We propose a coherent unified approach to the study of the linkages among economic growth, financial structure, and inequality. Of course, the relationship between financial structure and economic growth has long been studied both empirically and theoretically. Yet, on the one hand, empirical studies have been mainly focused on statistical relationships without a serious study of underlying mechanisms that generate the observations. On the other hand, most of theoretical studies have depicted clean but simple mechanisms without serious consideration having been given to the models’ quantitative predictions. The same dichotomy between theory and empirical work exists in the literature on inequality and growth.
Early seminal empirical contributions focusing on growth and financial structure are Goldsmith (1969), Shaw (1973), and McKinnon (1973). A more recent empirical treatment is King and Levine (1993). This body of empirical work establishes that financial deepening is at least an intrinsic part of the growth process and may be causal—that is, repressed financial systems harm economic growth. Theoretical efforts at modeling growth and endogenous financial deepening include Townsend (1978, 1983) and Greenwood and Jovanovic (1990) (hereinafter referred to as GJ). These models posit costly bilateral exchange or intermediation costs—for example, a fixed cost to enter the formal financial system and marginal costs to subsequent transactions. Other theoretical contributions such as Bencivenga and Smith (1991) turn intermediation off and on exogenously and have an external effect that makes growth with intermediation higher. Saint-Paul (1992) features limited diversification and multiple equilibrium growth paths, some with developed financial systems and specialized technologies and others with the opposite. In turn, Acemoglu and Zilibotti (1997) show that capital accumulation is associated with increasing intermediation and that better diversification, which comes with higher levels of wealth, reduces the variability of growth.
Likewise well known are seminal contributions on growth and inequality. Kuznets (1955) posited that growth is associated with increasing and eventual decreasing inequality. Interest and controversies, especially with respect to cross-country regressions, have continued ever since. A recent paper, Forbes (2000), confirms previous studies that high (initial) inequality is associated with low subsequent long-run growth but finds that the relationship is the opposite for the medium term. The micro mechanics that underlie Kuznets’s assertion have been made clear in the contributions of Mookherjee and Shorrocks (1982). As with income growth, a Theil index of inequality can also be decomposed into within- and across-group changes. It consists of changes in inequality within groups, population shifts between low- and high-income groups, and changes in income differentials across these groups. These decompositions have established, for example, that in periods of nontrivial growth in Mexico and Brazil, increasing inequality (and, eventually, decreasing inequality in Brazil) are associated with changing returns to education and changing regional or urban-rural income differentials (and, in Brazil, inflation). See Bouillon, Legovini, and Lustig (1999) and Ferreira and Litchfield (1999), respectively. In Taiwan and Chile, growth coupled with an apparently stable income distribution appears to be the result of offsetting structural forces. See Bourguignon, Foumier and Gurgand (2001) and Bravo, Contreras, and Urzua (1999), respectively. Resting separately from this strand of the empirical literature are the deservedly well-known theoretical contributions more motivated by Kuznets’s original assertion that growth may bring increasing, and eventually decreasing, inequality—namely, Aghion and Bolton (1997), Piketty (1997), Banerjee and Newman (1993), and Lloyd-Ellis and Bernhardt (2000).
We have a concern about this dichotomy between theories and empirical studies. Although most of the theoretical models characterize economic growth with financial deepening and changing inequality as transitional phenomena, typical empirical research employs regression analysis to find a coefficient capturing the effect of financial depth or inequality on growth. The implicit assumptions of stationarity and linearity are incorrect, even after taking logs and lags, if the variables of actual economies lie on complex transitional growth paths, as they do in the theoretical models. Using artificial data generated by a canonical model that by construction displays transitional growth with financial deepening and increasing inequality, we can sometimes replicate the typical empirical results of the literature: financial deepening appears to lead to subsequent higher growth, and inequality to subsequent lower growth. But statistical significance is weak and sensitive to initial history, time frame, the inclusion of covariates, and so on. Evidently regression coefficients are not informative about the underlying true relationships. In pointing this out, we add to the list of concerns which have been raised in recent literature—for example, in Banerjee and Duflo (2000).
Taking a more constructive tack, we show how to conduct quantitative research on transitional growth paths that is, how to test a model and learn something about actual economies from potential rejections. Our canonical model is based on GJ.2 As a prerequisite for a numerical study, we need to characterize analytical properties as much as possible. GJ characterizes primarily the initial and asymptotic economy, but leaves the all-important transitions somewhat unclear. GJ also studies only the log utility function and does not take its characterization of the log case to data. Here we extend the model to include a wider class of CRRA utility functions. We characterize much of the transitional dynamics analytically and, in so doing, provide new results. The seemingly nonconvex technology of participation is shown, under some conditions, to be convexified by the optimal choice of portfolio shares between risky and safe assets. Thus, ironically, those risk-averse households and businesses without access are not condemned to low -yield, if safe technologies but rather shift toward risky enterprizes, especially as their wealth approaches a critical value. Related, the derived single-valuedness of portfolio choice and savings facilitates further research into transitional dynamics using numerical methods. These make clear the rich and potentially complicated dynamics not obvious in original GJ formulation. For example, overall inequality movement is not necessarily monotonic on the growth path—it can increase, then decrease, and then increase again as it moves slowly toward its asymptotic steady state. Financial deepening and growth are not monotonic either.
With the model made tractable, we take it to an application, namely, Thailand during 1976–96, an emerging economy that was in a phase of economic expansion with uneven financial deepening and increasing (and then decreasing) inequality. The Thai economy serves as a prototypical example of the growth and inequality phenomena, pervasive in other countries, that motivate the growth, financial deepening, and inequality literature. Using the Thai Socio-Economics Survey (SES), Jeong (2000) finds that growth and inequality are strongly associated with financial deepening. We emphasize, however, that our methods are not peculiar to Thailand and we hope to extend the analysis to other countries.
In the spirit of the business-cycle literature, we calibrate the parameter values. Thus, the benchmark parameters are set from several sources. Data on the yields of relatively safe assets or occupations—5.4 percent per year for agriculture—and idiosyncratic shocks for business come from the Townsend-Thai data.3 Risk aversion is set at values typically found in the financial economics literature, and the real rate of interest implied by a preference for current consumption is set at 4 percent. Aggregate shocks are difficult to pin down, and we utilize the observed GDP growth rate over the sample. The marginal costs of utilizing the financial system are set at low values, but the higher fixed cost of entry is such that, as in the Thai Socio-Economic Survey (SES), 6 percent of the Thai population would have had access to the financial system in 1976, using a distribution of wealth estimated from the same 1976 SES data. The model is simulated at these and nearby values to deliver predicted paths, which can be compared to the actual Thai data. Note that we are calibrating a non-steady-state stochastic model. The business-cycle literature concerns stochastic steady states, not transitions. There is a literature on transitions or out-of-steady-state dynamics devoted to the study of depressions, but the models are deterministic (e.g., Hayashi and Prescott (2002)).
First, we look at the average prediction of the model. It is broadly consistent with the actual pattern of growth with increasing inequality along with financial deepening. However, although the simulated expected participation rate and the Theil index (an inequality measure) in the model almost trace out a smoothed version of the actual Thai data, the simulated expected GDP growth rate is lower than the actual Thai path. We vary the key parameters and conduct robustness experiments.
Because the actual path of the Thai economy is imagined here to be just one realization of many possible histories of the model economy, the actual Thai path should differ from the expected path of the model. We construct a metric of closeness between a simulated path and the actual data, considering the growth rate, financial deepening, and inequality over the entire 1976–96 period as one particular realization. We pick the best-fit simulation under this metric. The best-fit simulation is not an outlier among all the simulations. Therefore, in this sense, the model does replicate the actual data well. However, the best-fit simulation shows a reasonable match with GDP growth rate and financial deepening, while missing some of sharp upturns of the data. Inequality in the model misses the eventual downturn of the data and has more erratic movement.
Next, we examine whether the actual Thai data lie within a confidence region generated by the model, using a χ2 statistic constructed from means and covariances of model-generated histories. The model imposes sharp restrictions on the data. Indeed, at the benchmark parameter values, the model is rejected. We turn our attention to the source of the rejection. We decompose the relative successes and failures into within-variable and across-variable components. The model fits Thai financial participation the best and Thai inequality the least closely. Likewise, the model does better with the observed cross-variable relations between growth and financial deepening, and between inequality and financial deepening, but does worse with the observed relationship between growth and inequality.
Overall, our findings are favorable to our canonical model but not completely satisfactory. Of course, we are not unaware of the tension in this paper between the working out of the details of a structural and well articulated but highly abstract version of reality, on the one hand, and its serious application to Thai data on the other. But the goal here is to show that these two pieces can be brought together. This, we believe, is the way to make progress toward understanding the true relationships among endogenous financial deepening, economic development, and changing inequality.
