The effects of asymmetric information on economic growth.

I. Introduction

It has been widely recognized that asymmetric information affects investment, and investment is of course in turn important to economic growth. Knowledge of the mechanisms by which asymmetric information affects economic growth, however, appears incomplete. There are two types of asymmetric information, namely, hidden information and hidden action [1]. Although Azariadis and Smith have studied how hidden information (adverse selection) may affect economic growth with a monetary growth model [3] and a non-monetary growth model [2], the effects of hidden action and costly state verification have been ignored.

There have been many studies which at least suggest links between asymmetric information and economic growth. Diamond [6], Boyd and Prescott [5], and Williamson [16](1) show that asymmetric information between borrowers and lenders gives rise to financial intermediation. Diamond [6] and Williamson [16] show how financial intermediation can promote economic growth by reducing monitoring costs and channeling funds to the most efficient users. Greenwood and Jovanovic [9] show how collection and analysis of information by financial intermediaries can help investors' resources to flow to the place in the economic system where the funds will yield the highest social return. Bencivenga and Smith [4] show how a reduction of the fraction of savings held in the form of unproductive liquid assets is favorable to capital accumulation and economy growth. While contributing to the literatures of financial intermediation and economic growth greatly, these studies, however, do not specifically examine the relationship of asymmetric information to economic growth.

This paper attempts to explore how hidden action (moral hazard) and costly state verification affect economic growth. Hidden action is incorporated into a neoclassical growth model, in which people invest for future consumption, and investment leads to capital accumulation and, thus, economic growth. Borrowers have an informational advantage over lenders ex post concerning the outcome of investment project. Though verification can overcome such asymmetry, it is costly.

The results of this paper suggest that the effect of hidden action is to alter the path of economic growth. In the neoclassical growth model without asymmetric information, the economy grows at a constant rate. With hidden action, however, the growth path depends upon the stage of economic development. In the early stages of development, when capital is generally inadequate, verification of outcomes is relatively expensive. Agents cannot afford verification, so entrepreneurs issue only debt, not equity. In this case, lenders absorb all risks, interest rates tend to be low, and entrepreneurs tend to have high incomes. The tendency is thus for rapid capital accumulation and high economic growth.

In intermediate stages of development, verification is less prohibitive, and the growth rate is reduced because verification takes resources away from capital accumulation. In advanced stages of growth, however, verification costs are likely to be small relative to accumulated capital. Thus, the effect of hidden action steadily declines through the intermediate and advanced stages of growth.

The "poverty trap" phenomenon is a possibility. This happens when investment is insufficient to maintain desired levels of economic growth. Once in the trap, poverty may persist and external capital appears to be the only solution.

The results of this paper conflict with the usual conclusions of economic growth theory: this paper finds that low interest rates can promote economic growth, whereas in the literature of economic growth, high growth rates are usually associated with high interest rates. The usual argument in growth theory is that high interest rates convince households to consume more in the future, leading to high level of capital accumulation and economic growth [10]. The finding of this paper, however, suggest that low interest rates encourage entrepreneurs to borrow, which tends to lead to increased entrepreneurial income and capital accumulation. The apparent contradiction between the findings of this paper and growth theory suggests that further study is needed.

The rest of this paper continues as follows: Section II spells out the model. Section III contains a discussion of the effect of hidden action on the short-run growth rates. Section IV traces the growth path of an economy. Section V considers the long-run impact of hidden action. Section VI contains concluding remarks.

II. The Model

Consider an overlapping generations model with agents who enjoy a three-period life span. In each generation, there are N members, where N is an even and a fixed number (i.e., there is no population growth in this economy).

There is a single good, which can be called a capital good when used in production but is otherwise consumed. Young agents have no capital endowments, but are endowed with a project that transforms time t's good into time t + 1's good. Thus, they must borrow capital goods when they are young to produce consumption goods.

Middle-aged agents possess the output derived from the production initiated when they were young. Middle-aged agents will partition their output pie into four pieces: a piece to repay what was borrowed from the current old, a piece as investment or savings for future consumption, a piece as a bequest to the current young,(2) and a piece for their own current consumption. It is assumed that each initial middle-aged agent is endowed with a certain amount of the consumption good.

An old agent consumes only what is returned from the current middle-aged agents and then dies. It is assumed that the initial old agents are endowed with a certain amount of the consumption good.

Agents in the same generation are identical and are risk-averters. A typical agent's preferences are represented by the following utility function:

U([C.sub.t,t+1) + [Beta]U([C.sub.t,t+2]) + dU([D.sub.t,t+1])

where C is consumption, D is the amount of the bequest, [Beta] [is greater than] 0 is a discount factor for future consumption, d [is greater than] 0 also is a discount factor for leaving bequest, and the first subscript and the second subscript denote the generation and the time period, respectively. Notice that the utility is a function of bequest.(3) This is justified by the fact that people do leave bequests and enjoy doing so, and that people give money away to charities as a part of their financial plans. Without real loss of generality, we assume that U([multiplied by]) takes a logarithmic form.

