Economic development, flow of funds, and the equilibrium interaction of financial frictions

Micro Data and Financial Obstacles.

Here we briefly describe a series of studies using data from the Townsend Thai project that document that even within a given economy individuals face different types of financial frictions depending on location and urban/rural status. In particular, several studies using a variety of data, variables, and approaches reach the same conclusion, namely that MH problems are more pronounced in the central region and in urban areas whereas LC is the dominant constraint in the northeast region and in rural areas. For want of space we spare the reader a detailed description of the Townsend Thai project and its data, although this is available in SI Appendix, section A and in ref. 6.

Several studies make use of these data to infer financial obstacles on the ground. The working-paper version (5) describes these in detail, and we here only provide a brief summary. Paulson et al. (7) estimate the financial/information regime in place in an occupation choice model and find that MH fits best in the more urbanized central region whereas LC or a mixed regime fits best in the more rural northeast region. Karaivanov and Townsend (8) estimate the regime for households running businesses and find that an MH constrained financial regime fits best in urban areas and a more limited savings regime in rural areas. Finally, Ahlin and Townsend (9), with alternative data on joint liability loans, find that information seems to be a problem in the central area, with LC in the northeast.

Meso Data and Factor Flows.

Direct and indirect evidence suggests large flows of capital and labor.

Household’s Problem.

Households can either be entrepreneurs or workers. We denote by xit=1 the choice of being an entrepreneur and by xit=0 that of being a worker. First, consider entrepreneurs. An entrepreneur hires labor ℓit at a wage wt and rents capital kit at a rental rate rt+δ, where δ is the depreciation rate, and produces some output. His observed productivity has two components: a component, zit, that is known by the entrepreneur in advance at the time he decides how much capital and labor to hire and a residual component, εit, that is realized afterward. We will call the first component “entrepreneurial” ability and the second “residual productivity.” The evolution of entrepreneurial talent is exogenous and given by some stationary transition process μ(zit+1|zit). Residual productivity instead depends on an entrepreneur’s effort, eit, which is potentially unobserved, depending on the financial regime. More precisely, his effort determines the distribution p(εit|eit) from which residual productivity is drawn, with higher effort making good realizations more likely. We assume that intermediaries can ensure residual productivity εit. In contrast, even if entrepreneurial ability, zit, is observed, it is not contractible and hence cannot be ensured. An entrepreneur’s output is given byzitεitf(kit,ℓit),where f(k,ℓ) is a span-of-control production function.

Next, consider workers. A worker sells efficiency units of labor εit in the labor market at wage wt. Efficiency units are observed but are stochastic and depend on the worker’s true underlying effort, with distribution p(εit|eit).^¶ The worker’s true underlying effort is potentially unobserved, depending on the financial regime. A worker’s ability is fixed over time and identical across workers, normalized to unity.

Putting everything together, the income stream of a household isyit=xit[zitεitf(kit,ℓit)−wtℓit−(rt+δ)kit]+(1−xit)wtεit.As specified above, each household’s wealth (deposited with the intermediary) accumulates according to Eq. 1.

The timing is illustrated in Fig. 1 and is as follows. The household comes into the period with previously determined savings ait and a draw of entrepreneurial talent zit. Then, within period t, the contract between household and intermediary assigns occupational choice xit, effort, eit, and—if the chosen occupation is entrepreneurship—capital and labor hired, kit and ℓit, respectively. All these choices are conditional on talent zit and assets carried over from the last period, ait. Next, residual productivity, εit, is realized, which depends on effort through the conditional distribution p(εit|eit). Finally, the contract assigns the household’s consumption and savings, that is, functions cit(εit) and ait+1(εit). The household’s effort choice eit may be unobserved depending on the regime we study. All other actions of the household are observed. For instance, there are no hidden savings.

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Fig. 1.

Timing.