There is a continuum of agents in the economy as if with names indexed on the interval [0,1]. At an initial date t = 1, they are all identical in preferences and technology, with, for expositional purposes, identical initial wealth k1 > 0. Below this will be generalized to include an initial nondegenerate distribution of wealth drawn from the data. The basic decision problem of each period can be simply stated. An individual who owns assets or wealth kt at the beginning of period t will decide consumption ct and savings or investment st (into various assets or occupations) in period t with ct =kt− st. Utility is then given by:
with discount rate4β, 0 < β < 1. While GJ restricts attention to the log contemporaneous utility, u(ct) = log ct, we analyze as well the constant relative risk aversion (CRRA) utility function
There are two technologies of investment in the economy. A safe technology returns output next period at constant rate δ. A risky project returns output at a per unit but variable stochastic rate of θt + ∊t, where θt is a common shock across technologies and ∊t is an independently and identically distributed (i.i.d.) project-specific idiosyncratic shock with the mean of zero. One can think of the safe technology as subsistence agriculture and the risky one as nonfarm entrepreneurship, for example. We often use ηt ≡ θt + ∊t to denote the total per unit return on the stochastic technology.6
The cumulative distributions of θt and ∊t are time invariant and denoted as F(θt) and G(∊t), respectively. The supports of these distributions are assumed to be compact.7 Thus we specify the lower and upper bounds and the domains and ranges of the distribution functions:
The cumulative distribution of the total return
If an individual is on his own, not a member of the financial sector, then he invests some portion ϕt of total savings8st at date t into the risky asset or occupation, and his capital stock at the beginning of the next period t + 1 will be
Of course, the total savings is nonnegative, st ≥ 0, and the fraction invested in the risky asset is between zero and one, 0 ≤ ϕt ≤ 1. Note also that this makes capital kt+1 and consumption ct+1 functions of history of shocks through date t, ωt, not contemporary shocks at t + 1.
If an individual is a participant of the financial sector, then that individual deposits savings st in a bank or purchases shares of a financial institution, and then lets the institution make decision about the type of his project. But before the bank pays interest (next period), the bank discovers the true aggregate return θt (as if by sampling a very large but finite number of projects, then using the law of large numbers). In addition the bank can pool the returns on the individual projects it operates to smooth away completely9 variation from the idiosyncratic shocks ∊t. The returns on shares of projects then vary only with the aggregate state. An alternative interpretation is that the household puts money on deposit but then borrows to finance a project under advice from the bank. Average repayment is determined by either θ or δ. But if risky projects are undertaken, low ∊ households repay less, as if receiving insurance, and high e households repay more, as if paying premia, so as to repay θ on average. That is, the debt repayment is allowed to vary with idiosyncratic shocks.
A key decision at the very beginning of each date t is whether to enter the financial system. There are transaction costs as in Townsend (1978). First, there is a one time entry fee or a fixed cost q > 0 incurred at t. Second, in each period, a depositor or a shareholder gets per unit return r(θt)=γmax(δ,θt), where (1 − γ) is a variable cost with γ ∈ [0,1]; hence the variable cost is a cost proportional to the rate of return. Thus when an individual deposits savings st in the bank at the end of period t, his capital stock kt+1 of the next period will be
Greenwood and Jovanovic (1990) and Townsend (1978) show this return is consistent with that offered by a competing set of financial intermediaries. We do not pursue that decentralized interpretation further. Of course, this specification of transactions costs begs for serious generalization and extension, to other kinds of costs, to distinguish among various possible kinds of involvement in the financial system. We can allow heterogeneous costs across different households/businesses, and indeed below we do allow costs to vary with education and urban/rural status.
We assume with Greenwood and Jovanovic (1990) necessary conditions that assure the bank has indeed a real informational advantage despite marginal costs, and that the risky asset is potentially profitable enough to attract positive investment, that is, the expected risky return dominates the safe return, even without advance information.10
We define dt as the binary participation or selection decision at period t: dt = 0 if a household stays outside the financial system at t, and dt = 1 if it participates in the system. Thus a policy for a household is a vector xt = (dt, st, ϕt) at period t over that entry decision as well as over savings and technology choice. Note again that portfolio choice ϕt of a participant, dt = 1, is trivially determined (as if recommended by the bank with θ known). The policy space X is thus given by
As noted, the initial capital level at t, kt, is a function of decision xt−1 and shocks ωt−1 in the previous period, that is,
We assume the initial capital level is positive, k1 (x0, ω0) > 0. Savings cannot exceed wealth, and thus the feasible set for savings at t is not all of ℝ+ but rather [0, kt] for all t. More generally, the feasible set Г for policies at t is written as
This set Г is nonempty and compact valued, helping to ensure the existence of the optimal policies.11
Consumption at t is denoted as ct ∈ ℝ+. Consumption and savings must satisfy the resource constraint, respecting the incidence of transaction costs for new entrants,
We define utility function U as the discounted sum of instantaneous utilities for policy sequence x = (x1, x2,⋯)
We thus write the consumer’s infinite horizon problem as
We will analyze the effect of financial development in this unbounded growth model. However, we restrict the economy not to explode: U(x0, ω0, x) < ∞. This is assured by the following assumption, limiting the expected return to be smaller than 1/β, adjusted by risk aversion14,
βE[(r(θ))1−σ] < 1.
Note that assumption 3 applies to participants. By the same argument, βE [η1−σ] < 1 is the analogue condition15 for nonparticipants. This latter assumption is not necessary, however, because everyone eventually participates in the financial system, as we show below.
Since we focus on perpetual growth, it seems natural to ensure that the optimized utility16 has a real value bounded from below, U*(x0, ω0) > −∞. A sufficient condition is indeed to make the safe return sufficiently high, that is, greater than 1/β, equivalently,
βδ > 1.
B. Value Functions
Following the notation of Greenwood and Jovanovic (1990), we define V(k) as the value for those who have already joined financial intermediaries today, and W(k) as the value for those who have not joined today but have an opportunity to do so tomorrow. Also, we introduce W0(k) as the value for those who are restricted to never joining.17 Explicit forms of these value functions are:
subject to equation (3),
subject to equation (2), and
subject to equation (2).
W0(kt) ≤ W(kt)<V(kt); in particular; V is the only relevant branch on the right-hand-side of the functional equation (12).
Equivalently, we also introduce the value for a nonparticipant who faces the entry decision today as19
The value for new participants today can be written as
Thus, using value Z(kt), W(kt) can be also written as
C. Solutions of Value Functions and Policies
Below we display analytical, closed-form solutions for value functions V(k) and W0(k) and their associated policy functions.20 These will then be used in computation. However, only numerical solutions are available for W(k) or Z(k). The numerical algorithm is described in the appendix.
Solution of V(k), the Participant’s Value Function, and the Associated Policies
A participant’s value function is easy to solve for under log utilities. By guessing and verifying as in Greenwood and Jovanovic (1990), we get the analytical formula satisfying the functional equation (12).
The saving rate μ, defined as st =μkt, is equal to β for the log utility case.
More generally for CRRA utilities (σ ≠1), we can also get the analytical formula of the value function V(k) and the saving rate μ*:
See the derivation in the working paper, Townsend and Ueda (2001).
Solution of Wo(k), the Value Function for Those Never Allowed to Join the Bank, and the Associated Policies
Similarly, we can obtain analytical solutions21 for W0(k), the associated optimal saving rate μ** and the optimal portfolio share in the risky technology ϕ**. For CRRA utility,
where e**(η) = ϕ**η + (1 − ϕ**)δ is the optimized per unit return. We can show the uniqueness of the optimal portfolio choice ϕ**, which can take on the boundary values. Conditions for the boundary values are given by
- ϕ** = 0 if E[η] < δ, i.e., the safe return is sufficiently high (sufficient condition), and
- ϕ** = 1 only if E[l/ησ] ≤ 1/βδ, if the safe return is sufficiently low (necessary condition).
For the log utility, CRRA at σ = 1,
and the optimal savings is μ** = β.
III. Analytical Characterization
The entry cost is a one-time fixed cost and this introduces a fundamental nonconvexity. In particular, the value function might not be concave. Put differently, the value function V(k) is strictly concave after entry and the value function W(k) may be strictly concave before entry, but still the region of outer envelope (the value function Z(k)) that determines the entry point might not be concave (see22figure 1). If Z(k) is not concave, W(k) is no longer assured to be concave by definition (18). Further, if the value function has this nonconcave part, the optimization problem is not strictly concave and it is not apparent that the policy functions are single-valued.23
Figure 1.Nonconcave Function Z(k)
But the risky asset naturally allows some convexification. Indeed, we show that under quite general assumptions the optimal portfolio choice by an individual makes the value function concave and thus policy functions are single-valued. That is, households want to eliminate the nonconcave part, and any nonconcave part could be smoothed out by a lottery. The risky asset is a natural lottery. Sufficient assumptions to make this mechanism work are hence large variations of the risky return and strong risk averseness.
B. Transitional Value Functions and the Associated Policies
The participant’s saving rate μ*satisfies
This assumption is sufficient to assure proposition 2, which follows below, that the value function Z(k) is concave. For σ =1 (log case), the right hand side of (25) is log(l − β), but the left hand side is also log(l − β) because μ* = β in the log utility case. Hence this assumption is always satisfied in the log utility case.
Assumption 5 is another restriction on the growth rate. It says that the saving rate cannot be too high, given the risk aversion parameter σ, the discount rate β and the return from the safe asset δ. Since the saving rate is an endogenous variable, assumption 5 essentially pins down the curvature of utility function and variance of shocks.