The outcome of a young agent's project is uncertain. Suppose for simplicity that half of the N available projects are poor because they transform k input units into bk units of output instead of a more desirable gk units of output. Assume here that b [is less than] g and 1 + b [is greater than] 0. Given the uncertain production technology, the young agent can determine whether the project is good or bad only when output becomes available. He can do this simply by measuring the amount of output produced. We assume for the moment that information about the outcome of production is available to everyone free of cost. Due to the uncertainty, every young agent faces two possibilities: having a good project (gk) or a bad project (bk). Young agent's economic decisions are therefore project-dependent. Let

[B.sub.j,t](s) = individual bond holdings of generation j in period t when a holding project is s;

[k.sub.j,t](s) = individual capital holdings of generation j in period t when the project is s;

[r.sub.t](s) = rate of return on bond holdings from t to t + 1 when the project is s;

[C.sub.j,t](s) = individual consumption of generation j in period t when the project is s;

[D.sub.j,t](s) = bequest from generation j to generation t when the project is s.

Under the above assumptions and notations, the agent's utility maximization problem is thus

(1) max[U([C.sub.t,t+1](g)) + [Beta]U([C.sub.t,t+2](g)) + dU([D.sub.t,t+1](g))]/2 +[U([C.sub.t,t+1(b)) + [Beta] U([C.sub.t,t+2(b)) + dU([D.sub.t,t+1(b))]/2 s.t. [B.sub.t,t] + [k.sub.t,t] = [D.sub.t-1,t]

(2) [C.sub.t,t+1(g) + [B.sub.t,t+1](g) + [D.sub.t,t+1(g) = (1 + g)[k.sub.t,t] = (1 + [r.sub.t](g))[B.sub.t,t]

(3) [C.sub.t,t+1](b) + [B.sub.t,t+1(b) = (1 + b)[k.sub.t,t] + (1 + [r.sub.t](b))[B.sub.t,t]

(4) [C.sub.t,t+2](g) = (1 +[r.sub.t+1])[B.sub.t,t+1](g)

(5) [C.sub.t,t+2](b) = (1 + [r.sub.t+1][B.sub.t,t+1](b)

[C.sub.t,t+1](s) [is greater than or equal to] 0, [C.sub.t,t+2(s) [is greater than or equal to] 0, [k.sub.t,t] [is greater than or equal to] 0, [D.sub.t,t+1](s) [is greater than or equal to] 0, given [D.sub.t-1,t] [is greater than] 0, where

[D.sub.t-1,t] = 0.5[D.sub.t-1,t](g) + 0.5[D.sub.t-1,t](b), [r.sub.t+1] = 0.5[r.sub.t+1](g) + 0.5[r.sub.t+1](b).

In the above utility maximization problem, (1) is the agent's first-period budget constraint, (2) and (3) are the second-period budget constraints, and (4) and (5) are the third-period budget constraints. [D.sub.t-1,t] takes the above form because each agent within the same generation is identical, and gets the same bequest amount. When a risk averse agent invests, the agent diversifies wealth among investment projects. Since each project has an equal possibility being good or bad, and agents do not know the types of projects in advance, they will have to diversify fully. Therefore, [r.sub.t+1] has the above form.

The market clearing conditions for bonds and goods are, respectively,

(6) [B.sub.t,t] + 0.5[B.sub.t-1,t](g) + 0.5[B.sub.t-1,t(b) = 0,

(7) 0.5([C.sub.t-2,t(g) + [C.sub.t-2,t](B)) + 0.5([C.sub.t-1,t(g) + [C.sub.t-1,t](B)) + [k.sub.t,t] = (1 + 0.5(g + b)[k.sub.t-1, t-1].

As assumed, U(c) = ln c, it follows from the first-order conditions, that

(8) [r.sub.t](s) = s, s + or g or b

[C.sub.t,t+1](s) = (1 + s)[[D.sub.t-1,t]/(1 + [Beta] + d), s = g or b

[C.sub.t,t+2](s) = [Beta] (1 + [r.sub.t+1) (1 + s) [D.sub.t-1,t]/(1 + Beta + d), s = g or b

[B.sub.t,t+1](s) = [Beta (1 + s)[D.sub.t-1,t]/(1 + [Beta] + d), s = g or b

[D.sub.t,t+1](s) = d(1 + s)[D.sub.t-1,t]/(1 + [Beta] + d), s + g or b.

(8) shows that the market interest rate will equal the rates of return of the projects--as expected of a neoclassical growth model with constant returns to scale technology.

To find [k.sub.t,t], apply (1) and (6), and use the above results, obtaining,

[k.sub.t,t] = [D.sub.t-1,t] + [Beta] (1 + 0.5)(g + b))[D.sub.t-2,t-1]/(1 + [Beta] = d).

Since [D.sub.t,t+1] = 0.5([D.sub.t,t+1](g) + [D.sub.t,t+1(b)),

[D.sub.t,t+1] = d(1 + 0.5(g + b))[D.sub.t-2,t-1]/(1 + [Beta] + d), or

[D.sub.t-1,t] = [d(1 + 0.5(g + b))[D.sub.t-2,t-1]/(1 + [Beta] + d).

Therefore, [k.sub.t,t] = ([Beta] + d)[D.sub.t-1,t]/d. Capital accumulation in period t is thus

(9) [D.sub.t,t+1] = d(1 + 0.5(g + b))[D.sub.t-1,t]/(1 + [Beta] + d). [k.sub.t+1,t+1] = d(1 + 0.5(g + b))[k.sub.t,t]/(1 + [Beta] + d].

