Liquidity Choice and Misallocation of Credit

This paper studies a novel type of misallocation of credit between investments of varying liquidity. One type of investment is more liquid, i.e., its return is more pledgeable, and the other is more productive. Low liquidities of both investment types imply that the allocation of credit is constrained inefficient and that there is overinvestment in the liquid type. Constrained inefficient equilibria feature non-positive, i.e., one less than or equal the economy’s growth rate, and yet too high interest rate, too much investment and too little consumption. Financial development can reduce long-term welfare and output in a constrained inefficient equilibrium if it raises the liquidity of the liquid type. I show a maximum liquid asset ratio or a simple debt tax can achieve constrained efficiency. Introducing government bonds can make Pareto improvement whenever it does not raise the interest rate.

Abstract

This paper studies a novel type of misallocation of credit between investments of varying liquidity. One type of investment is more liquid, i.e., its return is more pledgeable, and the other is more productive. Low liquidities of both investment types imply that the allocation of credit is constrained inefficient and that there is overinvestment in the liquid type. Constrained inefficient equilibria feature non-positive, i.e., one less than or equal the economy’s growth rate, and yet too high interest rate, too much investment and too little consumption. Financial development can reduce long-term welfare and output in a constrained inefficient equilibrium if it raises the liquidity of the liquid type. I show a maximum liquid asset ratio or a simple debt tax can achieve constrained efficiency. Introducing government bonds can make Pareto improvement whenever it does not raise the interest rate.

1 Introduction

Financial frictions can distort the allocation of credit in the economy, resulting in low output and welfare. One type of credit misallocation is the overextending of credit to and overinvestment in liquid1 but low-productivity assets. Previous academic and policy research suggest that this type of misallocation pertains to many different contexts. Examples include allocating too much credit to large firms with low productivity, firms in construction and real estate sectors and government-owned enterprises with especial access to credit. Despite low productivity, high pledgeability of investment is a potentially important factor contributing to overinvestment in these examples (Gopinath et al. (2017), Dell’Ariccia et al. (2019) and Zheng et al. (2011)).

This paper studies a novel type of constrained inefficiency in credit allocation in an economy consisting of high-return/low-liquidity and low-return/high-liquidity projects, which involves overinvestment in the latter. The allocation of credit in this environment is clearly not the first best but also not even the second best: a planner who faces the same liquidity constraints as the private agents can still improve upon the equilibrium allocation and raise welfare. Symptoms of this inefficiency are too-high interest rate and investment relative to the constrained optimum, i.e., the second best. As a result, constrained Pareto improving policies reduce the interest rate and investment. This is in stark contrast to the unconstrained inefficiency in the same model where the interest rate and investment are too low relative to the unconstrained optimum, i.e, the first best, and the unconstrained Pareto improvement raises the interest rate and investment.

This new inefficiency also has important implications for financial development. I show that financial development that is conventionally thought to bring economies with financial frictions closer to the first best may in fact reduce output and welfare because it can increase the extent of misallocation. Such adverse effects are present when the allocation of credit is constrained in-efficient, and financial development raises the liquidity of the low-return/high-liquidity projects. These adverse effects can speak to the new evidence on the non-monotonic relationship between financial development, output, and welfare (Levine (2005)).

A brief description of the model is as follows. The economy consists of overlapping generations of entrepreneurs who live for three periods: young, middle-aged and old. There is a continuum of each generation present in each period, and there are no aggregate or idiosyncratic risks in the economy. When young, entrepreneurs receive a fixed endowment of perishable consumption goods (and nothing thereafter) that cannot be stored. Middle-aged entrepreneurs have an opportunity to invest in a portfolio of investments. There are two types of constant return to scale investment technologies. One type is more productive and has a higher return per unit of investment (productive type). The other type of investment has a higher pledgeable return per unit of investment (liquid type). That is, a bigger part of its return can be credibly promised to be paid back. The middle-aged use their wealth which is the principal and interest on the loan they made when young to the middle-aged in the previous period (the old in the current period) plus funds they borrow from the young in the current period. There is a competitive credit market in every period in which borrowing by the middle-aged is constrained by the total amount of pledgeable return to their investment portfolio. Higher wealth, more liquid portfolio, and a lower interest rate allow the middle-aged to borrow more from the young. Hence a liquid portfolio allows for a bigger investment size while a productive one raises the average return per unit of investment. Finally, entrepreneurs consume only when they are old.

When liquidities of both types are low, entrepreneurs might end up investing too much in the liquid type of investment due to a pecuniary externality leading to inefficiently liquid equilibria. The constrained efficient allocation in such cases requires investing only in the productive type at the steady state. The portfolio choice of the middle-aged entrepreneurs at each date depends on the prevailing interest rate: all else equal, a higher interest rate implies lower leverage and less investment in the liquid type. An additional unit of investment in the liquid type by an entrepreneur bids up the interest rate and raises the debt payments for other middle-aged entrepreneurs. But low liquidity of both investment types implies that the initial wealth of the middle-aged will be low since a low fraction of the returns to investment in any period can be invested by future entrepreneurs. Hence, given the borrowing constraint, the additional increase in the interest rate will be small and cannot sufficiently discourage other entrepreneurs from investing in the liquid type. In fact, when liquidities are high enough, investment in the liquid type by entrepreneurs would bid up the interest rate so high that it would make them switch to the productive type; therefore, investment in the liquid type cannot be an equilibrium. Hence, when liquidities are low, the negative effect of an additional unit of investment in the liquid type on debt payments more than offset the reallocation away from the liquid type, leaving other agents worse off. Inefficiently liquid equilibria feature non-positive, i.e., lower or equal to the growth rate, and yet too-high interest rate, too much investment and too little consumption.

The constrained inefficiency introduced in this work is more likely to arise in financially underdeveloped economies, e.g, low-income and emerging markets, where most types of investment have low liquidity. It is applicable to a wide range of environments in which investment projects differ in both their productivity and liquidity. Examples include but are not limited to real estate versus non-real estate, small and medium enterprises (SME) versus big mature firms, public versus private firms, and capital-intensive versus labor-intensive production technologies. Inefficiently liquid equilibria in the context of these examples would feature credit and investment booms accompanied by overinvestment in large firms, firms with more tangible assets such as land and physical capital, and state-owned firms.

This paper provides a new justification for public intervention in the financing of young firms and SME. A growing body of research studies the misallocation of credit especially in economies with underdeveloped financial markets. In contrast to this line of work, this paper argues that the allocation of credit to young firms and SME is not only not the first best, but that it even may not be the second best. Given that the second best allocation may be achieved using simple regulations such as a debt tax, this implies a stronger case for public support of young firms and SME financing than what was previously understood.

