Ngoc-Sang Pham
email: pns.pham@gmail.com

Impacts of (individual and aggregate) productivity and credit shocks on equilibrium aggregate production *

Keywords: JEL Classifications: D2, D5, E44, G10, O4 Productivity shock, financial shock, credit constraint, heterogeneity, productivity dispersion, distributional effects, efficiency, general equilibrium

In a market economy, the aggregate production level depends not only on the aggregate variables but also on the distribution of individual characteristics (e.g., productivity, credit limit, ...). We point out that, due to financial frictions the equilibrium aggregate production may be non-monotonic in both individual productivity and credit limit. By consequence, the emergence of some firms (for example, improving productivity or relaxing credit limit) may not necessarily be beneficial to economic development.

Introduction

The paper aims to investigate two basic questions in economics: what are the impacts of (individual and aggregate) productivity and financial shocks on the aggregate output?

Let us start by a simple framework: there are m agents and a singe good (which can be consumed or used to produce). Each agent i is endowed S i units of good and has a production function A i f i (k) where A i > 0 is the productivity, and f i is the autonomous production function. In an economy without market imperfections, the output must be

Y perf ect ≡ max (k i )≥0 i A i f i (k i ) subject to : i k i ≤ S ≡ i S i . (1)
Of course, the output Y perf ect is increasing in each productivity A i . In other words, the productivity has a positive effect on the output. This insight is well-known in economics textbooks or classical papers [START_REF] Solow | Technical change and the aggregate production function[END_REF][START_REF] Romer | Increasing returns and long-run growth[END_REF][START_REF] Romer | Endogenous technological change[END_REF]. Now, consider a competitive economy with financial market imperfections: each agent has a credit limit γ i . The equilibrium aggregate production is

Y = i A i f i (k i )
, where (k i ) is the allocation of inputs in general equilibrium. (2) Since the equilibrium allocation of capital (k i ) depends on the economy structure, the equilibrium production Y depends on productivities A ≡ (A i ) m i=1 and credit limits γ ≡ (γ i ) m i=1 of agents. So, we write Y = Y (A 1 , . . . , A m , γ 1 , . . . , γ m ). We aim to investigate conditions under which the aggregate production increases or decreases in productivity A i and credit limit γ i . Our novel point is that the aggregate output (and hence aggregate TPF) may decrease even the productivity and credit limit of all agents increase, i.e., Y (A 1 , . . . , A m , γ 1 , . . . , γ m ) < Y (A 1 , . . . , A m , γ 1 , . . . , γ m )

(3) for some (A i ), (γ i ) such that A i > A i , γ i > γ i , ∀i = 1, . . . , m.

First, we explain the role of productivity. The productivity is widely viewed as one of the most important determinants of economic growth. Our contribution is to argue that whether a positive productivity shock generates a positive (or negative) effect on the aggregate output depends on the distribution of productivity, the size of productivity shock, and the level of financial imperfection.

If the productivity of all agents increases (resp., decreases) at the same rate, then this change has a positive (resp., negative) effect on the aggregate production. However, when the productivity of agents increases at different rates, the aggregate production may decrease. This may happen if the TFP of less productive agents increases faster than that of more productive agents. Indeed, in such a case, less productive agents absorb more capital and produces more. Since the aggregate capital is limited, other producers (who are more productive) get less capital (because of market imperfections) and so they produce less. By consequence, the net effect may be negative. This happens if (1) the TFP of less productive agents is far from that of more productive producers, i.e., the productivity dispersion is high, 1 (2) the positive shock is quite small, (3) the credit constraint is tight.

Second, we focus on the effects of a change in credit limit (γ i ) on the aggregate production Y . One can expect that relaxing credit limits would have positive impact on the aggregate output as shown in [START_REF] Khan | Credit Shocks and Aggregate Fluctuations in an Economy with Production Heterogeneity[END_REF] (section VI. C), [START_REF] Midrigan | Finance and Misallocation: Evidence from Plant-Level Data[END_REF] (section II.B), [START_REF] Moll | Productivity Losses from Financial Frictions: Can Self-financing Undo Capital Misallocation?[END_REF] (Proposition 1), and Catherine, Chaney, 1 Andrews, Criscuolo and [START_REF] Andrews | Frontier Firms, Technology Diffusion and Public Policy: Micro Evidence from OECD Countries[END_REF] use a harmonised cross-country dataset, based on underlying data from the OECD-ORBIS database [START_REF] Gal | Measuring Total Factor Productivity at the Firm Level using OECD-ORBIS[END_REF], to analyze the characteristics of firms that operate at the global productivity frontier and their relationship with other firms in the economy. [START_REF] Andrews | Frontier Firms, Technology Diffusion and Public Policy: Micro Evidence from OECD Countries[END_REF] document growing productivity dispersion for several developed countries over the 2000s. [START_REF] Bouche | News from the Frontier: Increased Productivity Dispersion across Firms and Factor Reallocation[END_REF] present empirical evidence showing an increase in productivity dispersion between French firms during the period 1991-2016, with a growing productivity gap between frontier and laggard firms. See [START_REF] Goldin | Why is productivity slowing down?[END_REF] for an excellent review on the slowdown in productivity growth. [START_REF] Catherine | Aggregate Effects of Collateral Constraints[END_REF]. Our novel point is that, while a homogeneous positive credit shock improves the aggregate output, an asymmetric positive credit shock can reduce the output.

The intuition is similar to that in the case of productivity shocks we have mentioned above: If credit limits of less productive agents increase faster that those of more productive ones, less productive agents get more capital and more productive agents get less capital, hence the aggregate output may decrease. It should be noticed that although the aggregate output is not necessarily monotonic in credit limits of producers, it does not exceed that in the frictionless economy.

After having worked in a static model, we investigate our above questions in infinitehorizon models à la Ramsey. We prove that the non-monotonic effect of productivity and credit limit on the aggregate output cannot appear at the steady state in infinitehorizon models à la Ramsey. The reason is that the steady state interest rate only depends on the rate of time preferences of agents.

Therefore, we focus on the global dynamics of intertemporal equilibrium. First, our findings suggest that a permanent increase of productivity of less productive agents improves the aggregate output in the long run. However, when this productivity shock is quite small and credit constraints are tight, the aggregate output may decrease in the short-run and then increase from some period on.

Second, we look at the effects of credit limits. In the static model, an increase in the most productive agent's credit limit is always beneficial for the aggregate output. However, along intertemporal equilibrium, we show that an increase of the most productive producer's credit may reduce the output at every period. The intuition behind is that when her(his) credit limit goes up, the equilibrium interest rate increases, and hence, her(his) repayment also increases. This in turn reduces her(his) net worth in the next period. By consequence, her(his) saving and hence the production decrease. The mechanism can be summarized by the following schema:

Credit limit ↑ ⇒ Interest rate ↑ ⇒ Agent's net worth ↓ ⇒ ⇒ Saving ↓ ⇒ Production ↓ ⇒ • • • (4)
As in the static model, this mechanism can happen because the credit limit of the most productive agent remains low and the productivity dispersion is high. Our article is related to a growing literature on general equilibrium models with heterogeneous producers and financial frictions.2 Let us mention some of them. 3 Midrigan and Xu (2014) consider a two-sector model with a collateral constraint that requires the debt of producer does not exceed a fraction of its capital stock. They focus on balanced growth equilibrium to study the role of collateral constraint in determining TFP. Their parameterizations consistent with the data imply fairly small losses from misallocation, but potentially sizable losses from inefficiently low levels of entry and technology adoption. [START_REF] Khan | Credit Shocks and Aggregate Fluctuations in an Economy with Production Heterogeneity[END_REF] develop a dynamic stochastic general equilibrium with a representative household and heterogeneous firms facing a borrowing constraint (slightly different from ours) and focus on recursive equilibrium. They find that a negative shock to borrowing conditions can generate a large and persistent recession through disruptions to the distribution of capital. [START_REF] Buera | Financial Frictions and the Persistence of History: A Quantitative Exploration[END_REF] develop a model with individual-specific technologies and collateral constraints to investigate the role of the misallocation and reallocation of resources in macroeconomic transitions. [START_REF] Buera | Financial Frictions and the Persistence of History: A Quantitative Exploration[END_REF] find that collateral constraints have a large impact along the transition to the steady state. [START_REF] Moll | Productivity Losses from Financial Frictions: Can Self-financing Undo Capital Misallocation?[END_REF] studies the effect of collateral constraints on capital misallocation and aggregate productivity in a general equilibrium with a continuum of heterogeneous firms and financial frictions (modeled by a collateral constraint). Proposition 1 in [START_REF] Moll | Productivity Losses from Financial Frictions: Can Self-financing Undo Capital Misallocation?[END_REF] shows that the aggregate TFP is increasing in the leverage ratio which is the common across firms. 4 Our paper differs from this literature in two points. First, the credit limit is individualized in our model while all credit parameters in the above studies are common across firms. Second, we argue that this credit heterogeneity plays an important role in the distribution of capital and of income as well as in the aggregate output. Indeed, we prove that the aggregate output and the aggregate TFP in our model may not be monotonic functions of the credit limits which are different across agents; they may display an inverted-U form. 5 However, we show that, if agents have the same credit limit, the aggregate output and the aggregate TFP are increasing functions of this common credit limit; this finding is consistent with the above literature.

