I. INTRODUCTION

One of the persistent problems developing economies face is their inability to make substantial progress in raising the average level of human capital. Low levels of human capital investment persist despite the fact that human capital investments, through education, have been shown to have high returns in low-income countries where human capital is relatively scarce. Typical explanations for the lack of optimal, or even sufficient human capital investment on the part of low-income households usually begin with income and credit constraints. However, the failure of unconditional cash transfers, microcredit, and the lowering of school costs to increase educational investments suggest that income and credit constraints provide an insufficient explanation. Low school quality is also an incomplete explanation as it fails to fully reconcile the fact that returns to education are consistently estimated to be high even in areas where school quality appears to be relatively low. The failure of low income households to optimally invest in education, then, is one of the persistent puzzles in development.

New research has begun to shed light on another factor that appears to have a significant impact and that might solve the puzzle: low-income families may not be aware of the rate of return to investments in human capital (Jensen 2010). Information scarcity may, in fact, be one of the key hallmarks of poor households in low-income countries. For example, other research has found that agricultural households fail to use profitable fertilizers, and that demonstrating their effectiveness can increase utilization rates (Duflo et al. 2004).

Uninformed actors have been found in other studies of different types of economic decision making. Credit markets are one example where a number of studies have found that individuals underestimate the costs of borrowing (e.g., Stango and Zinman 2007). Other studies have found that workers are not well-informed of their pension or social security benefits (e.g., Chan and Stevens 2008; Mitchell 1988), or do a poor job estimating the risk of smoking (Viscusi 1990).

The fact that information can affect behavior has been supported by a number of different studies. For example, Duflo and Saez (2003) find that providing information about retirement benefits affects decisions about retirement plans, and the study of Dupas (2006) finds that information on age and HIV infection rates can influence the risky sexual behavior of Kenyan girls.

Human capital investment decisions are based on information identifying (or at least shedding some light on) the potential returns to education; and if, for poor households in low-income countries, this information is scarce, resisted, or is perceived to be of poor quality, then understanding and modeling the process through which households update their information--that is, learn about education--are essential in determining why certain outcomes, such as low human-capital investment traps, arise. Incorporating learning into theoretical models becomes all the more critical when the economy includes complicated feedback, as is the case with human capital investment: the actions of the households themselves affect the very returns they are trying to understand. (1) Thus modeling the mechanism through which households learn about the true returns to education is critically important in understanding the economic development of a country.

This paper builds a dynamic model of household investment in human capital, which exhibits both private and social returns. Under rationality, the model may have multiple equilibria including those corresponding to high and low education outcomes. The rational model is modified by assuming agents are boundedly rational and hold potentially heterogeneous beliefs; a set of learning mechanisms are then incorporated into the model to examine the process through which households learn about the real returns to education. Our findings are threefold:

* The low education rational equilibrium may be stable under learning: if agents use our simple learning mechanism then, under certain quite general conditions, the economy will converge (in a natural, probabilistic sense) to the low human-capital investment rational equilibrium. This result should be viewed with some surprise: coordination on beliefs-driven equilibria is known to be quite difficult in a general equilibrium setting, yet here we find the outcome robust even to agent-level heterogeneity in beliefs.

* The agent-level heterogeneity appears to only strengthen the power of pessimism: even if the initial distribution of beliefs includes the high-education outcome and does not include the low-education outcome, the model's nonlinear dynamics and expectational feedback may still lead to a poverty trap.

* Regardless of whether the specification of the model presents multiple equilibria, the nature of the learning algorithm, as influenced by beliefs and cultural norms, may prevent rapid convergence. Importantly, if the agent-level heterogeneity in beliefs is initially biased towards pessimism, along the resulting time path of the economy the vast majority of agents will persistently underestimate the benefits to education.

The goal of this theoretical exercise is not to exclude other possible mechanisms through which human capital investment is constrained. Educational quality, income and credit constraints, and fertility and household labor force participation, among other factors, might all play a role. To our knowledge, this is the first model that explicitly incorporates learning into a model of household human capital investment and growth. It is the goal of this paper to highlight how, consistent with empirical evidence, learning can play a significant role in household investment decisions.

This paper is organized as follows: Section II discusses the motivation for our interest in developing and studying a model that can explain the discussed features of the data; Section III develops the rational version of the model and characterizes the conditions under which equilibrium multiplicity arises; Section IV incorporates adaptive learning into the model; Section V catalogs the analytic and numerical results; and Section VI concludes. All technical arguments are relegated to the Appendix.

II. BACKGROUND

Average levels of education in low-income countries remain well below those of high-income countries despite large-scale educational expansion efforts over the last few decades. Barro and Lee (2001) estimated that the average years of education in developing countries for 2000 was 4.9 years whereas in advanced countries the average years of schooling was 9.8 years. They also find that in developing countries only 19.7% of the population over 25 years old have attained some secondary education and only 7.2% have attained some tertiary education whereas in advanced countries the figures are 39.4% and 29.1%, respectively. They also estimate that the gains in educational attainment through time are no faster in low-income countries: from 1960 to 2000, advanced countries and developing countries had similar growth trajectories in terms of average years of schooling, in other words, low-income countries are not catching up.

These low investment levels are in stark contrast to the relatively high returns to education experienced in developing countries. In a meta-study of the received empirical evidence, Psacharopoulos and Patrinos (2004) find an average year of schooling effect on income of 10.9% for low-income countries as opposed to 7.4% for high-income countries. Returns to investment in education are similarly divergent: private returns to investment in secondary education in low-income countries are 19.9% whereas they are 12.2% in high-income countries. The returns for higher education are 26% for low-income countries and 12.4% for high-income countries.

Our model assumes both private and social returns to education--the benefits to individual educational investment that accrue to non-investors. Psacharopoulos and Patrinos (2004) show that many studies have found substantial social returns to education in low-income countries: 21.3% for primary education, 15.7% for secondary, and 11.2% for higher education, on average (though they sound a word of caution about the reliability of these estimates given the challenges involved). Hall and Jones (1999) also find significant total factor productivity (TFP) and growth effects from average education levels as do de la Fuente and Domenech (2001). Also while Acemoglu and Angrist (2000) do not find evidence of social returns from high school education in the United States, Moretti (2004) finds sizable externalities associated with college education in the United States. Both of these findings are reinforced by Iranzo and Peri (2009) who find, using U.S. data, social returns from high school education in the 0%-1% range but in the 6%-9% range for college education. Considering the low average level of education in developing countries, estimates of positive social returns to lower levels of educational attainment do not seem unreasonable. Additionally, Borjas (1992, 1995) finds that human capital accumulation depends not only on the skill level of the parents but also on average skills of the previous generation in the same ethnic community and neighborhood in the United States. Wantchekon et al. (2015) find large village-level intergenerational human capital externalities in postcolonial Benin.