These results suggest that policies such as developing private bond markets, loan guarantees, and development of primary and secondary markets for asset securitization should be pursued with more caution in low-income and emerging market economies. Bond financing, for example, can mostly benefit large established firms that tend to have more liquid but low-productivity investment projects. Securitization and loan guarantees can be harmful if the underlying assets, e.g., residential mortgages, are more liquid relative to other investment opportunities in the economy. These policies may lower long-term output and welfare if they end up benefiting low-productivity but liquid investments in an already inefficiently liquid equilibrium. Policies that facilitate seizure of collateral can also be harmful to the long-term output and welfare when there is overinvestment in tangible assets in the economy. The reason is that these policies mainly increase the liquidity of investments with a high share of tangible assets. Examples of such policies include the creation of public property registries and the improvement of creditor rights in bankruptcy laws. In contrast, policies which raise the liquidity of the more productive but less liquid investment projects always enhance long-term output and welfare in an inefficient equilibrium.

Given an inefficient equilibrium, a planner can achieve efficiency by regulating the fraction of resources that is invested in the liquid type by entrepreneurs. This regulation may be implemented as a maximum liquid asset ratio, e.g., a cap on the ratio of real estate loans to total assets, within a perfectly competitive banking sector. Pareto efficiency can also be obtained by a simple debt tax. I study the welfare effects of government bonds, which are assumed to be fully liquid due to the ability of government to tax. Government bonds can make Pareto improvement only for inefficient equilibria where there is strictly positive investment in both types in the long run. In this case, government bonds crowd out the liquid type and crowd in the productive type so that the demand for funds and hence the interest rate remains unchanged in equilibrium. Entrepreneurs can substitute fully liquid government bonds for the liquid type to gain an extra amount of pledgeable return that can be used to borrow more funds that can be invested in the productive type. Since the interest rate does not increase, entrepreneurs’ return goes up while their debt payment stays the same. This increases entrepreneurs’ consumption and leads to Pareto improvement.

I characterize the competitive equilibrium and the steady state, and do comparative statics with respect to liquidities and returns of the two types of investments. Contracting technology, contract enforcement, corporate governance, and bankruptcy laws are among the factors that can affect the liquidity of investment. I show that an increase in the liquidity or return of the productive type of investment leads to a lower steady-state interest rate. This interest rate effect can be understood as follows. An increase in the liquidity or return of the productive type has two effects. First, for any given investment portfolio, it increases the liquidity of that portfolio which, in turn, raises the investment demand and interest rate. Second, an increase in the liquidity or return of the productive type makes the productive type more attractive to investors. This leads the investors to substitute the liquid for productive but still less liquid type, which reduces the interest rate. It turns out that the second effect dominates the first one, and so the steady state interest rate falls as the liquidity or return of the productive type increases.

The above interest rate effect can be useful in understanding the patterns of capital outflows in emerging market economies such as China in the past few decades. Consider an open-economy version of this model with two countries, home and foreign. The liquid type represents investment in large mature firms with easy access to external finance, e.g., state-owned firms in China, while the productive type is investment in highly productive entrepreneurial firms in the private sector with limited access to external finance. In such an open-economy version, an increase in the return of the productive type at home, e.g., higher productivity of private entrepreneurial firms in China, results in outlows of funds to the foreign country. These outlows are accompanied by a reallocation of credit toward the productive type at home. Moreover, if the home country is not small, this reallocation of credit lowers the world interest rate. The above narrative resembles the one suggested by Zheng et al. (2011).

Finally, there are three differences worth noting between the constrained inefficiency discussed in this paper and the conventional dynamic inefficiency in models with overlapping generations, e.g. Diamond (1965). First, there exist inefficiently liquid equilibria with a zero interest rate in steady state, while an equilibrium with zero interest rate is eicient in the traditional models. Second, in contrast to traditional models, a negative2 (lower than the growth rate) interest rate can be constrained eicient in this model. Third, a Pareto improvement in an inefficiently liquid equilibrium induces an even more negative interest rate. In traditional models, however, an interest rate below the growth rate of the output must be raised to make a Pareto improvement.

1.1 Related Literature

Farhi and Tiróle (2012) is closely related to this paper and is used as a benchmark for the analysis. The model structure of my paper is similar to that in Farhi and Tirole (2012). The only difference is that in this paper, entrepreneurs have access to different types of investments with different liquidity. Heterogeneity in investment types leads to different results that are complementary to the ones obtained in Farhi and Tirole (2012). The way limited liquidity is modeled in this paper is similar to Farhi and Tirole (2009, 2010) and Matsuyama (2007) while it also has a close connection to Holmström and Tirole (1998) and Kiyotaki and Moore (2002, 2005, 2008).

Normative results in this paper are in contrast with those of Woodford (1990) and Holmström and Tirole (1998). Low liquidity generated by the private sector is at the heart of inefficiency in Woodford (1990) and Holmström and Tirole (1998). In my paper, too much liquidity is what makes the decentralized allocation inefficient. The nature of inefficiency also differs from that of Samuelson (1958) and Diamond (1965). A negative interest rate, i.e., one that is less than economy’s growth rate, in Samuelson (1958) and Diamond (1965) implies that the interest rate is too low. In contrast, a non-positive interest rate in an inefficient equilibrium in this paper indicates that the interest rate is in fact too high.

Similar to this paper, Kehoe and Levine (1993) and Lorenzoni (2008) feature pecuniary externality as the source of inefficiency in the credit market. Pecuniary externality in these models arises because asset prices, spot prices, or the interest rate appear in constraints other than the budget constraint. The inefficient sale of productive assets in an environment with aggregate uncertainty is the key in Lorenzoni (2008) that leads to an externality, while here, it is the demand for investible resources in an economy without any uncertainty that entails an inefficient outcome.

Matsuyama (2007) studies a model with heterogenous assets of different liquidity. Matsuyama (2007) focuses on the dynamics of aggregate credit and capital stock when investment composition plays an important role for given returns and liquidities of investment. The goal of this paper, however, is to study the credit misallocation resulting from heterogeneity in investment liquidity as well as the effects of financial development, i.e., exogenous changes in liquidity of investment, on the economy. There are also two different types of assets with different liquidity in Giglio and Severo (2012), namely tangible and intangible capital. Besides having a different focus, Giglio and Severo (2012) does not feature any portfolio choice between liquid and illiquid investments. There is a high degree of complementarity between liquid and illiquid capitals in Giglio and Severo (2012) due to the Cobb-Douglas production technology. In this paper, however, the liquid and illiquid types are perfect substitutes. Hence Giglio and Severo (2012) is closer to an economy with one type of asset, e.g., Farhi and Tirole (2012), than this paper.

Similar to Bianchi (2011), in any inefficient equilibria a debt tax can restore efficiency in this model. The inefficiency in Bianchi (2011) is due to distortions in the relative price of non-tradable to tradable goods in an small open economy with a ixed interest rate. In contrast, the pecuniary externality in this paper works through the interest rate.

On the empirical side, Gopinath et al. (2017) show that the declining interest rate on borrowing has led to a reallocation of credit toward larger firms with less binding borrowing constraint in Spain. Dell’Ariccia et al. (2019) show that construction sector grows significantly more than other sectors during credit booms and especially so during the bad booms, i.e., those that end in crisis or subpar growth. Reis (2013) argues that the reallocation of credit toward low productivity firms in the nontradable sector has been the main cause of the economic stagnation and subsequent slump in Portugal between 2000 and 2012. In Reis (2013), and in contrast to my model, too many resources are invested in the less liquid nontradable sector and especially in its less productive firms at the expense of the firms in the tradable sector.