Our paper is related to [START_REF] Baqaee | Productivity and Misallocation in General Equilibrium[END_REF] who build a general equilibrium model where productivity and wedge are exogenous parameters to study how the impact of (productivity and wedge) shocks can be decomposed into a pure technology effect and an allocative efficiency effect. There are some differences between [START_REF] Baqaee | Productivity and Misallocation in General Equilibrium[END_REF] and the present paper. First, [START_REF] Baqaee | Productivity and Misallocation in General Equilibrium[END_REF] model frictions by wedge while we model frictions by a credit constraint and the credit limit is our exogenous parameter. Second, [START_REF] Baqaee | Productivity and Misallocation in General Equilibrium[END_REF] provide a quantitative analysis by applying their approach to the firm-level markups in the U.S. but they do not provide conditions (based on exogenous parameters) under which the aggregate output is increasing or decreasing in productivity and friction level (wedge in their framework). Although we do not provide quantitative applications of our results, we show several conditions (based on exogenous parameters) under which the aggregate output is increasing or decreasing in productivity and friction level (credit limit in our framework). We also run some simulations and extend our analyses in infinite-horizon models while [START_REF] Baqaee | Productivity and Misallocation in General Equilibrium[END_REF] do not do this exercise.

Our paper also concerns the literature on the welfare effects of financial constraints. Jappelli andPagano (1994, 1999) consider overlapping generations models with liquidity constraints and households living for three periods and argue that liquidity con-4 In both [START_REF] Buera | Financial Frictions and the Persistence of History: A Quantitative Exploration[END_REF], [START_REF] Moll | Productivity Losses from Financial Frictions: Can Self-financing Undo Capital Misallocation?[END_REF], the collateral constraint, which is slightly different from ours, states that the capital of a firm does not exceed a leverage ratio of its financial wealth.

5 Our finding is related to [START_REF] Aghion | The Inverted-U Relationship Between Credit Access and Productivity Growth[END_REF]. They consider a model of firm dynamics and innovation with entry, exit, and credit constraints, based on [START_REF] Klette | Innovating firms and aggregate innovation[END_REF], [START_REF] Aghion | Lessons from Schumpeterian growth theory[END_REF]. They assume that intermediate firms (monopolist) cannot invest more than µ times their current market value in innovation. They argue that the credit access may harm productivity growth because it allows less efficient incumbent firms to remain longer on the market, which discourages entry of new and potentially more efficient innovators.

straints may increase or decrease welfares. The central point in Jappelli andPagano (1994, 1999) is that liquidity constraints have two opposite effects on welfare: "they force the consumption of young below the unconstrained level but raise their permanent income by fostering capital accumulation". Obiols-Homs (2011) considers a general equilibrium with heterogeneous households (whose borrowings are bounded by an exogenous limit) and a representative firm. He argues that the borrowing limit has a negative on the welfare of borrower if its quantity effect dominates its price effect. As in Jappelli andPagano (1994, 1999), the mechanism of Obiols-Homs (2011) relies on the role of supply of credit to households who need to smooth their consumption. By contrast, our mechanism focuses on credit to firms who need credit to finance their productive investment. Moreover, Obiols-Homs (2011) considers exogenous borrowing limits while we focus on credit constraints and our model has endogenous borrowing limits.

Catherine, Chaney, Huang, Sraer, and Thesmar (2017) build a dynamic general equilibrium model with heterogeneous firms and collateral constraints. They focus on the steady state and provide estimates suggesting that lifting financial frictions (modeled by collateral constraints) would increase aggregate welfare by 9.4% and aggregate output by 11%. Our paper differs from [START_REF] Catherine | Aggregate Effects of Collateral Constraints[END_REF] in two aspects. First, although we also find that the aggregate output in the frictionless economy is higher than that in the economy with financial frictions, it is not a monotonic function of the degree of financial friction. Second, both individual and social welfares may not be monotonic in the degree of financial friction. Interestingly, lifting credit constraint may decrease the welfare of some agents.

Last but not least, our paper contributes to the debate concerning the slowdown in aggregate productivity growth that has been documented by several studies such as [START_REF] Andrews | Frontier Firms, Technology Diffusion and Public Policy: Micro Evidence from OECD Countries[END_REF], [START_REF] Bouche | News from the Frontier: Increased Productivity Dispersion across Firms and Factor Reallocation[END_REF], [START_REF] Goldin | Why is productivity slowing down?[END_REF]; see Footnote 1. Our above analyses suggest that the interplay between credit constraints, high heterogeneity of productivity, asymmetry of productivity and financial shocks may generate a slowdown in aggregate productivity growth. The aggregate productivity growth rate may be far from that of most productive firms. It may be even lower than the smallest productivity growth rate of firms. Our approach, which is different from those in the literature, is based on the general equilibrium theory with financial frictions.

The rest of our article is organized as follows. Section 2 presents a two-period general equilibrium framework. Section 3.1 studies the effects of productivity shocks while Section 3.2 investigates the role of credit shocks. Section 4 extends our analyses in Sections 3.1 and 3.2 to infinite-horizon general equilibrium models à la Ramsey. Section 5 concludes. Formal proofs are gathered in the appendices.

A static framework

We consider a deterministic two-period economy with a finite number (m) of heterogeneous agents. There is a single good (numéraire) which can be consumed or used to produce. Each agent i has exogenous initial wealth (S i units of good) at the initial date. To keep the model as simple as possible, we assume that agents just maximize their consumption in the second period and we focus on the output in this period.

Agents have two ways for investing. On the one hand, agent i can buy k i units of physical capital at the initial date to produce F i (k i ) units of good at the second date, where F i is the production function. 6 Since we are interested in the effect of individual productivity, we write

F i (x) = A i f i (x), (5)
where A i > 0 represents the productivity of agent i while f i is the original production function.

On the other hand, she can invest in a financial asset with real return r which is endogenous. Denote b i the asset holding of agent i. She can also borrow and then pay back Rb i in the next period. However, there is a borrowing constraint. The maximization problem of agent i can be described as follows:

(P i ) : π i = max k i ,b i [F i (k i ) -Rb i ] (6a) subject to: 0 ≤ k i ≤ S i + b i (budget constraint) (6b) Rb i ≤ γ i F i (k i ) (borrowing constraint) (6c)
where γ i ∈ (0, 1) is an exogenous parameter. Borrowing constraint (6c) means that the repayment does not exceed the market value of the borrower's project. 78 This is similar to the collateral constraint (4) in [START_REF] Kiyotaki | Credit and business cycles[END_REF] or the so-called earningsbased constraint in [START_REF] Lian | Anatomy of Corporate Borrowing Constraints[END_REF]. The better the commitment, the higher value of γ i , the larger the set of feasible allocations of the agent i. [START_REF] Kiyotaki | Credit and business cycles[END_REF] interprets γ i as the collateral value of investment. In our paper, we call γ i the credit limit of agent i.

The following table from the Enterprise Surveys (2018)'s panel datasets suggests that borrowing and collateral constraints matter for the development of firms.

An economy E with credit constraints is characterized by a list of fundamentals

E ≡ (A i , f i , γ i , S i ) i=1,...,m . Definition 1. A list (R, (k i , b i ) i ) is an equilibrium if (1) for each i, given R, the allocation (b i , k i
) is a solution of the problem (P i ), and (2) financial market clears

i b i = 0.
6 We can interpret the one-factor production function F i as a reduced form for a setting with other factors of production. Indeed, suppose that the firm has a two-factor production function, say capital and labor, G i (k, N ). For a given level of capital k i , the firm chooses labor quantity

N i to maximize its profit max Ni≥0 [G i (k i , N i ) -wN i ]. The first order condition writes ∂Gi ∂N (k i , N i ) = w. This implies that N i = N i (k i , w). So, G i (k i , N i ) = G i (k i , N i (k i , w)). We now define F i (k i ) ≡ G i (k i , N i (k i , w)).
7 Here, we follow [START_REF] Kiyotaki | Credit and business cycles[END_REF] by assuming that the debtor is required to put her project as collateral in order to borrow: If she does not repay, the creditor can seize the collateral. Due to the lack of commitment (or just because the debtor is not willing to help the creditor take the whole value of the debtor's project), the creditor can only obtain a fraction γ i of the total value of the project. Anticipating the possibility of default, the creditor limits the amount of credit so that the debt repayment will not exceed a fraction γ i of the debtor's project value.

8 Matsuyama (2007) (Section 2) considers a model with heterogeneous agents, which corresponds to our model with k i = 1, S i = w, b i = 1 -w. However, different from our setup, investment projects in [START_REF] Matsuyama | Aggregate Implications of Credit Market Imperfections[END_REF] are non-divisible.

We require standard assumptions on the production function.

Assumption 1. The production function f i is concave, strictly increasing, f i (0) = 0. The credit limit γ i belongs the interval (0, 1) for any i.

Lemma 1. Under Assumption 1, there exists an equilibrium.

We can prove the equilibrium existence by using the standard argument (see, for instance, Bosi, Le [START_REF] Bosi | Intertemporal equilibrium with heterogeneous agents, endogenous dividends and collateral constraints[END_REF])

Remark 1. Some authors [START_REF] Buera | Financial Frictions and the Persistence of History: A Quantitative Exploration[END_REF][START_REF] Moll | Productivity Losses from Financial Frictions: Can Self-financing Undo Capital Misallocation?[END_REF] set k i ≤ θw i , where w i ≥ 0 is the agent i's wealth and interpret that θ measures the degree of credit frictions (credit markets are perfect if θ = ∞ while θ = 1 corresponds to financial autarky, where all capital must be self-financed by entrepreneurs). In our framework, S i plays a similar role of wealth w i in [START_REF] Buera | Financial Frictions and the Persistence of History: A Quantitative Exploration[END_REF], [START_REF] Moll | Productivity Losses from Financial Frictions: Can Self-financing Undo Capital Misallocation?[END_REF]. Another way to introduce credit constraint is to set that b i ≤ θk i . This corresponds to constraint (3) in [START_REF] Midrigan | Finance and Misallocation: Evidence from Plant-Level Data[END_REF]. Other authors [START_REF] Kocherlakota | Bubbles and constraints on debt accumulation[END_REF][START_REF] Obiols-Homs | On borrowing limits and welfare[END_REF] consider exogenous borrowing limits by imposing a short sales constraint: b i ≤ B for any i. Under these three settings, the asset holding b i is bounded from above by an upper bound which does not depend on the interest rate R. 9

Equilibrium aggregate production and aggregate TFP

Given an equilibrium (R, (k i , b i ) i ), the aggregate output is Y = i A i f i (k i ). Since Y depends on fundamentals, we write Y = Y (A i ), (f i ), (S i ), (γ i ) . ( 7
)
The production in the market economy with financial imperfections depends on the distributions of productivity, initial wealth and credit limits.