Given the relatively low level of education in developing countries and the relatively high returns, researchers have been left to puzzle over explanations for the lack of investment in human capital. Explanations such as income and credit constraints, high discount rates, or simply errors of bias in the measurement of returns have been explored but recent empirical research has left them wanting. (2)

If income and credit constraints are the explanation for the low investment levels in education in low-income countries, easing them should yield substantial returns. However, Banerjee et al. (2015) found evidence from a large experiment on microcredit in India that providing families with credit did not increase educational investment or outcomes. The experience with unconditional cash transfer programs also suggests that income and credit constraints are insufficient to explain the human capital gap, as de Janvry et al. (2006) state: "... unconditional transfers have small effects on school choices compared to conditional transfers where the condition for the transfer is on school attendance."

A question that then arises is: do families in low-income countries have complete information about the returns to human capital? Jensen (2010) finds that perceived returns to education in the Dominican Republic are very low, particularly relative to actual returns as measured with earnings data. He then uses an experiment to study the effect of information about the true return to education on investment behavior. He finds that relative to students not provided with information about returns, informed students perceived dramatically increased returns. Such informed students were more likely to be enrolled in school the next academic term, and when observed 4 years later, those students had completed on average about 0.20 more years of schooling. He also found some evidence suggesting that students rely heavily on the earnings of workers in their own community when they formed their own expectations of earnings. There is strong evidence then that perceptions about the return to investments in human capital are both substantially low and a constraint on human capital investment in low-income countries.

The link between human capital and growth is now well-established in the empirical literature. Mankiw, Romer, and Weil (1992), Barro (1991, 1997), Benhabib and Spiegel (1994), and Hanushek and Kimko (2000) all find evidence of a strong causal relationship between a country's human capital levels and its growth rates. Human capital and growth theory dates back to the endogenous growth literature (see Aghion and Howitt 1997 for a good overview) and has been adapted to development most notably by Azariadis and Drazen (1990) which incorporates a threshold human capital externality in a model of economic growth.

Learning has a well-established place in the macroeconomics literature (e.g., Evans and Honkapohja 2001) but has not been widely adopted in growth models despite the considerable evidence of incomplete information. There are now, however, a handful of papers that incorporate learning in models of economic growth. Evans, Honkapohja, and Romer (1998) use a simple learning algorithm in an endogenous growth model with research and development to explain cycles between periods of high and low growth. Arifovic, Bullard, and Duffy (1997) use a genetic algorithms approach to explain the transition from low to high growth. Also perhaps the work most closely related to ours is that of Steiger (2009): she adapts learning to the Azariadis and Drazen (1990) framework and finds that while the low (in her case, zero) education steady state is learnable, the high education steady state is not; interestingly, however, she finds that complex stochastic dynamics may lead the economy away from the low education steady state and induce endogenous fluctuations.

III. KNOWING ABOUT EDUCATION

The modeling environment is a fairly general two-period, overlapping generations setting with agglomeration effects, and our results could be developed within this general framework; how ever, to facilitate exposition and to focus attention on the deeper mechanisms driving our results, we restrict attention to a highly stylized version of the model.

We first consider a nonstochastic environment and adopt the rational expectations viewpoint. We characterize, both graphically and analytically, the model's rational expectations equilibria (REE). We find that some parameterizations of the model's agglomeration effects lead to equilibrium indeterminacy, that is, multiple REE, including those associated with low human-capital investment traps.

Consider a two-period overlapping generations model. We assume that, at time t, the economy is populated by a continuum (unit mass) of identical young agents and a continuum of identical old agents: population is constant. Utility is obtained from consumption each period, and for simplicity, both young and old own a production technology and consume what they produce. (3)

Each agent is endowed with a unit of labor each period. Young agents may convert this labor endowment into goods and/or human capital using a simple linear technology. Old agents convert labor into goods using an advanced technology F. Formally, let [omega], [member of] [[OMEGA].sub.t], be the index of an agent born in time t. This agent's problem is given by

[mathematical expression not reproducible

where [C.sub.1r]([[omega].sub.t]) is the consumption level of agent [[omega].sub.t] when she is young, [n.sub.t],([[omega].sub.t]) is the quantity of labor directed toward good production when she is young, and [h.sub.t],([[omega].sub.t]) is the amount of human capital she acquires. Her consumption when old depends, through the advanced technology F, on the aggregate level of human capital in period t + 1, which we denote by [H.sub.t+1]. To choose the time t optimal level of human capital investment, agent [[omega].sub.t] must forecast [H.sub.t+1]: we denote this forecast by [H.sup.e.sub.t+1] ([[omega].sub.t]). It follows that her point expectation of consumption when old is given by F ([h.sub.t] ([[omega].sub.t]), [H.sup.e.sub.t+1] ([[omega].sub.t])). (4) The properties of this production function are discussed in detail below. (5)

In general we will allow for the possibility that agents are heterogeneous either in the information available to them or in how they form expectations. In an REE, homogeneity will prevail, and under homogeneity the consideration of expectations may appear superfluous in this nonstochastic setting: like the Diamond growth model, the return to human capital investment is predetermined--the consumption of an old agent in period t + 1 is entirely determined by decisions made in period t. Care must be taken, however, because, as alluded to earlier, this model may have multiple equilibria driven by expectational feedback. The potential for equilibrium multiplicity in this model arises because of the simultaneous determination of the h([[omega].sub.t]) and [H.sub.t + 1], : the optimal human capital investment decision of a given agent depends on the aggregate level of investment, which itself depends on the sum of the individual levels. The expectational feedback mechanism will play a central role in our analysis.

Under appropriate assumptions on F and u, as discussed in the Appendix, and given her forecast [H.sup.e.sub.t+1]. ([[omega].sub.t]), the behavior of agent co, is characterized by the following first-order condition (FOC):

u' ([C.sub.1+] ([[omega].sub.t])) = [F.sub.h] ([h.sub.t] ([[omega].sub.t]), [H.sup.e.sub.t+1] ([[omega].sub.t])) u' ([c.sup.e.sub.2t+1] ([[omega].sub.t]))

To simplify our analysis, we adopt a log-utility specification here and in the sequel which allows us to rewrite this FOC as

(1) [mathematical expression not reproducible]

Now observe that, by dropping time-subscripts and co indexes, Equation (1) determines the level of human capital chosen by any young agent in any period, given that agent's forecast of the next period's level of aggregate human capital. In particular, Equation (1) implicitly defines a behavioral function [??] : [0,1] [right arrow] [0,1] so that [h.sub.t] ([[omega].sub.t]) = [??] ([H.sup.e.sub.t+1] ([[omega].sub.t])).

The function [??] can be naturally interpreted as capturing the model's expectational feedback. At the individual level, this expectational feedback is clear. To clarify the aggregate mechanism, observe that

[mathematical expression not reproducible]

If all agents have the same expectations, as they would in an REE, then

(2) [H.sub.t+1] = [??] ([H.sup.e.sub.t+1]).