Defining [Q.sub.t] as total production in period t and using (9), the rate of production growth from t to t + 1 is

(10) [Q.sub.t+1]/[Q.sub.t] = d(1 + 0.5(g + b))/(1 + [Beta] + d).

Assuming d(1 + 0.5(g + b))/(1 + [Beta] + d) [is greater than] 1 (to assure that growth is positive), note that the right side of (10) is independent of t. Thus, without asymmetric information, production grows at the same constant rate as capital. This is not unusual since the model becomes equivalent to a neoclassical growth model without asymmetric information and the neoclassical growth models produce similar result.

III. Short-Run Economic Growth with Asymmetric Information

We now consider the economic growth in an environment in which the young can observe the outcome of their production, but the others can observe outcomes only by costly verification. Verification is costly because it requires forfeiture of some of the real product.(4)

We assume that a borrower asks for verification whenever desired, thereby incurring a verification cost.(5) We also assume that the verification cost is an increasing function of the level of debt.(6) For simplicity, we assume that verification cost follows the simple linear function,

VC = [micro] + ek,

where [micro] [is greater than] 0 and e [is greater than] 0 are constants, and k is the level of debt. This reflects the reasonable assumption that the firm has to pay a fixed retainer to initiate a bankruptcy process, but that the total cost increases with the level of debt. This simple linear form could also be thought of as a linear approximation of a monotonically increasing nonlinear verification cost function.

It has been established that with such an information structure, investment decision is part of a financial contract which specifies the level of investment, the payoff to the lender, and a verification region. Moreover, an optimal contract is a form of debt contract, as described in Townsend [15] and in Gale and Hellwig [8]. Townsend's results could be directly applied to the problem at hand, but another approach seems to lead clearer conclusions for the purposes of this paper. In this paper, we choose first to characterize the verification region and corresponding payoff schedule (the major characteristics of the debt contract). These are then incorporated into the young agent's maximization problem.

The characterizations of the verification region and corresponding payoff schedule are summarized in Lemma 1 in Appendix A. Lemma 1 shows that verification will take place only in bad state when verification cost is small. This is consistent with the results by Townsend [15] and Gale and Hellwig [8], which show that an optimal contract is a form of debt contract in which the rate of return is state contingent for some states (bad states) and state independent for the others (good states); verification takes place only in the bad states.

Lemma 1 implies that agents will always ask for verification only when a bad project is realized and verification cost is relative small. Incorporating this result into an agent's decision making process, the agent's utility maximization problem becomes:

max[U([C.sub.t,t+1(g)) + [Beta] U([C.sub.t,t+2](g)) + dU([D.sub.t,t+1](g))]/2

+ [U([C.sub.t,t+1](b)) + [Beta] U([C.sub.t,t+2(b)) + dU([D.sub.t,t+1(b))]/2

s.t. [B.sub.t,t] + [k.sub.t,t] = [D.sub.t-1,t],

[C.sub.t,t+1(g) + [B.sub.t,t+1](g) + [D.sub.t,t+1](g) + (1 + g)[k.sub.t,t] + (1 + [r.sub.t](g))[B.sub.t,t],

[C.sub.t,t+1](b) + [B.sub.t,t+1](b) + [D.sub.t,t+1](b) + ([micro] + [e[k.sub.t,t]) = (1 + b)[k.sub.t,t] + (1 + [r.sub.t](b))[B.sub.t,t],

[C.sub.t,t+2](g) = (1 + [r.sub.t+1)[B.sub.t,t+1](g),

[C.sub.t,t+2](b) = (1 + [r.sub.t+1][B.sub.t,t+1](b),

[C.sub.t,t+1](s) [is greater than or equal to] 0, [C.sub.t,t+2](s) [is greater than or equal to] 0, [D.sub.t,t+1](s) [is greater than or equal to] 0, given [D.sub.t-1,t], [is greater than] 0, where

[D.sub.t-1,t] = 0.5[D.sub.t-1,t(g) + 0.5[D.sub.t-1,t](b), [r.sub.t+1](g) + 0.5[r.sub.t+1](b).

Due to the resource consuming nature of the verification process, the market clearing condition (7) needs to be replaced by

0.5([C.sub.t-2,t](g) + [C.sub.t-2,t](b)) + 0.5([C.sub.t-1,t](b) + [k.sub.t,t] + 0.59[micro] + e[k.sub.t,t]

= (1 + 0.5(g + b))[k.sub.t-1,t-1].

The lower bound of b also needs to be modified too. We assume that b satisfies 1 + b-e [is greater than or equal to] [micro][D.sub.0,1] which guarantees that (1 + b)[k.sub.t,t] + (1 + [r.sub.t](b))[B.sub.t,t] - VC [is greater than or equal to] 0.

Solving for the modified problem, the capital accumulation is now

(11) [k.sub.t+1,t+1] = d(1 + 0.5(g + b - e))[k.sub.t,t]/(1 + [Beta] + d) - 0.5(d +[Beta])/ [micro] /(1 + [Beta] + d)

and the rate of production growth from t to t + 1 is

(12) [Q.sub.t+1]/[Q.sub.t] = d(1 + 0.5(g + b - e)/(1 + [Beta] + d) - [0.5d [micro] /(1 + [Beta] + d)]/[D.sub.t-1,t].