In an economy with perfect financial market, the aggregate production is simply determined by

Y perf ect ≡ max (k i )≥0 i F i (k i ) subject to : i k i ≤ S ≡ i S i . ( 8
)
9 [START_REF] Carosi | Endogenous restricted participation in general financial equilibrium[END_REF] present a two-period general equilibrium model with uncertainty, numéraie assets, and participation constraints described by functions of agent's choices and prices. [START_REF] Carosi | Endogenous restricted participation in general financial equilibrium[END_REF] prove the existence of equilibrium and study indeterminacy but do not provide comparative statics.

Y perf ect is increasing in A i , ∀i, and in S.

In equilibrium, we have i k = S. So, we have that Y ≤ Y perf ect . This is consistent with a number of studies on the macroeconomic effects of financial constraints [START_REF] Buera | Financial Frictions and the Persistence of History: A Quantitative Exploration[END_REF][START_REF] Karaivanov | Dynamic Financial Constraints: Distinguishing Mechanism Design from Exogenously Incomplete Regimes[END_REF][START_REF] Midrigan | Finance and Misallocation: Evidence from Plant-Level Data[END_REF][START_REF] Moll | Productivity Losses from Financial Frictions: Can Self-financing Undo Capital Misallocation?[END_REF][START_REF] Catherine | Aggregate Effects of Collateral Constraints[END_REF].

If we assume that F i (k) = A i f (k) where A i represents the individual productivity of agent i and f is a production function, then we can define the aggregate production function G and the aggregate TFP A by the aggregate TFP:

A ≡ Y F (S) (9)
the aggregate production function:

G(S) ≡ Y = Af (S). ( 10
)
Our main goal is to explore conditions under which the aggregate output Y given by ( 7) and TFP A given by ( 9) are increasing or decreasing in individual productivity A i and credit limit γ i .

Individual choice and characterization of equilibrium 2.2.1 Individual optimal choice

We firstly study the individual optimal allocation of agents. When technologies are linear (where Inada condition does not hold), it is easy to obtain the following result Lemma 2 (individual choice -linear production function). Assume that F i (K) = A i K. Let R > 0 be given. The solution for agent i's maximization problem is described as follows.

1. If R ≤ γ i A i , then there is no solution (k i = ∞).

2. If A i > R > γ i A i , then agent i borrows from the financial market and the borrowing constraint is binding. We have

k i = R R-γ i A i S i , a i = γ i A i R-γ i A i S i , π i = A i k i -Rb i = R(1-γ i ) R-γ i A i A i S i .
3. If A i = R, then the solutions for the agent's problem include all pairs

(k i , b i ) such that -S i ≤ b i ≤ γ i 1-γ i S i and k i = b i + S i . 4. If A i < R,
then agent i does not produce goods and invest all her initial wealth in the financial market:

k i = 0, b i = -S i .
Next, we consider the case of strictly concave technology.

Assumption 2. For any i, the function f i is strictly increasing, strictly concave, twice continuously differentiable,

f i (0) = 0, f i (∞) = ∞, f i (0) = ∞, f i (∞) = 0.
Before present the properties of individual optimal choice, we introduce some notations:

Definition 2. Given R, γ i , A i , S i , denote k n i = k n i (R/A i ) the unique solution to the equation A i f i (k) = R and k b i = k b i ( R γ i A i , S i ) the unique solution to R(k-S i ) = γ i A i f i (k).
k b i (k n i , respectively) represents the capital level of agent i when her borrowing constraint is binding (not binding, respectively). Under assumptions in Lemma 2, we can verify that: (1)

k n i is strictly decreasing in R/A i . Moreover, lim R/A 1 →0 k n i = +∞, and lim R/A 1 →∞ k n i = 0. (2) k b i is strictly increasing in S i but strictly decreasing in R γ i A i . Moreover, lim R/A i →0 k b i = +∞, and lim R/A i →∞ k b i = S i .
The following result characterizes the solution of the problem (P i ).

Lemma 3 (individual choice -strictly concave production function). Under Assumption 2, there exists a unique solution to the problem (P i ). The optimal capital k i is increasing in TFP A i , credit limit γ i but decreasing in the interest rate R.

1. If R k n i (R/A i )-S i A i f i (k n i (R/A i )) ≥ γ i , then credit constraint is binding and the capital level is k i = k b i . Moreover, k i = k b i ≤ k n i . 2. If R k n i (R/A i )-S i A i f i (k n i (R/A i )) < γ i , then credit constraint is not binding and k = k n i . In this case, we have k i = k n i < k b i . Proof. See Appendix A.
Lemma 3 leads us to introduce the following assumption.

Assumption 3. The function

kf i (k) f i (k) is increasing in k.
Under Assumptions 2 and 3, the function

(k-S i )f i (k) f i (k)
is strictly increasing in k.

Therefore, the function

G i (x) ≡ (k n i (x)-S i )x f i (k n i (x))
is strictly decreasing in x. Moreover, we k) . By consequence, we obtain the following result.

can check that lim x→+∞ G i (x) = -∞, lim x→0 G i (x) = lim k→∞ kf i (k) f i (
Corollary 1. Let Assumptions 2 and 3 be satisfied. So, if agent i's borrowing constraint is binding, we must have γ i ≤ lim x→∞ xF i (x) x) . By consequence, when F i (k) = A i k α i and γ i > α i , then agent i's borrowing constraint is not binding.

F i (
The following assumption and result show the interaction between interest rate, credit limit γ i and borrowing constraint.

Assumption 4. γ i < lim k→∞ kf i (k) f i (k) .
Corollary 2. Let Assumptions 2, 3 and 4 be satisfied. We can define R i the unique value satisfying

H i (R i ) ≡ R i k n i (R i /A i ) -S i A i f i (k n i (R i /A i )) = γ i . (11)
Then, we have that:

1. Agent i's borrowing constraint is binding if and only if

H i (R) ≥ γ i which is equivalent to R ≤ R i ≡ H -1 i (γ i ).
2. R i /A i does not depend on A i , and

lim A i →∞ R i = ∞, lim A i →0 R i = 0. R i is
increasing in productivity A i but decreasing in γ i and in S i .

3. We also have

k n i (R i /A i ) = k b i (R i /A i ).
The threshold R i is exogenous. It represents the subjective interest rate of agent below which agent borrows so that her(his) borrowing constraint is binding. Point 2 of Lemma 2 indicates that the credit constraint of agent i is more likely to bind if the interest rate, her initial wealth and credit limit are low, and/or her productivity is high.

Remark 2. Under Cobb-Douglas technology, i.e., F i (k) = A i k α , we can compute that

H i (R) = α 1 - R αA i S α-1 i 1 1-α is decreasing in R and H i (0) = α. So, if α i < γ i , then borrowing constraint is not binding, whatever the level of interest rate R. When H i (0) > γ i , i.e., α > γ i , we have R i = αA i S α-1 i 1 -γ i α 1-α .

Characterization of general equilibrium

Thanks to above analyses, we can fully characterize the general equilibrium in two cases: (1) linear technology and (2) Assumptions 2, 3 and 4 are satisfied. We only state here our result for the case of strictly concave technologies. See Theorem 2 in Appendix A for the case of linear technology Theorem 1 (characterization of general equilibrium: strictly concave technologies). Under Assumption 2, there exists a unique equilibrium. Assume, in addition, that Assumption 3 and 4 hold and R 1 < R 2 < . . . < R m , then the unique equilibrium is determined as follows:

1. In the regime R m , i.e., when

S < m i=1 k n i (R m /A i ), ( 12
)
credit constraint of any agent is not binding. In this case, the equilibrium coincides to that of the economy without credit constraints, and the interest rate is

R = R * > R m . Agent i borrows (k i ≥ S i ) if and only if F i (S i ) ≥ R * .
2. In the regime R n (with 1 ≤ n ≤ m -1), i.e., when

n i=1 k n i ( R n A i ) + m i=n+1 k b i ( R n γ i A i , S i ) > S ≥ n+1 i=1 k n i ( R n+1 A i ) + m i=n+2 k b i ( R n+1 γ i A i , S i ), ( 13
)
the equilibrium interest rate and agents' capital are determined by

n i=1 k n i ( R A i ) + m i=n+1 k b i ( R γ i A i , S i ) = S ≡ i S i (14a)
k i = k n i ( R A i ) if i ≤ n k b i ( R γ i A i , S i ) if i ≥ n + 1. ( 14b
)
Notice that R n < R ≤ R n+1 in this case. Any agent i (i ≥ n + 1) borrows and her credit constraint is binding. The credit constraint of any agent i ≤ n is not binding. Moreover, agent i (i ≤ n) borrows if and only if F i (S i ) ≥ R.

Proof. See Appendix A.1.

Impacts of productivity and credit shocks

This section aims to address our main question: How the aggregate output changes when productivities (A i ) and credit limits (γ i ) vary. First, given the interest rate R, the optimal quantity of physical capital k i of agent i depends on R, her productivity A i and credit limit γ i . So, we write k i = k i (A i , γ i , R). Second, in equilibrium, the equilibrium interest rate depends on agents' characteristics, including (A i ) and (γ i ). So, we can write R = R(A 1 , A 2 , . . . , A m , γ 1 , . . . , γ m ). By consequence, the physical capital of agents and the aggregate output are

k i = k i A i , γ i , R(A 1 , A 2 , . . . , A m , γ 1 , . . . , γ m ) , Y = i A i f i (k i )

Impacts of productivity shocks

In this section, we investigate conditions under which the aggregate production is increasing or decreasing when productivities (A i ) change.