Equation (2) characterizes our model's aggregate expectational feedback, which will, in turn drive the dynamics under adaptive learning. Because expectations are homogeneous in an REE, that is, [H.sub.t+1] = [H.sup.e.sub.t+1] ([[omega].sub.t]) for all [[omega].sub.t] [member of] [[OMEGA].sub.t], a solution [H.sup.*] = [??] ([H.sup.*]) identifies a REE, and if this equation has multiple solutions then, as a result of expectational feedback, the model has multiple REE. In Section IV we model more carefully how agents arrive at the forecasts [H.sup.e.sub.t+1], and therefore how a particular solution to Equation (2) might naturally arise.

For completeness, we provide the following definition:

DEFINITION 1. A rational expectations equilibrium of the stylized model is a solution [H.sup.*] to the equation [??] ([H.sup.*]) = [H.sup.*].

The possibility of equilibrium multiplicity arises when F exhibits increasing social returns. To provide a laboratory for analysis and to gain insight into the existence of multiple equilibria, we specify a functional form for F. Our specification is guided by the following desired characteristics:

1. [F.sub.h] > 0: individual human capital is a productive input;

2. [F.sub.hh] < 0: diminishing marginal returns to individual human capital;

3. [F.sub.H] > 0: aggregate human capital is a productive input;

4. [F.sub.hH] > 0: the marginal productivity of individual human capital is benefited by increased aggregate human capital.

To capture these characteristics in a flexible function form, as well as to facilitate analysis, we take F to be given by

(3) F(h, H) = [GAMMA] (H) [(h - [bar.h]).sup.[alpha](H)],

where [GAMMA] and a are suitably differentiate, non-negative functions, defined on [R.sub.+] (nonnegative reals), and [bar.h], [alpha] (H) [member of] (0, 1). The scalar [bar.h] captures the minimum education necessary to access the advanced technology. A detailed analysis of the production function, as well as a proof that it has the requisite properties, is contained in the Appendix. (6)

The function [GAMMA] captures the TFP effect of aggregate human capital accumulation. Because of our log utility specification, [GAMMA] plays no decision-making role, see Equation (1): it is used here to guarantee F has the properties listed above while allowing us to separate level effects from relative marginal effects. On the other hand, a plays a central role in our analysis. To interpret a, mentally set [bar.h] = 0; then the function a captures the output elasticity of individual human capital. We think of a as nondecreasing in H; thus a higher aggregate human capital stock increases the responsiveness of output to individual education levels.

Combining the functional form Equation (3) with the equilibrium condition [??]([H.sup.*]) = [H.sup.*] yields

(4) [alpha] ([H.sup.*]) = [H.sup.*] - [bar.h]/1 - [H.sup.*].

Existence and uniqueness of equilibria evidently depend on the behavior of [alpha]. If agglomeration effects are weak (or nonexistent, i.e., a is a constant) then there will be a unique equilibrium, whereas if a exhibits more involved behavior, multiple equilibria may arise.

Figure 1A provides the former case. Here [alpha] has a modest, roughly constant slope, and the model has a unique REE. In Figure 1B we depict a as increasing, but with varying rate. This particular shape for a may reflect a threshold effect: for low values of aggregate human capital, the elasticity is low and essentially constant, but as H crosses a threshold level, representing, say, a certain percentage of the population having a high school education, the elasticity jumps sharply, before again leveling off. In this case, there are three REE, though as we will find in Section IV, [H.sup.*.sub.U]. should not be considered a reasonable outcome.

Figure 1B depicts the potential for this model to explain a low human capital steady state. If agents predict that H - [H.sup.*.sub.L] then they will choose their education levels to sustain and hence rationalize precisely this outcome. The presence of equilibrium indeterminacy reflects the importance in this type of general equilibrium environment of expectations feedback: low human-capital investment traps exist because the belief that education is not valuable is self-fulfilling. Similarly, the high human-capital investment equilibrium is also self-fulfilling. However, and importantly, the rational expectations hypothesis does not speak to how agents come to have their pessimistic or optimistic beliefs: there is no selection mechanism associated to the theory.

One more point is worth emphasizing: regardless of which equilibrium is realized, and in fact regardless of whether multiple equilibria are even present, in an REE agents are not making forecast errors: they have a complete and correct understanding of the returns to their human capital investment. We conclude that even if a selection mechanism is specified, it is not possible for our model to explain the discrepancy, evidenced in the introduction, between agents' perceptions of the returns to education and the realizations of these returns, so long as we adhere to the paradigm of rational expectations.

IV. LEARNING ABOUT EDUCATION

The rational model exhibits two primary defects: knowledge of, and coordination on, the equilibrium is imposed, not modeled, even when multiple equilibria are present; and even if coordination is taken as given, no equilibrium of the model can explain the data. To address these concerns, we explicitly model expectations formation by taking our cue from the adaptive learning literature: see Evans and Honkapohja (2001) for a general reference. This literature dictates that we back away from the assumption that agents are omniscient, forward-looking actors; instead, we model them as using simple forecasting rules to form expectations. The agents' expectations then predicate their actions, which, in turn generate new data; agents use these data to update their forecasting models and the process repeats. In this way, the model is coupled with a learning dynamic which may then be analyzed to assess transitions to equilibria as well as selection issues. Importantly, expectations coordination is neither assumed nor required: rationality arises as an emergent outcome.

We begin by modifying the model to allow for uncertainty in returns to education. While there are many ways to go about this modification, our intention here is to remain as close to the rational model as possible, while retaining analytic tractability. In a rational expectations equilibrium, agents have perfect foresight: the young agent in time t knows the aggregate level of human capital in time t + 1, and bases her human-capital investment decision on this knowledge. Here we back off that assumption, and instead allow that agent [[omega].sub.t]'s forecast [H.sup.e.sub.t+1] ([[omega].sub.t]) may be incorrect.

Because the representative young agent's problem is otherwise unchanged, her behavior is still characterized by the function [??]. (7) However, now we must take a stand on what information is available to the agent and how she uses this information to forecast aggregate human capital, and this becomes somewhat involved. We assume that there is a finite number N of agent types I = 1, ..., N, and all agents of a given type hold the same forecasts of aggregate human capital. For simplicity, we assume that the offspring of an agent of type i is also an agent of type i, and we assume that there is a positive measure [[mu].sub.i] of agents of type i. (8) We assume that each young agent of type i in time t observes aggregate human capital [H.sup.0.sub.t] (i) up to a type-specific measurement error,

(5) [H.sup.0.sub.t] (i) = [H.sub.t] + [[sigma].sub.i][[epsilon].sub.t](i). (i).