To make the matter interesting, we assume d (1 + 0.5(g + b - e))/(1 + [Beta] + d)] [is greater than] 1. Comparison of (12) with (10) shows that hidden action negatively affects the growth rates, when [D.sub.t-1, t] [is greater than] 0. Intuitively, hidden action requires resources to overcome informational friction, leaving less capital. The economy thus grows at a lower rate. However, we may have a different story when VC is relatively large. As the next section shows, the effects can be positive under certain circumstances.

IV. Transition

When a country is undeveloped, VC can be large relative to investment and verification may not take place. In this section, we explore the conditions under which a young agent is likely to ask for verification. Let

[R.sub.g] = payments to lenders when the project is good, and

[R.sub.b] = payments to lenders when the project is bad.

In the event of a bad project, a young agent will ask for verification only if

(13) [R.sub.b] + VC [is less than] [R.sub.g]

The left-hand side of (13) is the amount a young agent must pay with verification. Note that

[R.sub.g] = - (1 + [r.sub.t](g))[B.sub.t,t] and [R.sub.b] = - (1 + [r.sub.t](b))[B.sub.t,t],

[k.sub.t,t] = (d + [Beta])[D.sub.t-1,t]/d, and - [B.sub.t,t] = [Beta][D.sub.t-1,t/d.(7)

(13) can be rewritten as

[micro] [is less than] [([r.sub.t](g) - [r.sub.t](b)) [Beta] /d - e(d + [Beta])/d][D.sub.t-1,t].

When the young do ask for verification, in solving the utility maximization problem in the previous section, we have [r.sub.t](g) = g, [r.sub.t](b) = b - e. So (13) becomes

(14) [D.sub.t-1,t] [is greater than] d[micro]/d[micro][(g - b)[Beta] - ed].

When the bequest a young agent receives from middle-aged agents satisfies (14), the young agent will have incentive to ask for verification, and it appears that (14) may not hold if [micro] is large enough. Interestingly, it can be shown that (14) will eventually be satisfied regardless of the size of [micro].

Suppose that [micro] is so large that [D.sub.0,1] does not satisfy (14). Investors will perceive the fact that the young agent does not ask for verification as a "good project" signal and therefore demand payment of [R.sub.g]. Thus, the young agent's utility maximization problem must be modified by simply replacing (3) with

[C.sub.t,t+1](b) + [B.sub.t,t+1](b) + [D.sub.t,t+1](b) = (1 + b)[k.sub.t,t] + (1 + [r.sub.t](g))[B.sub.t,t].

Because there is only one interest rate regardless of the projects, we can write [r.sub.t](g) as [r.sub.t]. Solving the modified problem, capital accumulation is now

(15) [k.sub.t+1,t+1] = 0.5(g + b)(d + [Beta] + d - [r.sub.t] [Beta]][k.sub.t,t]/(1 + [Beta] + d)

where r, is a solution of the following quadratic equation with b [is less than] [r.sub.t] [is less than] g:

(16) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Since the parameters in (16) are all independent of t, [r.sub.t] is constant over time. Therefore, we can omit subscript t from [r.sub.t]. Moreover, the following lemma establishes that r is close to b.

LEMMA 2. r [is less than] 0.5(g + b).

The proof of Lemma 2 is placed in Appendix B.

Lemma 2 shows that without verification, the interest rate is always less than 0.5(g + b). Recall that the expected interest rate is equal to 0.5(g + b) in the case of full information, and 0.5(g + b - e) in the case of hidden action with verification. Since r [is less than] 0.5(g + b), it is possible that r [is less than] 0.5(g + b - e). This means that it is possible that the interest rate with hidden action and without verification may be lower than that with verification. This is possible because when young agents who do not call for verification absorb all risks. Investors get the same interest rate regardless of states, and therefore are risk free, but they must pay a risk premium. The reduction in interest rates reflects the risk premium.

With Lemma 2, it can be shown that for any VC [is greater than] 0, there is always a time T after which verification will take place. In expression (15), consider the rate of capital accumulation

[0.5(g + b)(d + [Beta]) + d - r [Beta]]/(1 + [Beta] + d) = (1 + 0.5(g + b))d/(1 + [Beta] + d) + (0.5(g + b) - r)[Beta]/(1 + [Beta] + d). Since (1 + 0.5(g + b))d/(1 + [Beta] + d) [is greater than] 1 under Lemma 2,

[0.5(g + b)(d + [Beta])+ d - r [Beta]]/(1 + [Beta] + d) [is greater than] 1.

Define [Delta] = [0.5(g + b)(d + b) + d + r[Beta]/(1 + [Beta] + d), then [k.sub.t+1,t+1] = [Delta][k.sub.t,t] = [[Delta].sup.t][k.sub.1,1] and [D.sub.t-1,t] = [[Delta].sup.t-1] [D.sub.0,1]. Given [D.sub.0,1], for any [micro] [is greater than] 0, since [Delta] [is greater than] 1, there exists T such that for all t [is greater than or equal to] T,

[[Delta].sup.t-1] [D.sub.0,1] [is greater than] d[micro]/ [(g - b) [Beta] - ed].