A motivating example

Let us start by considering a simple economy having two agents with linear production functions

F i (k) = A i k with γ 2 A 2 < A 1 < A 2 .
In this case, we can explicitly compute the equilibrium interest rate and aggregate output (see Theorem 2 in Appendix A.2):

Y =    A 2 (S 1 + S 2 ) if A 1 < γ 2 A 2 S 1 +S 2 S 1 A 1 S 1 + A 2 S 2 A 1 (1 -γ 2 ) A 1 -γ 2 A 2 if A 1 ≥ γ 2 A 2 S 1 +S 2 S 1 (15)
This allows us to fully investigate the effects of productivity shocks. First, the aggregate output is always increasing in A 2 -the productivity of the most productive agent.

Second, looking at the effect of A 1 , we see that when γ 2 A 2

S 1 +S 2 S 1 < A 1 , we have ∂Y ∂A 1 = S 1 - (1-γ 2 )γ 2 A 2 2 S 2 (A 1 -γ 2 A 2 ) 2 , and ∂Y ∂A 1 ≥ 0 ⇔ S 1 S 2 A 1 A 2 -γ 2 2 ≥ (1 -γ 2 )γ 2 (16)
So, the aggregate output displays an U-shape as a function of the least productive agent's credit limit. It is increasing in A 1 if the productivity ratio A 1 /A 2 is higher than a threshold (or, equivalently, the productivity gap A 2 /A 1 is lower than a threshold). Figure 1 illustrates an example. In this numerical simulation, we set

S 1 = 1, S 2 = 0.7, A 2 = 1, γ 2 = 0.2, and let A 1 vary from γ 2 A 2 S 1 +S 2 S 1 = 0.34 to A 2 = 2.
Then the output, as a function of A 1 , is decreasing on the interval (0.34, 0.54] and then increasing in the interval (0.54, 1)

We now consider the case where both productivities A 1 and A 2 change.

Proposition 1 (productivity shocks). Consider a two-agent economy having linear technologies

F i (k) = A i k ∀i = 1, 2 with A 1 < A 2
, and borrowing constraints:

Rb i ≤ γ i A i k i .
Figure 1: Non-monotonic effect of the agent 1's productivity.

Assume that there is a productivity shock that changes the productivity of agents from

(A 1 , A 2 ) to (A 1 , A 2 ). Assume that A 2 > A 1 . Assume that the credit constraint of agent 2 is low so that γ 2 < A 1 A 2 S 1 S 1 +S 2 and γ 2 < A 1 A 2 S 1 S 1 +S 2 . Then, the output change is Y (A 1 , A 2 ) -Y (A 1 , A 2 ) = (A 1 -A 1 )S 1 + A 2 S 2 (1 -γ 2 ) A 1 A 2 -A 1 A 2 (A 1 -γ 2 A 2 )(A 1 -γ 2 A 2 ) (17)
(1) We have that:

If A 2 A 2 ≥ A 1 A 1 ≥ 1, then Y (A 1 , A 2 ) ≥ Y (A 1 , A 2 ) ( 18
)
(2) Assume that

S 2 A 2 (1 -γ 2 ) γ 2 A 2 (A 1 -γ 2 A 2 ) 2 -S 1 > 0, i.e., S 1 S 2 A 1 A 2 -γ 2 2 < (1 -γ 2 )γ 2 (19)
Then, there is a neighborhood B of (A 1 , A 2 ) such that

Y (A 1 , A 2 ) -Y (A 1 , A 2 ) A 1 -A 1 < 0 (20a) ∀(A 1 , A 2 ) ∈ B satisfying A 2 A 2 -1 A 1 A 1 -1 < γ 2 A 2 A 1 - S 1 (A 1 -γ 2 A 2 ) 2 S 2 A 1 A 2 (1 -γ 2 ) and A 1 = A 1 . (20b) Proof. See Appendix B.
Condition (18) says that the aggregate output increases if the productivity of both firms increases and the productivity of the most productive agent increases faster than that of the less productive one.

Let us now focus on point 2 of Proposition 1. Here, condition (19) plays a very important role. It is satisfied if the ratio

A 1 A 2 is low in the sense that A 1 A 2 < γ 2 + γ 2 (1-γ 2 )S 2 S 1
0.5 . This can be interpreted as a high productivity dispersion. Under this condition, we see that

γ 2 A 2 A 1 -S 1 (A 1 -γ 2 A 2 ) 2 S 2 A 1 A 2 (1-γ 2 )
∈ (0, 1). According to conditions ( 19) and (20a), under a positive shock that improves the TFP of all agents, the aggregate output may decrease:

Y (A 1 , A 2 ) < Y (A 1 , A 2 ), ∀A 1 > A 1 , A 2 > A 2 , (A 1 , A 2 ) ∈ B satisfying (20b).
Let us explain the economic intuition behind this result. Assume that the productivity dispersion is high and let us consider a small positive shock (both the TFP of both agents increases). If the productivity of the less productive agent increases faster than that of the most productive agent (i.e.,

A 2 A 2 is low with respect to A 1 A 1 , see condition (20b))
, the first agent absorbs more physical capital and the most productive agent gets less capital (i.e., k

2 (A 1 , A 2 ) < k 2 (A 1 , A 2 ))
. By consequence, the aggregate output may decrease.

Individual productivity shocks

We now work with more general models. Consider an equilibrium and agent j whose capital k j is strictly positive. Assume that we have the differentiability. In this case, condition i k i = S implies that

∂k j ∂A j + ∂R ∂A j m i=1 ∂k i ∂R = 0. Thus, we have ∂R ∂A j > 0 and ∂k j ∂R ∂R ∂A j + ∂k j ∂A j = - i =j ∂k i ∂R ∂R ∂A j > 0 (21)
where note that ∂k j ∂A j > 0, ∂k i ∂R < 0 ∀i = j. We see that an increase in the TFP A j of agent j would increase the equilibrium interest rate and the capital of agent j but decrease the capital of other agents. We can now explore the effect of the individual productivity A j on the aggregate production.

Proposition 2 (effects of productivity shocks -general decompositions). Consider an equilibrium with k j > 0. Let A j vary and assume that the equilibrium outcomes are differentiable functions. We have

∂Y ∂A j = f j (k j ) + A j f j (k j ) ∂k j ∂R ∂R ∂A j + ∂k j ∂A j ≥ 0
Added production of agent j

+ i =j A i f i (k i ) ∂k i ∂R ≤ 0 ∂R ∂A j ≥ 0
Production losses of other agents

(22) ∂Y ∂A j = f j (k j ) + A j f j (k j ) ∂k j ∂A j ≥ 0 Quantity effect + i A i f i (k i ) ∂k i ∂R ≤ 0 ∂R ∂A j ≥ 0 Price effect (23) ∂Y ∂A j = f j (k j ) Productivity effect + i =j A j f j (k j ) -A i f i (k i ) -∂k i ∂R ≥ 0 ∂R ∂A j ≥ 0 Allocation effect . ( 24
)
By consequence, ∂Y /∂A j ≥ 0, ∀j ∈ I, where I = arg max i>n {A i f i (k i )}. The aggregate output increases in A i if the firm i has the highest total marginal factor productivity:

While we directly get ( 22) and ( 23) by taking the derivative of Y with respect to A j , ( 24) is a consequence of (21).

Proposition 2 provides different interpretations of the effects of individual productivity shocks. For instance, when productivity A j increases, the production of agent j increases but that of other agents decrease. This mechanism is represented by ( 22).

However, conditions in Proposition 2 are based on endogenous variables. The following result, based on exogenous parameters, shows the role of credit limit on the effect of productivity change.

Corollary 3. 1. Consider the case of strictly concave technology and let Assumptions 2 and 3 be satisfied. If one of the following conditions a The credit limit of agent any is high, in the sense that

γ i > lim x→∞ xf i (x) f i (x) , ∀i. b γ i < lim x→∞ xf i (x) f i (x) , ∀i, R 1 < R 2 < • • • < R m , and S < m i=1 k n i (R m /A i ).
is satisfied, the equilibrium coincides to that in the economy without frictions, and hence, the equilibrium aggregate output is increasing in each individual productivity A i 2. Consider the linear technology: x) ensures that agent i's borrowing constraint is not binding (see Corollary 1). By consequence, the equilibrium coincides to that in the economy without frictions. Therefore, the output is increasing in each productivity.

F i (k) = A i k with A 1 < • • • < A m . We have Y ≤ Y perf ect ≡ A m i S i . Moreover, Y = Y * if and only if f m A m ≥ A m-1 (1 -Sm S ). The intuition of point 1.a is that condition γ i > lim x→∞ xf i (x) f i (
Under conditions in point 1.b of Corollary 3, Theorem 1 implies that the equilibrium coincides to that in the economy without frictions. Observe that m i=1

k n i (R m /A i ) is increasing in γ m because R i /A i does not depend on A i and k n i (R/A i ) is decreasing in R/A i . So, condition S < m i=1 k n i (R m /A i )
is more likely to be satisfied if the credit limit γ m of the agent m who needs the credit the most (in the sense that R m > R i , ∀i) is quite high, then the credit constraints of this agent and all other ones are not binding. This is consistent with point 2 of Corollary 3.