The paper proceeds as follows. Section 2 describes the model and characterizes competitive equilibria and steady states. Section 3 discusses properties of equilibria and their interpretations and applications. Section 4 studies the efficiency of competitive equilibria and how a planner can Pareto improve the competitive equilibrium allocation when it is inefficient. In Section 5, I introduce government bonds and analyze their welfare implications, and in Section 6 I conclude.

2 Model

2.1 Agents, Preferences and Technology

The model economy is comprised of overlapping generations of entrepreneurs with no uncertainty. Each individual lives for three periods, and there is a unit measure ofyoung, middle-aged, and old cohorts in each period. Entrepreneurs receive a fixed endowment e > 0 of non-storable and homogenous consumption goods when young and no endowment thereafter, and consume only when they are old.

The choice ofoverlapping generations is mainly for simplicity and tractability. One can think of the agents in this economy as firms in the real or financial sectors facing alternating investment opportunities and borrowing constraints. The main feature of the model is the ability of these firms to choose a portfolio of investment projects while pledging the return to their portfolio to outside investors, i.e., non investing firms with otherwise idle resources. In this sense, this model is similar to Kiyotaki and Moore (2002) and Kiyotaki and Moore (2005) and has a close connection with Woodford (1990).

In any period, the middle-aged have the opportunity to invest in two types of projects which pay off in the next period. Projects differ in their return and liquidity. A project of type j ∈ {1, 2} has a constant return to scale Rj where θjRj can be pledged to the outside investors. Limited liquidity of return can arise in many contexts and for a number of different reasons, including asymmetric information, moral hazard, and limited commitment. Following Kiyotaki and Moore (2002, 2005, 2008) and Farhi and Tirole (2009, 2012, 2010), I summarize all these frictions in the variable θj, j ∈ {1, 2}. Given (R1, R2), I refer to θj, j ∈ {1, 2} as well as θjRj, j ∈ {1, 2} as liquidity of type j. I make the following assumption about the return and liquidity of projects:

Assumption 1. R1 > R2 > 1 and θ1R1 < θ2R2 < 1.

Assumption 1 captures the trade-off between liquidity and return across the two types of projects; type 1 is more productive (productive type) while type 2 is more liquid (liquid type). This type of trade-off between liquidity and return can be observed in both real and financial sectors. Large and more mature firms in the real sector tend to have lower cost of external financing, that is, their investment is more liquid, than SME and young firms. This may be due to the availability of extensive records and accounts, their reputation, or the higher value of their collateral. Stateowned firms in many emerging market economies with easy access to finance tend to be less productive than the financially constrained entrepreneurial firms. Investment in tangible assets is more liquid than investment in intangibles, which are typically hard to liquidate in the event of bankruptcy. Liquid assets such as real estate versus other types of assets with lower liquidity but higher return, e.g., machinery and human capital, provide another example. Financial securities with different haircuts3 can serve as yet another example in the financial sector.

It is helpful for future analysis to define a benchmark economy in which there is no trade-off between the two types:

Definition 1. The Benchmark Economy is an economy where R1 > R2 > 1 and 1 > θ1R1 > θ2R2.

Note that in the benchmark economy, type 1 projects dominate type 2 projects in terms of both liquidity and return. This implies that entrepreneurs never invest in type 2 in equilibrium, such that the economy collapses to one with a single type of investment project, similar to Farhi and Tirole (2012). The benchmark economy is used throughout the current paper to provide better understanding of the results.

2.2 The Problem of Middle-Aged Entrepreneurs

In each period a competitive credit market opens up in which young and middle-aged entrepreneurs can lend and borrow. The young born in period t > 0 inelastically supply all their endowments in the capital market. The middle-aged entrepreneur at time t, who has transferred funds from period t − 1 by investing in the projects of the middle-aged at t − 1, can borrow additional funds from the young. But this borrowing by the middle-aged entrepreneur is constrained by the limited liquidity of her investment portfolio. She chooses her optimal investment portfolio given the ongoing interest rate rt and resources that have been transferred from period t − 1 to period t.

Let x1t and x2t denote investments in types 1 and 2 and let it denote the new funds raised by the middle-aged entrepreneur at t using the resources of the young entrepreneurs in period t.

Given the interest rate rt, a middle-aged entrepreneur at t solves the following problem:

ct+1omaxit,x1t,x2t0R1x1t+R2x2t(1+rt)it(I)s.t.x1t+x2t(1+rt1)e+it ,(1+rt)itθ1R1x1t+θ2R2x2t .

The first constraint in the maximization above is the resource constraint of the middle-aged entrepreneur. (1 + rt − 1)e is the wealth transferred from period t − 1 to t by the middle-aged entrepreneur through investing her endowment e, in the projects of middle-aged entrepreneurs in period t − 1. The second term, it, is the total external funds that the middle-aged entrepreneurs borrow from the young entrepreneurs at t. The second constraint is the manifestation of the limited liquidity of the investments; the middle-aged entrepreneur cannot borrow more than what she can credibly commit to pay in period t + 1. For type j ∈ {1, 2}, the maximum that can be credibly promised to the lenders is θjRjxjt, and so the total amount of pledgeable return is given by the right-hand side of the second constraint. Finally, ct+1o denotes the consumption of the old entrepreneur in period t + 1.

The implicit assumption in Problem I, that the middle-aged can cross-pledge, i.e., pledge the return to one type of project to invest in the other, is not essential. There are at least two other variations which produce the same results as in this setup. In one variation, the middle-aged entrepreneurs have to decide first how much of their initial wealth they want to invest in each type of project (which cannot be altered later), and then they can pledge the return of each type only for investment in that type. In the second alternative, each middle-aged entrepreneur can invest in only one type of project. One can show that both of these alternatives lead to results similar to this model and that what matters is how the aggregate investment portfolio is determined and not whether firms, i.e., entrepreneurs, actually hold any portfolios.

The resource constraint is always binding in Problem I. If the interest rate is not too high, the borrowing constraint has to be binding as well. In this case one can eliminate x1t and x2t in the above problem and reach the following reduced form:

Lemma 1. In any competitive equilibrium where 1 + rt < R1 for all t, the borrowing constraint of the middle-aged entrepreneur binds in every period. Moreover, the problem of the middle-aged entrepreneur can be written in the following form:

maxitΛ(θ,R;rt)it+Φ(θ,R;rt1)e(II)s.t.(θ1R1(1+rt1)1+rtθ1R1)eit(θ2R2(1+rt1)1+rtθ2R2)e,

where,

Λ(θ,R;rt)((θ2θ1)R1R2θ2R2θ1R1)((1θ1)R1(1θ2)R2θ2R2θ1R1)(1+rt),Φ(θ,R;rt1)((θ2θ1)R1R2θ2R2θ1R1)(1+rt1).