That is [D.sub.t-1,t] [is greater than] d[micro]/[(g - b)[Beta] - ed] and condition (14) holds for all t [is greater than or equal to] T. Therefore, no matter how large VC is, there is always a T after which verification will take place, regardless of the size of VC. The intuition behind this is straightforward. When agents start with low endowments, they can not afford verification because the cost is relatively high. As the economy grows, the agent's wealth increases, and verification cost becomes small relative to investment. Investment increases as the economy grows, the difference between [R.sub.g] and [R.sub.b] increases and eventually exceeds the verification cost.

The above discussion reveals an interesting path of economic growth. Note that

[Q.sub.t+1] = N(1 + 0.5(g + b))[k.sub.t+1,t+1] and

[k.sub.[[t].sub.1],t+1] = [0.5(g + b)(d + [Beta]) + d - r [Beta]] [k.sub.t,t]/(1 + [Beta] + d),

then

[Q.sub.t+1]/[Q.sub.t] = [0.5(g + b)(d + [Beta]) + d - r[Beta]]/ (1 + [Beta] + d)

= (1 + 0.5(g + b))d/(1 + [Beta] + d) + (0.5(g + b) - r) [Beta](1 + d).

It is useful to denote the growth rate under hidden action but without verification as [[Lambda].sub.3], the rate with full action as [[Lambda].sub.1], and the rate with hidden action and verification as [[Lambda].sub.2](t). By Lemma 2,

[[Lambda].sub.3] [is greater than] (1 + 0.5(g + b))d/(1 + [Beta] + d).

On the other hand, from (10),

[[Lambda].sub.1 = (1 + 0.5(g + b))d/(1 + [Beta] + d)

and from (12),

[[Lambda].sub.2](t) = (1 + 0.5(g + b - e))d/(1 + [Beta] + d) - [0.5 [micro] d/(1 + [Beta] + d)]/[D.sub.t-1,t].

Therefore, we have

[[Lambda].sub.3] [is greater than] [[Lambda].sub.1] [is greater than] [[Lambda].sub.2](t).

That is, the growth rate under hidden action but no verification ([[Lambda].sup.3]) is higher than the rate with full information ([[Lambda].sub.2]), which is higher than the rate under hidden action and verification ([[Lambda].sup.2](t)).

[[Lambda].sub.1] is higher than [[Lambda].sub.2](t) because hidden action and informational friction necessitates verification, which consumes resources. With less resources available for capital accumulation, the growth rate is lower than under full information. Surprisingly, however, the growth rate under hidden action without verification is even higher than the full informational growth rate [[Lambda].sub.1].

[[Lambda].sub.3] exceeds [[Lambda].sub.1] because of a lower interest rate. As Lemma 2 showed, the interest rate is less when agents have no incentive to spend resources on overcoming informational frictions. This lower interest rate increases opportunities for young agents to borrow capital and make profits. Recall that under full information, young agents can only make zero profits. However, Lemma 2 shows that when information is asymmetric and young agents do not call for verification, the lower interest rate makes positive profits possible, and young agents will borrow more. A lower interest rate also means that middle-aged agents have become richer through projects initiated when they were young. Greater wealth thus means that middle-aged agents save more, even at fixed rates of saving. Thus, both the rate of capital accumulation and the rate of economic growth are higher than under symmetric information.

We can now trace the growth path, which is shown in Figure 1. The solid curve expresses the growth path of an economy affected by hidden action, and the dashed line shows the growth path of a neoclassical economy without asymmetric information. The solid curve shows that when a country is undeveloped, the country may grow at a high rate because the lack of verification leading to a low interest rate. At some point, entrepreneurs start to request verification, and the growth rate will be reduced dramatically. As growth continues, the rate of growth will increase as the ratio of available capital to verification cost rises. In sum, growth shows a high-low-moderate path, quite different from that obtained in a neoclassical growth models and from some endogenous growth models [14]. The growth path shown in Figure 1 is consistent with the observation that a richer country can grow faster than a poorer country but a poorer country can also grow faster than a richer country, as has been observed [12; 13]

[Figure 1 ILLUSTRATION OMITTED]

V. Long-Run Growth with Hidden Action

To see the long-run effect of hidden action on the growth rate, we need the following lemma:

LEMMA 3. Let [X.sub.n] = a[X.sub.n-1] - [a.sub.1], where a and [a.sub.1] are constants. If a [is greater than] 1, then as n [right arrow] [infinity], [X.sub.n] [right arrow] + [infinity] when [X.sub.0] [is greater than] [a.sub.1]/(a - 1), and [X.sub.n] [right arrow], and [X.sub.n] [right arrow] - [infinity] when [X.sub.0] [is less than] [a.sub.1]/(a - 1).

The proof of Lemma 3 is in Appendix B.

