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.

References

  • Alles, L. (2001). Asset securitization and structured inancing: Future prospects and challenges in emerging market countries. IMF Working Papers.

    • Search Google Scholar
    • Export Citation
  • Beck, T., E. Feyen, A. Ize, and F. Moizeszowicz (2008). Benchmarking inancial development. World Bank Working Paper.

  • Bianchi, J. (2011). Overborrowing and systemic externalities in the business cycle. American Economic Review 101(7), 34003426.

  • Brunnermeier, M. K. and L. H. Pedersen (2009). Market liquidity and funding liquidity. Review of Financial Studies 22(6), 22012238.

  • Caballero, R. J. and A. Krishnamurthy (2006). Bubbles and capital low volatility: Causes and risk management. Journal of Monetary Economics 53(1), 3553.

    • Search Google Scholar
    • Export Citation
  • Calvo, G. A., F. Coricelli, and P. Ottonello (2014). The labor market consequences of financial crises with or without inflation: Jobless and wageless recoveries. NBER Working Paper (18480).

    • Search Google Scholar
    • Export Citation
  • Chiquier, L., O. Hassler, and M. Lea (2004). Mortgage securities in emerging markets. World Bank Policy Research Working Paper (3370).

  • Dell’Ariccia, G., E. Ebrahimy, D. Igan, and D. Puy (2019). Discerning good from bad credit booms: The role of construction. IMF Working Papers.

    • Search Google Scholar
    • Export Citation
  • Diamond, P. A. (1965). National debt in a neoclassical growth model. American Economic Review 55(5), 11261150.

  • Dietsch, M. and J. Petey (2004). Should sme exposures be treated as retail or corporate exposures? a comparative analysis of default probabilities and asset correlations in french and german smes. Journal of Banking and Finance 28(4), 773788.

    • Search Google Scholar
    • Export Citation
  • Farhi, E. and J. Tirole (2009). Leverage and the central banker’s put. American Economic Review, Papers and Proceedings 99(2), 589593.

    • Search Google Scholar
    • Export Citation
  • Farhi, E. and J. Tirole (2010). Collective moral hazard, maturity mismatch and systemic bailouts. Forthcoming in American Economic Review.

    • Search Google Scholar
    • Export Citation
  • Farhi, E. and J. Tirole (2012). Bubbly liquidity. Review of Economic Studies 79(2), 678706.

  • Ghate, A. and R. L. Smith (2005). Infinite linear programs: I. characterizing extreme points through basic variables. Unpublished manuscript.

    • Search Google Scholar
    • Export Citation
  • Giglio, S. and T. Severo (2012). Intangible capital, relative asset shortages and bubbles. Journal of Monetary Economics 59(3), 303317.

    • Search Google Scholar
    • Export Citation
  • Gopinath, G., e. Kalemli-Ozcan, L. Karabarbounis, and C. Villegas-Sanchez (2017). Capital allocation and productivity in south europe. Quarterly Journal of Economics 132(4), 19151967.

    • Search Google Scholar
    • Export Citation
  • Gulde, A.-M., J. C. Nascimento, and L. M. Zamalloa (1997). Liquidity asset ratios and financial sector reform. IMF Working Paper.

  • Holmstrom, B. and J. Tirole (1997). Financial intermediation, loanable funds, and the real sector. Quarterly Journal of Economics 112(3), 663691.

    • Search Google Scholar
    • Export Citation
  • Holmström, B. and J. Tirole (1998). Private and public supply of liquidity. Journal of Political Economy 106(1), 140.

  • Hsieh, C.-T. and P. J. Klenow (2009). Misallocation and manufacturing tfp in china and india. Quarterly Journal of Economics 124(4), 14031448.

    • Search Google Scholar
    • Export Citation
  • Kehoe, T. J. and D. K. Levine (1993). Debt-constrained asset markets. Review of Economic Studies 60(4), 865888.

  • Kiyotaki, N. and J. Moore (2002). Evil is the root of all money. American Economic Review, Papers and Proceedings 92(2), 6266.

  • Kiyotaki, N. and J. Moore (2005). Financial deepening. Journal ofthe European Economic Association 3(2-3), 701713.

  • Kiyotaki, N. and J. Moore (2008). Liquidity, business cycles and monetary policy. Unpublished Manuscript.

  • Lee, I. H., M. Syed, and L. Xueyan (2012). Is china overinvesting and does it matter? IMF Working Paper (WP/12/277).

  • Levine, R. (2005). Finance and growth: Theory and evidence. Handbook of Economic Growth, Chapter 12 1-Part A, 865934.

  • Liberti, J. M. and A. R. Mian (2010). Collateral spread and inancial development. Journal of Finance 65(1), 147177.

  • Lorenzoni, G. (2008). Inefficient credit booms. Review of Economic Studies 75(3), 809833.

  • Matsuyama, K. (2007). Credit traps and credit cycles. American Economic Review 97(1), 503516.

  • Qian, J. and P. E. Strahan (2007). How laws and institutions shape financial contracts: The case of bank loans. Journal of Finance LXII(6), 28032834.

    • Search Google Scholar
    • Export Citation
  • Reis, R. (2013). The portuguese slump and crash and the euro crisis. The Brookings Papers in Economic Activity, 143210.

  • Samuelson, P. A. (1958). An exact consumption-loan model of interest with or without the social contrivance of money. Journal ofPolitical Economy 66(6), 467482.

    • Search Google Scholar
    • Export Citation
  • Stevenson, H. (2010). Can development finance institutions finance development? International Financial Corporation (World Bank Group).

  • Woodford, M. (1990). Public debt as private liquidity. American Economic Review 80(2), 382388.

  • Zheng, S., K. Storesletten, and F. Zilibotti (2011). Growing like china. American Economic Review 101(1), 202241.

A Appendix: Implementation of Pareto Improving Reallocation

Suppose that there is free entry into the banking sector that lasts only for one period and is owned by the young. Free entry implies zero profits in equilibrium. Banks are funded by deposits from the young and lend to middle-aged entrepreneurs. In period t, banks are required to keep the share of the liquid-type investment on their balance sheet less than or equal to αtsp, i.e., the share chosen by the planner. Under such regulation, middle-aged entrepreneurs choose the maximum possible share αtsp, and the resulting allocation coincides with the social planner allocation. To see why, assume that this is the case for all periods up to t− 1. Then the initial wealth at t is wt1sp, which is no greater than wt1ce (initial wealth in the competitive equilibrium) by Proposition 6. But 6 implies that middle-aged would choose a share αℓt not less than αtce at a lower level of initial wealth which itself is not less than αtsp. Hence the middle-aged choose the maximum possible share of αtsp under regulation which implies that the wealth in the next period is wtsp. Since w0sp=w0ce, the same logic applies to the initial period. The claim is thus proven by induction.

B Appendix: Proofs

Proof of Lemma 1. In any equilibrium, the resource constraint binds, and so I can solve for the value of x2t in the above and rewrite the problem as:

maxit,x1t0(R1R2)x1t+(R2(1+rt))it+R2(1+rt1)es.t(1+rtθ2R2)it(θ1R1θ2R2)x1t+θ2R2(1+rt1)e,0x1t(1+rt1)e+it

One can immediately see from the above that 1 + rt > θ2R2. Otherwise, it must be that 1 + rtθ2R2 < R2, in which case it can be raised unboundedly and there may not be any maximum to the objective function. One must also have 1 + rtR1. Otherwise, the optimal solution to the problem requires that it = 0. To see why rewrite the above with x2t in the objective function. If 1 + rt > R1, then by Assumption 1 both coefficients of x2t and it are strictly negative, and so the best an entrepreneur can do is to set both to zero. This cannot be an equilibrium since market clearing in the capital market cannot be satisfied.