The bold symbols (θ, R) are the vector of liquidities and returns of the two types of investments, i.e., (θ1, θ2, R1, R2).

In Lemma 1 the term Λ is the net marginal (and average) return of external funds it when borrowing constraint is binding. The two bounds in the constraint of Problem II corresponds to the two limits; when it hits the lower (upper) bound, the entrepreneur invests only in the productive (liquid) type, depending on the sign of Λ. Define rΛ(θ, R) as the interest rate in period t that makes the entrepreneurs indifferent between the two types:

1+rΛ(θ,R)(θ2θ1)R1R2(1θ1)R1(1θ2)R2.(1)

Then the entrepreneurs’ optimal demand for funds is characterized as follows:

{it=(θ2R2(1+rt1)1+rtθ2R2)e,if rt<rΛ(θ,R),it[(θ1R1(1+rt1)1+rtθ1R1)e,(θ2R2(1+rt1)1+rtθ2R2)e]if rt=rΛ(θ,R),(2)it=(θ1R1(1+rt1)1+rtθ1R1)e,if rt>rΛ(θ,R).

Figure 1 is an illustration of the middle-aged demand for funds given by 2 and the inelastic supply of funds by young entrepreneurs at time t.

Figure 1:
Figure 1:

Supply (red) and demand (blue) for funds at any period t as a function of the interest rate. wt − 1 denotes the wealth of the middle-aged, i.e. (1 + rt − 1)e. The two arms on the demand curve correspond to investing only in type 1 or 2. The flat segment in between corresponds to rt = rΛ (θ, R), where entrepreneurs mix. A higher period t − 1 interest rate, i.e. higher wt − 1, makes the two arms of the demand curve shift to the right but has no effect on the demand curve’s flat segment.

Citation: IMF Working Papers 2019, 284; 10.5089/9781513521480.001.A001

2.3 Competitive Equilibrium

In each period there is a fixed supply of funds e. Market clearing condition dictates:

it=e,t0.(3)

Combining 2 and 3 yields the equilibrium path of the interest rates:

{θ2R2(2+rt1)1if θ2R2(2+rt1)1<rΛ(θ,R),θ1R1(2+rt1)1if θ1R1(2+rt1)1>rΛ(θ,R),(4)rΛ(θ,R)otherwise

Given 2, the dynamic upper and lower bounds on the interest rate are:

θ1R1(2+rt1)1rtθ2R2(2+rt1)1.(5)

I can now define a competitive equilibrium as follows:

Definition 2. A competitive equilibrium is a sequence {it,x1t,x2t,rt}t=0 of investments and interest rates and an initial value of r−1 that satisfy conditions 1 to 5, in which x1t and x2t solve problem I and 1 + rt < R1 for all t > 0.4

Using 2 and 3, the aggregate investment portfolio at any date is:

{x1t=0,x2t=(2+rt1)eif rt<rΛ(θ,R),x1t=(θ2R2(2+rt1)(1+rΛ(θ,R))θ2R2θ1R1)e,x2t=((1+rΛ(θ,R))θ1R1(2+rt1)θ2R2θ1R1)eif rt=rΛ(θ,R),(6)x1t=(2+rt1)e,x2t=0if rt>rΛ(θ,R).

Entrepreneurs specialize in the productive (liquid) type when the interest rate is relatively high (low). To characterize competitive equilibrium, it is useful to define the following three regions in the parameter space:

Definition 3. Define F as the set of (θ, R) that satisfies Assumption 1 and also θ1R11θ1R1<R1. Then the three regions of F are defined as follows:

The Liquid Region is defined as F={(θ,R)F|(θ1R11θ1R1)<(θ2R21θ2R2)1+rΛ(θ,R)}.The Mixed Region is defined as Fm={(θ,R)F|(θ1R11θ1R1)<1+rΛ(θ,R)<(θ2R21θ2R2)}.The Illiquid Region is defined as Fi={(θ,R)F|1+rΛ(θ,R)(θ1R11θ1R1)<(θ2R21θ2R2)}.

Notice that all three regions, F, Fm, and Fi, have nonempty interiors. I require θ1R11θ1R1<R1 to ensure that the borrowing constraint is binding in the steady state.5

Lemma 2. Each of the three regions in Definition 3 has a unique and stable steady state equilibrium. More specifically:

{rss=(θ2R21θ2R2)1if (θ,R)F.rmss=rΛ(θ,R)if (θ,R)Fm.riss=(θ1R11θ1R1)1if (θ,R)Fi.

Moreover, at the steady state, the entrepreneurs specialize in the liquid and productive type of investments in regions F and Fi, respectively. Entrepreneurs invest in both types in Fm, where the amounts of each type are given by 6.

Using Lemma 2, the following proposition establishes the existence and uniqueness of competitive equilibrium:

Proposition 1. Given any (θ, R) ∈ F and an initial condition 1 + r−1 < R1, there exists a unique competitive equilibrium that converges to the steady state corresponding to (θ, R), given by Lemma 2.

3 Properties of Equilibria

In this subsection I analyze the three regions in Lemma 2 and do comparative statics with respect to θ = (θ1, θ2) for a given vector of returns R = (R1, R2). Given R, values of θ correspond to different liquidities of the two investment types, which reflect different institutional environments, i.e., contract enforcement, contracting technology, bankruptcy laws, corporate governance, etc. The following lemma summarizes general properties of the three regions for any given vector of returns.

Proposition 2. For a given vector of returns R the following are correct:

Figure 2:
Figure 2:

Image of F, Fm and Fi for R1 = 4 and R2 = 3 over the space of (θ1, θ2). The white area below the positively sloped straight line where Assumption 1 is violated corresponds to the benchmark economy in Definition 1.

Citation: IMF Working Papers 2019, 284; 10.5089/9781513521480.001.A001

  • 1- When θ is small enough (close to the origin), one can have all three types of steady-state equilibria.

  • 2- For any θ in the liquid region, θ(11+R1,11+R1).

  • 3- For any value of θ1, the values of θ2 for which (θ1, θ2) belongs to the liquid region lies strictly above the respective values of θ2 for which (θ1, θ2) belongs to the illiquid region.

  • 4- The boundary of the liquid region is a non-monotonic curve cutting the θ1 = 0 line twice: once at the origin and again at θ=(0,11+R1).

  • 5- The inner boundary of the illiquid region is a strictly increasing and convex function of θ1 which reaches the maximum possible of θ2=1R2.

  • 6- The top right corner of F in the space of liquidities, that is, θ=(11+R1,1R2), belongs to the illiquid region.

Figure 2 suggests that the allocation of credit is non-monotonic in θ2. The following lemma establishes the non-monotonicity of credit allocation and interest rate with respect to changes in θ.

Lemma 3. In the mixed region, the steady state interest rate is strictly decreasing in θ1 and R1 while it is strictly increasing in θ2 and R2. Moreover, the fraction of total funds invested in the liquid type at the steady state, i.e., x2ssx1ss+x2ss, is non monotonic in θ2 and has an interior maximum for relatively low values of θ1. In contrast, this ratio is always weakly decreasing in θ1 and strictly decreasing in θ1 and R1 in Fm.