Let a = d(1 + 0.5(g + b - e))/(1 + [Beta] + d), [a.sub.1] = 0.5d[micro]/(1 + [Beta] + d). It can be shown that, [D.sub.t-1,t] = a[D.sub.t-2,t-1] - [a.sub.1]. Since a [is greater than] 1, the limit of [D.sub.t-1,t] depends upon whether [D.sub.0,1] [is greater than] [a.sub.1]/(a - 1) or [D.sub.0,1] [is less than] [a.sub.1]/(a - 1). By Lemma 3, if [D.sub.0,1] [is greater than] [a.sub.1]/(a - 1) [D.sub.t-1,t] [right arrow] + [infinity], and if [D.sub.0,1] [is less than] [a.sub.1]/(a - 1), [D.sub.t-1,t] [right arrow] - [infinity] as t [right arrow] [infinity].

First consider [D.sub.0,1] [is greater than] [a.sub.1]/(a - 1). As t [right arrow] [infinity] [D.sub.t-1,t]. Therefore, as t [right arrow] [infinity], equation (12) becomes

(17) [Q.sub.t+1]/[Q.sub.t] [right arrow] d(1 + 0.5(g + b - e))/ (1 + [Beta] + d).

(17) indicates that the growth rate under hidden action and state verification will converge to a constant rate of d(1 + 0.5(g + b - e))/(1 + [Beta] + d) from below. That is, the long-run growth rate is about d(1 + 0.5(g + b - e))/(1 + [Beta] + d), higher than the short-run growth rate but lower than the growth rate with full information. This is because the fixed part of the verification cost becomes insignificant in the long-run, thus reducing some of the influence of hidden action. The variable part of verification cost, however, still remains.

Perhaps a more interesting long-run effect is with [D.sub.0,1] [is less than] [a.sub.1]/(a - 1). Lemma 3 shows that if [D.sub.0,1] [is less than] [a.sub.1]/(a - 1), [D.sub.t-1,t] [right arrow] -[infinity]. From a purely algebraic point of view, when we start with a small [D.sub.0,1] [D.sub.0,1] [is less than] [a.sub.1], then there exits such a period after which all [D.sub.t-1,t]'s will be negative. Of course, negative bequests are not allowed by assumption, and, in fact, will not occur as we shall show. It is possible, however, that an economy may fall into a "poverty trap".

Recall that the dynamics of [D.sub.t-1,t] is governed, under verification, by

(18) [D.sub.t,t+1] = a[D.sub.t-1,t] - [a.sub.1]

and no verification, by

(19) [D.sub.t,t+1] = [Delta][D.sub.t-1,t].

Whether (18) or (19) prevails depends on the value of [D.sub.t-1,t]. As has been shown in section IV, if [D.sub.0,1] [is less than] d[micro]/[(g - b) [Beta] - ed], [D.sub.t,t+1] = [Delta][D.sub.t-1,t]. Since [infinity] [is greater than] [D.sub.t,t+1] increases over time. On the other hand, if [D.sub.0,1] [is less than] [a.sub.1]/(a - 1) [D.sub.t,t+1] = a[D.sub.t-1,t] - [a.sub.1], [D.sub.t,t+1] and declines over time. The result depends upon whether d[micro]/[(g - b) [Beta] - ed] or [a.sub.1]/(a - 1) is larger. Since [a.sub.1] = 0.5 d[micro]/(1 + [Beta] + d) and a = d(1 + 0.5(g + b - e))/(1 + [Beta] + d),

[a.sub.1]/(a - 1) = d[micro]/[d(g + b) - 2(1 + [Beta] - ed].

Thus, to compare d[micro]/[(g - b) [Beta] - ed], [a.sub.1]/(a - 1), we need only compare (g - b)[Beta] with d (g + b) - 2(1 + [Beta]). Consider d (g + b) - 2(1 + [Beta]) - (g - b)[Beta]. Since it is reasonable to assume that d [is less than] [Beta] [is less than] 1 end b [is less than] 2, we have

d(g + b) - 2(1 + [Beta]) - (g - b)[Beta] = g(d - [Beta]) + b(d + [Beta]) - 2(1 + [Beta]) [is less than] 0.

It follows d(g + b) - 2(1 + [Beta]) [is less than] (g - b)[Beta], and thus

d[micro]/[(g - b)[Beta] - ed] [is less than] [a.sub.1]/(a - 1).

If on the other hand [D.sub.0,1] [is less than] d[micro]/[(g - b) [Beta] - ed], [D.sub.t,t+1] = [Delta][D.sub.t-1,t] rises over time. In this situation there must exist a time point to after which mature agents begin to ask for verification. This condition is

[D.sub.[t.sub.0] - 1,[t.sub.0]] [is greater than] d[micro]/[(g - b)[Beta] - ed]

[D.sub.[t.sub.0] - 2,[t.sub.0] - 1] [is less than] d[micro]/[(g - b)[Beta] - ed].

Bequests follow from condition (18). Since d[micro]/[(g - b)[Beta] - ed] [is less than] [a.sub.1]/(a - 1) it is possible that

[D.sub.[t.sub.0] - 1,[t.sub.0]] [is less than] [a.sub.1]/(a - 1)

By Lemma 3 [D.sub.t,t+1] begins to decline at [t.sub.0]. There must exist a [t.sub.1] with [t.sub.1] [is greater than] [t.sub.0] again by Lemma 3 such that

[D.sub.[t.sub.1] - 1,[t.sub.1]] [is greater than] d[micro]/[(g - b)[Beta] - ed]

[D.sub.[t.sub.1] - 2,[t.sub.1] - 1] [is less than] d[micro]/[(g - b)[Beta] - ed].