To better understand 1.b, we look at the case where

F i (k) = A i k α , ∀i, ∀k, with α > γ i ∀i. According to Remark 2, we have R m = αA m S α-1 m (1 -γm α ) 1-α , and hence m i=1 k n i ( R m A i ) = m i=2 ( αA i R m ) 1 1-α = m i=1 ( A i A m S α-1 m (1 -γm α ) 1-α ) 1 1-α = m i=1 ( A i A m ) 1 1-α S m 1 -γm α . So, condition S < m i=1 k n i (R m /A i ) becomes m i=1 S i < m i=1 ( A i Am ) 1 1-α Sm 1-γm α
. This can be satisfied if γ m is high in the sense that it is closed to α. 10 It would be important to explore condition under which ∂Y /∂A j may be negative. First, we focus on the case of strictly concave production functions.

10 For instance, we can take γ i = γ < α, S i = s, ∀i, and

A 1 < • • • < A m . Then R 1 < • • • < R m . Moreover, S < m i=1 k n i (R m /A i ) becomes m(1 -γm α ) < m i=1 ( Ai Am ) 1 1-α , which is satisfied if γ is closed to α.
Proposition 3 (effects of productivity shocks -strictly concave technology). Consider the case of strictly concave technology and let Assumptions 2 and 3 be satisfied. Assume also that R

2 < R 3 < • • • < R m .
1. There exists Ā1 > 0 such that the equilibrium output Y is increasing in A 1 on the interval ( Ā1 , ∞).

Consider the case when

A 1 is small. Denote D 2 = k n 2 ( R 2 A 2 ) + m i=3 k b i ( R 2 γ i A i , S i ), D 3 = 3 i=2 k n i ( R 3 A i ) + m i=3 k b i ( R 3 γ i A i , S i ), . . . D m = m i=2 k n i ( R m A i ) Since R 2 < R 3 < • • • < R m , we have D 2 > D 3 > • • • > D m > 0. (a) If S < D m , then the output is increasing in A 1 when A 1 is small enough. (b) Assume that D n > S > D n+1 (25)
γ i f i (k) kf i (k) < S 1 S 1 + t≥n+1 S t , ∀i = n + 1, . . . , m, ∀k ∈ (0, S) (26)
lim x→+∞ x f 1 (x) < 0 (27)
Then, for any A 1 small enough, we have that ∂Y ∂A 1 < 0.

Proof. See Appendix B.

Proposition 3 shows the role of two important factors: credit limits (γ i ) and productivity A 1 . According to point 1, when the productivity A 1 is high, a positive productivity shock is good for the aggregate output. The intuition behind is that when A 1 is high enough, the marginal productivity A 1 f 1 (k 1 ) is the highest total marginal factor productivity, and hence, decomposition (24) ensures that ∂Y ∂A 1 > 0. Regarding part 2.a of Proposition 3, condition S < D m is non-empty and can be satisfied with a large class of parameter.

11 Observe that D m is increasing in A 2 , . . . , A m-1 but decreasing in A m because R i /A i does not depend on A i and k n i (R/A i ) is decreas- ing in R/A i . Moreover, D m
is increasing in agent m s credit limit γ m . In other words, condition S < D m is likely to be satisfied if γ m is quite high. In such a case, point 2.a ensures that, when A 1 is small enough, the credit constraints of all agents are not binding and hence the aggregate output is increasing in

A i , ∀i ≥ 1. 11 Indeed, let F i (k) = A i k α , ∀i, ∀k, with α > γ i . We have R m = αA m S α-1 m (1 -γm α ) 1-α , and hence D m = m i=2 ( Ai Am ) 1 1-α Sm 1-γm α . When S i = s, γ i = γ, ∀i, and A 2 < • • • < A m , then we have R 2 < • • • < R m . Condition S < D m is equivalent to m(1 -γ α ) < m i=2 ( Ai Am )
1 1-α which can be satisfied.

Let us now look at point 2.b. Condition D n > S > D n+1 ensures that when A 1 is low enough the credit constraint of any agent i ≥ n + 1 is binding while that of any agent i ≤ n is not. Condition (26) means that agents whose credit constraints are binding have a very low credit limit. In such a case, the aggregate output may be decreasing in productivity A 1 when A 1 is small enough. The fact that A 1 is very small ensures that the productivity dispersion is high. This is consistent with condition ( 16) in a two-agent model with linear production functions.

In the case of a two-agent model, we have the following result with more details and intuitive conditions. Proposition 4. Consider a two-agent model.

1. Let Assumptions 2, 3 and 4 be satisfied. Assume also that

k n 2 ( R 2 A 2 ) < S, γ 2 < S 1 S 1 + S 2 Sf 2 (S) f 2 (S) , lim x→+∞ x f 1 (x) < 0
Then, for any A 1 small enough, we have that ∂Y ∂A 1 < 0. 2. By consequence, in a two-agent economy with Cobb-Douglas production functions

(F i (k) = A i k α ) and γ 2 < α S 1 S 1 +S 2 , we have ∂Y ∂A 1 < 0 for A 1 small enough. Proof. See Appendix B.
With linear production functions, we have the following result which is a consequence of Theorem 2 in Appendix A.

Proposition 5 (effects of productivity shocks -linear technology). Assume that

F i (k) = A i k ∀i, ∀k with A 1 < • • • < A m . Assume that A n > max i (γ i A i ) and m i=n+1 AnS i An-γ i A i ≤ S ≤ m i=n AnS i An-γ i A i .
Then, the equilibrium interest rate R * = A n and the aggregate output is

Y = A n n i=1 S i + m i=n+1 A n (1 -γ i ) A n -γ i A i A i S i . ( 28
)
We also have that ∂Y ∂A j > 0, ∀j > n, and

∂Y ∂A n = n i=1 S i - m i=n+1 (1 -γ i )γ i A 2 i S i (A n -γ i A i ) 2 (29)
We can see clearly that ∂Y ∂An may have any sign. Since ∂Y ∂An is increasing in A n , it can be negative when A n is low and positive when A n is high. This is consistent with our insights mentioned above.

General productivity shocks

Assume now that the TFP of all agents depends on an exogenous variable x ∈ R in the sense that A i = A i (x) where A i is a differentiable function of x and A i (x) > 0, ∀i. We wonder how the aggregate output changes when x varies. First, we have that

Y (x) = i A i (x)f i k i (x) = i A i (x)f i k i A i (x), R(A 1 (x), . . . , A m (x)) ∂Y ∂x = i A i (x)f i k i (x) + i A i (x)f i (k i (x))k i (x). ( 30
)
Since the physical capital k i depends on x, we simply write k i = k i (x). We have

k i (x) = ∂k i ∂R ∂R ∂x + ∂k i ∂A i ∂A i ∂x , ∂R ∂x = j ∂R ∂A j ∂A j ∂x Notice that ∂k i ∂A i ≥ 0, ∂k i ∂R ≤ 0, ∂A i ∂x ≥ 0, ∂R ∂x ≥ 0.
So, we can expect that k i (x) may have any sign. However, we have i k i (x) = 0 because i k i = S in equilibrium.

We have two decompositions which help us to understand why the aggregate output may be increasing or decreasing in the exogenous productivity shock x.

∂Y ∂x

= i A i (x)f i k i (x) + i:k i (x)≥0 A i (x)f i (k i (x))k i (x)
Added production of agent j

+ i:k i (x)<0 A i (x)f i (k i (x))k i (x)
Production losses of other agents (31)

∂Y ∂x = i A i (x)f i k i (x) + i A i (x)f i (k i (x)) ∂k i ∂A i > 0 ∂A i ∂x > 0 Quantity effect + i A i (x)f i (k i (x)) ∂k i ∂R < 0 ∂R ∂x > 0 Price effect (32)
When the TFP of all agents changes at the same rate, we have the following result.

Proposition 6 (homogeneous productivity shocks). Consider an equilibrium. Assume that an exogenous shock makes the individual TFP vary from A i to A i (x) = xA i , ∀i, where x > 0. Then, for this new economy, there is an equilibrium where Y (x) = xY , i.e., the aggregate output changes at the same rate.

Proof. Denote (R, (k i , b i )) an equilibrium for the economy E ≡ (A i , f i , γ i , S i ) i=1,...,m with borrowing constraints:

Rb i ≤ γ i A i f i (k i ). We can check that (R(x), (k i , b i )), where R(x) ≡ xR, is an equilibrium for the new economy E(x) ≡ (A i (x), f i , γ i , S i ) i=1,...,m . In equilibrium, the new aggregate output is Y (x) = i A i (x)f i (k i ) = xY.
Next, we consider the case where productivity changes are not proportional. In such a case, we argue that positive productivity shocks may reduce the aggregate output. Indeed, by using Taylor's theorem and Proposition 3, we obtain the following result.

Proposition 7 (asymmetric productivity shocks). Consider an economy which satisfies conditions in case 2.(b) in Proposition 3, and A 1 > 0 small enough. Then, there exist g ∈ (0, 1) and a neighborhood G of (A 1 , . . . , A m ) such that

Y (A 1 , . . . , A m ) -Y (A 1 , . . . , A m ) A 1 -A 1 < 0, (33)
∀(A 1 , . . . , A m ) ∈ G satisfying | A i -A i A 1 -A 1 | < g, ∀j.
Proof. Denote A ≡ (A 1 , . . . , A m ) and A ≡ (A 1 , . . . , A m ). By Taylor's theorem, we have

Y (A ) -Y (A) = ∂Y (A) ∂A 1 (A 1 -A 1 ) + i≥2 ∂Y (A) ∂A i (A i -A i ) + i h i (A, A )(A i -A i )
where lim A →A h i (A, A ) = 0. We can choose < 0, g < 1 and (A i ) such that

A i -A i A 1 -A 1 | < g and ∂Y (A) ∂A 1 + i≥2 ∂Y (A) ∂A i A i -A i A 1 -A 1 < < 0.
In this case, we get (33).