Now suppose that the borrowing constraint does not bind for some t ≥ 0. Since the coefficient of x1t in the objective function of the above problem, which is R1R2, is strictly positive by Assumption 1, x1t must be at the highest possible value, which is (1 + rt −1)e + it . At this value the objective function can be written as (R1 − (1 + rt))it + Dt −1 where Dt −1 is determined at t − 1.

Moreover, the borrowing constraint at this value of x1t is:

(1+rtθ1R1)it<θ1R1(1+rt1)e.

Since the constraint is not binding, one can raise it by an small amount ϵ > 0 so that the constraint is still satisfied and the value of the objective function is increased by (R1 − (1 + rt))ϵ. This contradiction shows that the borrowing constraint must always be binding. The rest of the lemma is straightforward by using the borrowing constraint to eliminate x1t.

Proof of Lemma 2. First, I show that these are the only steady-state equilibria for the three regions. Suppose that rzss is a steady-state interest rate for z ∈ {, m, i}. Consider n steady state of the liquid region. If θ2R21θ2R2<(1+rss), by 5 both of the upper and lower bounds on the next period interest rate will be strictly smaller than rzss. Nor can it be that (1+rss)<θ2R21θ2R2. In that case using 5, the upper bound for the interest rate in the next period θ2R2 (2+rss), will be strictly bigger than 1+rss but strictly less than θ2R21θ2R2 and so strictly less than 1 + rΛ (θ, R) (since (θ, R) ∈ F). Hence given 4 the next period interest rate will be the upper bound itself, which is a contradiction given that it is strictly bigger than θ2R21θ2R2. Hence one must have (1+rss)=θ2R21θ2R2. In a similar fashion, I can show that if there exists an steady state for Fi, it must be (1+riss)=θ1R11θ1R1. Finally, suppose that (1+rmss)<(1+rΛ(θ,R)) in the mixed region. Then using 4 and the fact that the economy is at the steady state, one must have (1+rmss)=θ2R21θ2R2 which gives θ2R21θ2R2<(1+rΛ(θ,R)). This is a contradiction given that the economy is in Fm. Similarly one cannot have (1+rΛ(θ,R))<(1+rmss), and so (1+rmss)=(1+rΛ(θ,R)).

It only remains to check that these steady states exist. As I showed above, the trajectories of the interest rates are consistent with the equilibrium conditions 4 and 5 given an initial interest rate 1 + r−1 equal to the steady state. The values of it are exogenously given and equal to e, and the values of x1t and x2t can be derived from 6. The only condition that remains is that 1 + rt < R1 for all t ≥ 0. To see this, note that under Assumption 1:

1+rΛ(θ,R)<min(R1,R2).

Hence, the remaining condition is satisfied for the liquid and mixed regions. The condition is also satisfied in the illiquid region since I assumed that θ1R11θ1,R1<R1 in Definition 3. Local stability of the steady states in F and Fi follows from the fact that θ1R1 < θ2R2 < 1 by Assumption 1. If 1+rzss1+rΛ(θ,R) where z ∈ {, i}, suppose without loss of generality that 1+rzssϵ<1+rt< 1+rzss. For small enough ϵ > 0, the whole interval [1+rzssϵ,1+rzss] is either strictly below or above 1 + rΛ (θ, R). In either case, 4 and 5 imply 1+rt<1+rt+1=θ2Rz(2+rt)<1+rzss, and hence by Assumption 1, the interest rates starting from a point in the interval [1+rzssϵ,1+rzss] converge to the steady state value. For the case in which 1+rzss=1+rΛ(θ,R) where z ∈ {, i} or the mixed region, Fm, where the steady state interest rate is 1 + rΛ(θ, R), suppose without loss of generality that 1 + rt − 1 < 1 + rΛ(θ, R) (the proof for the case 1 + rt − 1 > 1 + rΛ(θ, R) is very similar).

If 1 + rΛ(θ, R) ∈ [θR1(2 + rt − 1), θ2R2 (2 + rt − 1)], then 4 gives 1 + rt = 1 + rΛ(θ, R). Otherwise, suppose that θ2R2 (2 + rt − 1) < 1 + rΛ(θ, R). Then 5 implies 1 + rt − 1 < 1 + rt = θ2R2(2 + rt − 1) < 1 + rΛ(θ, R). Hence, 1 + rt+k, k = 1, 2, 3, ... converges to 1 + rΛ(θ, R). The proof is very similar

when 1 + rΛ(θ, R) < θ1R1 (2 + rt − 1), and so this completes the proof.

Proof of Lemma 3. For the steady-state interest rate observe that:

rΛ(θ,R)θ1=(1θ2)R1R2(R2R1)((1θ1)R1(1θ2)R2)2<0,(11)
rΛ(θ,R)θ2=(1θ1)R1R2(R1R2)((1θ1)R1(1θ2)R2)2>0.(12)

Let sj(θ,R)=xjssx1ss+x2ss be the share of type j ∈ {1, 2} in total investment at the steady state. By 6:

s1(θ,R)=θ2R2(1θ2R2)(1+rΛ(θ,R))(θ2R2θ1R1)(2+rΛ(θ,R)),s2(θ,R)=(1θ1R1)(1+rΛ(θ,R))θ1R1(θ1R2θ1R1)(2+rΛ(θ,R)).

Now one can rewrite s1 (θ, R) as:

s1(θ,R)=12+rΛ(θ,R)(1θ2R2)θ2R2θ1R1.

The numerator of the above is strictly increasing in θ1 by Proposition 3, and the denominator is strictly decreasing in θ1. This implies that s1(θ, R) is strictly increasing in θ1 when (θ, R) ∈ Fm and hence monotone in θ1 in all three regions. For s2(θ, R) one has:

s2(θ,R)θ2=(θ2R2θ1R1)rΛ(θ,R)θ2((1θ1R1)(1+rΛ(θ,R))θ1R1)R2(2+rΛ(θ,R))((θ2R2θ1R1)(2+rΛ(θ,R)))2.

Arranging terms in the numerator, the above can be written as:

s2(θ,R)θ2=(a(θ1)θ22+b(θ1)θ2+c(θ1))R2((θ2R2θ1R1)(2+rΛ(θ,R))((1θ1)R1(1θ2)R2))2,

where a(θ1), b(θ1) and c(θ1) are:

a(θ1)=R1R22(1+R1)(1(1+R1)θ1),b(θ1)=R1R22(1+R1)+R1R2((R1R2)(1+2R1)R1(R1R21))θ1R12R2(1+R1)(2+R2)θ12,c(θ1)=R1R2(R1R2)R1R2(R1(2+R1)R2)θ1+R12(1+R2)(R1R2R1+R2)θ12+R13(1+R1)θ13.