We can see this point more clearly by simplifying assumption 5. Notice that the right hand side of (25) is just a linear combination of log(l −β) and
By the closed solution of μ* in (21), this condition is equivalent to
Note that sufficient condition (27) is only slightly stronger condition than assumption 3, which restricts the saving rate so that the life time utility is bounded from above. Also, the impact of (28) is such that, given the discount rate β and the safe return δ, either the curvature of the utility function σ or the variance of aggregate shocks should be sufficiently large so that (28) is satisfied.24 The first parameter σ makes people risk averse so that the outer envelope in figure 1 has a lot of curvature in such a way that a small amount of randomness allows convexification. The second provides a sufficient amount of randomness in nature, to span the otherwise nonconcave part.25
Through the choice of his portfolio share in the risky asset, an individual can “control” the randomness of his capital at the next period and hence his lifetime utility from the next period on. That is, if the aggregate shock has sufficiently large variance, he can choose his desired level of randomness over next period’s capital. If there remained a nonconcave part in the value function next period, random returns from his investment would make his expected lifetime utility today bigger relative to nonrandom investment. This assumes the savings are sufficiently low that the household is operating both today and tomorrow in a similar neighborhood of capital. Another way to see this result is to note that the outer envelope is supposed to reflect discounted expected utility for each given wealth k, but evidently higher values are possible, if the envelope is nonconcave, by a little more randomization in k today, something possible with greater stochastic investment. This need for randomization would imply that we have not found the optimal policy yet and thus that further iteration of the value function would be called for, so as to eliminate the nonconcave part eventually. In sum we have,
(i) Z(k) is monotonically increasing, and under assumption 5, is concave
(ii) Z(k) is continuous and differentiable on ℝ++.
(iii) Given kt−1, dt−1and dt the optimal policy (μt, ϕt) is single-valued, and μt(k) and ϕt(k) are continuous Junctions on ℝ++.
The remaining question is whether the participation decision is single-valued.
People join the financial system whenever they are indifferent26: dt(k) = 1 when W(k) = V(k−q).
We thus let k* denote the critical capital level, that is, the least capital level at which the values of W(k) and V(k−q) coincide. This level is, of course, uniquely determined. Under assumption 6, the participation decision for those whose initial asset level is less than the critical capital level becomes single-valued and is monotonically increasing27 in k. That is, if a household starts with small capital, it will be outside the financial system until he accumulates wealth greater than or equal to the critical capital level k*, and then it will join the financial system and stay inside forever.
C. Characterization of the Saving Rate and the Portfolio Share
Since nonparticipants prepare to pay the future fixed entry fee, their saving rate will be higher than participants. This is an extension of proposition 2 in GJ, and the proof is the same and is omitted here. Let the participant’s saving rate be denoted μ* as in equation (21).
Although it will be shown that everyone eventually joins the bank, those who have very little wealth act as if they would never be able to join the bank. Very poor people have almost the same value and policies as those who never have the opportunity to join the bank. These policies are the saving rate μ** and the portfolio share ϕ** derived earlier. That is, in the short run, agents with very small capital spend little effort to accumulate capital to join the bank. We can extend proposition 3 of Greenwood and Jovanovic (1990) to include the CRRA utility case, but the proof is the same and we omit it.
(i) μ > μ*for nonparticipants.
(ii) For all ∊ > 0, there exists some k∊sufficiently small such that
D. The Asymptotic Economy
Because the return is high compared to the saving rate, in the long run, roughly every household becomes rich and its wealth grows unboundedly.
(i) Participants in the bank almost surely accumulate their wealth to exceed any K < ∞ in the long run as t → ∞.
(ii) Those who never join the bank almost surely accumulate their wealth to exceed any K < ∞ in the long run as t → ∞.
Everyone eventually participates in the bank, almost surely, and hence W(k) > W0(k) for all k.
This proposition implies that no one actually takes the value of W0.
E. Population Distributions
Although each household’s return is not affected by others’ choices, it does depend on its wealth. As a consequence, macroeconomic variables such as the growth rate and inequality measures vary with the wealth distribution.
The cumulative transition function from period t capital k to period t + 1 capital k′ is defined as
If we know the optimal policies (dt, St, ϕt), which are functions of k, and if we are given k and k′, we can then analytically construct Ψ(k’; k) defined in (29). For example, if we focus on the transition of nonparticipants only, this cumulative probability should be truncated at the critical capital level k* at which people join the financial intermediary. That is, given this period’s capital level k, we should integrate the above cumulative transition function with respect to next period’s capital over [0, k′] for those who could never have enough capital to join the bank next period, and over the restricted range [0, k*] for those who have a positive chance to join. We can thus derive the cumulative transition function for nonparticipants Ψ(k’; k).
Then given the initial distribution of the wealth, M0, the wealth distribution at each period t ≥ 1 for nonparticipants is recursively derived. Let Mt(k′) measure the size of the population (cumulative distribution) in period t, who are outside of the intermediated sector at t and will have a capital stock kt+1 ≤ k′ in period t + 1. Then,
The fraction of agents who stay out of the bank at t is thus written in terms of distribution of nonparticipants at t − 1 with capital level at t less than the critical level to join, that is, Mt−1(k*).
In a similar manner, we define
Now with these distributions, we can find the ex post gross growth rate of capital kt+1/kt from t to t + 1, namely (31) below for those who are outside of financial system both in period t − 1 and t,
(32) below for participants both in period t − 1 and t,
The economy-wide gross growth rate is easily obtained as the evolution of the population average wealth level. Let Kt denote average wealth level for the economy. It is defined as:
From this, the economy-wide gross growth rate is calculated essentially a Kt+1/Kt apart from entry costs. That is, the economy-wide growth rate is basically a wealth weighted average of growth rate of each household, so it reflects rich households’ growth rates more than poor households’. This is clear if we write Kt+1 in terms of period t’s population distributions Mt−1 and
The expected economy-wide gross growth rate E[Kt+1]/Kt in transition can be analytically derived from formulas (34) and (35). We compare this statistic to the GDP growth rates in Monte Carlo simulations below. Note that, as t → ∞, it approaches μ*E[r(θt)], the long-run, steady-state growth rate in which (almost) everyone participates in the financial system.
Again, these population distributions and statistics are analytically obtained, once we find the optimal policies (dt, st.ϕt) and initial distribution M0. In other words, we do not need to draw large samples to construct these statistics in computation. We do need to approximate the transition matrix numerically, however.30
IV. The Thai Economy: Growth, Inequality, and Financial Deepening
The economy to which we take the model is the Thai economy, concentrating on its emerging market growth phase, 1976–96, prior to the financial crisis of 1997. Needless to say, we do not attempt in this paper to analyze the crisis itself. Rather, we concentrate on what seems to have been the prior transition period, to see if we can understand this through the lens of the model, as a financial transition. We also take the view that to understand the crisis and prospects for recovery one must understand the growth that preceded it.
We use various, multiple sources of data for calibration, estimation, and general discussion of the Thai economy. The national income accounts are constructed by the National Economic and Social Development Board, the NESDB. Credit and monetary aggregates as well as savings are provided by the Bank of Thailand. A complete village census with interviews of headmen was conducted by the Community Development Department, the CDD, biannually starting in 1986. A nationally representative Socio-Economic Survey covering income and expenditures, the SES, has been implemented at a substantial scale starting in 1976 with over 11,000 households, then repeated in 1981, and finally biannually after 1984.31 In addition, we draw on the Townsend-Thai data,32 a specialty cross-sectional May 1997 survey of 2880 households in the central and northeastern regions, with measures of income, wealth, financial sector participation, savings and credit, and other items.
The Thai economy displays growth, financial deepening, and increasing (and eventually decreasing) inequality. Before the financial crisis of 1997, the Thai economy had grown rapidly. The NESDB numbers show growth in gross domestic product, ranging from 3 to 7 percent from 1976 to 1986, then a relatively high and sustained average growth of 8.3 percent from 1986–94, especially high in the 1987–89 period, and finally tailing off somewhat to 4 percent by 1996. The per capita income numbers from the SES are similar, with an overall average at 4.96, relatively low at 1.98 from 1976-86, then high at 8.78 from 1986–92, and lower but still high at 6.94 from 1992–96.
Overall financial deepening is apparent in macro aggregates. The ratio of M2/GDP rises steadily, surpassing the US by 1992. Similar movements are apparent in M3/GDP, rising even faster in the 1988–94 period and also total credit/GDP, as reported in Klinhowhan (1999). Total credit extended by commercial banks increased with a particular surge in credit received by firms in the 1986–90 period. Likewise, restricting attention to rural areas, credit from the Bank for Agriculture and Agricultural Cooperatives (BAAC, a rural development bank) as a percentage of agricultural output increased throughout the 1981–97 period.
At a more micro level, from the CDD data, the fraction of village headmen reporting access to commercial bank credit rises from 0.26 to 0.41 in the 1986 to 1994 period, and to the BAAC from 0.80 to 0.92. At the level of households in the Socio-Economic Survey, respondents are asked whether they had changes in assets and/or liabilities from various financial institutions due to a transaction by any member in the previous month. Though no doubt noisy and off in levels, this measures rises from 6 to 26 percent, 1976–96. This is the measure of changing participation we use below. Further, we can stratify by occupation, education, and urban/rural status and see the same upward trend on this extensive margin for all groups. Though the access numbers are higher for those in urban areas, those with college education, and nonfarm groups (wage earners, business owners), the expansion is particularly evident for villages, from 8 to 35 percent, those with primary education, from 1 percent to 25 percent, and farmers, 6 to 24 percent.