There are two key points that ensure (33). The first condition is ∂Y (A) ∂A 1 < 0, i.e., the output is decreasing in A 1 in a neighborhood of (A 1 , . . . , A m ); notice that this may happen only if A 1 is small enough. Of course, we have

Y (A 1 ,...,A m )-Y (A 1 ,...,Am) A 1 -A 1 > 0 if ∂Y (A) ∂A i > 0, ∀i. The second condition is | A i -A i A 1 -A 1 | < g, i.
e., the productivity does not change at the same rate and that the productivity of the less productive agent (agent 1) increases faster than that of the most productive agents. This implies that agent 1 absorbs more capital than other ones.

Impacts of credit shocks

In this section, we investigate the effects of credit limits (γ t ) on the aggregate production, which help us to understand better the relationship between finance and economic growth. A meaningful question is whether financial development has positive effects on the economic growth. In our model, relaxing credit limit (i.e., increasing γ i ) can be interpreted as reduction of financial friction or improvement of the financial sector.

Consider an equilibrium and an agent j whose borrowing constraint is binding. Then, the equilibrium interest rate R is increasing in γ j . Assume that we have the differentiability. Then,

m i=1 ∂k i ∂R ∂R ∂γ j + ∂k j ∂γ j = 0, which imply that ∂k j ∂R ∂R ∂γ j + ∂k j ∂γ j = - i =j ∂k i ∂R ∂R ∂γ j ≥ 0 ( 34
)
where ∂k j ∂γ j ≥ 0, ∂k i ∂R ≤ 0.

Impacts of individual credit shocks

We firstly look at the effects of credit limits (γ i ) i .

Proposition 8 (effects of credit shocks). Consider an equilibrium. The equilibrium outcomes do not depend on credit limits γ i of agents whose borrowing constraints are not binding. For any agent j whose borrowing constraint is binding, let γ j vary and assume that the equilibrium outcomes are differentiable functions. Then, we have decompositions:

∂Y ∂γ j = F j (k j ) ∂k j ∂R ∂R ∂γ j + ∂k j ∂γ j ≥ 0
Added production of agent j

+ i =j F i (k i ) ∂k i ∂R ≤ 0 ∂R ∂γ j ≥ 0
Production losses of other agents

(35) ∂Y ∂γ j = F j (k j ) ∂k j ∂γ j ≥ 0 Quantity effect + i F i (k i ) ∂k i ∂R ≤ 0 ∂R ∂γ j ≥ 0 Price effect (36) ∂Y ∂γ j = ∂R ∂γ j ≥ 0 i =j F j (k j ) -F i (k i ) -∂k i ∂R ≥ 0 (37)
While we directly get ( 35) and ( 36) by taking the derivative of Y with respect to γ j , condition (37) is a consequence of ( 35) and (34).

In (35), the first term represents the marginal added production of agent j when her credit limit γ j is relaxed while the two last term represent the marginal production loss of other agents. The aggregate output increases in γ j if the marginal added production exceeds the marginal production loss.

Condition (37) represents the redistributive effect of a change in the credit limit γ j . According to (37), we obtain the following result.

Corollary 4. Denote I n = arg max i {F i (k i )}. Thus, we have that ∂Y /∂γ j ≥ 0 ∀j ∈ I n , i.e., the aggregate output is increasing in the credit limit of agents having the highest marginal productivity.

We now provide conditions under which the aggregate output may be decreasing in credit limits.

Proposition 9 (individual credit shock). Assume that F i (k) = A i k ∀i, k. Assume that max i (γ i A i ) < A 1 < • • • < A m .
Consider the case where the equilibrium interest rate is belong to the interval (A n , A n+1 ). Then, we have that:

1. ∂Y ∂γ n+1 < 0 < ∂Y ∂γm if n + 1 < m. 12
2. Consider an entrepreneur i with n + 1 < i < m, we have that:

∂Y ∂γ i > 0 if A i is high enough, i.e., A i -A i-1 A m -A i > m t=i+1 γtAtSt (An-γtAt) 2 i-1 t=n+1 γtAtSt (A n+1 -γtAt) 2 (38a) ∂Y ∂γ i < 0 if A i is low enough, i.e., A i -A n+1 A i+1 -A i < m t=i+1 γtAtSt (A n+1 -γtAt) 2 i-1 t=n+1 γtAtSt (An-γtAt) 2 . ( 38b
)
Proof. See Appendix C.

Condition ∂Y ∂γ n+1 < 0 indicates that an increasing of the credit limit of the least productive producer harms the aggregate output while condition ∂Y ∂γm > 0) has a similar interpretation as in Corollary 4.

According to (38a) and (38b), the aggregate output is more likely to be increasing (resp., decreasing) in the credit limit of an agent if the TFP of this producer is quite close to those of more productive entrepreneurs (resp., that of the least productive entrepreneur) or/and credit limits and initial wealths of more productive agents (γ t ) t>i are low.

We complement our above points by a numerical example.

Example 1. Consider a three-agent economy with linear production functions F i (k) = A i k, ∀i, ∀k, and borrowing constraints are Rb i ≤ γ i A i k i . In Appendix C, we completely compute the equilibrium. Assume now that fundamentals are given by S 1 = 4, S 2 = 4,

S 3 = 3, A 1 = 1, A 2 = 1.2, A 3 = 1.5, γ 1 = 0.2.
First, we set γ 3 = 0.3 and we let γ 2 vary. Figure 2 shows the effects of the agent 2's credit limit γ 2 on the equilibrium interest rate and the aggregate output. When γ 2 varies from 0.15 to 0.45, the interest rate varies from A 1 = 1 to A 2 = 1.2. The aggregate output is not monotonic functions of γ 2 : It is increasing in γ 2 in the regime A 1 where the interest rate R = A 1 , but decreasing in γ 2 in the regime R 1 where the interest rate R = R 1 (consistent with Proposition 9), and then constant in the regime A 1 where R = A 2 .

Figure 2: Non-monotonic effects of credit limit γ 2 .

Second, we set γ 2 = 0.3 and let γ 3 vary. Figure 3 shows the effects of the most productive agent's credit limit γ 3 on the equilibrium interest rate and the aggregate output. The output is increasing in γ 3 (this is consistent with point 1 of Proposition 9).

Figure 3: Monotonic effects of credit limit γ 3 .

Impacts of general financial shocks

Assume that the credit limit of all agents depends on an exogenous variable x ∈ R in the sense that γ i = γ i (x) where γ i is a differentiable function of x and γ i (x) > 0.

We wonder how the aggregate output changes when x varies. We have that

Y (x) = i A i f i k i (x) = i A i f i k i γ i (x), R(γ 1 (x), . . . , γ m (x)) . ( 39
)
Since the physical capital k i depends on x, we have

k i (x) = ∂k i ∂γ i ∂γ i ∂x + ∂k i ∂R ∂R ∂x , ∂R ∂x = j ∂R ∂γ j ∂γ j ∂x (40) Recall that ∂k i ∂γ i ≥ 0, ∂k i ∂R ≤ 0, ∂γ i ∂x ≥ 0, ∂R ∂x ≥ 0 because ∂R ∂x = j ∂R ∂γ j ∂γ j
∂x and ∂R ∂γ j ≥ 0, ∀j. So, we see that k i (x) may have any sign. However, we know i k i (x) = 0 because

i k i = S in equilibrium.
We have two decompositions which help us to understand why the aggregate output may be increasing or decreasing in the exogenous shock x.

∂Y ∂x = i:k i (x)≥0 A i f i (k i (x))k i (x)
Added production of agent j

+ i:k i (x)<0 A i f i (k i (x))k i (x)
Production losses of other agents (41)

∂Y ∂x = i A i f i (k i (x)) ∂k i ∂γ i > 0 ∂γ i ∂x > 0 Quantity effect + i A i f i (k i (x)) ∂k i ∂R < 0 ∂R ∂x > 0 Price effect (42)
We consider the case of homogeneous credit shock.

Proposition 10 (homogeneous credit shock). Assume either F i (k) = A i k, ∀i, ∀k or Assumption 2 is satisfied. Assume also that γ i = γ ∈ (0, 1), ∀i. Then the equilibrium aggregate output is an increasing function of the credit limit γ.

Proof. See Appendix C.

The intuition of the result is simple: all credit-constrained firms, who have higher marginal productivity, can borrow more from other agents who have lower marginal productivity, and hence produce more. This point is consistent with those in in Khan and Thomas (2013) (section VI. C), Midrigan and Xu (2014) (section II.B), [START_REF] Moll | Productivity Losses from Financial Frictions: Can Self-financing Undo Capital Misallocation?[END_REF] (Proposition 1), and [START_REF] Catherine | Aggregate Effects of Collateral Constraints[END_REF].

We now assume that there is an aggregate shock on credit limits under which the new credit limits are (γ i ) i . Our novel point is that, even γ i > γ i ∀i, the new aggregate output Y = Y (γ 1 , . . . , γ m ) may be lower than Y = Y (γ 1 , . . . , γ m ). Formally, we have the following result.

Proposition 11 (general credit shocks). Assume that

F i (k) = A i k ∀i, k, and max i (γ i A i ) < A 1 < • • • < A m . Consider the regime R n in which the equilibrium interest rate is in the interval [A n , A n+1 ).
Consider an agent i such that n + 1 < i < m and assume that condition (38b) holds. Then there exist g ∈ (0, 1) and a neighborhood G of (γ 1 , . . . , γ m ) such that

Y (γ 1 , . . . , γ m ) -Y (γ 1 , . . . , γ m ) γ i -γ i < 0, (43)
∀(γ 1 , . . . , γ m ) ∈ G satisfying | γ j -γ j γ i -γ i | < g, ∀j = i.
We can apply the same argument used in Proposition 7 to prove Proposition 11.