To show that s2(θ, R) has at most one (interior) maximum, it is enough to show that given any θ1,a(θ1)θ22+b(θ1)θ2+c(θ1) has at most one root as a quadratic polynomial of θ2 inside F. By Proposition 2, θ111+R1, and therefore a(θ1) ≤ 0. In the next step, I show that c(θ1) > 0 for all θ1 by proving that c˜(θ1)=R11(c(θ1)R13(1+R1)θ13)>0 inside F. Ifc˜(θ1)=0 has no roots, then c˜(θ1)>0 since c˜(0)>0. Therefore, suppose θ1* is the smallest root of c˜(θ1)=0:

θ1*=R2(R1(2+R1)Δ)2R2(R1R2),Δ=(R2(R1(2+R1)R2))24R1R2(R1R2)(1+R2)(R1R2R1+R2).

Now one has:

θ1*=R2(R1(2+R1)R2)Δ2R2(R1R2)>11+R1(1+R1)2(R2(R1(2+R1)R2))24R1R2(R1R2)(1+R2)(R1R2R1+R2)<(R2(1+R1)(R1(2+R1)R2))2R2(R1R2))2(1+R1)(R1(1+R1)(1+R2)(R1(R21)+R2)R2(R1(2+R1)R2)>R2(R1R2).

The last inequality holds since:

R1(1+R1)(1+R2)(R1(R21)+R2)R2(R1(2+R1)R2)>R1R2(1+R1)(1+R2)R1R2(2+R1)>R12R2>0>R2(R1R2).

Hence θ1*>11+R1, and since θ111+R1 in F, one must have c(θ1) > 0 in F. Now since a(θ1) ≤ 0 and c(θ1) > 0 in F, at least one root of a(θ1)θ22+b(θ1)θ2+c(θ1) for any given θ1 has to be non-positive. Therefore, a(θ1)θ22+b(θ1)θ2+c(θ1) has at most one root in F for any θ1, and consequently s2(θ, R) has at most one (interior) maximum. Note that when θ is on the boundary of Fm and Fi, s2(θ, R) = 0 and hence s2(θ,R)θ2>0 given any θ1. Now suppose θ˜1 is the value for which the vertical line θ1=θ˜1 is tangent to the boundary of F. Observe that when θ2 increases along θ1=θ˜1 line, s2(θ, R) reaches the maximum of one at the point of tangency. Therefore, beyond the point of tangency s2(θ, R) must be strictly decreasing in θ2. This implies that for the particular value of θ1 = θ˜1, there is a unique maximum for s2(θ, R). Hence, by continuity there must be a unique maximum for s2 (θ, R) over the range of θ2 given any θ1 in a neighborhood of θ˜1, which completes the proof.

PROOF OF PROPOSITION 1. First I show that for all t ≥ 0, 1 + rt < R1. Suppose 1 + rt − 1 < R1 for some t ≥ 0. Consider the window defined by 5, where 1 + rt ∈ [θ1R1 (2 + rt − 1), θ2R2 (2 + rt − 1)]. If θ2R2 (2 + rt − 1) ≤ 1 + rΛ(θ, R), 4 implies 1 + rt = θ2R2 (2 + rt − 1) ≤ 1 + rΛ(θ, R) < R1. The last inequality holds by Assumption 1. If θ1R1 (2 + rt − 1) < 1 + rΛ(θ, R) < θ2R2 (2 + rt − 1), by 4 I get 1 + rt = 1 + rΛ(θ, R) < R1. Finally, consider the case 1 + rΛ(θ, R) < θ1R1 (2 + rt − 1). The equilibrium path of interest rate, given by 4, implies that 1+rt=θ1R1(2+rt1)max(1+rt1,θ1R11θ1R1)<R1.

The first inequality holds because the value of θ1R1 (2 + rt − 1) is always between 1+rt1andθ1R11θ1R1

The second inequality is obtained by the assumption that 1 + rt − 1 < R1 and definition of F. By induction, 1 + r−1 < R1 implies 1 + rt < R1 for all t ≥ 0. This proves the necessary condition for the interest rates in the competitive equilibrium.

In the second step, I prove the existence and uniqueness. I show that given (θ, R) ∈ F and the initial condition 1 + r−1, a unique path of interest rates is defined by 4 and 5. Note that given the path of interest rates, I can simply solve for (x1t, x2t, it) for all t ≥ 0 using 3 and 6 in each period. Suppose I have determined the unique interest rate 1 +rt −1 for t − 1. Consider the window, defined by 5, where 1 + rt ∈ [θ1R1(2 + rt − 1), θ2R2(2 + rt − 1)]. If θ2R2 (2 + rt − 1) ≤ 1 + rΛ(θ, R) or 1 + rΛ(θ, R) ≤ θ1R1 (2 + rt − 1), using 4 gives 1 + rt = θ2R2 (2 + rt − 1) and 1 + rt = θ1R1 (2 + rt − 1) respectively. Finally, suppose θ1R1(2 + rt − 1) < 1 + rΛ(θ, R) < θ2R2(2 + rt − 1). Then, if 1 + rt > 1 + rΛ(θ, R), by 4 one must have 1 + rt = θ1R1(2 + rt − 1) < 1 + rΛ(θ, R). Similarly, if 1 + rt < 1 + rΛ(θ, R), by 4 one must have 1 + rt = θ2R2(2 + rt − 1) > 1 + rΛ(θ, R). The two contradictions show that one must have 1 + rt = 1 + rΛ(θ, R). Hence, I have shown that given 1 + rt − 1, there is a uniquely determined interest rate at time t, that is, 1 + rt. Therefore, by induction, I have shown that given an initial condition 1 + r−1, there is a unique path of interest rates for all t ≥ 0.

In the third and final step, I show that the unique equilibrium path of the interest rates defined in step two converges to the unique steady state characterized in Lemma 2, for given (θ, R) ∈ F and an initial condition 1+r−1. Consider the case (θ, R) ∈ F first. Note that if 1+rt − 1 ≤ 1+rΛ(θ, R), using 4 and 5 implies 1+rt=θ2R2(2+rt1)max(1+rt1,θ2R21θ2R2)1+rΛ(θ,R). Hence, if 1 + r−1 ≤ 1 + rΛ(θ, R), the path of interest rates is defined as 1 + rt = θ2R2(2 + rt − 1) for all t ≥ 0. This path is clearly convergent to 1+rss=θ2R21θ2R2. Now suppose 1 + r− 1 > 1 + rΛ(θ, R), which implies1+r1>θ2R21θ2R2. Define the series {1+r¯1}t=1>as 1+r¯t=θ2R2(2+r¯t1) for all t ≥ 0 and 1+r¯1=1+r1. If 1+rt11+r¯t1, 5 implies 1+rtθ2R2(2+rt1)θ2R2(2+r¯t1)=1+r¯t. Hence by induction one must have 1+rt1+r¯t for all t ≥ 0. Since by Assumption 1 θ2R2 < 1, this immediately implies that there is a finite t0 for which 1 + rt0 ≤ 1 + rΛ(θ, R). Therefore, this case is similar to the previous part of the proof and so convergence is established.