We also emphasize the apparent effect of wealth constraints within each of these categories. We use the SES data to extract a latent-index measure of wealth and thus plot estimated wealth against that same SES measure of financial participation. Specifically, as in Jeong (2000), the SES provides ownership information on twenty household assets, and the latent variable is constructed to try to best explain the cross sectional variance. This is given a crude value in Thai baht by multiplying against the rental value of the respondent’s house, though one should not take the assigned values literally (and we rescale below relative to the transactions cost). Then average access (again, transaction in the previous month) can be plotted by wealth deciles. In every survey year, even as late as 1996, these profiles are distinctly upward sloping with somewhat higher slopes at relatively high wealth levels. The theory with no heterogeneity in costs would predict a 0–1 jump up at some threshold wealth level but mitigated locally by the incurred fixed costs. We take these figures as prima facie evidence of wealth constrained choice of access to the financial system, as the theory would suggest. Of course the theory is giving us the structure to interpret the data. Causality cannot be inferred from the data alone.
As the model would suggest, beneath the growth of income and financial deepening lies relatively high and increasing inequality. The Gini measure of income inequality computed from the SES rises slowly but steadily from 0.42 in 1976 to 0.54 in 1992, then falls in the end to 0.50 by 1996. This is high for Asia, but lower than in some Latin American countries. This inequality reflects disparities in regional and rural/urban growth rates and appears to be related to factors discussed in various literatures, in particular, as we emphasize here, wealth-constrained access to the formal financial system. Logically, we now focuses on the financial transition. That is, we formally view the Thai economy through the lens of the model to see if we can interpret some of these data and document anomalies.
V. Setup for Numerical Analysis
In the following sections we report on quantitative properties of the model-based simulations. In this section, we fix reasonable values for the parameters, as in calibration exercises of the business cycle literature. Then we use numerical methods33 to compute the value and policy functions. In the following section, based on the data from Monte Carlo simulations, we carry out linear regressions and report the results. Afterward, we proceed to examine the model in three ways: average predictions, best-fit simulations, and confidence intervals. Finally we report on more detailed, micro level descriptions of our simulated data, decomposing inequality change and allowing heterogeneity in education and urban/rural status with a focus on participation rates.
A. Setting Parameters
We use income to capital ratios from the Townsend Thai data (Townsend and others (1997)) to estimate the technology parameters for those not in the financial system. The survey shows that the net return from capital investment in subsistence agriculture, which we regard here as a crude approximation to the safe project δ at 5.4 percent in 1997. For the risky project we use the income/capital ratios for those in nonagricultural business (and again without access) at 9.7 percent on average (see below for more discussion). We use the 1997 cross section data to set the support of the idiosyncratic shocks ∊ as [−0.6,0.6] to mimic the difference between the top and bottom one percent returns in that year. That is, we used the range of returns or income to capital ratios from the bottom 1 percent to the top 99 percent (the more extreme tails no doubt contain noisy outliers).
We set the value of the discount rate at β = 0.96, following the business cycle literature.34 We report the log utility case primarily, but in a later section, we also report σ = 1.5 case as a robustness check.
Pinning down the range of aggregate shocks turned out to be somewhat difficult. However, we know that the difference between the minimum and maximum of the real per capita growth rate from 1976 to 1996 is about 8.7 percent, and according to the model with its transactions cost, underlying variation of the aggregate shocks would be yet larger. Thus we set the range for the aggregate shocks θ at 10 percent.35 For the mean of θ, we vary the mean and pick one which minimized the sum of square errors, the squared sum of the difference between the actual Thai growth rate and the predicted analytic path of the model.36
The fixed cost q is a free parameter, and we take it to be q = 5 in model units of capital. But by comparing the critical capital level k* in model units and k* in the actual data in Thai baht, we will find a scalar or “exchange rate” between model units and actual Thai baht. The critical capital level in model units is obtained by computing the value functions, namely k* = 15.37 The critical capital level in the actual data is estimated using the Socio-Economic Survey and the observed fraction participating in 1976. That is, we use the wealth distribution of 1976 from Socio-Economic Survey (SES) of Thailand as the initial condition38 (in 1990 baht), again, following Jeong (2000). We also use information about participation in the financial system from the same Socio-Economic Survey. According to that survey, the fraction of the population who had access to the financial system was 6 percent in 1976. The estimated cumulative distribution of wealth of 1976 shows that people who had wealth of more than 195,000 baht39 were 6 percent of the population in 1976. Since the critical level of the model is k* = 15, we set the scalar or “exchange rate” as 13,000 baht per model unit capital (in 1990 baht) to generate 6 percent participation in 1976.40
These parameters must satisfy assumptions described in the model section. For example, we made assumption 2, E[r(θ)] > E[θ] > δ. This condition implies that the variable cost 1 − γ cannot be large, and so we assume a zero variable cost, 1 − γ = 0 for the benchmark case, but run some robustness checks for other values (e.g., 2 percent). Also, we would like to see some variation in the function ϕ(k), so interior solutions of ϕ** for W0(k) are desirable. This is checked numerically, but essentially, the mean risky return E[θ] with some adjustment for risk aversion cannot be much larger than safe return δ.41
All these numbers are summarized in table 1.
B. Value and Policy Functions
Computed value functions are shown in figure 2. W(k) is always between V(k) and W0(k). It approaches W0(k) as k goes to zero, and approaches V(k−q) as k goes to ∞, as discussed earlier. The critical level of capital to join the bank is k* = 15, where W(k) and V(k − q) cross.
Figure 2.Value Functions
The saving rate of nonparticipants increases with their wealth level up to near the critical level of capital that determines the entry decision, and then decreases slightly (see figure 3). This is due to consumption smoothing, preparing for the payment of the fixed fee.
The portfolio share of risky assets varies in figure 3 as expected around the optimal level ϕ** under W0 (k), the value function of those who never enter the bank. It is increasing first and then decreasing. It is, however, almost always larger than ϕ**. Nonparticipants put their wealth in the risky asset as a natural lottery to convexify their life-time utility (value function).
For small levels of capital, the figures show that the saving rate and portfolio share approach those of those who never join the bank. This illustrates proposition 3.
The simulated wealth evolution over 30 periods is shown in a three dimensional graph,42figure 4. This is not a particular realized sample path but rather is the expected evolution, based on the analytically derived transition function (29), period by period, using a grid43 for possible capital values and the computed, numerical approximations to the nonparticipants’ optimal policies (dt, μt, ϕt) as a function of wealth kt.
Figure 4.Wealth Evolution
VI. Growth, Inequality, and Financial Deepening: Spurious Regressions
The advantage of any formal, structural model of growth is that the mechanism or “drivers” are made clear. Here for example, given initial inequality in the wealth distribution and the parameters of technology and preferences, the drivers are the realized draws of idiosyncratic and aggregate shocks. There are stationary aspects to the model: household savings and portfolio decisions at date t, and hence the likelihood of financial participation at date t + 1, are all determined by current wealth and current participation status. But aggregate growth, inequality, and overall financial deepening are not stationary time series even after taking logs and lags. They are all endogenous and all determined by these underlying shocks and decisions in complex and nonlinear ways.44 Note that these complex dynamics are not only found in our canonical model, but also in many other theoretical models that depict endogenous financial deepening, inequality, and growth.
We ask here whether our canonical model can generate data consistent with the findings in the empirical literature. We fix the benchmark economy, populate it with 1002 households respecting the initial 1976 Thai wealth distribution, and then draw idiosyncratic shocks in the population and aggregate temporal shocks for 30 years. We do this experiment 1000 times, with different shocks, generating in effect panel data for 1000 (artificial) countries. We then revisit these countries to examine their status in later years.45
King and Levine (1993) report that there is a robust positive relationship between “initial” 1960 financial depth and subsequent growth, averaged over 1960 to 1989. They conclude that financial services stimulated growth. Here we regress 20 year average growth rates on “initial” 1980, or 1985, level of financial depth, controlling for the initial log level of GDP (as created by 5 or 10 years of early model history).46 Likewise Forbes (2000) replicates a typical finding in the empirical literature: a robust negative relationship between “initial” inequality in 1965 and average growth from 1965 to 1990. Here we regress 20 year average growth rates on initial 1980, or 1985, levels of inequality. Finally, we include both inequality and financial deepening on the right hand side. In sum, with error term vm for the m-th simulation or country, we run various versions of cross-country regressions:
Table 2 reports on these long-run, 20 year growth regressions. As shown in the first column, the higher is the “initial” level of financial deepening, the higher is the subsequent 20 year average growth rate, though the significance level is marginal. The positive sign is of course consistent with King and Levine (1993). Also, column (2) reports that the lower is the “initial” level of inequality, the higher is the subsequent 20 year average growth rate. The significance level is marginal but the sign is consistent with Forbes (2000). However, when we include both financial depth and inequality as right-hand side variables, in column (3), the negative sign on inequality is reinforced while the sign on financial depth is reversed—both are now quite significant.47
|1985 as initial period||1980 as initial period|
We repeat these regressions with 1980 as the initial year. The results on both financial depth and inequality are weakened, except for the financial depth in column (6). Note that we did not take 1976 as an “initial” period, because all 1000 simulations share the same wealth distribution in 1976. In other words, the true initial condition is the same for all the 1000 simulated economies, meaning there should not exist any meaningful relationship between initial financial depth or inequality and subsequent growth. The substantial changes of sign, size, and significance of coefficients are consistent with underlying transitional growth model, that is, there is no stable relationship among these variables.