Suppose that θ2 increases while θ1 is held constant. On the one hand, this increase encourages middle-aged entrepreneurs to invest more in the liquid type at any given interest rate. On the other hand, this increase in liquidity of the liquid type increases the demand for funds and raises the interest rate at the steady state, which discourages the entrepreneurs from investing in the liquid type. Following Lemma 3, the second effect dominates the first one for high enough values of θ2. In this case, any further increase in θ2 bids up the interest rate so much that entrepreneurs are forced to lower the share of the liquid type in their portfolios.

An increase in θ2 or R2 raises the average liquidity of any portfolio and so raises the demand for funds at a given interest rate. This tends to bid up the steady-state equilibrium interest rate. In contrast, an increase in θ1 or R1 makes the productive type more attractive for a given interest rate. This, in turn, encourages the entrepreneurs to substitute the productive for the liquid type. Since the productive type is still the less liquid project, this substitution lowers the interest rate. It turns out that the second effect dominates the first one in the mixed region, which results in a strictly lower interest rate.

In a more general case, where financial development affects both liquidities or both returns, the direction of change in the interest rate and allocation of funds depends only on the relative change in liquidities or returns, e.g., if Δθ2Δθ1 is less (more) than the slope of the isoline in Figure 3, the interest rate decreases (increases).

Lemma 3 can be useful in thinking about capital outflows in emerging market economies such as China. These outflows have been puzzling because in a neoclassical world, capital has to flow to countries with the highest marginal product of capital, which seems to be a feature of fastgrowing emerging market economies. In particular, consider an open-economy version of this model with two countries, home and foreign. The middle-aged entrepreneurs at home and abroad can borrow from young entrepreneurs both at home and abroad. Suppose that the liquid type at home represents large mature firms with easy access to external finance, e.g., state-owned firms in the case of China, while the productive type represents more productive entrepreneurial firms with limited access to external finance. To simplify the exposition, assume that initially home and foreign have the same liquidities and returns but different endowments e and e* where e < e*, so that the net flows of funds is zero. In autarky an increase in the productivity of the productive type R1 at home leads to a lower interest rate and a higher fraction of resources invested in the productive type. This implies that in the two-country version, higher R1 at home induces the funds to flow out of the economy. The reallocation of credit toward the more productive type is still present in the two-country case but its magnitude is somewhat dampened relative to the autarky. If the home economy is large enough, i.e., ee+e* is not small, the outflow of funds lowers the equilibrium world interest rate.

Figure 3:
Figure 3:

Contour plot of the steady-state interest rates (red lines) for the three regions. Interest rates are highest at the top left corner.

Citation: IMF Working Papers 2019, 284; 10.5089/9781513521480.001.A001

The above narrative is similar to the one suggested by Zheng et al. (2011). They build an OLG model to reconcile high growth and high return to capital with a growing foreign surplus in China over the past three decades. The reallocation of labor from the less productive state-owned firms to the more productive but financially constrained private firms makes the economy look like an AK model during the transition. The constant returns to investments in state-owned and private firms during the transition are reminiscent of R1 and R2 in this model.6

4 Welfare and Efficiency

In this section I first study the efficiency of competitive equilibria and investigate policies that can Pareto improve upon the inefficient allocations. Next, I examine the effects of financial development on long-term welfare and discuss implications for measurement and policy.

4.1 Efficiency of Competitive Equilibria

I start by defining the notion of efficiency that I use throughout:

Definition 4. An allocation in the overlapping generations economy is called constrained Pareto efficient if a social planner cannot reallocate the resources to make at least one entrepreneur strictly better of while keeping all others at least as well off and if the reallocation respects the liquidity7 constraint in I. More formally, an allocation {ct*,x1t*,x2t*}t=0 is constrained Pareto efficient if it is feasible, i.e., it satisfies the following series of constraints for all t ≥ 0:

{ct+x1t+x2tR1x1t1+R2x2t1+e,x1t+x2tθ1R1x1t1+θ2R2x2t1+e(7)

and there does not exist any feasible allocation {ct,x1t,x2t}t=0 such that ctct* for all t ≥ 0 with at least one strict inequality, given initials xj,1=xj,1* for j ∈ {1, 2}.

The following proposition about the benchmark economy can help the reader understand the results more clearly:

Proposition 3. Any competitive equilibrium in the benchmark economy is constrained Pareto efficient.

Consider steady-state equilibria where investment in the liquid type is strictly positive. Suppose the planner reduces the aggregate debt payments of all middle-aged entrepreneurs in every generation to the young by an amount of δ > 0 by substituting the productive for the liquid type. Let the increase in the productive type be > 0; then the resource constraint implies that investment in the liquid type has to be reduced by + δ. Given δ, the maximum possible is determined when the borrowing constraint binds. The change in the consumption level of the old at t ≥ 1 is ΔVss = R1 − ( + δ)R2 + δ. Note that the initial middle-aged entrepreneur is strictly better off because the planner reduces her debt payments to the young while, in contrast with the future middle-aged, her receipts from the initial old do not change.8 Therefore if ΔVss ≥ 0, the steady-state allocation is constrained inefficient. The above reallocation can also work outside the steady state. The following proposition characterizes the constrained inefficient equilibria:

Figure 4:
Figure 4:

An illustration of inefficiently liquid equilibria (light gray region) and the line corresponding to rΛ = 0 for R1 = 4 and R2 = 3.

Citation: IMF Working Papers 2019, 284; 10.5089/9781513521480.001.A001

Proposition 4. Consider any competitive equilibrium with liquidities and returns given by (θ, R) ∈ FFm. If rΛ (θ, R) ≤ 0, the competitive equilibrium is constrained inefficient. Moreover, the equilibrium interest rate at the steady state is strictly negative when (θ, R) lies in the interior of the inefficient region, i.e., {(θ, R) ∈ FFm | rΛ(θ, R) ≤ 0}, and zero on part of its boundary that lies in Fm.9

Comparing Proposition 4 and Proposition 3 implies that entrepreneurs’ portfolio choices are the key feature causing inefficiency. Inefficiency arises because middle aged entrepreneurs’ portfolio choices entail a pecuniary externality. In order to better understand this pecuniary externality, the following lemma summarizes some of the properties of the constrained inefficient equilibria:

Lemma 4. For any R satisfying Assumption 1, the following are correct. The sets of inefficiently liquid competitive equilibria in F are nonempty with a strictly positive measure. There are inefficiently liquid equilibria in any arbitrarily small neighborhood of the origin. At θ1 = 0, the maximum value of θ2 that results in constrained inefficient equilibria is increasing in R1. The set of inefficiently liquid equilibria in F is a proper subset of F if and only if R1R2R211. Finally, let Si denote the unique intersection of rΛ (θ, R) = 0 with the boundary of Fi. Then all (θ, R) which correspond to inefficiently liquid equilibria have liquidities, i.e., θ, less than Si.