Proposition 11 shows that the aggregate output may be reduced even the credit limits of all agents increase (i.e., γ i > γ i , ∀i). It complements Proposition 10, Proposition 9, and those in [START_REF] Buera | Financial Frictions and the Persistence of History: A Quantitative Exploration[END_REF], [START_REF] Khan | Credit Shocks and Aggregate Fluctuations in an Economy with Production Heterogeneity[END_REF], [START_REF] Midrigan | Finance and Misallocation: Evidence from Plant-Level Data[END_REF], [START_REF] Moll | Productivity Losses from Financial Frictions: Can Self-financing Undo Capital Misallocation?[END_REF], [START_REF] Catherine | Aggregate Effects of Collateral Constraints[END_REF]. Recall that these studies provide conditions under which relaxing credit limits has positive impact on the aggregate output.

Impacts of both productivity and financial shocks

Assume that the TFP and credit limit of all agents depend on an exogenous variable x ∈ R + in the sense that A i = A i (x), and γ i = γ i (x) where A i , γ i is a differentiable function of x. The equilibrium aggregate production is

Y (x) = i A i (x)f i k i A i (x), γ i (x), R A 1 (x), . . . , A m (x), γ 1 (x), . . . , γ m (x) . (44)
We have

∂k i ∂x = ∂k i ∂A i ∂A i ∂x + ∂k i ∂γ i ∂γ i ∂x + ∂k i ∂R ∂R ∂x and ∂R ∂x = i ∂R ∂A i ∂A i ∂x + i ∂R ∂γ i ∂γ i ∂x
If A i (x) ≥ 0 and γ i (x) ≥ 0, then we have ∂R ∂x ≥ 0. Since ∂k i ∂R ≤ 0 (price effect), the capital stock k i can be decreasing or increasing in the exogenous shock x. However, we have i k i (x) = 0 since i k i = i S i is exogenous. By consequence, we have the following decomposition of the effect of the exogenous shock x:

∂Y ∂x = i A i (x)f i k i (x) + i:k i (x)≥0 A i (x)f i (k i (x))k i (x)
Added production of agent j

+ i:k i (x)<0 A i (x)f i (k i (x))k i (x)
Production losses of other agents

. ( 45
)
We can also decompose ∂Y ∂x into quantity and price effects:

∂Y ∂x = A i (x)f i k i (x) + i A i (x)f i (k i (x)) ∂k i ∂A i ≥ 0 ∂A i ∂x ≥ 0 + ∂k i ∂γ i ≥ 0 ∂γ i ∂x ≥ 0 Quantity effect + i A i (x)f i (k i (x)) ∂k i ∂R ≤ 0 ∂R ∂x ≥ 0 Price effect (46)
From these decompositions and our analyeses above, we can see that the aggregate output Y and the aggregate productivity defined by ( 9) may be increasing or decreasing in the shock x, depending on the nature of shocks and the distributions of productivity, credit limit and endowments.

Productivity growth, productivity dispersion and credit constraint

Consider the case F i (k) = A i f (k), ∀i, ∀k. The aggregate productivity T F P is defined by T F P = Y /f (S). Assume that there is a shock that changes productivity from A i to A i and credit limit from γ i to γ i . The new TFP of the economy is T F P = Y /f (S).

We have

T F P T F P = Y (A 1 ,...,A m ) f (S) Y (A 1 ,...,A 2 ) f (S) = Y (A 1 , . . . , A m ) Y (A 1 , . . . , A 2 )
We aim to understand the relationship between the aggregate productivity growth

T F P T F P and individual ones

A 1

A 1 , . . . , A m Am . In the economy without frictions, by using the definition (8) we have that

T F P T F P = max{ i A i f (k i ) : k i ≥ 0, i k i ≤ S} max{ i A i f (k i ) : k i ≥ 0, i k i ≤ S} Observe that min i { A i A i }A i f (k i ) ≤ A i f (k i ) ≤ max i { A i A i }A i f (k i ). So, obtain that min i { A i A i } ≤ T F P T F P ≤ max i { A i A i }.
However, when we consider economies with credit constraints, our above analyses (see Propositions 1,3,7,11) show that the aggregate productivity growth T F P T F P may be less than min i { A i A i }. Indeed, for instance, we can choose (A i ) and (A i ) so that all conditions in Proposition 7 are satisfied and min i {

A i A i } > 1.
In this case, we have Y (A 1 , . . . , A m ) -Y (A 1 , . . . , A m ) < 0, or, equivalently, T F P T F P < 1. We summarize our points in the following result.

Proposition 12 (productivity growth, productivity dispersion and credit constraint). Consider the case F i (k) = A i f (k), ∀i, ∀k. Assume that there is a shock that changes productivity from A i to A i and credit limit from γ i to γ i .

1. In the economy without frictions, we always have

min i { A i A i } ≤ T F P T F P ≤ max i { A i A i } (47)
2. Consider economies with credit constraints E ≡ (A i , f i , γ i , S i ) i=1,...,m .

(a) If

A i

A i = g > 0, ∀i, then Proposition 6 implies that T F P T F P = g. (b) However, under some situations as in Propositions 1, 3, 7, 11, we may have that

T F P T F P < min i { A i A i }. ( 48
)
By consequence, the aggregate productivity growth rate may be far from that of most productive firms. It may be even lower than the smallest productivity growth rate of firms. Our points contribute to the debate concerning the slowdown in aggregate productivity growth. For instance, by using data in 23 OECD countries over the 2000s, [START_REF] Andrews | Frontier Firms, Technology Diffusion and Public Policy: Micro Evidence from OECD Countries[END_REF] document a slowdown in aggregate productivity growth, a rising productivity gap between the global frontier and other firms, and that productivity growth at the global frontier remained robust.

As recognized by [START_REF] Goldin | Why is productivity slowing down?[END_REF], there is no single reason for the slowdown in aggregate productivity growth. We provides a supplyside point of view by using a general equilibrium model with credit constraint. Our above analyses suggest that the interplay between credit constraints, high heterogeneity of productivity, asymmetry of productivity and financial shocks may generate a slowdown in aggregate productivity growth.

Extension: Models à la Ramsey

We now consider an infinite-horizon model with credit constraints. Agent i maximizes her intertemporal utility subject to budget and borrowing constraints max

(c i ,k i ,b i ) ∞ t=0 β t i u i (c i,t ) (49a)
subject to:

c i,t + k i,t + R t b i,t-1 ≤ F i,t (k i,t-1 ) + b i,t (49b) R t+1 b i,t -γ i F i,t (k i,t ) ≤ 0 (49c)
where we assume that b i,-1 = 0 ∀i and denote w i,0 = F i,0 (k i,-1 ).

We make standard assumptions.

Assumption 5. u i is concave, continuously differentiable, u i (0) = +∞.

Definition 3. An equilibrium is a list ((c i,t , k i,t , b i,t ) i , R t ) t≥0 such that (1) given (R t ), the allocation (c i,t , k i,t , b i,t
) is a solution of the above maximization problem, and (2) markets clear:

i b i,t = 0, i (c i,t + k i,t ) = i F i,t (k t-1
), ∀t.13 Remark 3 (steady state analysis). Let Assumptions 1, 5 be satisfied. Assume that F i,t = F i , i.e., does not depend on time. Consider a steady state equilibrium with k i > 0, ∀i.

1. The steady state interest rate is R = 1/ max i {β i }.

Assume, in addition, that

β 1 > β i , ∀i ≥ 2. Then A 1 f 1 (k 1 ) = R = 1/β 1 , agent
1's borrowing constraint is not binding, and for any i ≥ 2,

R -γ i A i f i (k i ) R = A i f i (k i )(1 -γ i )
Hence, k i is increasing in A i . Since Rβ i ≤ 1, the value k i is increasing in credit limit γ i . By consequence, the steady state output Y = i A i f i (k i ) is increasing in TFP A i and credit limit γ i for any i.

Proof. See Appendix D.

According to this observation, the non-monotonicity of the aggregate output only appears in the transitional dynamics of the economy. Therefore, we will focus on transitional dynamics.

Impact of productivity along transitional dynamics

In general, it is difficult to comparing intertemporal equilibrium in infinite-horizon models. For the sake of tractability, we assume that there are two agents with utility function u i (c) = ln(c) and the linear production function F i,t (k) = A i,t k with A 1,t < A 2,t , ∀t. In some cases, we can explicitly compute the equilibrium interest rate and aggregate output.

Lemma 4. Consider an infinite-horizon two-agent model with utility function u i (c) = ln(c) ∀c, ∀t, ∀i = 1, 2 and production functions F i,t (k) = A i,t k, ∀k, ∀t, ∀i = 1, 2. Assume that γ 2 A 2,t < A 1,t < A 2,t ∀t, and

β t 1 s 1,0 -β t 2 γ 2 A 2,t+1 A 1,t+1 -γ 2 A 2,t+1 (1 -γ 2 )A 2,t A 1,t -γ 2 A 2,t • • • (1 -γ 2 )A 2,1 A 1,1 -γ 2 A 2,1 s 2,0 > 0∀t ≥ 0.
where s i,0 = β i w i,0 . Then there exists an equilibrium with R t = A 1,t ∀t. At equilibrium, the aggregate output at date t, (t ≥ 1), is

Y t =β t-1 1 A 1,t A 1,t-1 • • • A 1,1 s 1,0 (50)
+ β t-1 2 (1 -γ 2 ) t A 2,t A 1,t A 1,t -γ 2 A 2,t A 2,t-1 A 1,t-1 A 1,t-1 -γ 2 A 2,t-1 • • • A 2,1 A 1,1 A 1,1 -γ 2 A 2,1 s 2,0
Proof. See Appendix D.

Lemma 4 allows us to investigate the impacts of productivity shocks. First, we look at the effects of individual productivity growth.