We now write the problem of a household that contracts with the intermediary in recursive form. The two state variables are wealth, a, and entrepreneurial ability, z. Recall that z evolves according to some exogenous Markov process μ(z′|z). It will be convenient below to denote the household’s expected continuation value by 𝔼z′v(a′,z′)=∑z′v(a′,z′)μ(z′|z), where the expectation is over z′. A contract between a household of type (a,z) and an intermediary solvesv(a,z)=maxx,e,k,ℓ,c(ε),a′(ε)∑εp(ε|e){u[c(ε),e]+β𝔼z′v[a′(ε),z′]}s.t.∑εp(ε|e){c(ε)+a′(ε)}=∑εp(ε|e){x[zεf(k,ℓ)−wℓ−(r+δ)k]+(1−x)wε}+(1+r)a[2]and also is subject to regime-specific constraints specified below.

The contract maximizes a household’s expected utility subject to a break-even constraint for the intermediary. Note that the budget constraint in Eq. 2 averages over realizations of ε; it does not have to hold separately for every realization of ε. This is because the contract between the household and the intermediary has an insurance aspect. Such an insurance arrangement can be “decentralized” in various ways. The intermediary could simply make state-contingent transfers to the household. Alternatively, intermediaries can be interpreted as banks that offer savings accounts with state-contingent interest payments to households.

In contrast to residual productivity ε, talent z is assumed to not be insurable. Before the realization of ε, the contract specifies consumption and savings that are contingent on ε, c(ε), and a′(ε). In contrast, consumption and savings cannot be contingent on next period’s talent realization z′.^# As we explain above, one reason for introducing uninsurable talent shocks (besides realism) is to guarantee the existence of a stationary distribution in the presence of MH.

The contract between intermediaries and households is subject to one of two frictions: private information in the form of MH or LC. Each friction corresponds to a regime-specific constraint that is added to the dynamic program Eq. 2. For sake of simplicity and to isolate the economic mechanisms at work, the only thing that varies across the two regimes is the financial friction. It would be easy to incorporate some differences, say in the stochastic processes for ability z and residual productivity ε at the expense of some extra notation. Most studies in the existing macro development literature work with collateral constraints that are either explicitly or implicitly motivated as arising from an LC problem. In contrast, there are relatively fewer studies that model financial frictions as arising from an asymmetric information problem like in our MH regime. Notable exceptions are refs. 13⇓–15. We specify our two financial regimes in turn.

Urban Areas: MH.

In this regime, effort e is unobserved. Because the distribution of residual productivity, p(ε|e), depends on effort, this gives rise to a standard MH problem: Full insurance against residual productivity shocks would induce the household to shirk, that is, to exert suboptimal effort. The contract takes this into account in terms of an incentive-compatibility constraint:∑εp(ε|e){u[c(ε),e]+β𝔼z′v[a′(ε),z′]}≥∑εp(ε|e^){u[c(ε),e^]+β𝔼z′v[a′(ε),z′]} ∀e,e^.[3]This constraint ensures that the value to the household of choosing the effort level assigned by the contract, e, is at least as large as that of any other effort, e^. The optimal dynamic contract in the presence of MH solves Eq. 2 subject to the additional constraint Eq. 3. As already mentioned, to fix ideas, we would like to think of this regime as representing the prevalent form of financial contracts in urban and industrialized areas.

Relative to existing theories of firm dynamics with MH, our formulation in Eq. 3 is special in that only entrepreneurial effort is unobserved. In contrast, capital stocks can be observed and a change in an entrepreneur’s capital stock does not change his incentive to shirk. More precisely, the distribution of relative output obtained from two different effort levels does not depend on the level of capital. This is a result of two assumptions: that output depends on residual productivity ε in a multiplicative fashion and that the distribution of residual productivity p(ε|e) does not depend on capital (i.e., it is not given by p(ε|e,k)). We focus on this instructive special case because—as we will show below—it illustrates in a transparent fashion that MH does not necessarily result in capital misallocation but that it can nevertheless have negative effects on aggregate productivity, gross domestic product (GDP), and welfare.

The existing literature on optimal contracting subject to MH typically makes use of an alternative formulation for problems like the one used here. In particular, the relevant dynamic programming problem is typically written with “promised utility” as a state variable and features a “promise-keeping” constraint (16, 17). We here instead develop an alternative approach: We invert the Pareto frontier between households and intermediaries, thereby replacing promised utility as the relevant state variable by household wealth. This formulation has two advantages. First, the contracting problem in terms of wealth “communicates” more seamlessly with the rest of the model, in particular when we later embed the contracting problem in general equilibrium, which features a market-clearing condition in terms of wealth. Second, our alternative formulation can be mapped to the data more directly: Our ultimate interest is in flow of funds across households and regions, which is more naturally thought of in terms of wealth rather than promised utilities.