As Figure 4 and Lemma 4 suggest, the economy becomes inefficiently liquid when liquidities of both types are relatively low.10 The portfolio choice of middle-aged entrepreneurs at each date depends on the prevailing interest rate: all else equal, higher interest rate implies lower leverage and less investment in the liquid type. An additional unit of investment in the liquid type by an entrepreneur bids up the interest rate and raises the debt payments for other middle-aged entrepreneurs. But low liquidity of both investment types implies that the initial wealth of the middle-aged will be low since a small fraction of the returns to investment in any period can be invested by future entrepreneurs. Given the borrowing constraint, the additional increase in the interest rate will be small and cannot sufficiently discourage other entrepreneurs from investing in the liquid type. In fact, when liquidities are high enough, investment in the liquid type by entrepreneurs would bid up the interest rate so high that it would make them switch to the productive type; therefore, investment in the liquid type cannot be an equilibrium. Hence, when liquidities are low the negative effect of an additional unit of investment in the liquid type on debt payments more than offsets the reallocation away from the liquid type, leaving other agents worse off. Low liquidities of both investment types are also the reason the inefficiently liquid steady states have a non positive interest rate.

Figure 5:
Figure 5:

This figure shows the expansion of the inefficiently liquid region when R = (4, 2). In contrast with the case of R = (4, 3), all competitive equilibria in liquid region F are constrained Pareto inefficient. For low values of θ1 and compared to R = (4, 3), higher values of θ2 can lead to inefficiency.

Citation: IMF Working Papers 2019, 284; 10.5089/9781513521480.001.A001

The above lemma suggests that countries with a low level of financial development but high growth opportunities, i.e., a large R1, may be more prone to this type of constrained inefficiency (Figure 5).11 These economies suffer from shortages of stores of value due to low liquidity of return to investment. Investment in real estate is a liquid but relatively unproductive investment that has served as an important store of value in these countries. The analysis above suggests that these countries may be investing too much in real estate.

Investment in liquid assets such as real estate can take the form of bubbly equilibria in models with borrowing constraint. Absent uncertainty, these bubbly equilibria are Pareto efficient because they help agents transfer resources across periods.12 To my knowledge, this is the first paper to show that investment in liquid assets may be inefficient even in an environment without uncertainty.

Countries with underdeveloped financial institutions may also be investing too little in illiquid but highly productive projects such as young firms and small and medium enterprises (SMEs) with high growth potential as opposed to old and large firms. Young firms and SMEs are commonly believed to face severe frictions in financing their operations through credit markets. Therefore, SME finance is a prevalent concern among policy makers in both developed and developing countries.13 A growing body of research studies the misallocation of credit especially in economies with underdeveloped financial markets. In this line of work, misallocation of credit is the result of the limited net worth of some firms with high marginal product of capital which are facing binding borrowing constraints. Allocation of credit across firms, e.g., young firms or SMEs versus old and large firms, in such environments is not first best.14 This paper shows that the allocation of credit between SME and large firms or young and old firms may not even be the second best since young firms and SME not only have limited net worth to use as collateral but may also be subject to higher collateral requirements. The higher collateral requirements are due to inadequate records and accounts to document firm performance, as well as, higher levels of credit risk.15 The latter notion of inefficiency assumes that the planner faces the same contractual, informational, and institutional constraints in reallocating the resources as do private agents. Hence, this paper can provide a stronger case for supporting young firms and the SME sector than what was previously understood.

Investment in tangible assets in financially underdeveloped economies may also be constrained inefficient. Firms with a higher share of tangible assets or inputs such as land and machinery find it easier to pledge collateral and hence enjoy a higher borrowing capacity. As another example, capital-intensive production technologies can pledge a higher fraction of their output than labor-intensive technologies because labor cannot be pledged as collateral.16

I close this section by characterizing the set of constrained Pareto efficient allocations in the following proposition:

Proposition 5. Let (θ, R) ∈ F. If rΛ(θ, R) > 0, any allocation {ct,x1t,x2t}t=0 that satisfies 7 with equality for all t ≥ 0 is constrained Pareto efficient. Consequently, any competitive equilibrium corresponding to (θ, R) is constrained Pareto efficient. If rΛ (θ, R) ≤ 0, any allocation {ct,x1t,x2t}t=0 that satisfies 7 with equality for all t ≥ 0 and has x2t = 0, tT for some T ≥ 0 is constrained Pareto efficient. Hence, any competitive equilibrium in Fi is constrained Pareto efficient.

4.2 Regulated Economy

In this section, I discuss policies that can implement the Pareto improving reallocation proposed in Section 4.1.

Consider any competitive equilibria and suppose that the social planner can dictate the fraction αℓt of total funds that are invested in the liquid type. In this case, the entrepreneur only chooses the level of new funds raised it, and, the maximization problem of the middle aged entrepreneurs takes the following form:

maxit0((1αt)R1+αtR2)(it+(1+rt1)e)(1+rt)it(IV)s.t.(1+rt)it(θ1(1αt)R1+θ2αtR2)(it+(1+rt1)e).

The following proposition shows that this type of policy can implement the Pareto improving reallocations in Section 4.1.

Proposition 6. Any Pareto improving reallocation of the type analyzed in Section 4.1, when δ is small enough in absolute value, can be implemented by regulating the investment portfolios of the entrepreneurs. In an inefficiently liquid equilibrium, a planner chooses a lower liquid investment-tototal investment ratio, and the regulated interest rate is lower than in the unregulated equilibrium. Moreover, given any inefficiently liquid equilibria where rΛ(θ, R) < 0 or one where rΛ(θ, R) = 0 in the mixed region, this regulation can implement a Pareto improvement reallocation that results in a constrained Pareto efficient allocation.

The above regulation is akin to a maximum liquid asset ratio in a perfectly competitive banking sector that lends out the funds deposited by the young to the middle-aged entrepreneurs.17 The banks should be required to keep the fraction of their assets invested in the liquid type less than or equal to what the social planner chooses in Proposition 6.1819

It is worth noting that the overinvestment in liquid assets is accompanied by too much investment relative to the constrained optimum. To see why, recall that the aggregate investment in any period t is (2 + rt − 1)e. Given that the interest rate is too high in a constrained inefficient equilibrium, any Pareto improvement would reduce aggregate investment at all periods. Some emerging market economies, e.g., China, are likely examples of such inefficient investment booms.20

The fact that the interest rate is too high in constrained inefficient equilibrium is in contrast with the conventional dynamic inefficiency in the overlapping generations models first studied in Samuelson (1958) and Diamond (1965). The conventional dynamic inefficiency implies an interest rate that is too low, which has to be raised by a planner to achieve efficiency.

For this regulation to work, banks should be able to observe and monitor investments by the entrepreneurs in the two types. The following lemma shows that one can reach the Pareto frontier via a simpler and less demanding instrument:

Lemma 5. Given an inefficiently liquid competitive equilibrium, a social planner can make a Pareto improvement that reaches the Pareto frontier by levying a debt tax (and reimbursing via a lump sum transfer) where the middle-aged entrepreneur has to pay (1 + τ) (1 + rt)it at t + 1 for all tT for some T ≥ 0 and a constant τ > 0.