This instability can be observed more clearly when we replicate medium-term regressions reported in Forbes (2000). She reports robust positive relationship between lagged inequality and five year average growth rate over 1965 to 1995, contrary to her long-run regressions. We construct medium-term, five year average variables48 and conduct panel estimation of the effect of lagged financial deepening and inequality on the GDP growth rate, controlling for country fixed effects, time dummies, and the lagged GDP levels in logarithms. In sum, we run the following regression on the panel, though abusing the notation above,
where αm is the fixed effect and αt is the time dummy.49
We also confirm in table 3 that the regression results of medium-term panel regressions are quite different from long-run regressions. While the sign on inequality is now positive in some instances, consistent with the results of Forbes (2000), inequality is never significant. Indeed, none of the regressors are significant.50
|1976-80 as initial period||1981-85 as initial period|
VII. Growth, Inequality, and Financial Deepening: Examining the Model’s Predictions
A. Average Prediction
We examine the model based on its average prediction. The solid lines of Figure 5 show the numerically constructed expectation of the growth rate and the participation rate, using equation (35) and the critical value of capital k*. They are almost identical to the average of the Monte Carlo simulations51 shown as the dot-dash lines. We can see that both expected growth and participation increase over time. The predicted growth rate is however low relative to the Thai data (the dashed line), and participation goes through the middle of the Thai data, while missing the S-upturn that began around 1986.
Figure 5.Benchmark Average Paths
A Theil index of inequality is constructed and displayed as the solid line in the bottom part of Figure 5. It shows a more or less steady increase over time, and while centered around the Thai data in early years, is misses the downturn in Thai inequality around 1994.52 It is higher than the Monte Carlo average (the dot-dash line), because it is an analytic upper bound to inequality by keeping track of all possible realizations of the aggregate and idiosyncratic shocks under their specified distributions and the derived saving and portfolio policy functions.53 The evolving distribution of wealth has a wider support than the Thai data, as might be predicted given its construction. More revealing perhaps of the model’s mechanics, it has twin peaks, as those who gain entry separate themselves from the rest, leaving a dent in the wealth distribution.
With alternative, larger variance aggregate shocks, Ө = [1.022, 1.172], the figures are almost the same (see figure 6). But simple averages over time of the three variables show higher levels as reported in Table 4. In other words, the model’s average prediction of the growth rate becomes closer to the actual Thai data, but the model’s average prediction for financial deepening and inequality are further away. This trade-off among three variables seems consistent among many of our experiments. Thus, there is something in the structure of the actual Thai economy which is not present in structure of the model.
Figure 6.Alternative Average Paths
|Actual Thai Data||5.97||15.43||8.27|
B. Robustness Experiments
We check if the model’s average predictions are robust to other variations around the chosen benchmark parameters. We also see in the way the mapping from the parameters of the model to policy functions and dynamic paths, and so understand better the capabilities and limitations of the model. Figure 7 and 8 show two of these experiments, with policy functions on the left hand side (benchmark are dotted lines) and analytic, expected paths on the right hand side (actual Thai economy are dashed lines). We also report on others experiments, beyond these in the figures.
Figure 7.Lower Variance of Idiosyncratic Shock Figure 8.Higher Safe Return Figure 9.Accuracy of Simulation
Raising the risk aversion parameter to 1.5 lowers the saving rate initially, (though savings jumps up with higher wealth), lowers the fraction invested in risky assets, and lowers k* (the critical level of wealth to join the financial system). The dynamic paths of the model show slightly less growth and less participation but put the rising trend of inequality squarely though the Thai data path. Lowering the variance of the idiosyncratic shock to [−0.3, 0.3] lowers the saving rate, causes the portfolio allocation to go to the obvious risky extreme, and raises k* (figure 7). This accelerates participation well above observed Thai rates. A well-fit dynamic path comes from assuming a high mean return on the safe asset, to 8.49, so the share of the risky asset in the portfolio goes very low (figure 8). This raises k* considerably, and of even greater interest, captures the initially slow and subsequent fast rise in participation in the Thai data. Higher marginal transactions costs at 2 percent lower both savings and growth.
C. Best-Fit Simulation
Our goal here is to investigate the simulation that best fits the actual Thai path. Since the actual path of the Thai economy is imagined here to be just one realization of many possible histories of the model economy, the actual Thai path should differ from the above expected path. Further, the null hypothesis is that the Thai economy is a transition economy, so we cannot use steady state calculations as in the real business literature. Instead, we use the entire history as one sample, construct a metric that measures closeness of simulated variables to the actual Thai data, and pick the best-fit simulation based on this metric.
We choose household sample size N = 1002, which is sufficiently large so that the population average of idiosyncratic shocks is virtually zero and so that we are able to replicate the initial 1976 distribution of wealth as observed in the SES data.54 We simulate from 1976 to 1996 (T = 21) by drawing idiosyncratic shocks and aggregate (common) shocks. Specifically, we let θm = (θm1, θm2,⋯, θmT) denote a time path of the aggregate shock as a realized sequence of draws from the specified distribution F(θ). For every sequence of aggregate shocks, we simulate the economy S = 100 times for the N = 1002 people, thus obtaining various possible realized sample distributions of idiosyncratic shocks drawn from the underlying distribution G(∊), i.i.d. over the (finite) population and over time. We do this for M = 40 aggregate shocks.55
Let xNm denote a column vector containing p time and type variables in a particular simulation, one out of one hundred simulations, under the fixed m-th aggregate shocks θm path with N people. Let am denote the expected value of macro variable xNm with respect to idiosyncratic shocks given specific realization of aggregate shocks. Let bm denote the cross sectional average of individual variance (the second moment around the mean conditional on the initial wealth level and specific realization of aggregate shocks).56 Because we can write the GDP growth rate, the financial sector participation rate, and the Theil index as weighted averages of corresponding individual variables with normalization, we can apply the strong law of large numbers and the central limit theorem to the realizations xNm relative to the mean am as the number N of the population with i.i.d. draws of ∊ goes to infinity:
Under the m-th aggregate shock path, the macro variables asymptotically follow a normal distribution,57
This suggests that a squared error metric, as the square of variance-weighted normally distributed random variables, will have a χ2 distribution. We return to that below.
For now, however, we note that squared deviations, not from the model average but from the Thai path, also have a nice distribution. We can thus construct a natural squared-error metric. Specifically, let
The asymptotic distribution of the closeness metric is the following noncentral χ2:
The goodness-of-fit of a particular model path can be calculated as the inverse function of the cumulative distribution of this noncentral χ2 distribution at the value ξNm0. Thus high χ2 values are associated with unlikely outliers. With this criterion, we pick the best-fit simulation out of 100 for each realization m, and then pick the overall best over m among the M = 40 possible aggregate shock paths.58
Figures 10 and 11 show the best-fit simulation from among the 4000 possibilities for the benchmark economy and the alternative with higher variance of the aggregate shock. The growth rate is still low relative to the Thai path, but the model path now has considerable variability, while missing the big upturn in 1986. The best-fit with higher aggregate variability produces as anticipated a yet more erratic growth path, and a higher average path as well. The best-fit participation rate mirrors its analytic and average counterparts, here perhaps with slow-moving small deviations from trend. The Theil measure of inequality under the benchmark parameter values is much more erratic and even produces occasional decreases in inequality along its overall upward trend. Under the alterative higher-variance path, inequality is higher and diverges dramatically away from the observed Thai levels.
Figure 10.Benchmark best-fit Figure 11.Alternative best-fit
The best-fit simulation is not an outlier among all possible model paths at the specified parameter values. To establish this, we return to equation (38). It seems natural to construct a metric for deviations of a particular model path from the mean under a particular shock sequence θm. That is, let
This asymptotically follows a χ2(p) distribution with p = 57 degree of freedom (i.e., the three macro variables over 19 periods),
Note that the right hand side of (42) does not depend on the specific aggregate shock θm, as seems natural since we are fixing the aggregate shock and taking the limit only as the number N of households gets large. One does need the conditional mean and variance, am and bm, to construct the left hand side of (42), however.
After we carefully construct these moments,59 we obtain ψNm and plot it in Figure 9 using the dashed line, while we also plot the true χ2(57) distribution using the solid line. These two lines illustrate the overall accuracy of our simulations. In the tails, where the error would be at a maximum, at the theoretical 5 percent level, the simulated distribution has 4.8 percent of its mass, and similarly at the 95 percent, the simulated distribution has 95.1 percent of its mass.
To return to the issue at hand, the best-fit simulation lies within an 83.2 percent confidence region, that is, well within a standard 95 percent confidence region. It is thus not an outlier among possible model paths. Under the alternative higher θ variance experiment, the best-fit path is within a 78.3 percent confidence region. Both these best-fit paths capture important aspects of the Thai economy as described above.
D. Confidence Region
Although we have shown that the average simulation is capable of producing a smoothed-out version of Thai data and the best-fit simulations trace the actual Thai paths quite well, we have yet to test whether the actual Thai economy path itself could be a likely realization of the model at given parameter values. The problem in doing this is that we do not know the aggregate shock path which underlies the realized Thai data, on the null hypothesis that the model is true. Furthermore, we can not simply fix θ and run 100 simulations, and then repeat for each of the 40 aggregate shocks, since the idiosyncratic and aggregate shock variables to do not independently affect the three variables in question. Thus, to pick up the joint variation and co-determination, we construct 1000 simulated economies with sets of unconditional and randomly chosen aggregate and idiosyncratic shocks.