From period [t.sub.1] on [D.sub.t,t+1] will follow (19) and rise over time. Thus [D.sub.t,t + 1] cycles perhaps forever. This is the poverty trap.

Figure 2 shows the dynamics of bequest and a possible poverty trap for a special case in which the initial bequest [D.sub.0,1] [is less than] d[micro]/[(g - b)[Beta] - ed]. In the figure the horizontal axis is [D.sub.t-1,t], and the vertical axis is [D.sub.t,t+1]. D' represents equation (19) D" represents (18) A = [D.sub.0,1] d[micro]/[(g - b)[Beta] - ed], and C = [a.sub.1]/(a + 1) The bequest starts at A which is lower then d[micro]/[(g - b)[Beta] - ed], so it follows (19) first. It rises and reaches point 1. At point 1 the bequest is larger than B, it is now governed by (18). Because the bequest is below C from point 2 the bequest falls to point 3 further to point 4 after which the bequest for the next period falls below B, after which future bequests are ruled by (19). From point 5 the bequest rises to point 6 and exceeds B. When (18) takes over at point 7 the bequest drops to point 8 and further to point 2 since it is still below C. A cycle then begins: 2 [right arrow] 3 [right arrow] 4 [right arrow] 5 [right arrow] 6 [right arrow] 7 [right arrow] 8 [right arrow] 2. In other words, in this particular situation, the bequest, goes up and down, but can never go beyond C. Therefore, the bequest is trapped in an interval of [0, C]. This implies that, from the expressions of consumption in section II, this economy is in a poverty trap.

[Figure 2 ILLUSTRATION OMITTED]

If we interpret the bequest as investment of parents in their children, the above result indicates that if current generation does not invest enough in future generations, a nation may fall into a poverty trap. Once in the trap, poverty will persist unless outside forces raise the investment level. Since the model developed in this paper had no government, forced saving could be one solution. Another solution would be foreign investment. The injection of outside funds certainly leads to higher level of capital accumulation. This may help a nation to get out of the poverty trap and enjoy economic prosperity. A good example is the recent success of some Asia countries, such as South Korea, Singapore, Taiwan, and Thailand. Although many factors contribute to the economic miracle of these countries, the outside investment certainly played an important role in their leaps from poverty to prosperity.

VI. Conclusion

In this paper, we have investigated the relationship between economic growth and asymmetric information. Due to the costs and the incentive problems of overcoming informational frictions, the effect of hidden action can be positive at early stages of economic development where agents can not afford to verify information, but negative in later stages once agents can afford to demand and exercise verification options in investment contracts. The negative effects of verification are, however, larger in the short-run than in the long-run.

This paper also presents an economic growth path that differs from those of the neoclassical growth theory and/or endogenous growth models. In this paper, it is shown that a country may follow a high-low-moderate growth path, or that it may fall into a poverty trap. This result indicates that, under normal conditions, a country can expect neither perpetually high nor perpetually low economic growth. A downturn of growth is a phenomenon of economic development, not necessarily a symptom of a bad economic conditions.

The poverty trap phenomenon underscores the importance of investment external to the domestic private economy. Forced saving by the government, or international investment, may be required to extricate a country from a poverty trap.

The results of this paper point to several topics for future exploration. Consideration of the effects of hidden action, along with hidden information, seems a natural extension of the theoretical framework developed in this paper. The results of this paper also seem to point to the desirability of empirical study of growth paths of various economies.

Appendix A

Consider a contract between a young agent and a middle-aged agent in any period. To facilitate the discussion, we shall drop the subscripts for a moment and need a few more notations. Denote K as the middle-aged agent's wealth at beginning of the period and k as lending level. Given k, let

p(s) = probability that state s occurs;

S(k) = prestate contractual choice of verification region;

S'(k) = prestate contractual choice of non-verification region;

R (k, s) = prestate contractual choice of payoff to the lender when s [Epsilon] S(k);

h (k) = prestate contractual choice of payoff to the lender when s [Epsilon] S'(k);(8)

y(k, s) = realized output level in state s.

According to Townsend [15], we need consider only consistent contracts in which any prestate contractual choice is same as the actual realization, since given any contract, there exists a consistent contract which achieves the same allocation of resources. In order words, we can treat the prestate contractual choices, S(k), S'(k), R(k, s), and h(k), as same as the actual realizations. Then the contract problem is for the middle-aged agent to find k to(9)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

subject to

(A.1) V [is greater than or equal to] [V.sub.0],

(A.2) R(k,s) + VC [is less than] h(k), s [element of] S(k)

(A.3) y(k,s) - h(k) [is greater than or equal to] 0, s [element of] S'(k),

(A.4) y(k,s) - R(k,s) - VC [is greater than or equal to] 0, S(k),

(A.5) 0 [is less than or equal to] k [is less than or equal to] K.

Here [V.sub.0] denotes middle-aged agent's autarky utility level and (A.1) is an incentive requirement for middle-aged agent to invest; (A.2) is a consistency requirement of a consistent contract with which agents have no incentive to misrepresent; (A.3), (A.4), and (A.5) are feasibility requirements.