The measurement errors [[epsilon].sub.t] (i) are taken to be serially and cross-sectionally independent, with zero mean and unit variance, with [[sigma].sub.i] [greater than or equal to] 0 scaling the measurement error. The observation [H.sup.0.sub.t] (i) is used to estimate a forecasting model. Because rational equilibria in our model are steady states, the natural forecasting model is regression on a constant, that is, a simple average. Written in recursive form, this estimation procedure may be expressed as

(6) [H.sup.e.sub.t+1] (i) = [H.sup.e.sub.t] (i) + [[gamma].sub.t], ([H.sup.0.sub.t] (i) - [H.sup.e.sub.t] (i)). (0).

Here [[gamma].sub.i] is the "gain" of the learning algorithm (Equation (6)), and measures how seriously agents take new data. (9) If [[gamma].sub.t], = 1/t then Equation (6) reduces to the recursive least squares estimator, which, in our case, is the sample mean; if [[gamma].sub.t] = [gamma] > 0 is a small constant, then Equation (6) is referred to as a "constant gain learning algorithm," and represents geometric discounting of past data.

Having characterized the behavior of the agents, we may specify the equilibrium concept.

DEFINITION 2. Given the processes [{[[epsilon].sub.t] (i)}.sup.N.sub.i=1], initial expectations [{[H.sup.e.sub.0] (i)}.sup.N.sub.i=1] and an initial stock of aggregate human capital [H.sub.0], a learning equilibrium is a collection of sequences {[H.sub.t], [{[H.sup.e.sub.t] (i)}.sup.N.sub.i=1]} satisfying Equations (5) and (6), as well as [H.sub.t+1] = [[summation].sup.N.sub.i=1] [[mu].sub.i] x [??] ([H.sup.e.sub.t+1] (i)).

We observe that if the measurement error is shut down ([[sigma].sub.i] = 0), and if homogeneity is imposed at time zero ([H.sup.e.sub.0] (i) = [H.sup.e.sub.0] (j) = [H.sup.*]), then this definition reduces to an REE.

V. RESULTS

The rich dynamic structure of the model's learning equilibria allows a number of different questions to be addressed. Perhaps most urgently, what is the relationship between rational equilibria and learning equilibria? If learning equilibria bear no resemblance to rational equilibria then we are led to question the relevance of this exercise; however if learning equilibria evolve to approximate rational equilibria in some natural sense then we may view the learning mechanism as an equilibrium selection device, providing justification for the rational requirement of expectations coordination. Second, and perhaps more ambitiously, are the transitions dynamics that result from the learning mechanism able to succeed where rationality failed, by potentially explaining the systematic agent-level underestimations of returns to education?

A. Asymptotic Rationality

The number and nature of the stylized model's rational equilibria are determined as solutions to [H.sup.*] = [??] ([H.sup.*]), and are depicted graphically in Figure 1, whereas learning equilibria have a considerably more complex characterization, given in Definition 2. Can we assess the relationship between these creatures? The problem is an a priori difficult one: the rational equilibria are straightforward to understand and analyze; but the learning equilibria are processes representing solutions to nonlinear stochastic recursive algorithms (SRAs), and, generally speaking, SRAs are quite difficult to understand. Fortunately, the forms of the SRAs relevant for adaptive learning in economic models are amenable to asymptotic and short sequence analysis using the theory of Ljung (1977): again, see Evans and Honkapohja (2001) for applications to macroeconomics.

Our first result provides the relationship between rational and learning equilibria in case of decreasing gain.

PROPOSITION 1. Let [H.sup.*] represent a rational equilibrium: [H.sup.*] = h([H.sup.*]). Let y, = j. If

(7) [alpha]' ([H.sup.*]) < [[(1 + [alpha] ([H.sup.*])).sup.2]]/[1 - [bar.h]]

then locally [H.sup.e.sub.t] (i) [right arrow] [H.sup.*] with probability 1, for i = 1, ..., n.

A formal statement of this result, together with its proof, are contained in the Appendix.

Proposition 1 provides a simple stability condition which, when satisfied, imparts the asymptotic equivalence of rational and learning equilibria: if the rational equilibrium satisfies Equation (7), if initial beliefs are within the rational equilibrium's basin of attraction, and if agents form expectations by simply averaging their initial beliefs with realized data, then the learning equilibrium will converge in a natural sense to the rational equilibrium. If the stability condition is satisfied, we say that the associated rational equilibrium is stable. Somewhat remarkably, the stability condition (7) is directly related to the graphical depiction of rational equilibria (10): stability obtains precisely when the graph of [alpha] crosses the upward-sloping curve from above. Note that in Figure 1 B, both the upper and lower rational equilibria are stable: in each case, agents will learn to be rational.

For stable rational equilibria, the coordination problem is solved: agents are not required to carefully and accurately forecast the behavior of other agents (which would require forecasting the forecasts of other agents, etc.); instead, they simply view past data and compute the mean. Furthermore, Proposition 1 establishes equilibrium stability as a selection mechanism in two distinct senses: first, unstable rational equilibria, such as the middle equilibrium depicted in Figure 1, should be discarded--agents cannot learn to coordinate on them; and second, given two distinct stable rational equilibria, selection by learning agents will depend on initial beliefs.

Proposition 1 together with Figure 1 provides a simple explanation of poverty traps: agglomeration effects impart equilibrium multiplicity, and the economy is stuck in a rational equilibrium characterized by coordination failure. While this story is well-understood, the novelty of our treatment is that now we know why agents are stuck in the low human capital equilibrium: they were initially skeptical of the value of education and their simple adaptive algorithm, together with the associated forecast-based behaviors, reinforced and coordinated their skepticism.

B. Transition Dynamics: Analytic Results

Our second result characterizes the finite horizon behavior in case of constant gain. The result is more technical in nature, but is useful in that it allows for precise statements about transition dynamics and allows for a good approximation to asymptotic behavior. In the Appendix it is shown that a stable rational equilibrium H corresponds to a Lyapunov stable fixed point of the following differential equation:

(8) d[H.sup.e]/d[tau] = [??] ([H.sup.e]) - [H.sup.e].

Let [[??].sup.e.sub.t] ([H.sup.e.sub.0]) be the solution to Equation (8) corresponding to the initial condition [H.sup.e.sub.0]. The following result provides a relationship between this solution and the associated learning equilibrium.

PROPOSITION 2. Let [H.sup.*] represent a stable rational equilibrium and let [{[H.sup.e.sub.0] (i)}.sup.n.sub.i=1] be initial beliefs which are near [H.sup.*]. Fix T, and let [{[H.sup.e.sub.t] (i) ([H.sup.e.sub.0] (i), [gamma])}.sup.T.sub.t=0] be the finite-length time path of beliefs associated to the constant gain [gamma] and initial beliefs [H.sup.e.sub.0] (i), as determined by the recursion (Equation (6)). Then, as [gamma] [right arrow] 0, [{[H.sup.e.sub.t] (i) ([H.sup.e.sub.0] (i), [gamma])}.sup.T.sub.t=0] converges weakly to a random process with mean [{[[??].sup.e.sub.t] ([H.sup.e.sub.0] (i))}.sup.T.sub.t=0] and vanishing variance.