We calculate the
To make headway on the sources of rejection under the current specification, we examine the goodness-of-fit statistics of each variable and combinations of variables. Numbers in the column “Growth” in table 5 show the calculated
Replaced by Actual
First three columns in table 5 show sound rejections for growth, financial deepening, and Theil, each taken separately, one at a time. Of these, the model fits financial deepening the best, and inequality the worst. The alternative economy does better on all three dimensions. As was evident from table 4, the overall time-averaged mean values of the benchmark and alternative (average over shocks and best-fit) paths are not much different from the corresponding Thai averages, especially so for financial deepening and many of the Theil indexes.Therefore, these rejections are driven by failure of the model to pick up the auto-covariance structure of the data. Note that the structural model is not a model of business cycle fluctuations.
We can also look at the goodness-of-fit of two variables at a time, and more to the point, see if the model’s prediction concerning covariance across variables is helping or hurting. If the sum of any two columns for each variable one at a time on the left-hand side of Table 5 is greater than the corresponding sum-column on the right hand side of the Table 5, then covariance across variables is helping. We conclude that cross variable covariance between growth and financial deepening is helping to reduce distances from the mean, and similarly for financial deepening and Theil, but the opposite is true for growth and inequality. With the three variables together, the bottom line is unclear: across-variable covariance is hurting in the benchmark case, but helping in the alternative case.
The failure of the model stems in part from the auto-covariance structure that the model generates, in line with the problem we saw in the best-fit simulations (high oscillations in growth rates and in inequality). This shortcoming partly reflects our criterion that matches first moments or paths of the variables, not second moments, and partly our time independent assumptions on the idiosyncratic and aggregate shocks, contrary to the auto-correlated shocks of the real business cycle literature. The model, as a theory of growth, explains the growth facts well (e.g., the average trend of the actual Thai data) but less so the cyclical components.
VIII. Further Micro Evidence
A. Inequality Decompositions
Jeong (2000) documents that, in the SES, participation in the financial sector, along with occupation and education, are key driving variables, explaining 39 percent of the change in levels of income (the residual is income growth within the categories themselves) and, using the mean log deviation, 72 percent of the change in inequality (the residual is change in inequality within categories). Further, financial participation alone can account for 20 percent of change in income and 41 percent of change in inequality, 1976–96. Put another way, education and occupation shifts account for 45 percent and 30 percent of the change in income in the 1976–86 and 1994–96 periods, respectively, higher than other categories, but financial access accounts for 27 percent in the 1986–92 periods, higher than any other category in that period. Recall also that financial deepening was particularly evident in that period. Likewise, financial sector participation in the 1986–92 period can account for 76 percent of inequality change: 35 percent by the composition change to the higher-income access group, 28 percent by the divergence of the income gap across access, no-access groups, and 13 percent by the shift to the higher inequality group. The financial access numbers are nontrivial in other periods, as well.
As noted, the benchmark economy tracks the overall trend of inequality and financial participation reasonably well, and not terribly bad with respect to growth. We can further decompose this growth and inequality change. The composition effect on growth of population shifts from the low income nonparticipation sector to the higher income participation sector accounts for 30 percent of income change in Thailand and 60 percent in the best-fit benchmark model simulation for 1976–96 periods. This is because the model overpredicts the income gap across these two groups, approximately 3–1 in the data versus 5–1 in the model. Similarly, the composition effect of population shifts on inequality accounts for 35 percent of the inequality change in Thailand and 69 percent in the best-fit benchmark simulation. Evidently the model at benchmark parameter values over emphasizes the role of financial participation, as might seem natural given that this is the focus of the model itself. This is not a pre-ordained feature of the model, however. At an alternative best-fit, higher-aggregate-shock/higher mean simulation, for example, we do better on the compositional effect, down to 54 percent of growth, somewhat nearer to the data, and only 12 percent of inequality change, less than in the data. Unfortunately, this comes at the expense of an increased (and unobserved) divergence of income across the two sectors, while also missing the absolute magnitudes of overall trends. It thus seems the model at various parameter values can deliver a rich range of possible inequality phenomena, but there are nontrivial tradeoffs, depending on which one we try to fit. This is reminiscent of early works and debates on real business cycle models.
B. Heterogeneity in Participation
The model seemingly assumes uniform entry costs q, and makes predictions for all households together, but this too can be examined further. Specifically, the model with its linear returns (and no endogenous prices) allows us to calibrate and simulate for various of the key education and residence groups, one at a time.
For example, we can distinguish SES households by the completed level of education of the head (elementary and advanced secondary). We continue to fix technology and preference parameters at their benchmark values, including the model version of q, hence k*. But for each group separately we center the initial (SES estimated) wealth distribution so that the initial participation rates of the Thai data match the predictions of the model (on the false assumption that everyone above the threshold is participating). Those initial participation rates in 1976 are 10 and 34 percent for the two chosen education groups and 8 and 25 percent for rural and urban households, respectively. We then simulate economies from 1976 to 1996 one group at a time.
Two related findings emerge from this analysis. First, the model at benchmark values tends to over predict the subsequent participation rates of the advanced secondary and urban groups. That is, the model predicts that access would have been higher for them over time relative to what we observe in the data (at the benchmark parameters, which, again, match reasonably well the overall rates.) Ironically, one begins to wonder if there are barriers to participation, not for the poorly educated, village residents but rather for their highly educated urban counterparts. Of course we could introduce heterogeneity in preference and technology parameters across these groups, but the direction of needed change might seem surprising a priori, that is, higher risk aversion for those educated/urban groups, higher variance of idiosyncratic shocks, or lower mean returns.
Second, and related, we can fix the exchange rate between the model units and Thai baht using one group, and thus compare in common currency units the transactions costs q across the various groups. Even though the educated have higher access, for example, they have an even higher, right-shifted distribution of wealth, so the threshold wealth k* for them is relatively high, making their transaction costs q relatively high also. Thus we find that participation costs are higher (not lower) for the educated and urban groups. Again, to overturn this, we would need to raise (not lower) risk aversion, raise the variance of the idiosyncratic shocks, or lower the mean return, for example.
We follow the research agenda laid out in Lucas’s Presidential Address to the American Economic Association in 2003 (Lucas (2003)): “For us, today, value theory refers to models of dynamic economies subject to unpredictable shocks, populated by agents who are good at processing information and making choices over time. ⋯ [This involves] ⋯ formulating explicit models, computing solutions, comparing their behavior quantitatively to observed time series and other data sets. As a result, we are able to form a much sharper quantitative view⋯”
In this paper, we take this agenda to the study of economies in transition, a departure from typical calibration exercises on steady-state dynamics. That is, we have proposed a quantitative research methodology consistent with the widely held view that financial deepening and changing inequality, along with economic development, are transitional phenomena. We also show that, consistent with this view, simple regression studies would not be able to capture the true linkages among growth, financial deepening, and inequality.
The model we use is a relatively simple prototype which emphasizes a fixed cost of financial participation, but this generates complicated nonlinear nonstationary dynamics. We overcome apparent nonconvexities. In particular, we point out the possible usage of a risky asset or occupation as a natural lottery which can convexify the frontier of lifetime utility (the value function). We then establish the monotonicity of the participation decision with wealth and single-valuedness of savings and portfolio choices. These analytical results enable us to study the model further with numerical methods.
We apply the model to Thai data, calibrate, and make predictions. We look at the analytic and average prediction of the model. It is broadly consistent with the actual pattern of growth occurring with increasing inequality and financial deepening. However, while the expected participation rate and the expected inequality measure generated by the model almost trace out a smoothed version of the actual Thai data, the simulated expected GDP growth rate is lower than the actual Thai path. We vary the key parameters and conduct robustness experiments.
Because the actual path of the Thai economy is imagined here to be just one realization of many possible histories of the model economy, we construct a χ2 metric of closeness and examine the best-fit simulation. The best-fit simulation shows a reasonable match with the actual Thai data and is not an outlier among all the simulations. Therefore, in this sense, the model does replicate the actual data well. However, it misses some of sharp upturns of the GDP growth rate and of financial deepening in the data. Inequality in the model misses the eventual downturn of the data and has more erratic movement.
We also construct from a set of Monte Carlo simulations a confidence region and formally test the model. The model imposes sharp restrictions on the data. Indeed, at the benchmark parameter values, the model is rejected. We turn our attention to the sources of the rejection. We decompose the relative successes and failures into within-variable and across-variable components. The model fits Thai financial participation the best and Thai inequality the worst. Likewise, the model does better with the observed cross-variable relations between growth and financial deepening, and between inequality and financial deepening, but does worse with the observed relationship between growth and inequality.
With the model in hand, we return to a more detailed examination of Thai micro data. The model rightly focuses attention on the composition effect, the population shift from relative autarky to financial participation, but it is missing other components of growth and inequality decompositions. We also introduce heterogeneity and examine financial participation by education and urban/rural status.
A. Proof of Proposition 2 (i)
Monotonicity is easy to show (see the working paper, Townsend and Ueda (2001)).
The proof for concavity is shown by contradiction, supposing Z(k) is not concave, that is, suppose Z(k) is locally convex in the neighborhood of
average of capital next period is larger than the capital this period, and
Z(·) is locally convex at
Consider now the value for the nonparticipant at the capital level
Since the policy
By the condition (A2) and,
We can rewrite this as
However, since V(k) > Z(k) by proposition 1,
By substituting this into the inequality (A5), we get
The point in the domain of the second Z term in the right hand side of (A11) is
is larger than or equal to
Note that the point in the first Z term, (A12), can be rewritten with
Now we need to show
Rewrite this as
Hence we can take 1/E[ẽ(η)] as
Now we prove the right hand side of (A 15) is greater than or equal to 1/δ:
By rearranging terms, we get
By taking the logarithm,
By dividing both sides by σ, we get condition (25) in the assumption 5. Therefore
B. Proof of Proposition 2 (ii)
Proof. Continuity of Z(k) in ℝ++ is immediate by concavity. Concavity also implies differentiability of Z(k). The proof is essentially the same as those for theorems 4.10 and 4.11 of Stokey and others (1989), pages 84–85. □
C. Proof of Proposition 2 (iii)
First, we would like to show that the range of u(c) + β ∫ Z(k) is ℝ (excluding ±∞). By the Inada condition, we can restrict the range of instantaneous utility functions in ℝ without affecting the optimal choice of savings. the range of this value function is ℝ by assumptions 3 and 4. Hence the range of u(c) +β ∫ Z(k) is also ℝ.