Now consider the case (θ, R) Fm where Definition 3 implies that θ1R11θ1R1<1+rΛ(θ,R)<θ2R21θ2R2. Without loss of generality, suppose 1 + r−1 > 1 + rΛ(θR). Define the series {1+rt}t=1 as 1+rt=θ1R1(2+rt1) for all t ≥ 0 and 1+rt=1+r1. It is easy to see that there is a finite and unique t0 ≥ 0 such that 1+rt01+rΛ(θ,R)<1+rt01. Now note that if 1 + rt − 1 > 1 + rΛ(θ, R) for some t ≥ 0, one must have θ2R2(2+rt1)min(1+rt1,θ2R21θ2R2)>1+rΛ(θ,R) and therefore 4 and 5 give 1 + rt = max(θ1R1(2 + rt − 1), 1 + rΛ(θ, R)). Using this observation and by induction, for −1 ≤ tt0 − 1 one must have 1+rt=1+rt>1+rΛ(θ,R), and so θ2R2(2+rt)min(1+rt1,θ2R21θ2R2)>1+rΛ(θ,R). Using 4 and the definition of t0, this implies that 1 + rt0 = max(θ1R1(2 + rt0 −1), 1 + rΛ(θ, R)) = 1 + rΛ(θ, R). Therefore, the path of interest rates converges to the steady state-interest rate, 1 + rΛ(θ, R), in finite periods. The proof for the case 1 + r−1 < 1 + rΛ(θ, R) is very similar. Finally, if 1 + r−1 = 1 + rΛ(θ, R), the economy is already in the steady state, and all future interest rates will be the same.

The proof for the illiquid region is very similar to the case of liquid region, and so I do not provide it here.

PROOF OF PROPOSITION 2. First, I compute the boundaries of the illiquid and liquid regions as functions of θ1. For the illiquid region the defining boundary is characterized by:

1+rΛ(θ,R)=(θ1R11θ1R1).

Using 1 and solving the above as a function of θ1, I get:

θ2i(θ1)=(θ1R1(1θ1(1+R2))R2(1θ1(1+R1))).

This function is strictly increasing and convex in θ1 since θ1R1 is increasing and:

ddθ1((1θ1(1+R2))(1θ1(1+R1)))=R1R2(1θ1(1+R1))2>0.

Also observe that θ2i(0)=0, and so no matter how close to the origin, there are illiquid equilibria in any neighborhood of the θ = 0. Now the characterizing equation for the liquid region is:

1+rΛ(θ,R)=(θ2R21θ2R2).

Collecting terms involving θ1 or θ2 on different sides, I obtain two distinct curves:

θ¯2(θ1)=((θ1R1(1+R2)+R2)+(θ1R1(1+R2)+R2)24θ1R1R2(1+R1)2R2(1+R1)),θ¯2(θ1)=((θ1R1(1+R2)+R2)(θ1R1(1+R2)+R2)24θ1R1R2(1+R1)2R2(1+R1)).

Note that obviously θ2(θ1)θ¯2(θ1) and θ2(0)=0, and so the lower boundary characterizing the liquid region passes through the origin. This means that there are liquid steady-state equilibria at any neighborhood of the origin.

Now let ∆(θ1) ≡ (θ1R1 (1 + R2) + R2)2 − 4θ1R1R2(1 + R1). Then the two curves θ2(θ1) and θ¯2(θ1) touch each other when ∆(θ1) = 0. This equation has two roots:

θ¯1=(R2((2(1+R1)(1+R2))+4(1+R1)(R1R2))R1(1+R2)2)θ1=(R2((2(1+R1)(1+R2))4(1+R1)(R1R2))R1(1+R2)2)

The smaller root is less than 11+R1 since:

θ1<11+R1(1+R1)R2((2(1+R1)(1+R2))4(1+R1)(R1R2))<R1(1+R2)2(1+R1)R2(2(1+R1)(1+R2))R1(1+R2)2<(1+R1)R24(1+R1)(R1R2)(R1R2)(2R1R2+R21)<(1+R1)R24(1+R1)(R1R2).

If I square both sides, cancel R1R2, and collect the terms, I get:

R1<4R12R23+8R12R22+4R1R23+7R1R22+R23+4R12R2+2R22+2R1R2+R2.

This is obviously the case given Assumption 1. The bigger root is greater than 11+R1 since:

θ¯1>11+R1(1+R1)R2((2(1+R1)(1+R2))4(1+R1)(R1R2))>R1(1+R2)2(1+R1)R2(2(1+R1)(1+R2))R1(1+R2)2>(1+R1)R24(1+R1)(R1R2)(R1R2)(2R1R2+R21)>(1+R1)R24(1+R1)(R1R2).

The last inequality is obvious given that one term is positive and the other is negative. Therefore the point at which the two curves θ¯2(θ1) and θ2(θ1) touch each other inside F is θ1. the fact that θ1<11+R1 proves that for high θ1 there is no liquid steady state.

Next, I prove that θ¯2(θ1) is strictly decreasing and θ2(θ1) is strictly increasing. The derivatives are:

dθ2(θ1)dθ1=C0(R1(1+R2)+(R1(1+R2)(θ1R1(1+R2)+R2)2R1R2(1+R1))Δ(θ1)12).dθ2(θ1)dθ1=C0(R1(1+R2)(R1(1+R2)(θ1R1(1+R2)+R2)2R1R2(1+R1))Δ(θ1)12).

C0 is just a constant. It is easy to see that the term in parentheses just before Δ(θ1)12 is always negative for θ1θ1. Hence, dθ2ϵ(θ1)dθ1 should be strictly positive. Now for the other case:

dθ¯2(θ1)dθ1<0R12(1+R2)2Δ(θ1)<(R12(1+R2)2θ1R1R2(2(1+R1)(1+R2)))2(1+R2)2Δ(θ1)<(R1(1+R2)2θ1R2(2(1+R1)(1+R2)))2(1+R2)<2(1+R1)(1+R2).

The last statement is correct given Assumption 1. In the last step I have used the definition of ∆(θ1) to cancel out all terms. What I proved shows that for any θF one must have θ(11+R1,11+R1). This is because I showed that θ1<11+R1 and thatdθ¯2(θ1)dθ1 is strictly decreasing while θ¯2(θ1) stays above θ2(θ1) and intersects with θ1=0at11+R1.

In the next step I want to prove that the liquid region lies above the illiquid region. First, I observe the following:

rΛ(θ,R)θ2=(1θ1)(R1R2)R1R2((1θ1)R1(1θ2)R2)2>0.

Now suppose that rΛ(θ1,θ2,R)θ2R21θ2R2 and rΛ(θ1,θ2,R)θ1R11θ1R1 where (θ1, θ2, R) and (θ1,θ2,R) are in F. Then, if θ2θ2, by the derivation above rΛ(θ1,θ2,R)rΛ(θ1,θ2,R) and hence:

θ2R21θ2R2rΛ(θ1,θ2,R)(θ1,θ2,R)θ1R11θ1R1.

This is not possible since it implies that θ2R2θ1R1 and hence (θ1, θ2, R) cannot be in F. In the last step of the proof, I show that (1R2,11+R1)Fi. First, note that:

rΛ(θ,R)θ1=(1θ2)(R2R1)R1R2((1θ1)R1(1θ2)R2)2<0.

Second, observe that θ2i(θ1) is strictly increasing, passes through the origin, and also converges to infinity when θ1 gets close to 11+R1. This means that θ2i(θ1) cuts the horizontal border of F that is θ2=1R2 at an interior point, say, (θ¯1,1R2) where θ¯1<11+R1. At this point θ2i(θ¯1)=θ¯1R11θ¯1R1 since I have proven above that rΛ(θ,R)θ1<0, for any θ1(θ¯1,11+R1) I obtain:

θ2i(θ1)<θ2i(θ¯1)=θ¯1R11θ¯1R1<θ1R11θ1R1.