A precise statement of the result, together with a formal proof is given in the Appendix. (11)

Intuitively, Proposition 2 states that for any time horizon T, the expected time path of beliefs will be well-approximated by the solution to the differential Equation (8), provided that the gain is small enough; and, since this differential equation locally directs paths of beliefs to a stable rational equilibrium, it follows that, locally, agents' beliefs will get near the rational equilibrium and stay near it for a long time. The implication of this Proposition--and this is what we need for the paper--is that for small gain [gamma] and finite horizon T, the solution to the differential Equation (8) provides a good approximation to the expected time path of the recursive algorithm capturing the evolution of households' beliefs: so, instead of studying the recursive algorithm (which is hard), we study the differential equation (which is easy).

Just as Proposition 1 provided a theoretical explanation for poverty traps, Proposition 2 provides an analytic framework to address evidenced misperceptions about the returns to education. We denote by [xi] our measure of misperceptions, and take it to be the ratio of expected to realized marginal returns to individual education:

[xi] ([H.sup.e] (i)) [equivalent to] [F.sub.h] (h ([H.sup.e] (i),) [H.sup.e] (i))/[F.sub.h] (h ([H.sup.e] (i),) H)

where H is the realized level of aggregate human capital. Note that agent i underestimates the return to education when [xi]([H.sup.e] (i)) < 1.

To leverage Proposition 2 as an explanation of misperceptions the following observations are needed:

1. If [H.sup.e] (i) < H then [xi]([H.sup.e](i)) < 1.

2. If [H.sup.*] is a stable steady state and [H.sup.e](i) < [H.sup.*] is within its basin of attraction then [H.sup.e](i) < h([H.sup.e](i)).

Item 1 follows from the fact that [F.sub.hH] > 0; item 2 follows from standard stability analysis.

Now let [H.sup.*] be a stable steady state, and as a simple, first thought experiment, assume all agents hold identical beliefs [H.sup.e](i) = [H.sup.e] < [H.sup.*], with He within the basin of attraction of [H.sup.*]. According to Proposition 2, the ensuing timepath of beliefs is well approximated by the solution [[??].sup.e] ([H.sup.e.sub.0]) to the differential Equation (8) corresponding to the initial condition [H.sup.e](0) = [H.sup.e]; and qualitative analysis of Equation (8) shows that [H.sup.e.sub.t] [right arrow] [H.sup.*] from below. It follows from item 2 that

[H.sup.e.sub.t] [approximately equal to] [[??].sup.e.sub.t] < h ([[??].sup.e.sub.t]) [approximately equal to] h ([H.sup.e.sub.t]) = [H.sub.t],

where the last equality follows from the identical-beliefs assumption. Finally, by item 1, we conclude that along the entire time path, up to approximation, [xi] ([H.sup.e.sub.t]) < 1.

Allowing for local heterogeneity of initial beliefs only serves to strengthen the result. Under some specifications of the distribution of initial beliefs, some agent types may hold accurate forecasts of future human capital levels, while other types are pessimistic and subsequently underestimate the benefit of education. Provided the steady state is stable, the dynamics implied by Equation (8) impart persistence to the pessimism, as is evidenced in the data. The details of the trajectories of beliefs are parameter specific, and explored in more detail in the next section.

Transition Dynamics: Numerical Results. The analytic results established above provide explanations of poverty traps and persistent productivity misperceptions that are robust to parameter specifics; however, exposure of the rich detail of the transition paths, as well as the nuanced nature of the basins of attraction, requires numerical work. In this subsection, we consider a parametric specification of the model that yields multiple steady states, and use simulations to examine outcomes.

Stability, Heterogeneity, and Basins of Attraction. The concern here is the robustness of poverty traps to the location and distribution of initial beliefs. To explore this concern, we use a discrete-time approximation to the differential system Equation (8), solved against a variety of initial conditions, and then appeal to Proposition 2.

Consider a specification of the model consistent with Figure IB in which there are two stable steady states: [H.sup.*.sub.H] and [H.sup.*.sub.L]. Next, imagine that there are 20 agent types, with initial beliefs [{[[H.sup.e.sub.0] (i)}.sup.20.sub.i=1] distributed uniformly over each of four intervals of equal length, denoted by A, B,C, and D, respectively. Finally, assume that intervals B and D are centered on [H.sup.*.sub.H] and [H.sup.*.sub.L], respectively, while A and C lie fully below and above [H.sup.*.sub.L], respectively. Precisely this configuration is considered in Figure 2, which presents the corresponding time-paths of beliefs for each of these sets of initial conditions.

The important conclusion to draw from this figure is the robust stability of the poverty trap [H.sup.*.sub.L]: each of the sets of initial conditions A, B, and C lead to the poverty trap; and further, interval C includes only optimistic agents, at least in the sense that the asymptotic outcome is lower than initial expectations, and interval C includes the high-productivity outcome, [H.sup.*.sub.H].

Next we consider the same parametric specification, but instead of appealing to Proposition 2, we simulate the stochastic economy. Figure 3 presents the results of 100 simulations, each with 50 agent types using a decreasing-gain learning algorithm. (12) Each simulation was initialized with beliefs distributed uniformly over interval C. Note that the model's inherent stochasticity, together with the real-time learning behavior of agents, imparts considerable uncertainty regarding the asymptotic outcome: all simulations result in an initial contraction of the span of beliefs, but what happens next depends on the interaction of stochasticity, expectational feedback, and learning behavior. Approximately 60% of the simulations fell into the poverty trap, while the remainder eventually coordinated on the high-productivity equilibrium.

Coordination failure, as identified by a poverty trap, is predicted as a possible outcome of the standard, rational model with multiple equilibria; however, the rational-expectations hypothesis, and by consequence the standard rational model, is silent about how coordination is achieved. Our framework incorporates a natural learning algorithm to model expectations coordination, and yields a stark and sobering result: the poverty-trap equilibrium is a potentially robust, emergent outcome of the learning process; furthermore, the pessimistic views of some agents may overwhelm the optimism of others.

Underestimating the Benefit of Education. We return now to misperceptions: why do agents underestimate the return to education for any extended period of time? In Section V.B we used Proposition 2 to argue that learning about education takes time, and that along the transition path we should expect agent-level underestimation. Here we examine this argument numerically, using stochastic simulations. Figure 4 displays the time paths of [xi]([H.sup.e] (i)) for an economy with 1,000 agent types, all of whom hold the same initial, pessimistic beliefs. The dark curve provides the average value of [xi] and the gray-shaded area identifies the cross-sectional location of 95% of the outcomes. The dashed horizontal line at 1 characterizes rational beliefs. We see that it takes nearly six periods for the 95% band to even include agents that have nonpessimistic forecasts, and even after 10 periods approximately 75% of the agents still underestimate the marginal benefit of education. We note that imposing instead that agents have diverse initial beliefs yields a qualitatively similar conclusion; we focus on homogeneous initial beliefs to emphasize how the model's inherent stochasticity quickly imparts heterogeneous beliefs on the economy's agents.