We now show that S(k) = {0} and S'(k) = {1} when VC is relative small (the case that VC is relative large is discussed in section IV). Note that constraints (A.3) and (A.4) must be binding, otherwise the middle-aged agent can always improve his utility by raising in(k) or R(k, s). Then

(A.6) R(k,s) = y(k,s) - VC, s [element of] S(k),

(A.7) h(k) = min {y(k,s), s [element of] S'(k)}.

The young agent described in (A.7) will transfer the least amount possible because there is no verification in this situation.

When VC is relative small, consider S(k) = {0} and S'(k) = {1}. From (A.6) and (A.7),

R(k, 0) = y(k,0) - VC,

h (k) = min {y(k,s), s [Epsilon] {1}} = y(k, 1).

Then, by y(k, 0) = bk [is less than] gk = y(k, 1),

y(k, 0) - VC + VC = y(k, 0) [is less than] y(k, 1).

That is, constraint (A.2) is also satisfied. Therefore, we have the following result:

LEMMA 1. When VC is relatively small, S(k) = {0} and S'(k) = {1}.

Appendix B

Proof of Lemma 2

Proof: When solving for the maximization problem in section IV, r can be expressed as

r = [g[C.sub.t,t+1](b) + b[C.sub.t,t+1](g)/[C.sub.t,t+1](b) + [C.sub.t,t+1](g)].

Consider r - 0.5(g + b),

r - 0.5(g + b) = [g[C.sub.t,t+1](b) + b[C.sub.t,t+1](g)]/[[C.sub.t,t+1](b) + [C.sub.t,t+1](g) - 0.5(g + b)

= 0.5 [(g - b)([C.sub.t,t+1](b) - [C.sub.t,t+1] (g))]/[[C.sub.t,t+1](b) - [C.sub.t,t+1](g)].

In solving the problem, it can be shown that

[C.sub.t,t+1](b) - [C.sub.t,t+1](g) = (b - g)[k.sub.t,t]/(1 + [Beta] + d),

therefore,

r [is less than] 0.5(g + b). []

Proof of Lemma 3

Proof: [X.sub.n] = a[X.sub.n-1] - ([a.sub.n-1] + [a.sub.n-2] + ...+ a + 1) [a.sub.1]. Since [a.sub.n-1] + [a.sub.n-2] + ... + a + 1,

[X.sub.n] = [a.sup.n][X.sub.0] - ([a.sup.n] - 1)[a.sub.1]/(a - 1)

= [a.sup.n][[X.sub.0] - [a.sub.1]/(a - 1)] + [a.sub.1] + [a.sub.1]/

(a - 1).

By a [is greater than] 1, when n [right arrow] [infinity], [a.sup.n] [right arrow] [infinity] and [X.sub.n] [right arrow] + [infinity] if [X.sub.0] [is greater than] [a.sub.1]/(a - 1), and when n [right arrow] [infinity], [a.sup.n] [right arrow] infinity, and [X.sub.n] [right arrow] -[infinity] if [X.sub.0] [a.sub.1]/(a - 1). This paper is based upon one of the chapters of author's Ph.D. dissertation at the University of Iowa. The author would like to thank Lloyd Blackwell, III., Satyajit Chatterjee, Narayana Kocherlakota, and Shih-Yen Wu for helpful comments on earlier drafts of this paper. The author would also like to thank an anonymous referee for helpful comments.

(1.) These works [6; 15;16], contribute to establishing the link between asymmetric information and economic growth, although they do not deal with the issues of economic growth directly. (2.) It may he more natural to think of bequests as gifts left by the old. In fact, in the early draft of this paper, we did exactly that and had similar results. Yet, the algebraical expressions looked more intimidating. For this reason, we changed it to the current setting.

(3.) It is well-known that a single sector overlapping generations model with finite lived agents and convex technologies displays no long run growth. The modifications that will result in long run growth include: the existence of bequests, the use of taxation to redistribute income from the old to the young, and the addition of another sector such that the price of capital is not necessarily one [10]. For the purpose of this paper, we feel that it is best to include bequests.

(4.) The verification cost can be interpreted as the cost of bankruptcy proceeding. It has been shown that firms ask for verification when bad states (bankrupt) occur [15]. When a firm goes bankrupt, it files for legal protection, the true state of the firm's well-being is then revealed (verified). The firm, of course, bears the cost of bankruptcy proceeding.

(5.) It is not unusual for a borrower to incur a verification cost. For example, when a firm goes bankrupt, it declares bankruptcy and bears the cost of the bankruptcy proceeding. This cost is a example of a verification cost. Moreover, it can be shown that whoever explicitly pays the cost does not affect the results of this paper. In fact, bath parties pay the cost even though only one party is assigned to pay. This is similar to the situation of sales tax.

(6.) In the literature, it has been treated as either a constant or an increasing function of the debt level [15].

(7.) A detail derivation of these equations is available upon request. (8.) h(k) here is independent of s is due to the fact that lender does not have information on what true state is without verification.

(9.) It is equivalent to let the young agent maximize his or her utility subject to certain constraints.

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