SI Appendix, section D lays out our approach and its connection to the more standard formulation in detail. We here briefly summarize it. Consider first a special case with no ability (z) shocks and only residual productivity (ε) shocks. For this case Proposition 1 in SI Appendix, section D shows that the two formulations are equivalent if the Pareto frontier between households and intermediaries is monotone. In this case, one can invert the Pareto frontier and use a change of variables to express the problem in terms of household wealth rather than promised utility. In this sense, the insurance arrangement regarding ε-shocks is optimal (taking all paths of interest rates and wages as fixed). Next, consider the case with both z-shocks and ε-shocks. This case is then simply the problem just described but with uninsurable ability shocks “added on top.” That is, in this case it is no longer true that we solve a fully optimal contracting problem. This is because we rule out insurance against z-shocks by assumption, whereas an optimal dynamic contract would allow for such insurance. In contrast, the insurance arrangements regarding ε-shocks are optimal as shown by the equivalence with an optimal dynamic contract in the absence of z-shocks.

Given this equivalence between the two formulations, it is also easy to motivate why we assume that idiosyncratic shocks are partly uninsurable. Dynamic MH economies in which all shocks can be insured against often do not feature a stationary distribution of promised utilities (see e.g., refs. 18 and 19). In our formulation this would correspond to nonexistence of a stationary wealth distribution. Uninsurable shocks aid with ensuring stationarity and, indeed, our numerical results indicate that a stationary wealth distribution always exists. Besides realism, ensuring stationarity is another reason for making the assumption that ability shocks are uninsurable.

When solving the problem Eq. 2 to Eq. 3 numerically, we allow for lotteries in the optimal contract to “convexify” the constraint set as in ref. 19. See SI Appendix, section E for the statement of the problem, Eq. 2 to Eq. 3 with lotteries.

Rural Areas: LC.

In this regime, effort e is observed. Therefore, there is no MH problem and the contract consequently provides perfect insurance against residual productivity shocks, ε. Instead we assume that the friction takes the form of a simple collateral constraint:k≤λa, λ≥1.[4]This form of constraint has been frequently used in the literature on financial frictions (see, e.g., refs. 7 and 20⇓⇓⇓⇓–25). It can be motivated as an LC constraint. The exact form of the constraint is chosen for simplicity. Some readers may find it more natural if the constraint were to depend on talent k≤λ(z)a as well. This would be relatively easy to incorporate, but others have shown that this affects results mainly quantitatively but not qualitatively (24, 26). The assumption that talent z is stochastic but cannot be insured makes sure that collateral constraints bind for some individuals at all points in time. If instead talent were fixed over time, for example, individuals would save themselves out of collateral constraints over time (27).

The optimal contract in the presence of LC solves Eq. 2 subject to the additional constraint Eq. 4.

Factor Demands and Supplies.

Households, via the intermediaries they contract with, interact in competitive labor and capital markets, taking as given the sequences of wages and interest rates. Denote by kj(a,z) and ℓj(a,z) the common optimal capital and labor demands of households with current state (a,z) in regime j∈{MH,LC}. A worker supplies ε efficiency units of labor to the labor market, so labor supply of a cohort (a,z) isnj(a,z)≡[1−xj(a,z)]∑εp(ε|ej(a,z))ε.[5]Note that we multiply by the indicator for being a worker, 1−x, so as to only pick up the efficiency units of labor by the fraction of the cohort who decide to be workers. Finally, individual capital supply is simply a household’s wealth, a.

Equilibrium.