The problem in a constrained inefficient equilibrium is that the middle-aged raise too much debt. The excess borrowing bids up the interest rate by a socially inefficient amount. Hence a debt tax is a natural way to penalize the excess borrowing and internalize the pecuniary externality that leads to inefficiency.

4.3 Output and Welfare in the Long Run

In this part of the paper, I show how this model differs from the benchmark economy in terms of the effect of financial development on long-term welfare. As in the previous sections, financial development refers to improvements in contract enforcement, contracting technology, bankruptcy laws, and corporate governance that raise the liquidity of investment. The following proposition shows that in part of the inefficient region, financial development that raises the liquidity of the liquid type lowers long-term output and welfare:

Proposition 7. Let Vmss(θ,R) and Ymss(θ,R) denote the steady-state utility and aggregate output in the mixed region. Vmss is increasing in θ2 if and only if 1 + rΛ (θ, R) > 2θ1R1. Consequently, Vmss is always increasing in θ2 in the efficient part of the mixed region. Moreover, within the inefficient part of the mixed region, if θ2 is low enough for any given θ1, Vmss is decreasing in θ2. In the inefficient part of the mixed region, there exists a threshold θ1* such that given θ1<θ1*, Ymss is decreasing in θ2 if θ2 is low enough. Finally, for any economy (θ1, θ2), if Ymss is decreasing in θ2, then Vmss is decreasing in θ2 as well.

An increase in θ2 in the inefficient region has two effects. On the one hand, it increases the liquidity of any given portfolio and consequently the investment size. On the other hand, it makes investment in the liquid type more attractive, which encourages the entrepreneurs to substitute the liquid for the productive type. This second effect is detrimental to the output and welfare since investment in the liquid type is inefficient. Hence Vmss and Ymss decrease when the second effect is dominant.

Proposition 7 suggests that certain financial market policies may be harmful for long-term welfare, especially in economies where even most liquid investments are not very liquid. Developing corporate bond markets, loan guarantees, and asset securitization for residential mortgages, have been on the agenda of policy makers in emerging market countries as well as in international organizations such as the International Monetary Fund and the Bank of International Settlements.21 Proposition 7, however, implies that these policies may have negative welfare consequences, especially in less financially developed economies including many low-income and emerging markets. Low levels of financial development make these economies more likely to lie in the inefficient region. Mortgage loan guarantees, development of asset securitization, and corporate bond markets may raise the liquidity of the relatively more liquid investments such as home mortgages and investments in large and mature firms. Therefore, this type of financial development can reduce long-term output and welfare when there is overinvestment in the relative more liquid sectors of the economy.22

Policies that facilitate seizure of collateral by creditors may also end up worsening long-term welfare when an economy is inefficiently liquid. Creating public property registries and enhancing creditor rights in bankruptcy laws are among such policies. These policies result in higher liquidity for investments with a high share of tangible assets or inputs such as land and physical capital. Qian and Strahan (2007), for example, show that higher creditor rights affect collateral requirements more for firms with more tangible assets. Raising the liquidity of investment in tangible assets, however, may lower long-term output and welfare when the economy is overinvesting in tangible assets. In contrast, policies that increase the liquidity of investments in intangibles, e.g., human capital or labor-intensive production, raise long-term output and welfare when there is overinvestment in tangibles. Examples of such policies include improving accounting standards, creating credit records, and establishing information sharing platforms.

The results in this section have an important bearing on the measurement and benchmarking of financial development.23 Financial development may or may not correlate with higher output and welfare depending on its composition and the initial level of development: it may lead to constrained inefficient economies or may lower long-term output and welfare in a financially underdeveloped economy. Hence it is essential for any type of measurement or benchmarking to capture the different compositions of credit market developments. Moreover, for low values of liquidities, financial development can lead to lower aggregate output at the steady state. This can speak to some recent evidence on the non-monotonic relationship between financial development and growth, especially at low levels of financial development.24

5 Public Liquidity

In this section I study whether and how the introduction of government bonds can improve welfare in a constrained inefficient equilibrium. The effects of introducing government bonds on competitive equilibria can also help to empirically distinguish between this model and models with one type of investment such as Farhi and Tirole (2012), Holmström and Tirole (1998), and Woodford (1990).

5.1 Competitive Equilibrium with Government Bonds

Consider the model in Section 2 with only one difference: the young and middle-aged entrepreneurs at any time t ≥ 0 can purchase a one-period, risk free government bond sold at par, denoted by bty and btm. A unit of bond purchased at time t is a promise by the government to deliver one unit of consumption good plus the interest in period t + 1. The final form of the maximization problem of the middle-aged with government bonds (as long as 1 + rt < R1) that can be compared to II is as follows:

maxit,btm0Λ(θ,R;rt)(itbtm)+Φ(θ,R;rt1)eτt+1o(IIb)s.t.(θ1R1(1+rt1)1+rtθ1R1)e(itbtm)(θ2R2(1+rt1)1+rtθ2R2)e.

Φ and Λ are as before, given in II. The important assumption here is that investment in government bonds is perfectly liquid so that bond purchases reduce the total debt payments to (1+rt)(itbtm). τt+1o denotes the lump sum tax that is levied on the old entrepreneurs before consumption takes place.25 I suppose that government balances its budget every period:

(1+rt)bt=bt+1+τt+1o.(8)

Market clearings dictate that for all t ≥ 0:

it+bty=e,(9)
btm+bty=bt.(10)

In order to ensure the existence of competitive equilibrium in which borrowing constraint is still binding, one needs to restrict the supply of bonds. Let σt=bte be the normalized supply of bonds for all t ≥ 0;26 then one needs the following assumption:

Assumption 2. σt<min(1θ2R2,1θ1R11θ1) for all t ≥ 0 and σ=mintσt<min(1θ2R2,1θ1R11θ1) exists.

One can redefine the three regions as follows:

Definition 5. Define F(Σ) as the set of (θ, R) that satisfies Assumption 1 and Assumption 2, given the sequence Σ={σt}t=0. Then the three regions of F(Σ) are defined as:

The Liquid Region:F(Σ)={(θ,R)F(Σ)|((1σ)θ1R1(1σ)θ1R1)<((1σ)θ2R2(1σ)θ2R2)1+rΛ(θ,R)}.The Mixed Region:Fm(Σ)={(θ,R)F(Σ)|((1σ)θ1R1(1σ)θ1R1)<1+rΛ(θ,R)<((1σ)θ2R2(1σ)θ2R2)}.The Illiquid Region:Fi(Σ)={(θ,R)F(Σ)|1+rΛ(θ,R)((1σ)θ1R1(1σ)θ1R1)<((1σ)θ2R2(1σ)θ2R2)}.