This means that θ1Fi for θ1(θ¯1,11+R1).

PROOF OF PROPOSITION 3. When R1 > R2 > 1 and 1 > θ1R1 > θ2R2, entrepreneurs only invest in type 1 since type 2 is dominated in terms of both liquidity and return. Hence, this economy collapses to the economy in Farhi and Tirole (2012) with only one investment type, (θ1, R1), and no bubbles or outside liquidity. Farhi and Tirole (2012) show in their Proposition 5 that under the assumption that R1 > 1, all competitive equilibria are Pareto efficient and hence constrained Pareto efficient as well.

PROOF OF PROPOSITION 4. I proceed in two steps. First I prove some comparative statics regarding steady-state utility levels, and then I complete the proof by considering the transition dynamics.

Let Vss and Viss be the steady-state utility levels in the liquid and illiquid regions. For any values of (θ, R) ∈ F, the following statements are correct. Vss(θ,R)Viss(θ,R) and rΛ(θ, R) have the same sign. Vss(θ,R)Vmss(θ,R) is positive if and only if rΛ(θ, R) > 0 and (θ, R) ∉ F. Vmss(θ,R)Viss(θ,R) is negative if and only if rΛ(θ, R) < 0 and (θ, R) ∉ Fi.

The proofs of the above statements are as follows. First I show that the steady-state level of utility for any values of α for which (γα1γα)<Rα is given by:

Vαss=(Rαγα1γα)e=((1α)(1θ1)R1+α(1θ2)R2(1α)(1θ1R1)+α(1θ2R2))e.

Additionally, the steady state utility levels for the three regions in Definition 3 are Vss=((1θ2)R21θ2R2)e, Viss=((1θ1)R11θ1R1)e and:

Vmss=((θ2θ1)2R12R22(θ2R2θ1R1)((1θ1)R1(1θ2)R2))e.

Moreover suppose (θ, R) ∈ Fm and that α˜=(x2ss(θ,R)x1ss(θ,R)+x2ss(θ,R))e.. Then in the regulated economy corresponding to α˜ one has:

1+rΛ(θ,R)=1+rα˜ss=((1α˜)θ1R1+α˜θ2R2(1α˜)(1θ1R1)+α˜(1θ2R2)),Vmss(θ,R)=Vα˜ss=((1α˜)(1θ1)R1+α˜(1θ2)R2(1α˜)(1θ1R1)+α˜(1θ2R2))e.

To see the above, let Rαt = (1 − αℓt)R1 + αℓt R2 and γαt = θ1 (1 − αℓt)R1 + θ2αℓt R2 be the return and liquidity of the regulated portfolio at time t. Problem IV is the maximization problem of an entrepreneur that has access only to one type of investment project with a return of Rαt and liquidity of γαt. The optimal solution to IV is:

{it=((1+rt1)γαt(1+γt)γαt)eif Rαt1+rt,it=0if Rαt<1+rt.

Note that γαt < 1 by Assumption 1. Also note that for any (θ, R) ∈ F, there exists an ϵ > 0 such that (γαt1γαt)<Rαtfor allαt[0,ϵ). This is because the inequality holds for αℓt = 0 according to the definition of F, and so by continuity it holds in a neighborhood of zero. If {αt}t=0 are all set to α and this value is in the neighborhood above, the steady-state equilibrium of the regulated economy is

1+rαss=(γα1γα),

where variables without time subscript correspond to α. Using the values of steady-state interest rates in Lemma 2, market clearings, and the objective function in II, deriving Vss, Viss and Vmss is straightforward. For the regulated economy, recall that by IV and the above equation, the

objective function when the social planner sets α = α will be:

Vαss=(((1α)R1+αR2)(2+rαss)(1+rαss))e=(Rα(1+γα1γα)γα1γα)e=(Rαγα1γα)e.

Note that the numerator and denominator of Vαss are weighted averages of those of Vss and Viss. This implies that Vαss always lies between the two values of Vss and Viss. For last part of the proposition, observe that 6 gives:

α¯=(1+rΛ(θ,R))θ1R1(2+rΛ(θ,R))(θ2R2θ1R1)(2+rΛ(θ,R)).

The interest rate is:

1+rα˜ss=γα˜1γα˜=((1+rΛ)θ1R1(2+rΛ))θ2R2+(θ2R2(2+rΛ)(1+rΛ))θ1R1((1+rΛ)θ1R1(2+rΛ))(1θ2R2)+(θ2R2(2+rΛ)(1+rΛ))(1θ1R1)(θ2R2θ1R1)(1+rΛ(θ,R))(θ2R2θ1R1)=1+rΛ(θ,R).

Therefore the utility levels at the steady state should be the same. Note that 1+rΛ(θ,R)<min(R1,R2)Rα˜ by Assumption 1, and so the proof is complete.

In the second step, I prove the lemma. Consider Vss(θ,R)Viss(θ,R). Note that:

Vss(θ,R)>Viss(θ,R)(1θ2)R21θ2R2>(1θ1)R11θ1R1R211θ2R2>R111θ1R1(R21)(1θ1R1)>(R11)(1θ2R2)(R2θ1R1R2+θ1R1)>(R1θ2R1R2+θ2R2)(θ2θ1)R1R2>(1θ1)R1(1θ2)R21+rΛ(θ,R)=((θ2θ1)R1R2(1θ1)R1(1θ2)R2)>1.

Now define the following terms:

Ω(θ,R)((θ2θ1)R1R2((1θ1)R1(1θ2)R2)(θ2R2θ1R1))e,Γ(θ,R)(θ2R2(1θ1)R1(1θ1)R2)(1θ2R2)(θ2θ1)R1R2(1θ2R2)((1θ1)R1(1θ2)R2).

Note that the denominators of Ω (θ, R) and Γ (θ, R) are strictly positive. Moreover, one can easily see that the numerator of Γ (θ, R) is positive if and only if 1 + rΛ(θ, R) > 1 and that the numerator of Ω (θ, R) is positive if and only if 1+rΛ(θ,R)<θ2R21θ2R2=1+rss(θ,R) or equivalently (θ, R) ∉ F. Ω (θ, R) is the welfare gains per unit of reduction in x1 of investing the freed resources in x2, and Γ (θ, R) is the maximum amount of reduction in x1 that can possibly occur (see Section 4.1). Given that

Vmss=((θ2θ1)2R12R22(θ2R2θ1R1)((1θ1)R1(1θ1)R2))e.

Now I want to compute and simplify Vmss=(θ,R)+Ω(θ,R)Γ(,R)=DENNUM. The common denominator and the numerator are:

DEN=(1θ2R2)((1θ1)R1(1θ2)R2)(θ2R2θ1R1),NUM=(1θ2R2)(θ2θ1)R1R2((θ2θ1)R1R2((θ2θ1)R1R2((1θ1)R1(1θ2)R2)))+θ2R2((1θ1)R1(1θ2)R2)((θ2θ1)R1R2((1θ1)R1(1θ2)R2)),=((1θ1)R1(1θ2)R2))((θ2θ1)R1R2+θ2R22θ2R1R2θ2R2(θ2R2θ1R1)),=((1θ1)R1(1θ2)R2))(1θ2)R2(θ2R2θ1R1).