Second, we would like to show that u(c) + β ∫ Z(k) is strictly concave in (μ, ϕ). However, u is strictly concave, so we only need to show concavity of ∫ Z(k). Take α ∈ (0.1), (μ1, ϕ1) and (μ2,. ϕ2) both from [0,1] × [0,1] with s1 < s2.
Since Z(k) is concave by proposition 2,
Hence ∫ Z(k) is concave in (μ,. ϕ), and thus u(c) + β ∫ Z(k) is strictly concave.
Therefore, the optimal policies (μ (k), ϕ (k)) are single-valued and continuous functions on k. □
D. Proof of Lemma 1 (i)
Proof. Optimal capital evolves as follows.
We can rewrite this equation as follows, using the error term “normalized” by mean return
By taking the logarithm, we get
Recursive substitution backwards in time yields
Let gj≡ μ*E[r(θj)] and
Divide (A29) by t,
The first term goes to zero as t becomes large, and the third term converges to zero, almost surely, because it is the sum of an i.i.d. shock with bounded variance and zero mean.69 Hence,
E. Proof of Lemma 1 (ii)
Proof. By changing the variables, almost the same argument of the proof of lemma 1 applies. Denote (μ**, ϕ**) as the optimal saving rate and portfolio share of those who never joins the bank, and define e**(ηj) = ϕ**ηj + (1 − ϕ**) δ. Take
From here, the argument is the same as in lemma 1. □
F. Proof of Proposition 4
G. Proof of Proposition 5
This proof has five parts: approximating the initial wealth distribution, law of large numbers, central limit theorem, application to the three macro variables, and computation of moments.
Conditional on the aggregate shock vector θm for the m-th simulation, each household’s income at each period varies with idiosyncratic shocks and its own previous wealth level, indirectly through savings and portfolio choice. Note, however, that the wealth level is growing and not ergodic even for each household level. Thus we represent each household’s wealth in the beginning of period t as
Approximation of Wealth Distribution
From the SES data set, we obtain the initial cumulative distribution of wealth M0(k)72 by a step function with L steps. Specifically, it has L elements in its support,
A probability mass of
is assigned for support
In practice, we take
Note that we use the following notation for the Riemann integral with respect to the approximate distribution M0:
We pick N people to approximate this initial wealth distribution. Mass of nj ≡ round(Nlj) is assigned73 for support
and for whole distribution in sup norm,
Law of Large Numbers
Now we look at support
The convergence of the approximate weight of each element of the support, equation (A35), can be viewed as sure convergence of random variable nj/N that follow a degenerated (relative to Gt (·)) distribution. Since sure convergence is stronger than almost sure convergence, we can apply the Slutsky theorem for multiplication of the almost sure convergent sequences,
Moreover, we can also apply it for sum of the almost sure convergent sequences,
We define the true population mean of wealth level at t, conditional on aggregate shock
which by construction (equation (A34) and (A37)) is equal to the right hand side of equation (A39). Also, the left hand side of equation (A39) can be expressed as a simple average of wealth variables over individual i from 1 to N, instead of averaging over individual i within nj people sharing the same initial wealth and then averaging again over these j = 1, ⋯, L types. We define the simple sum as
By substituting the left hand side of equation (A39) by SNt and the right hand side by akmt, we obtain a law of large numbers including cross sectional variations,
Central Limit Theorem
We redefine the wealth variable as a deviation from the mean of the j-th support,
By construction, the true mean over idiosyncratic shock history ∊t is zero,
We also introduce true variance for individuals sharing the same initial wealth
This varies with the individual, depending on which support his initial wealth belongs to. Moreover, we introduce the sum of these variances, which is the true variance including cross sectional variation:
We define sum of wealth in this deviation form,
By applying the Lindeberg-Lyapounov theorem,75 we obtain a central limit theorem including cross sectional variation,
In the original, not deviation, form, this is expressed as
Note that the law of large numbers including cross sectional variation, equation (A41) assures that the mean of the left hand side is asymptotically zero.
Let gmit≡ kmit/kmit −1 − 1 denot i-th household’s income grow rate. The GDP growth rate xGmNt is a weighted average of individual growth rate,76
The individual variable in the numerator, kmit−1gmit, is, like the wealth variable, a Borel function of i.i.d.
where aGmt is the expected value of xGmNt..
As above, we obtain a law of large numbers including cross sectional variation,
where apmt is the expected value of xPmNt.
Theil index xImNt, an inequality measure taking values on [0,1], is defined as follows,
Again, we obtain a law of large numbers and use the Slutsky theorem for division and logarithm operations. Equation (A54) then converges to its expected value ajmt,
Note that 1/ln(N) is a factor to normalize the right hand side so that it will always belong to the [0,1] range.
Stacking all these variables in a column vector, we write
and then define a (57 × 1) column vector,
Similarly, we define the expected value column vector,
In sum we obtain a multivariate version of a law of large numbers,
To use a central limit theorem and appropriate variances, we redefine individual variables as
Equation (A60) says that this overall average of ziN converges almost surely to its true mean am.
For individuals sharing the same initial wealth level
The covariance matrix for individuals sharing
and overall covariance matrix bm (57 × 57 matrix) including cross sectional variation is defined as lj- weighted average of
As we did for a single variable (wealth), we can apply (multivariate version of) the Lindeberg-Lyapunov theorem and obtain the central limit theorem including cross sectional variation,
Proposition (5) directly follows from this.
Computation of Moments
We can estimate the simulated sample estimates for the mean am and the variance bm by sample analogue using individual variable ZiN in each simulation and then take an average of them over the simulations. However, an estimate for bm requires estimates for the individual variances for all the elements (i.e., 5000,15000, ⋯, 535000 baht) of the support of distribution of the initial wealth. The estimates for individual variances may not be accurate given the relatively small number of people assigned to each element.
Hence, we utilize yet another central limit theorem. With superscript s,
and the central limit theorem,
Using these, we can construct an estimate for the mean as
and for the variance,
Moreover, there are two possible strategies for estimates of the variance bm. The first uses the S = 100 simulations to construct the sample estimate for bm as defined above (A71). The second ignores the dependence on the realized sequence θm and estimates the overall unconditional variance using the (larger) sample of simulations over M = 40, thus using a common estimate b for each bm in ψNm.78 The second strategy actually displays less error in the approximation to the χ2 distribution and is used for the construction of ψNm shown in Figure 9.
H. Proof of Proposition 6
Proof. We transform the central limit theorem (A67) using moments around the actual Thai data x0,
Note that, to measure the deviation of each simulated macro variable from the unconditional mean a, replacing x0 by a will suffice. Further, if we replace each simulated macro variable xNm by actual Thai data x0 (but keep deviation from the unconditional mean a), then we can measure the deviation of the actual Thai data from the unconditional mean. □
The main program computes the optimal policy functions, the saving rate and the portfolio share, and the value functions. With these data, the simulation program computes the population dynamics of the economy.
The main program consists of six parts:
- Set the relevant parameters.
- Write the functions of V(k) and W0(k) in order to refer to these values in the following procedure and to take appropriate initial function of iteration.
- Computation of Z(k).
- Save the data of value functions and policy functions together with parameter values.
- Simulation of population dynamics on growth and inequality using the data of 4.
- Save the data of the simulation of 5.
B. The Construction of a Compact Domain for Z(k)
We use Z(k) in (15) instead of W(k) in (18) for iteration. Iteration on Z(k) has at least two advantages over iteration on W(k). One is that the Z(k) formulation involves simple integration, while the W (k) formulation requires an evaluation of the maximum operator inside the integrals. Essentially, the decision to join the financial intermediation is written explicitly for Z(k). Simple integration saves much computational time.
The other advantage is that since Z(k) takes the same value as V(k−q) when k is high, we can use V(k − q) as the value of Z(k) for k higher than some upper end point
Proposition 3 (ii) suggests that those who have very small wealth act approximately as if they do not expect to join the bank ever. This implies in turn that we can truncate the domain on the left at some small capital
C. Approximation and Iteration
We use the value function iteration method to obtain values and policies. Since the model uses continuous utility functions and continuous distributions of shocks, some computational difficulties arise. The computer can only handle discrete data, and approximation of the functions and integrations are necessary.
The following is the numerical procedure to obtain value functions and policy functions.
- First, we choose the initial given and known function Z0(k) on given
. Basically any continuous function that is between W0 and V is appropriate.81
- Second, we construct an approximation to that Z0. This is given notationally bywhere C denotes the approximation procedure and An is the parameter of that approximation at iteration number n. Here of course n = 0, since we have not yet done any iteration.We use the Chebyshev approximation method, which is more accurate than any approximation with the same number of nodes.82 This interpolates between special grid points by utilizing the information of all the points, and the fit is almost the best possible.