We use the saving policy functions a′(ε) and the transition probabilities μ(z′|z) to construct transition probabilities Pr(a′,z′|a,z;j) in the two regimes j∈{MH,LC}. In the computations we discretize the state space for wealth, a, and talent, z, so this is a simple Markov transition matrix. Given these transition probabilities and initial distributions gj,0(a,z), we then obtain the sequence {gj,t(a,z)}t=0∞ fromgj,t+1(a′,z′)=Pr(a′,z′|a,z;j)gj,t(a,z).[6]Note that we cannot guarantee that the process for wealth and ability in Eq. 6 has a unique and stable stationary distribution. Whereas the process is stationary in the z-dimension (recall that the process for z, μ(z′|z), is exogenous and a simple stationary Markov chain), the process may be nonstationary or degenerate in the a-dimension. That is, there is the possibility that the wealth distribution either fans out forever or collapses to a point mass. Similarly, there may be multiple stationary equilibria. In the examples we have computed, these issues do, however, not seem to be a problem and Eq. 6 always converges, and from different initial distributions.

Once we have found a stationary distribution of states from Eq. 6, we check that markets clear and otherwise iterate. Denote the stationary distributions of ability and wealth in regime j by Gj(a,z). Then, the labor and capital market clearing conditions areϑ∫ℓMH(a,z)dGMH(a,z)+(1−ϑ)∫ℓLC(a,z)dGLC(a,z)=ϑ∫nMH(a,z)dGMH(a,z)+(1−ϑ)∫nLC(a,z)dGLC(a,z),ϑ∫kMH(a,z)dGMH(a,z)+(1−ϑ)∫kLC(a,z)dGLC(a,z)=ϑ∫adGMH(a,z)+(1−ϑ)∫adGLC(a,z).The equilibrium factor prices w and r are found using the algorithm outlined in appendix A.1 of ref. 23.

Note that, in equilibrium, the demands and supplies of both capital and labor are equated in a frictionless manner and that this requirement determines the allocation of factors across regions. That is, we assume that there are no frictions to the movement of capital or labor across regions. In a counterfactual policy experiment, described later in this paper, we examine the effect of going from such an integrated equilibrium to the opposite extreme, namely autarky.

Calibration.

Due to space constraints, we relegated the discussion of functional form choices and calibration of parameter values to SI Appendix, section F. Our calibration targets various regional aggregates, namely income, consumption, capital, wealth, and the rate of entrepreneurship in both rural and urban areas (SI Appendix, Table 5).

Interregional Flow of Funds.

At these calibrated parameter values we compute the model’s steady state. See SI Appendix, section E for details on the computations. We feature in Table 1 the variables for each of the two regions separately, the overall economy-wide average, using population weights, and especially the flow of capital and labor across regions. As is evident in Table 1 the (urban) MH area has higher values of income, capital, labor, consumption, and wealth than the (rural) LC area.^∥ All variables are expressed as ratios to the corresponding first-best values, each line, one at time. The first-best economy eliminates the LC and MH constraints in rural and urban areas, respectively, so they are identical and thus have the same variable values—region labels lose any meaning in the first-best because one region is just a clone of the other one. In contrast, with the financial obstacles included, we see in Table 1 the additional implication that the urban area consistently has values higher than those of the rural area (i.e., more activity is concentrated there than in the first-best, and less in the rural area). The top part of the table is thus a tell-tale indicator of the relatively dramatic interregional flows at the bottom of the table. Urban areas are importing 23% of the overall capital used and 75% of the labor. Likewise, rural areas are exporting 39% of their capital and 86% of their labor. This is consistent with the direct and indirect evidence reported above. Equivalently urban areas are 79% of the economy’s capital and 65% of its labor even though they account for only 30% of the population.**

There are of course many other factors that distinguish cities from villages and industrialized from agricultural areas, and we listed some of these in the Introduction. Although we consider these other factors to be of great importance for explaining interregional flow of funds, we purposely exclude them from our theory and focus on differences in financial regimes only, in effect conducting an experiment that makes use of the model structure and answers the following question: How large are the capital and labor flows that arise from regional differences in financial regime alone? Our framework generates a number of observed rural–urban patterns by letting only the financial regime differ across these regions. In our model, without regional differences in the financial regimes, urban and rural areas would be identical with no factor flows occurring between the two regions.

To explain why this is happening we proceed in steps, first looking at the interest rate then the occupation choices and related variables in each region (at the equilibrium interest rate and wage and, of course, at our calibrated parameter values).

Determination of the Equilibrium Interest Rate.

The interest rate is depressed relative to the rate of time preference in both regions, but as we shall now see there are pressures for it to be far lower in the LC rural area, if the domestic economy were not open across regions.

