption and A, is the aggregate shock. (9) Thus, this condition says that every individual's marginal utility must grow in locksteps with the aggregate and, hence, with each other. No individual-specific term appears on the right-hand side, such as idiosyncratic income shocks, unemployment, sickness, and so on. All these idiosyncratic events are perfectly insured in this world. From here one can introduce a number of additional assumptions for empirical tractability.
Complete Markets and Cross-Sectional Heterogeneity: A Digression
So far we have focused on what market completeness implies for the study of aggregate phenomena in light of Constantinides's theorem. However, complete markets also imposes restrictions on the evolution of the cross-sectional distribution, which can be seen in (1). For a given specification of U, (1) translates into restrictions on the evolutions of [c.sub.t] and [d.sub.t] (possibly a vector). Although it is possible to choose U to be sufficiently general and flexible (e.g., include preference shifters, assume non-separability) to generate rich dynamics in cross-sectional distributions, this strategy would attribute all the action to preferences, which are essentially unobservable. Even in that case, models that are not bound by (1)--and therefore have idiosyncratic shocks affect individual allocations--can generate a much richer set of cross-sectional distributions. …
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