Our model's explanation of misperceptions lies in the specification and interpretation of the gain parameter [[gamma].sub.t]. This parameter measures the willingness of young agents to modify their beliefs; and in this sense it can be viewed as measuring the strength of the agents' priors: small values of the gain imply that the agent has strong priors and is thus unwilling to quickly alter his beliefs; large gain values imply weak convictions and the willingness to let new data dictate forecasts. We conclude that Jensen's results may be explained by either assuming agents have strong priors, or that they do not fully trust new information, and thus they choose small values of the gain for the updating algorithm.

VI. CONCLUSION

If households have inaccurate perceptions about the true returns to education, and these perceptions lead to underinvestment in human capital, then it becomes of critical importance to understand the mechanisms through which learning about the true returns takes place, and how this learning affects economic growth.

This paper explored these issues in a dynamic model that explicitly incorporates learning, as well as the possibility of expectations heterogeneity. It is found that multiple stable equilibria may exist, and regardless of the presence of equilibrium indeterminacy, learning dynamics, together with pessimistic beliefs can lead to persistent underinvestment in human capital.

APPENDIX

Here we justify the various assumptions made in the body of the paper, and provide the technical proofs. To facilitate exposition, this appendix is organized into two subsections: in the first, we focus on modeling assumptions and agent behavior, and we identify with greater care the associated rational equilibria; in the second we characterize learning equilibria more formally and provide the proofs to the propositions.

The Model under Rationality

Set [R.sub.+] = [0, [infinity]). Let r : [R.sub.+] [right arrow] (0, [infinity]) be differentiate and increasing. Let 0 < [[alpha].sub.L] < [[alpha].sub.H] < 1 and let [alpha]: [R.sub.+] [right arrow] [[[alpha].sub.L], [[alpha].sub.H] be continuously differentiate and increasing. Let [bar.h] [member of] [0, 1). The production function F : [0,1] X [R.sub.+] [right arrow] [R.sub.+] is given by

[mathematical expression not reproducible]

where [GAMMA](H) is a scalar which can be thought of as capturing TFP, and [alpha] is the potentially nonlinear response of elasticity to the expected agglomeration measure, which is assumed bounded between [[alpha].sub.L] and [[alpha].sub.H]. Finally, [bar.h] is the minimum education level needed to access the sophisticated technology F. Set

[??] = [[[alpha].sub.L] + [bar.h]]/[[[alpha].sub.L] + 1 > [bar.h]]

We make the following assumption on [GAMMA]:

(Al) [GAMMA]' (H)/[GAMMA] (H) > - log([??] - [??])[alpha]' (H).

We have the following result:

LEMMA 1. There exists a scalable family of functions increasing differentiable [GAMMA] satisfying Equation (9). Further, if [GAMMA] satisfies Equation (Al), and if h [member of] ([??], 1) and H [greater than or equal to] [??] then:

1. [F.sub.h] (h,H)> 0.

2. [F.sub.hh](h,H)< 0.

3. [mathematical expression not reproducible].

4. [mathematical expression not reproducible].

Proof. Let [delta] > 0 and consider the differential equation

(A2) [GAMMA]' = [GAMMA] x ([absolute value of log ([??] - [bar.h])] [alpha]' (H) + [delta]).

This ode is separable, and so may be solved using standard techniques to obtain

[GAMMA] (H) = [lambda]e ([|log([??]-[bar.h])][alpha](H)+[delta]-H),

where [lambda] is any positive scalar. Now simply notice that any solution to Equation (A2) also satisfies Equation (A 1).

Showing that F satisfies items (1) and (2) is trivial. To see that F also satisfies Equations (3) and (4), notice that

[mathematical expression not reproducible].

The signs of both of these partials are positive provided that

[GAMMA]' (H) + [GAMMA] (H) [alpha]'(H) log (h - [bar.h]) > 0,

which is equivalent to Equation (Al).

We conclude that provided we restrict attention to h [member of] ([??], 1) and H [greater than or equal to] [bar.h], F satisfies the desired properties.

Now that the production function has been defined with care, we turn to agent behavior and model equilibrium. For completeness and consistency, we assume that besides having access to the technology F, old agents may also use the same primitive technology they used when young. We make the additional assumption that using F is a full-time job: if the old agent directs any labor to the primitive technology, then he relinquishes access to the advanced technology. Since we assume that the agent must base all of his decisions on expectations (i.e., he must decide whether to use the advanced technology before knowing the value of H) we know that if the agent plans to use the primitive technology in the second period, he will necessarily choose h = 0. Therefore, since F(0, [H.sup.e]) = 0, we may model agent behavior as follows: let [delta] : [0,1] [right arrow] O, 1) be defined by [delta](l) = l and [delta](h) = 0 for 0 [less than or equal to] h < 1. The representative agent solves

[mathematical expression not reproducible].

As this formulation of the agent's problem makes clear, his choice of education, ft, depends on his expectations [H.sup.e]. Notice that since the marginal utility of consumption goes to infinity as consumption goes to zero, it follows that we can rule out the corner solution h = 1. We exogenously impose that [H.sup.e] [greater than or equal to] 0. There are two cases:

Case 1: [H.sup.e] < [bar.h]. If [H.sup.e] < [bar.h] then for any h, we have that F(h, [H.sup.e]) = 0. It follows that the representative agent sets h = 0.

Case 2: [H.sup.e] [greater than or equal to] [bar.h]. Now the agent has two choices. Either he can set h = 0, and thus receive total utility equal to zero, or the agent can choose some h > [bar.h]. Since we already know that h < 1, it follows that if h > [bar.h] then h will be chosen to satisfy the interior FOC given by

(A3) h [equivalent to] [??] ([H.sup.e]) + [bar.h]/ [alpha]([H.sup.e]) + 1 [member of] ([??], 1).

We conclude that the representative agent chooses h = [??] ([H.sup.e] if and only if [H.sup.e] [greater than or equal to] [bar.h] and the utility received by choosing h = [??] ([H.sup.e]) is greater than zero. But

0 < log (1 - [??] ([H.sup.e])) + log[GAMMA] ([H.sup.e]) + [alpha] ([H.sup.e]) log ([??] ([H.sup.e]) - [bar.h]) [??]