Fig. 2 graphically examines the aggregate demand for and supply of capital at various parametric interest rates, as if the regions were open to the rest of the world, and thus illustrates the determination of the equilibrium interest rate (as in ref. 29) for each region separately, where the curves cross, as if it were a closed economy (no regional or international capital flows).

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Fig. 2.

Determination of equilibrium interest rate: moral hazard (A); limited commitment (B).

Fig. 2A plots capital demand and supply for the MH regime (solid lines) and contrasts them with demand and supply in the “first-best” economy without MH (dashed lines). For each value of the interest rate, the wage is recalculated so as to clear the labor market. Fig. 2B repeats the same exercise for the LC regime. The first-best demand and supply (the dashed lines) are the same in the two panels and serve as a benchmark to assess the differential effects of the two frictions on the interest rate.

Consider first the MH economy in Fig. 2A. Relative to the first-best, MH depresses capital demand for all relevant values of the interest rate. This is because MH results in entrepreneurs and workers exerting suboptimal effort, which depresses the marginal productivity of capital. The effect of MH on capital supply is ambiguous and differs according to the value of the interest rate. It turns out that this ambiguity is the result of a direct effect and a counteracting general equilibrium effect operating through wages. For a given fixed wage, MH always decreases capital supply (i.e., capital supply shifts to the left). This is due to a well-known result: the inverse Euler equation of ref. 30, which states that the optimal contract under MH discourages saving whenever the incentive compatibility constraint Eq. 3 binds and hence results in individuals’ being saving-constrained (see also refs. 31 and 32). Lemma 1 in SI Appendix, section B derives the appropriate variant of this result for our framework and discusses the intuition in more detail. However, counteracting this negative effect on capital supply is a positive general equilibrium effect: Labor demand, and hence the wage, falls relative to the first-best, resulting in more entry into entrepreneurship, higher aggregate profits, and higher savings. The overall effect is ambiguous, and in our computations capital supply shifts to the right for some values of the interest rate and to the left for others.

Contrast this with the LC economy in Fig. 2B. Under LC, capital demand shifts to the left whereas capital supply shifts to the right. The drop in capital demand is a direct effect of the constraint Eq. 4, and it is considerably larger than the demand drop under MH. That capital supply shifts to the right is due to increased self-financing of entrepreneurs (refs. 23 and 26, among others). As a result, the interest rate drops considerably relative to the first-best, and more so than under MH. Obviously, the size of this drop depends on the parameter λ, which governs how binding the LC problem is. The value we use in Fig. 2 is the one we calibrate, 1.80, but our findings are qualitatively unchanged for many different values of λ.

The finding that the equilibrium interest rate is lower under LC than under MH is present in all our numerical experiments and under a big variety of alternative parameterizations we have tried.^††

This is not surprising, given that Fig. 2 suggests that there are some strong forces pushing in this direction. Foremost among these is that, under MH, individuals are savings-constrained, which, all else equal, pushes up the interest rate; in contrast, LC results in higher savings due to self-financing, which pushes down interest rates. Also going in this direction is that in practice LC results in a greater drop in capital demand than MH.

The bottom line from this analysis of the interest rate is that when the two regions are opened to capital (and labor) movements, capital flows toward what would have been the higher interest rate region, namely the MH urban area.^‡‡ Labor is complementary with capital and so the wage would have been higher in the MH urban area, too, if it were not for labor flows.

Are Different Financial Regimes Necessary?

In the working-paper version (5), we also show that if we had followed much of the macro development literature on financial frictions, and just assumed those frictions, rather than imposing what we “see on the ground” (i.e., infer from micro data), then we would not be able to simultaneously match salient features of both the meso and micro data. It is key that the type of financial regime varies, as opposed to urban/industrialized and rural/agrarian areas’ being subject to the same financial regime but with differing tightness of the financial constraint. To make this point, we conduct the following experiment. We suppose that, instead of MH, the central area is subject to the same form of LC as the northeast area but with a higher, more liberal maximum leverage ratio. We show that to do as well as our benchmark economy in terms of matching observed factor flows, we have to raise the central leverage ratio to well beyond reasonable levels (close to infinity).