Therefore:

Vmss(θ,R)+Ω(θ,R)Γ(θ,R)=θ2R21θ2R2=Vss(θ,R),Vss(θ,R)Vmss(θ,R)=Ω(θ,R)Γ(θ,R).

By the last equation, it is obvious that the sign of Vss(θ,R)Vmss(θ,R) is positive if and only if

rΛ (θ, R) > 0 and (θ, R) ∉ F . For the last case, define:

Ωi(θ,R)(((1θ1)R1(1θ2)R2)(θ2θ1)R1R2(θ2R2θ1R1))e,Γi(θ,R)(1θ1R1)(θ2θ1)R1R2θ1R1((1θ1)R1(1θ2)R2)(1θ1R1)((1θ1)R1(1θ2)R2).

Note that Ωi (θ, R) = −Ω (θ, R). Similar simplifications lead to:

Viss(θ,R)Vmss(θ,R)=Ωi(θ,R)Γi(θ,R).

Hence Viss(θ,R)Vmss(θ,R) is positive if and only if rΛ(θ, R) < 0 and (θ, R) ∉ Fi.

Now I complete the proof of the main theorem. By Proposition 2, the competitive equilibrium converges to a unique steady state corresponding to (θ, R) ∈ FFm. This implies that there exist T ≥ 0 and ε > 0 such that x2tε for tT. Suppose one reduces x2t for tT + 1 by δ + ϵ, increases x1t for tT + 1 by ϵ, and reduces x2T and increases x1T both by 1θ2R2θ1R1δ. Moreover, δ > 0, ϵ > 0 are such that ϵ + δ < ε and:

δ=(θ2R2θ1R1)ϵ+θ2R2δ,ϵ=1θ2R2θ2R2θ1R1δ.

Similar to what is shown in the text, this reallocation reduces the debt payments of each generation from T onward by δ and leaves all middle-aged entrepreneurs at or after T strictly better off when rΛ(θ, R) < 0. If rΛ(θ, R) = 0, the reallocation does not affect the utility of the middleaged after T but increases the utility of the middle-aged at T. This proves that the competitive equilibrium is constrained Pareto inefficient whenever rΛ(θ, R) ≤ 0.

If (θ, R) ∈ Fm, then by definition rΛ(θ, R) < 0 implies a strictly negative interest rate at the steady state. If (θ, R) ∈ F, by the first part of the proof on steady-state utility levels, rΛ(θ, R) < 0 implies:

θ2R21θ2R2<θ1R11θ1R11+rΛ(θ,R)<1.

Hence by Lemma 2, the steady-state interest rate is strictly negative, which completes the proof.

Proof of Lemma 4. The straight line corresponding to rΛ(θ, R) = 0 is θ2Λ(θ1)=(R1(R21)R2(R11))θ1+(R1R2R2(R11)). This line intersects horizontal line θ1 = 0 at θ2Λ(θ1)=R1R2R2(R11), which implies θ2Λ(0)>0. Hence, Proposition 2 and Proposition 4 imply that a strictly positive neighborhood of the origin, i.e., θ = 0, corresponds to inefficiently liquid equilibria. Since, by Proposition 2, any neighborhood of the origin contains liquid equilibria, it follows by Proposition 4 that there are inefficiently liquid equilibria in any small enough neighborhood of the origin. Note that by Proposition 2, the boundary of F cuts the vertical axis θ1 = 0 at the origin and θ=(0,11+R1) and also that the upper part of the boundary is negatively sloped in the (θ1, θ2) plane. Therefore, it follows that the rΛ(θ, R) = 0 line passes through F if and only if its intersection with θ1 = 0, that is (0,θ2Λ(0), lies below or at θ=(0,11+R1). This is the case whenever R1R2R211.

For the last part, let Si denote the unique intersection of rΛ (θ, R) = 0 with the boundary of Fi. Observe that by Proposition 2, the inefficiently liquid region lies above the convex inner boundary of Fi and below rΛ (θ, R) = 0. This completes the proof.

PROOF OF PROPOSITION 5. For part of this proof, I use some of the results in Ghate and Smith (2005), specially their Theorem 2.6. This theorem shows that complementary slackness conditions are sufficient for optimality in a linear programming with infinite variables and infinite number of constraints when feasible points, constraints, and objective functions of both primal and dual problems are elements of appropriate spaces. A necessary condition for this result is that the feasible points of the primal problem, i.e., feasible allocations {ct,x1t,x2t}t=0, lie in . To see this, note that by 7 and Assumption 1:

x1t+x2tθ1R1x1t1+θ2R2x2t1+eθ1R1(x1t1+x2t1)+e.

This together with 7 gives:

{x1t+x2ti1+e1θ1R1,ctR1(i1+e1θ1R1)+e.

i−1 is the total investment at t = −1, which is an initial condition to the problem. The above proves that {ct,x1t,x2t}t=0 for any feasible allocation.

Now let (θ, R) ∈ F and consider an allocation {ct*,x1t*,x2t*}t=0 that satisfies 7 with equality for all t ≥ 0. If there exists a series of strictly positive weights {λt}t=01 such that {ct*,x1t*,x2t*}t=0 solves:

max{ct,x1t,x2t}t=0Σt=0λtcts.t.ct+x1t+x2tR1x1t1+R2x2t1+ex1t+x2tθ1R1x1t1+θ2R2x2t1+ect0,x1t0,x2t0,

then {ct*,x1t*,x2t*}t=0 is constrained Pareto efficient. Let {ηt,γt,δ1t,δ2t,δct}t=0 be the Lagrange multipliers for resource constraint, borrowing constraint, and non-negativity constraints on x1t, x2t, and ct respectively. As discussed above, any feasible allocation is bounded. Hence the sufficient conditions for {ct*,x1t*,x2t*}t=0 to be a maximum are:

{λtηt+δct=0,(R1ηt+1ηt)+(θ1R1γt+1γt)+δ1t=0,(R2ηt+1ηt)+(θ2R2γt+1γt)+δ2t=0,ηt0,γt0,δ1t0,δ2t0,δct0,δ1tx1t=0,δ2tx2t=0,δctct=0.(SC)

for t ≥ 0, provided that {ηt,γt,δ1t,δ2t,δct}t=01. First consider the case where rΛ (θ, R) > 0. In this case, if I set {δ1t,δ2t,δct}t=0 to zero, solving the first three series of equations in SC, I obtain the following for t ≥ 0:

ηt=λt,γt+1=R1R2θ2R2θ1R1λt+1,λt+1=(1θ1)R1(1θ2)R2(θ2θ1)R1R2λt+1,λ1=θ2R2θ1R1(θ2θ1)R1R2(λ0+γ0).

The coefficient in the second difference equation above is (1 + rΛ(θ, R))−1. Therefore, for any positive λ0 and γ0, λ1 is given by the above and

λt=(1+rΛ(θ,R))(t1)λ1.

Since rΛ(θ, R) > 0, the resulting {λt}t=0 and consequently all {ηt,γt,δ1t,δ2t,δct}t=01 lie in 1. Therefore, all the conditions above which are sufficient for optimality are satisfied, and {ct*,x1t*,x2t*}t=0 is constrained Pareto efficient.