(A4) 0 < log (1 - [bar.h].1 + [[alpha].sub.H]) + [GAMMA] [bar.h] + [[alpha].sub.L]) log ([[alpha].sub.L]] x (1 - [bar.h])/1 + [[alpha].sub.L]),

where the implication uses the fact that [alpha] [member of] [[[alpha].sub.L], [[alpha].sub.H]] and [GAMMA]' > 0. Since r is scalable, we may assume that Equation (A4) holds. Thus, we assume here, and in the text, that if [H.sup.e] [greater than or equal to] [bar.h] then h = [??] ([H.sup.e]) as given by Equation (A3). We summarize these results in the following Lemma:

LEMMA 2. Assume Equations (Al) and (A4) hold, and let [A.sup.e] [greater than or equal to] 0. Representative agent behavior is given by

[mathematical expression not reproducible]

In the text we focused exclusively on the case [H.sup.e] [greater than or equal to] [bar.h]. Note that if [H.sup.e] [greater than or equal to] [bar.h] then h ([H.sup.e]) [greater than or equal to] [??]; thus, whenever the representative agent is accessing the advanced technology, the associated production function has the desired properties as listed in Lemma 1.

Now recall that a rational equilibrium is defined by any human capital level [H.sup.*] so that [??]([H.sup.*]) = [H.sup.*]. When [H.sup.*] = 0, the result is an "autarky" rational equilibrium: if agents expect that no one is accumulating human capital then his best response is also to forgo education and simply adopt the primitive technology in both periods. In the text, we focused on equilibria of the form [H.sup.*] [greater than or equal to] [bar.h]. Notice that, in this case, agent behavior is captured by their interior FOC, and h = [??] ([H.sup.*]) [greater than or equal to] [bar.h]: thus the production function has the desired properties.

Learning Equilibria

In the text, we identified learning equilibria with time paths of expectations generated by simple recursive algorithms capturing the weighted average of past observations. Recall that agent i"s learning mechanism is characterized by the following SRA:

(A5) [H.sup.e.sub.t+1] (i) = + [[gamma].sub.t] ([H.sub.t] + [[sigma].sub.i] [[member of].sub.t] (i) - [H.sup.e.sub.t] (i)),

where

[H.sub.t] = [N.summation over (j=1)] [[mu].sub.j] x [??] ([H.sup.e.sub.t] (J)).

To analyze this system we adopt the following notation:

(A6) [mathematical expression not reproducible]

(A7) T ([H.sup.e.sub.t]) = [N.summation over (j=1)][[mu].sub.j] x [??] ([H.sup.e.sub.t] (j))).

In aggregate, then, the system Equation (A5) may be written

(A8) [H.sup.e.sub.t+l] = [H.sup.e.sub.t] + [[gamma].sub.t] (T ([H.sup.e.sub.t]) [cross product] [l.sub.N] - [H.sup.e.sub,t] + [[xi].sub.t]).

To study the asymptotic behavior of Equation (A8), we appeal to Ljung's theory, as developed in Evans and Honkapohja (2001). The idea is to approximate the mean time path of [H.sup.e.sub.t] with the solution to a differential equation. To this end, let

g([H.sup.e]) = E (T ([H.sup.e]) [cross product] [1.sub.N] - [H.sup.e] + [[xi].sub.t]) = T([H.sup.e]) [cross product] [1.sub.N] - [H.sup.e],

and notice that g is continuously differentiable on ([??], 1). Ljung's theory provides conditions under which the differential equation

(A9) = d[H.sup.e]/d[tau] = g([H.sup.e])

offers locally a good approximation to the expected behavior of the stochastic process [H.sup.e.sub.t]. To make this statement precise, let [H.sup.*] identify a rational equilibria, and notice that, abusing notation somewhat by letting

[mathematical expression not reproducible]

is a rest point of the ode Equation (A9): g ([[??].sup.*]) = 0. Assume that the real parts of the eigenvalues of Dg ([[??].sup.*]) are negative, that is, [[??].sup.*] is a Lyapunov-stable rest point of Equation (A9). Then there exists an open set D [subset] [R.sup.N] containing [[??].sup.*] and a twice continuously differentiable Lyapunov function f : D [right arrow] [R.sub.+] so that, among other properties, f ([[??].sup.*]) = 0 and whenever [[??].sup.*] [not equal to] [H.sup.e] [member of] D it follows that f([H.sup.e])> 0 and g([H.sup.e])[not equal to] 0.

The function f identifies the local nature of the approximation, and tells us how to construct an appropriate projection facility: for c > 0, denote by 5(c) the lower contour set for f:

S(c)={[H.sup.e][member of] D :f ([H.sup.e]) [less than or equal to] c}.

Notice that there exists c so that c' [less than or equal to] c implies that S(c') is compact and int(S(c')) [not equal to] [empty set]. Because a is continuously differentiable, it follows that T is continuously differentiable, and so locally Lipschitz, which is a property necessary for our stability result. Therefore, we may choose c > 0 so that T is Lipschitz on S(c). Fix 0 < [c.sub.2] < [c.sub.1] < c, pick any [??] [member of] S ([c.sub.2]), and let P ([H.sup.e.sub.t], [C.sub.1]) be the statement

[H.sup.e.sub.t] + [[gamma].sub.t] (T [[H.sup.e.sub.t]] [cross product] [1.sub.N] - [H.sup.e.sub.t] + [[xi].sub.t])) [member of] s ([C.sub.l]).

Now augment the recursive algorithm Equation (A8) as follows:

(A10) [mathematical expression not reproducible].

The modified learning algorithm (Equation (A 10)) incorporates the projection facility: any time agents' expectations deviate too sharply from rationality--that is, fall outside the set S([c.sub.1]))--they are placed back to an arbitrary point within the set S([c.sub.2]). The employment of a projection facility is standard in the learning literature--see Evans and Honkapohja (2001) for discussion and details--and is necessary to obtain almost sure convergence: it is always possible that an unusual sequences of shocks will push expectations so far away from the steady state that return is not possible. On the other hand, the learning algorithm can be further modified so that the probability that the projection facility is used is arbitrarily small.

For the algorithm (Equation (A 10)) to be well-defined, we require that [[??].sup.*] is a Lyapunov-stable rest point of Equation (A9). We have the following lemma:

LEMMA 3. If

(A11) [alpha]' ([H.sup.*]) < (1 + [alpha] ([H.sup.*])).sup.2] 1 - [bar.h],

then the real parts of the eigenvalues of Dg ([[??].sup.*]) are negative, Proof. We compute

Dg ([[??].sup.*]) = DT ([[??].sup.*])[cross product][1.sub.N] - [I.sub.N].

Let eig(M) be the collection of eigenvalues of a given square matrix M. It suffices to show that the real parts of eig (DT ([[??].sup.*]) [cross product] [1.sub.N]) are less than unity. Since DT ([[??].sup.*]) [cross product] is an N x N matrix with N identical rows, it has rank 1; thus 0 is an eigenvalue of multiplicity N - 1. The remaining eigenvalue corresponds to the eigenvector [1.sub.N]:

(DT ([[??].sup.*])[cross product] [1.sub.N]) x [l.sub.N] = ([N.summation over (i=1)]T [H.sup.e](i)([[??].sup.*])) [cross product] [l.sub.N] = [lambda] x [1.sub.N],

where

[mathematical expression not reproducible]

Finally,

[mathematical expression not reproducible]

and the result follows.