- We set the Chebyshev interpolation nodes in the compact state space
for evaluating the function Z0(k). Given the degree of polynomials p and the choice of number of nodes m over , the nodes k(l) is given bywhere x(l) on [−1,1] (l = 1, ⋯, m, m > p + 1) is a Chebyshev interpolation node:
- Evaluate Z0 at the nodes k = k(l) for l = 1, ⋯, m:
- Then compute the Chebyshev coefficient A0≡ (A01, ⋯, A0p) by the least squares method:where Ti is the Chebyshev polynomial defined over [-1, 1] as
- Finally, we get the approximation over all k:
- We set the Chebyshev interpolation nodes in the compact state space
- Third, we take the appropriate extrapolation. This is for the entire range of k. Specifically, define
for all k ∈ ℝ as follows.
- Fourth, we calculate W1(k) at each grid point bywhere k+(k, μ, ϕ, η) = μk(ϕ η + (1−ϕ) δ).
- We change variables of integration. Let h(η) denote probability distribution of η (recall the cdf was defined as H(η)). Given (k, μ, ϕ), k+(k, μ, ϕ, η) is a function of η. Given (k, μ, ϕ) and the specific value for k+, η is calculated from inverse function of k+, η = k+(−1) (k, μ, ϕ). We change the variable from η to k+ and redefine the integrand as
, a function of k+ given the Chebyshev coefficient A0,
- Here, we use the Gaussian quadrature to get the approximate value of integral. The Gaussian quadrature utilizes the orthogonal polynomial approximation and calculates the integration with good accuracy and little time. Using specific discretization of k+, which is
, with associated weight wi, orthogonal approximation of the integral takes the form:These are specific to polynomial and degree of approximation, which one can get from a table in a textbook on computation.83
- Maximization over (μ, ϕ) on equation (A83) is conducted by a grid search with successive refinements and simplex method.
- We change variables of integration. Let h(η) denote probability distribution of η (recall the cdf was defined as H(η)). Given (k, μ, ϕ), k+(k, μ, ϕ, η) is a function of η. Given (k, μ, ϕ) and the specific value for k+, η is calculated from inverse function of k+, η = k+(−1) (k, μ, ϕ). We change the variable from η to k+ and redefine the integrand as
- Finally, we take the value of Z1(k) for each k = k(l).Then we approximate the Z1(k) as same as Z0(k); i.e.,
. From this we can calculate W2(k), and construct Z2(k). This makes .
- Iteration goes until Z(k) converges to a fixed point.
D. Numerical Approximation of Evolution of Population Density
After the optimal policies (μ (k), ϕ (k)) are obtained from the numerical computation, given the initial distribution of the wealth, M0(k1), the wealth distribution at each period for nonparticipants is recursively derived by equation (30). Then we approximate analytical distribution Ψ (k′;k), equation (29), given k and k′ with a computer that cannot handle continuous distribution. We thus use the step function approximation with finite grids. Given an initial distribution of k1 defined on a grid, we define the distribution k2 using the nearest point in the grid as the approximation for a particular k2.
Note that as economy grows, the wealth distribution will disperse more. This creates some difficulties that, in each tail end of distribution, mass become tiny and could not be captured by a computer. Here we face a trade-off, that is, making the grid space finer increases accuracy for early periods when the distribution is thick everywhere, but reduces accuracy for later periods when the distribution becomes wide-spread.
AcemogluDaron and FabrizioZilibotti1997 “Was Prometheus Unbound by Chance? Risk, Diversification, and Growth” Journal of Political EconomyVol. 105No. 4 (August) pp. 709–751.
AghionPhilippe and PatricBolton1997 “A Theory of Trickle-Down Growth and Development” Review of Economic StudiesVol. 64No. 2 (April) pp. 151–172.
ÁlvarezMaría J. and AntoniaDíaz2002 “Minimum Consumption and Transitional Dynamics in Wealth Distribution” (February). ManuscriptUniversidad Carlos III de Madrid.
BanerjeeAbhijit V. and Andrew F.Newman1993 “Occupational Choice and the Process of Development” Journal of Political EconomyVol. 101No. 2 (April) pp. 274–298.
BanerjeeAbhijit V. and EstherDuflo2000 “A Reassessment of the Relationship Between Inequality and Growth: Comment” (December). ManuscriptMIT.
BencivengaValerie and BruthSmith1991 “Financial Intermediation and Endogenous Growth” Review of Economic StudiesVol. 58.
BergeClaude1997Topological Spaces (Mineola: Dover).
BillingsleyPatrick1995Probability and Measure (New York: John Wiley & Sons, Inc.).
BouillonCesar B.AriannaLegovini and NoraLustig1999 “Rising Inequality in Mexico: Returns to Household Characteristics and the ‘Chiapas Effect’.” ManuscriptInter American Development Bank.
BourguignonFrancisMartinFournier and MarcGurgand2001 “Fast Development with a Stable Income Distribution: Taiwan, 1979–94” Review of Income and WealthVol. 47No. 2 (June) pp. 139–163.
BravoDavidDanteContreras and SergioUrzua2002 “Poverty and inequality in Chile 1990–1998: Learning from Microeconomic simulations” Universidad de Chile Working Paper No. 196 (October).
De NardiMariacristina2002 “Wealth Inequality and Intergenerational Links” Federal Reserve Bank of Minneapolis Research Department Staff Report No. 314 (Novemeber).
Den HaanWouter J. and AlbertMarcet1994 “Accuracy in Simulations” Review of Economic StudiesVol. 61No. 1 (January) pp. 3–17.
FerreiraFrancisco H.G. and Julie A.Litchfield1999 “Education or Inflation? The Roles of Structural Factors and Macroeconomic Instability in Explaining Brazilian Inequality in the 1980s” STICERD Discussion Paper (March). DARP 41.
ForbesKristin J.2000 “A Reassessment of the Relationship Between Inequality and Growth” American Economic ReviewVol. 90No. 4 (September) pp. 869–887.
GoldsmithRaymond W.1969Financial Structure and Development (New Haven: Yale University Press).
GreenwoodJeremy and BoyanJovanovic1990 “Financial Development, Growth, and the Distribution of Income” Journal of Political EconomyVol. 98 pp. 1076–1107.
HayashiFumio and Edward C.Prescott2002 “The 1990s in Japan: A Lost Decase” Review of Economic DynamicsVol. 5 (January) pp. 206–235.
HildebrandFrancis B.1987Introduction To Numerical Analysis (Mineola: Dover).
JeongHyeok2000 “Sources of Kuzunets Dynamics in Thailand.” PhD dissertation (Chicago: The University of Chicago).
JeongHyeok and Robert M.Townsend2001 “Evaluation of Models of Growht and Inequality.” ManuscriptThe University of Chicago.
JuddKenneth L.1998Numerical Methods in Economics (Cambridge: MIT Press).
KingRobert G. and RossLevine1993 “Finance and Growth: Schumpeter Might be Right” Quarterly Journal of EconomicsVol. 108No. 3 (August) pp. 717–737.
KlinhowhanUbonrat1999 “Monetary Transmission Mechanism in Thailand” Master’s thesis (Bangkok: Thammasat University).
KuznetsSimon1955 “Economic Growth and Income Inequality” American Economic ReviewVol. 45No. 1 (March).
KydlandFinn and Edward C.Prescott1982 “Time to Build and Aggregate Fluctuations” EconometricaVol. 50No. 6 (November) pp. 1345–1370.
Lloyd-EllisHuw and DanBernhardt2000 “Enterprise, Inequality and Economic Development” Review of Economic StudiesVol. 67No. 1 (January) pp. 147–168.
LucasRobert E.Jr.2003 “Macroeconomic Priorities” American Economic ReviewVol. 93No. 1 (March) pp. 1–14.
McKinnonRonald I.1973Money and Capital in Economic Development (Washington, DC: Brookings Institution).
MookherjeeDilip and Anthony F.Shorrocks1982 “A Decomposition Analysis of the Trend in UK Income Inequality” Economic JournalVol. 92No. 368 (December) pp. 886–902.
PikettyThomas1997 “The Dynamics of the Wealth Distribution and the Interest Rate with Credit Rationing” Review of Economic StudiesVol. 64No. 2 (April) pp. 173–189.
Saint-PaulGilles1992 “Technological Choice, Financial Markets and Economic Development” European Economic ReviewVol. 36No. 4 (May) pp. 763–781.
ShawEdward S.1973Financial Deepening in Economic Development (New York: Oxford University Press).
StokeyNancy L.Robert E.LucasJr. and Edward C.Prescott1989Recursive Methods in Economic Dynamics (Cambridge: Harvard University Press).
TownsendRobert M.1978 “Intermediation with Costly Bilateral Exchange” Review of Economic StudiesVol. 45 (October) pp. 417–425.
TownsendRobert M.1983 “Financial Structure and Economic Activity” American Economic Review (December) pp. 895–911.
TownsendRobert M.AnnaPaulsonSombatSakuntasathienTae JeongLee and MikeBinford1997 “Questionnaire design and data collection for NICHD grant Risk, Insurance and the Family, and NSF grants.” ManuscriptThe University of Chicago.
TownsendRobert M. and KenichiUeda2001 “Transitional Growth with Increasing Inequality and Financial Deepening” IMF Working Paper No. 01/108.
UedaKenichi2003 “Limit Theory on Aggregate Variables with Heterogeneous Types in an Incomplete Market Economy” (June). Manuscript International Monetary Fund.