Now let rΛ(θ, R) < 0 and consider a feasible allocation {ct*,x1t*,x2t*}t=0 for which there exists a T ≥ 0 such that x2t*=0 for tT. If one sets {δ1t,δct}t=0 to zero, the first three sets of sufficient conditions in SC give the following for t ≥ 0:

ηt=λt,γt+1=R1R2θ2R2θ1R1λt+11θ2R2θ1R1δ2t,λt+2=(1θ1)R1(1θ2)R2(θ2θ1)R1R2λt+1+θ2R2(θ2θ1)R1R2δ2t+11(θ2θ1)R1R2δ2t,λ1=θ2R2θ1R1(θ2θ1)R1R2(λ0+γ0+δ20).

Given any positive λ0 and γ0, suppose one sets δ2t = 0 for 0 ≤ tT − 1. This implies λt = ρt−1λ1 for 1 ≤ tT, where ρ = (1 + rΛ(θ, R))−1 > 1. Moreover, let δ2T = αλT and δ2t = αλt for tT + 1, where α and α′ are positive constants to be determined. For tT + 2, the above equations lead to the following difference equation:

λt+1=(ρ+θ2R2(θ2θ1)R1R2α)λt1(θ2θ1)R1R2αλt1.

This difference equation has a solution of the form λt+1 = t where m is the smallest root of the characteristic equation:

m=12(ρ+θ2R2(θ2θ1)R1R2(ρ+θ2R2(θ2θ1)R1R2)24α(θ2θ1)R1R2).

It is easy to see that

m<1α<(θ2θ1)R1R2(ρ)1θ2R2.

Hence, if α is small enough, and given the appropriate initial condition, i.e., λT+2 = T+1, one can generate {λt}t=01. For time T + 1 and T + 2, the difference equation becomes:

λt+1=(ρ+θ2R2(θ2θ1)R1R2α)λT,λt+2=(ρ+θ2R2(θ2θ1)R1R2α)λT+11(θ2θ1)R1R2αλT.

Therefore λT+2 = T+1 if and only if:

(ρ+θ2R2(θ2θ1)R1R2αm)(ρ+θ2R2(θ2θ1)R1R2α)=1(θ2θ1)R1R2α.

The above equation is linear in α′. Note that one always has θ2R2ρ < 1 and hence for small enough α there is a strictly positive solution for α′. Therefore a small enough α > 0 defines unique values of 0 < m < 1 and α′ > 0 such that {λt}t=0 and {ηt,γt,δ1t,δ2t,δct}t=01 are in 1 and satisfy SC. This proves that {ct*,x1t*,x2t*}t=0 is constrained Pareto efficient.

Finally, let rΛ(θ, R) = 0 and consider a feasible allocation {ct*,x1t*,x2t*}t=0 for which there exists T ≥ 0 such that x2t*=0 tT. Setting {ct*,x1t*,x2t*}t=0 and {δ2t}t=0T1 to zero implies λt = λ1 for 1 ≤ tT. Using SC for k ≥ 1, one can obtain

λT+k=λTζ(Σj=0k2δ2T+j)+vδ2T+k1,

where ζ=1θ2R2(θ2θ1)R1R2 and v=1(θ2θ1)R1R2. To satisfy the above condition, for any j ≥ 0 define

δ2T+j=(λTλT+ζ)j+1,λT+j=(λT+ζ+v)(λTλT+ζ)j.

It is easy to see that {λt}t=0 and {ηt,γt,δ1t,δ2t,δct}t=01 lie in 1 and satisfy SC. This proves that {ct*,x1t*,x2t*}t=0 is constrained Pareto efficient and completes the proof.

PROOF OF PROPOSITION 6. First, I prove that the proposed regulation can implement Pareto improving reallocations used in Proposition 4. Consider an inefficiently liquid equilibrium corresponding to rΛ(θ, R) ≤ 0. Similar to the proof of Proposition 4, there exists T ≥ 0 and ε > 0 such that x2tε for tT. Suppose one reduces x2t for tT + 1 by δ + , increases x1t for tT + 1 by , and reduces x2T and increases x1T both by 1θ2R2θ1R1δ. Moreover, δ > 0 and > 0 are such that + δ < ε and

δ=(θ2R2θ1R1)ϵ+θ2R2δ,ϵ=1θ2R2θ2R2θ1R1δ.

This reduces the debt payments of generations on or after T exactly by δ. It has already been shown in the text that the above reallocation is a Pareto improvement. Let {δt,κt,vt}t=0 be defined as the decrease or increase in (1 + rt)e, x1t and x2t respectively, as above. Then one has

δt=θ2R2vtθ1R1κt,δt1=vtκt.

Let {αt}t=0 be the fraction of the liquid type in total investment for the original competitive equilibrium. Now define {α˜t}t=0 as follows:

α˜t=x2tvtx1t+x2tδt1.

Suppose the planner regulates the portfolios according to {αt}t=0. Let {x˜1t,x˜2t,r˜t}t=0 be the prices and quantities in the regulated equilibrium. Define r1*=r1 and {r1*}t=1 recursively:

1+r1*=(α˜tθ2R2+(1α˜t)θ1R1)(2+rt1*).

By IV and market clearing, r1* is an upper bound for r˜t for all t. Now suppose (1+rt1*)e=θ1R1x1t1+θ2R2x2t1δt1, which is true for t = 0 by assumption. Then one has

α˜tθ2R2+(1α˜t)θ1R1=θ2R2(x2tvt)+θ1R1(x1tκt)x1t+x2tδt1.

However, by resource constraint of the original competitive equilibrium, and using recursive equations above defining {δt, κt, νt}:

θ2R2(x2tvt)+θ1R1(x1tκt)=θ1R1x1t+θ2R2x2tδt,x1t+x2tδt1=θ1R1x1t1+θ2R2x2t1δt1+e.

Hence, it must be that (1+rt*)e=θ1R1x1t+θ2R2x2tδt, and so by induction this holds for all t ≥ 0. Now note that in the original competitive equilibrium one must have 1 + rt < Rα,t ≡ (1−αℓt)R1 + αℓtR2 for all t. Whether entrepreneurs specialize in the liquid type or mix at time t, the interest rate has to be no bigger than 1 + rΛ(θ, R). Using Assumption 1, one has 1 + rΛ(θ, R) < R2 and so

1+rt1+rΛ(θ,R)<R2Rα,t.

Observe that 1+rt*<1+rt and Rα˜,tRα,t. The latter is true because α˜tαt by construction. Hence 1+rt*<Rα˜,t for all t, which immediately implies that the borrowing constraints are binding in the regulated equilibrium and that r˜t=rt* for all t. Thus, the allocation induced by the regulation coincides with the Pareto superior allocation given at the beginning.

In the second part, I show that this type of regulation can make a Praeto improvement that reaches the Pareto frontier given by Proposition 5. Consider an inefficiently liquid equilibrium corresponding to rΛ(θ, R) ≤ 0. Since it converges to the steady state by Proposition 1, for an arbitrarily small > 0 one can choose T such that the differences between equilibrium values of the interest rate, investments in the two types, and utility of the old generations and their steadystate values are all less than for tT. Similar to the first part, suppose {α˜t}t=0,{x˜1t,x˜2tr˜t}t=0 be the liquid fraction of investment, the prices and quantities in the regulated equilibrium. Now define α˜t=αt if tT − 1 and α˜t=0 if tT. In other words, let’s replicate the original competitive equilibrium allocation up to time T − 1 and then completely shut down investment in the liquid type on or after T.