We are now ready to state our result:

PROPOSITION 3. Assume [SIGMA][[gamma].sub.t] = [infinity] and [SIGMA] [[gamma].sup.2.sub.t] < [infinity]. If [H.sup.*] is a rational equilibrium of the model satisfying

(A12) [alpha] < ([H.sup.*]) < (1 + [alpha] ([H.sup.*])).sup.2]/1 - [bar.h],

then [H.sup.e.sub.t] as determined by Equation (A1O), converges to [[??].sup.*] almost surely.

Proof. Given our construction of the algorithm (Equation (A 10)), and the assumptions imposed on [alpha], the proof of this proposition follows from Lemma 3 and from Corollary 6.8, on page 136 of Evans and Honkapohja (2001).

Proposition 3 addresses the decreasing gain case: [[gamma].sub.t] [right arrow] 0. Now we analyze the behavior of the learning algorithm under constant gain:

(A13) [H.sup.e.sub.t+1] = [H.sup.e.sub.t] + [gamma] (T ([H.sup.e.sub.t]) [cross product] [1.sub.N] - [H.sup.e.sub.t] + [[xi].sub.t]),

where [gamma] > 0 is small. Since [H.sup.e.sub.t] = G ([H.sup.e.sub.t-1]) + [gamma][[xi].sub.t] for appropriate G, it follows that [H.sup.e.sub.t] will never "settle down," that is, converge to a degenerate random variable. However, results from Benveniste et al. (1990) provide for short sample analysis. In particular, the results provide the precise sense in which the solution to the ode Equation (A9) approximates the expected time path of [H.sup.e.sub.t]. To compare solutions to Equation (A9) with realizations of the stochastic process [H.sup.e.sub.t], two adjustments must be made: [H.sup.e.sub.t] must be defined for all real t, not just for integer values of t: and the time-scale for [H.sup.e.sub.t] must be adjusted. To this end, let [H.sup.e.sub.n] ([H.sup.e.sub.0]) be defined by Equation (A 13) with t replaced by n and with initial condition [H.sup.e.sub.0], and set

[[??].sup.e.sub.t] ([gamma], [H.sup.e.sub.0])= [H.sup.e.sub.n] ([H.sup.e.sub.0]) whenever [gamma]n [less than or equal to] t [less than or equal to] [gamma] (n + 1).

So [[??].sup.e.sub.t] ([gamma], [H.sup.e.sub.0]) is a step function with heights defined by [H.sup.e.sub.n] and with bins of width [gamma] identifying the adjusted timescale.

Now let [[??].sup.e.sub.t] ([H.sup.e.sub.0]) be the solution to the ode Equation (A9) corresponding to the initial condition [H.sup.e.sub.0]. Let 5(c) as above, with [[??].sup.e.sub.t] ([H.sup.e.sub.0]) [member of] S (c) for 0 [less than or equal to] t [less than or equal to] T. Finally, assume [alpha] is twice continuously differentiable. We have the following result:

PROPOSITION 4. If [H.sup.*] is a rational equilibrium of the model satisfying

[alpha]' ([H.sup.*]) < [(1 + [alpha]([H.sup.*])).sup.2]/1 - [bar.h],

then there is a continuous time stochastic process [y.sub.t] with y(0) = 0 so that the continuous time process

[[gamma].sup.-1/2] ([[??].sup.e.sub.t] ([gamma], [[??].sup.e.sub.0]) - [[??].sup.e.sub.t] ([H.sup.e.sub.0]))

converges weakly to [y.sub.t] for 0 [less than or equal to] t [less than or equal to] T. Furthermore, E[y.sub.t] = 0. (13)

This proposition follows immediately from our assumptions on a and from Proposition 7.8 of Evans and Honkapohja (2001, 163). Intuitively, we conclude that for small [gamma] and finite horizons, the solution to the ode Equation (A9) well-approximates the expected behavior of the process [H.sup.e.sub.t], provided that the timescale is appropriately adjusted.

ABBREVIATIONS

FOC: First-Order Condition

REE: Rational Expectations Equilibria

SRA: Stochastic Recursive Algorithm

TFP: Total Factor Productivity

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(1.) This self-reinforcing feedback dynamic is likely an accurate description of low-income countries, which are known to have large social returns to education.

(2.) For example, there has been some concern over potential mis-measurement of the true returns to education. However Duflo (2001) uses a policy experiment and finds an average return to a year of schooling of 7.8% in Indonesia, and Emerson and Souza (2011) use instrumental variables techniques and find average returns to a year of education in Brazil to be 13.4%.

(3.) This yeoman farmer assumption is a technical device which helps expose the salient features of the model; however, we could equally develop the arguments by assuming competitive goods and labor markets, and inelastic labor supply.

(4.) We focus on point expectations here because the model is nonstochastic.

(5.) In the Appendix, we allow the old agent to also access the primitive technology, and then establish conditions sufficient to guarantee that he will not choose to do so.

(6.) Also, it is shown in the Appendix that the young will choose h > [bar.h] so that F(h, H) is well-defined.

(7.) As is common in learning models of this type, here we are adopting "point expectations," that is, the agent makes decisions based on the assumption her estimate [H.sup.e.sub.t+1]. ([[omega].sub.t]) is correct.

(8.) These assumptions could be relaxed considerably without changing the results, but the details would be distracting.

(9.) For simplicity, we assume that all agent types use the same gain, but again, this is not essential for our results.

(10.) This is remarkable because the graphical depiction of rational equilibria has no a priori relation to the learning model's dynamics.

(11.) As the skeptical reader might have noticed, the formal statement requires scaling time so that the continuous process [[??].sup.e.sub.t] ([H.sup.e.sub.0] (i)) and the discrete process [H.sup.e.sub.i] (i) are comparable: see the Appendix for details.

(12.) The precise form of the production function used for simulations is given in the Appendix.

(13.) The process [y.sub.t] is determined as the solution to the stochastic differential equation dy (t) = h' ([[??].sup.e.sub.t] ([H.sup.e.sub.0]))y(t)dt + [[sigma].sub.[member of]] dw(t), where w(t) is a standard Wiener process.

Emerson: Professor, Department of Economics, Oregon State University, Corvallis, OR 97331. Phone 541 737 1479, Fax 541 737 5917, E-mail patrick.emerson@oregonstate.edu

McGough: Professor, Department of Economics, University of Oregon, Eugene, OR 97403. Phone 541 602 4122, E-mail bmcgough@uoregon.edu

doi: 10.1111/ecin.12487

Online Early publication September 1, 2017

Caption: FIGURE 1 Rational Expectations Equilibria

Caption: FIGURE 2 Convergence with Different Sets of Initial Beliefs

Caption: FIGURE 3 Convergence with the Same Sets of Initial Beliefs

Caption: FIGURE 4 Underestimating the Returns to Education