The existence and uniqueness of competitive equilibria and the steady states with government bonds are established as follows:

Lemma 6. For any (θ, R) and Σ satisfying Assumption 1 and Assumption 2 and for any given initial condition 1 + r−1 < R1, there is a unique competitive equilibrium that converges to a unique and stable steady state corresponding to the region in Definition 5 containing (θ, R). The steady-state interest rates for the three regions are:

{1+rss(Σ)=((1σ)θ1R1(1σ)θ1R1)if(θ,R)F(Σ),1+rmss(Σ)=1+rΛ(θ,R)if(θ,R)Fm(Σ),1+riss(Σ)=((1σ)θ2R2(1σ)θ2R2)if(θ,R)Fi(Σ).

Moreover, at the steady state, the entrepreneurs specialize in the liquid and productive type of investments in regions F(Σ) and Fi(Σ) respectively but invest strictly positive amounts in both types in Fm(Σ).

The following lemma shows the effects of government bonds on the allocation of credit at the steady state:

Lemma 7. Let ixssx1ss+x2ss denote the total amount of resources invested in the two types by entrepreneurs at the steady state. One has the following:

ixssσ|(θ,R)F=((θ2R2(1σ)θ2R2)21)e,ixssσ|(θ,R)Fm=e.

and:

x1ssσ|(θ,R)Fm=1+rΛ(θ,R)θ2R2θ2R2θ1R1e>0,x2ssσ|(θ,R)Fm=1+rΛ(θ,R)θ1R1θ2R2θ1R1e<e.

An increase in the long-term supply of public liquidity σ in the liquid region crowds out private investment when public liquidity is scarce and crowds in private investment when public liquidity is abundant.27 The marginal effect of public liquidity on private investment is strictly increasing in the level of public liquidity, i.e., 2ixssσ2>0. On the other hand, government bonds always crowd out private investment one for one in the mixed region and the marginal effect of public liquidity on private investment is constant, i.e., 2ixssσ2=0. Public liquidity crowds out the liquid type and crowds in the productive type in the mixed region. The crowding out of the liquid type happens more than proportionally so that the demand for funds and consequently the interest rate remain unchanged. In contrast to other models which feature the crowding out effect, government bonds crowd out private investment while having no effects on the interest rate.

5.2 Welfare Effects of Government Bond

The effects of government bonds on long-term welfare are characterized as follows:

Lemma 8. Let Vzss(Σ) denote the steady-state utility level for region z ∈ {, m, i} given (θ, R) ∈ Fz (Σ), when the long-run supply of government bonds is σ. Then one has:

Vss(Σ)σ|σ=0=(R211θ2R2)rsse,Viss(Σ)σ|σ=0=(R111θ1R1)risse,Vmss(Σ)σ|σ=0=rmsse.

rzss denotes the steady-state interest rate for region z ∈ {, m, i} when there is no government bond in the economy.

Introduction of government bonds in an economy which lies in the inefficient part of the liquid region F is harmful to long-term welfare. The reason is that government bonds crowd out the more productive (relative to government bond) investment in the liquid type. This negative long-term effect implies that government bonds cannot Pareto improve constrained inefficient equilibria in the liquid region. In contrast, Lemma 8 implies that the supply of government bonds enhances the steady state utility in the inefficient part of Fm. The following proposition shows that government bonds can Pareto improve the competitive allocation in the mixed region:

Proposition 8. For any (θ, R) in the inefficient part of F, there exists ∊ > 0 such that for any Σ satisfying Assumption 2 with a long-term supply of bonds no more than ∊, Σ cannot Pareto improve the competitive equilibrium corresponding to (θ, R). Moreover, for any inefficiently liquid equilibria in Fm and also for constrained inefficient equilibria corresponding to the unique point (θ*, R*) ∈ F where r (θ*, R*) = 0, i.e., where the steady-state interest rate is zero, there exists a small enough sequence of government bonds Σ, which Pareto improves the competitive equilibrium allocation.

As discussed in the previous subsection, public liquidity crowds out the liquid investment more than proportionally to keep the demand for funds and the interest rate unchanged in the mixed region. This crowding-out effect is also the reason why government bonds can make Pareto improvement in the mixed region. By substituting one unit of investment in government bonds for one unit in the liquid type, entrepreneurs can pledge more than before, and the borrowing constraint becomes less binding. Entrepreneurs can use that extra amount of liquidity to invest in the productive type while the interest rate does not increase. This raises their consumption and results in a Pareto improvement.

6 Conclusion

This paper introduces a new type of constrained inefficiency in the allocation of credit across investments with different liquidities and returns. Constrained efficiency can be achieved by a regulation akin to a maximum liquid asset ratio in a perfectly competitive banking sector or via a debt tax. The nature of this inefficiency is unconventional in that Pareto improvement reduces the interest rate. Comparative statics reveal non-monotonic effects of technological and financial development, i.e., higher return and liquidity, on the interest rate, credit allocation, and long-term output and welfare. These results have important bearings on the measurement and benchmarking of financial development.

There are many potentially insightful extensions of this stylized model. Bubbles may arise in this model where liquidities are low. Compared to the benchmark economy, bubbles, similar to government debt, may have a different effect on the interest rate, credit, and investment in the mixed region. It will be interesting to see whether bubbly equilibria are efficient or if bubbles can Pareto improve inefficient equilibria in this model. Additionally, the welfare loss of inefficient equilibria can be magnified in an extension with growth externalities. In an extension with endogenous growth where productive type represents the more knowledge-intensive technology that entails knowledge spillovers, inefficient equilibria may feature a lower long-term growth rate relative to the optimum. Finally, while I discussed some of its implications for capital flows, an open-economy version of this model deserves more exploration in future research. Financial openness increases the supply elasticity of funds and makes the interest rate less responsive to the decisions of domestic entrepreneurs. These effects can make inefficiency a less likely outcome. The welfare effects of financial openness both in a Pareto sense and in the long run are other issues which can be studied in a similar vein.

Liquidity Choice and Misallocation of Credit
Author: Mr. Ehsan Ebrahimy
  • View in gallery

    Supply (red) and demand (blue) for funds at any period t as a function of the interest rate. wt − 1 denotes the wealth of the middle-aged, i.e. (1 + rt − 1)e. The two arms on the demand curve correspond to investing only in type 1 or 2. The flat segment in between corresponds to rt = rΛ (θ, R), where entrepreneurs mix. A higher period t − 1 interest rate, i.e. higher wt − 1, makes the two arms of the demand curve shift to the right but has no effect on the demand curve’s flat segment.

  • View in gallery

    Image of F, Fm and Fi for R1 = 4 and R2 = 3 over the space of (θ1, θ2). The white area below the positively sloped straight line where Assumption 1 is violated corresponds to the benchmark economy in Definition 1.

  • View in gallery

    Contour plot of the steady-state interest rates (red lines) for the three regions. Interest rates are highest at the top left corner.

  • View in gallery

    An illustration of inefficiently liquid equilibria (light gray region) and the line corresponding to rΛ = 0 for R1 = 4 and R2 = 3.

  • View in gallery

    This figure shows the expansion of the inefficiently liquid region when R = (4, 2). In contrast with the case of R = (4, 3), all competitive equilibria in liquid region F are constrained Pareto inefficient. For low values of θ1 and compared to R = (4, 3), higher values of θ2 can lead to inefficiency.