Similar to the first part, it is easy to show that borrowing constraints are binding in the regulated equilibrium and that the regulated equilibrium converges to a new steady state with x˜1ss, x˜2ss and r˜ss. The new steady-state interest rate 1+r˜ss=θ1R11θ1R1 is below {1+r˜t}t=T and strictly so, at least for 1+r˜T for small enough . The reason is that the original steady-state interest rate is either 1 + rss = 1 + rΛ(θ, R) or 1+rss=θ2R21θ2R2 and in either case strictly bigger than 1+r˜ss=θ1R11θ1R1. Since is small, the whole sequence of {1+r˜t}t=T has to lie above 1 + rss and strictly so, at least for t = T.

This implies that utilities of the middle-aged for tT have to be above their new steadystate value, i.e., Viss(θ,R), and strictly above for t = T, in the regulated economy because initial wealth of the middle-aged is bigger than the steady state level. By the first part of Proposition 4, if rΛ(θ, R) < 0 one has Viss(θ,R)>Vzss(θ,R) for z ∈ {m, }, where Vzss(θ,R) is the steady-state utility for the original steady state. Hence if is small enough, all of the middle-aged at or after T are better off, while the middle-aged before T are left as well off. If rΛ(θ, R) = 0 and (θ, R) ∈ F, then after some T, the original equilibrium reaches the steady state level, and so it is still true that at least the middle-aged at T are strictly better off, while all others are at least as well off.

Proof of Lemma 5. Suppose that the social planner levies taxes {τt}t=0 on the old at t + 1. The middle-aged problem changes to:

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

Note that the tax is rebated back to the agent, i.e., Tt = τt(1+rt)it. This ensures that any allocation that solves the above problem for any sequence of {τt}t=0 satisfies 7 and hence is feasible for a constrained social planner. After simplifying the objective function using the budget and the binding borrowing constraints, as in the problem without taxes, we end up with

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

where

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

After solving the above linear maximization, we get

{it=(θ2R2(1+rt1)1+rtθ2R2)e,if rt<rΛ(θ,R;τt),it[(θ1R1(1+rt1)1+rtθ1R1)e,(θ2R2(1+rt1)1+rtθ2R2)e]if rt=rΛ(θ,R;τt),(13)it=(θ1R1(1+rt1)1+rtθ1R1)e,if rt>rΛ(θ,R;τt).

We also get

rt={θ2R2(2+rt1)1if θ2R2(2+rt1)<1+rΛ(θ,R;τt),θ1R1(2+rt1)1if θ1R1(2+rt1)>1+rΛ(θ,R;τt),rΛ(θ,R;τt)otherwise

where

1+rΛ(θ,R;τt)=(θ2θ1)R1R2(1(1+τt)θ1)R1(1(1+τt)θ2)R2.

Note that we have rΛ(θ,R;τ)τ<0 and that rΛ(θ, R; τ) converges to zero as τ gets large. Now consider an inefficiently liquid equilibrium corresponding to rΛ(θ, R; τ) ≤ 0. Since it converges to the steady state by Proposition 1, for an arbitrarily small > 0, one can choose T such that the differences between equilibrium values of the interest rate, investments in the two types, and utility of the old generations and their steady state values are all less than for tT. Similar to the second part of Proposition 6, the social planner can shut down any investment in the liquid type for all tT, this time using a debt tax. To do that, the social planner can set τt = 0 for t < T and τt = τ for tT, such that 1 + rΛ(θ, R; τ) < θ1R1(2 + rt−1) for all tT. Such a value for τ exists because T is large so that the sequence of interest rates {rt1}t=T stays close enough to their steady-state value and also so that rΛ(θ, R; τ) can be made small enough by choosing a large enough τ. Under this debt tax, the economy will specialize in the productive type for tT, and similar to Proposition 6, we can show that at least one agent is strictly better off while all agents are at least as well off. And since there is no investment in the liquid type in the new allocation for tT, by Proposition 5, the new allocation is constrained Pareto efficient.

PROOF OF PROPOSITION 7. By Proposition 4 we have:

Vmss=((θ2θ1)2R12R22(θ2R2θ1R1)((1θ1)R1(1θ2)R2))e.

Normalizing e = 1 to simplify the exposition and taking the derivative, we get:

Vmss(θ,R)θ2=Ω{2(θ2θ1)(θ2R2θ1R1)((1θ1)R1(1θ2)R2)(θ2θ1)2(R2((1θ1)R1(1θ2)R2)+R2(θ2R2θ1R1))}

where

Ω={R1R2(θ2R2θ1R1)((1θ1)R1(1θ2)R2)}2,

is the sign of Vmss(θ,R)θ2 is the sign of its numerator. Therefore, we have:

Vmss(θ,R)θ22(θ2θ1)(θ2R2θ1R1)((1θ1)R1(1θ2)R2)(θ2θ1)2(R2((1θ1)R1(1θ2)R2)+R2(θ2R2θ1R1))>02(θ2R2θ1R1)((1θ1)R1(1θ2)R2)(θ2θ1)R2(((1θ1)R1(1θ2)R2)+(θ2R2θ1R1))>01(θ2θ1)R2(1θ1)R1(1θ2)R2>θ1(R1R2)θ2R2θ1R1(θ1+θ2)((1θ1)R1(1θ2)R2)(θ2θ1)R2(θ2R2θ1R1)(1θ1)R1(1θ2)R2>2θ1R1(θ2θ1)R2(R1R2)(1θ1)R1(1θ2)R2+2θ1R2>2θ1R1(θ2θ1)R1R2(1θ1)R1(1θ2)R2=1+rΛ(θ,R)>2θ1R1.

As for the second part, note that the boundary of Fi and Fm is defined by 1+rΛ(θ,R)=θ1R11θ1R1. If the statement is true on the boundary, it will be true for the interior of the efficient region of Fm. The reason is that for any given value of 1 + rΛ(θ, R), θ1 reaches its maximum value on the boundary of Fi and Fm. We know that an equilibrium is efficient in Fm if and only if rΛ(θ, R) > 0. Hence an equilibrium on the boundary of Fi and Fm is efficient if and only if

θ1R11θ1R1>1θ1R1>12θ1R11θ1R1>2θ1R11+rΛ(θ,R)>2θ1R1.

For the third part, note that on the boundary of Fi and Fm we have 1+rΛ(θ,R)=θ1R11θ1R1<1, then θ1R11θ1R1<1, and hence 1 + rΛ(θ, R) < 2θ1R1. Therefore, for any given θ1, if θ2 is close enough to the boundary of Fi and Fm, one has 1 + rΛ(θ, R) < 2θ1R1, because 1 + rΛ(θ, R) is strictly increasing in θ2. We now prove the last two parts of the proposition about Ymss(θ,R). To compute Ymss(θ,R), we first use 6 and the value of rΛ(θ, R) to get

{x1ss=R2(θ2(θ2θ1)R1R2(1θ2)(θ2R2θ1R1))(θ2R2θ1R1)((1θ1)R1(1θ2)R2),x2ss=R1((1θ1)(θ2R2θ1R1)θ1(θ2θ1)R1R2)(θ2R2θ1R1)((1θ1)R1(1θ2)R2).

Hence, we have

Ymss