Proposition 13 (impact of individual productivity). Assume that there are two agents having production functions F i (k) = A i k ∀i with A 1 < A 2 , and utility function u i (c) = ln(c) ∀i. Assume that

A 1 -γ 2 A 2 > 0, β 1 > (1 -γ 2 )A 2 A 1 -γ 2 A 2 β 2 , s 1,0 - γ 2 A 2 A 1 -γ 2 A 2 s 2,0 > 0. (51)
Then, there is an equilibrium with the interest rate R t = A 1 ∀t. In this equilibrium, we have that:

1. If s 1,0 s 2,0 (A 1 -γ 2 A 2 ) 2 (1-γ 2 )γ 2 A 2 2 > 1, then ∂Yt ∂A 1 > 0 ∀t. 2. If s 1,0 s 2,0 (A 1 -γ 2 A 2 ) 2 (1-γ 2 )γ 2 A 2 2
< 1, then there exists a time t 0 such that ∂Yt ∂A 1 < 0 ∀t ≤ t 0 and ∂Yt ∂A 1 ≥ 0 ∀t > t 0 . Moreover, the time t 0 is decreasing in the productivity gap

A 1 /A 2 .
Moreover, we have that the output ratio Y 1,t Y 2,t → ∞, the share of net worth of the productive agent W 2,t Yt → 0, and the growth rate

G t ≡ Y t+1 Yt → β 1 A 1 . Proof. See Appendix D.
Condition (51) ensures that the less productive agent still produces, i.e., k 1,t > 0 ∀t. This happens if its TFP A 1 is not too low and the rate of time preference β 1 is high enough. Notice that these conditions imply that β 1 A 1 > β 2 A 2 . This ensures that agent 1 still produces and the growth rate Y t+1

Yt converges to β 1 A 1 . Proposition 13 allows us to understand the impact of a shock on the TFP of the less productive agent. Observe that, if A 1 increases, then the output will increase in the long run. However, point 2 of Proposition 13 indicates that, if A 1 increases but it is still low, the output may decrease in the short run and then increase in the long run.

Example 2. To complement our theoretical finding in Proposition 13, we run a simulation in a two-agent model with linear production function F i (k) = A i k, and s 1,0 = 200, s 2,0 = 100, β 1 = 0.99, β 2 = 0.4, A 1 = 1.5, A 2 = 2.25. The credit limit of agent 2 is γ 2 = 0.4. Let us denote Y t (A 1 ) the equilibrium aggregate output of the economy when the productivity of agent 1 is A 1 . The following graphics show how the difference between Y t (A 1 + h) -Y t (A 1 ) changes over time, where h is a productivity change.

First, when the productivity of agent 1 increases from 1.5 to 1.53 (a small productivity shock), the output goes down and then goes up. Precisely, Y t (1.5+0.03)-Y t (1.5) < 0 for t = 1, 2, 3, 4 and then Y t (1.5 + 0.03) -Y t (1.5) > 0, ∀t ≥ 5.

Second, when there is a big productivity shock so that the productivity of agent 1 increases from 1.5 to 1.95, the output goes up at any period: Y t (1.5+0.45)-Y t (1.5) > 0, ∀t ≥ 1. This is consistent with Proposition 1.

Impact of temporary productivity shocks

Let us look at the impact of temporary productivity shock. Assume that there is a productivity shock only at date 1, which affects the TFP of agent 1: A 1,1 . We would like to understand how the aggregate output changes when A 1,1 varies. According to Lemma 4, we can compute

∂Y t ∂A 1,1 = β t-1 1 A 1,t • • • A 1,2 s 1,0 -β t-1 2 (1 -γ 2 ) t A 2,t A 1,t A 1,t -γ 2 A 2,t • • • A 2,2 A 1,2 A 1,2 -γ 2 A 2,2 γ 2 A 2 2,1 (A 1,1 -γ 2 A 2,1 ) 2 s 2,0 (A 1,t • • • A 1,2 ) -1 ∂Y t ∂A 1,1 = β t-1 1 s 1,0 -β t-1 2 (1 -γ 2 )A 2,t A 1,t -γ 2 A 2,t • • • (1 -γ 2 )A 2,1 A 1,1 -γ 2 A 2,1 γ 2 A 2,1 A 1,1 -γ 2 A 2,1 s 2,0
Corollary 5 (impact of temporary productivity shocks). Under Assumptions in Lemma 4, there exists an equilibrium with R t = A 1,t ∀t. At equilibrium, we have that

∂Y t ∂A 1,1 > 0 ⇔ β t-1 1 s 1,0 -β t-1 2 (1 -γ 2 )A 2,t A 1,t -γ 2 A 2,t • • • (1 -γ 2 )A 2,1 A 1,1 -γ 2 A 2,1 γ 2 A 2,1 A 1,1 -γ 2 A 2,1 s 2,0 > 0. ( 52
)
To illustrate our finding, let us focus on a particular case where A 1,t = A 1 ∀t ≥ 2 and A 2,t = A 2 ∀t ≥ 1. In this case, we have that, for t ≥ 2,

∂Y t ∂A 1,1 > 0 ⇔ β t-1 1 s 1,0 -β t-1 2 (1 -γ 2 )A 2 A 1 -γ 2 A 2 t-1 (1 -γ 2 )γ 2 A 2 2 (A 1,1 -γ 2 A 2 ) 2 s 2,0 > 0. ( 53
)
By consequence, ∂Yt ∂A 1,1 > 0 if β 1 /β 2 , s 1,0 /s 2,0 , A 11 are high enough, A 2 /A 1 is low.

Impact of credit limits along transitional dynamics

In this section, we aim to show the effects of credit limits on the aggregate output in intertemporal equilibrium. The following result provides a characterization of intertemporal equilibrium in a tractable case.

Lemma 5. Consider a two-agent economy with linear production functions

F i (k) = A i k, ∀i = 1, 2, where γ 2 A 2 < A 1 . Assume that β 2 > β 1 , A 1 < γ 2 A 2 (1 + s 2,0 s 1,0 ) < A 2 (54)
where s i,0 ≡ β i w i,0 and w i,0 is the endowment of agent i at the initial date.

Then there exists an equilibrium with the interest rates

R 1 = γ 2 A 2 (1 + s 2,0 s 1,0 ) ∈ (A 1 , A 2 ) (55a) R t = A 2 , ∀t ≥ 2. ( 55b
)
Denote s i,t ≡ k i,t -b i,t the individual saving of agent i at date t. In such an equilibrium, we have Agent 1:

s 1,0 = β 1 w 1,0 , s 1,1 = β 1 γ 2 A 2 (s 1,0 + s 1,0 ), s 1,t = (β 1 A 2 ) t-1 s 1,1 , ∀t ≥ 1 k 1,t = 0, ∀t ≥ 0 Agent 2: s 2,0 = β 2 w 2,0 , s 2,1 = β 2 (1 -γ 2 )A 2 (s 1,0 + s 1,0 ), s 2,t = (β 2 A 2 ) t-1 s 2,1 , ∀t ≥ 1 k 2,0 = s 2,0 + s 1,0 k 2,t = s 2,t + s 1,t = A t 2 β t 1 γ 2 + β t 2 (1 -γ 2 ) (s 1,0 + s 2,0 )∀t ≥ 1.
Proof. See Appendix D.

In such an equilibrium, only the most productive agent produces. Notice that her borrowing constraint at date 1 is binding but her borrowing constraints from date 2 on are not. From date 2 on, the equilibrium interest rate equals the productivity of the most productive agent: R t = A 2 , ∀t ≥ 1. However, interest rate between the initial date and date 1 is different from the productivity of agent 2 because the credit limit γ 2 is not so high (in the sens that γ 2 < s 1,0 s 1,0 +s 2,0 ) and the productivity gap is high (in the sense that A 1 A 2 < γ 2 (1 + s 2,0 s 1,0 ); see condition ( 54)). Notice that Kiyotaki (1998)'s Section 2 only focuses on the case where the equilibrium interest rate equals the rate of return on investment of unproductive agents, i.e., R t = A 1 , ∀t.

Thanks to Lemma 5, we can observe the effect of the credit limit.

Proposition 14. Under assumptions in Lemma 5, the aggregate output is given by

Y 1 = A 2 k 2,0 = A 2 R 1 R 1 -γ 2 A 2 s 2,0 = A 2 (s 1,0 + s 2,0 ) ∀t ≥ 2, Y t = A 2 k 2,t-1 = A t 2 β t-1 1 γ 2 + β t-1 2 (1 -γ 2 ) (s 1,0 + s 2,0 )
The aggregate output at date 1 does not depend on the credit limit γ 2 of the most productive agent and

∂Y t ∂γ 2 < 0, ∀t ≥ 2. ( 56
)
It means that the aggregate output from date 2 on decreases when the most productive agent's credit limit increases. This is different from the standard view on the effects of financial constraints as shown in [START_REF] Buera | Financial Frictions and the Persistence of History: A Quantitative Exploration[END_REF], [START_REF] Khan | Credit Shocks and Aggregate Fluctuations in an Economy with Production Heterogeneity[END_REF], [START_REF] Midrigan | Finance and Misallocation: Evidence from Plant-Level Data[END_REF], [START_REF] Moll | Productivity Losses from Financial Frictions: Can Self-financing Undo Capital Misallocation?[END_REF], [START_REF] Catherine | Aggregate Effects of Collateral Constraints[END_REF].

Let us explain the insight of this finding. Denote W i,t the net worth of agent i at date t. We have W i,t = F i (k i,t-1 ) -R t b i,t-1 . In equilibrium in Proposition 14, the net worth of agent 2 at date 1 is

W 2,1 = A 2 k 2,0 -R 1 b 2,0 = (1 -γ 2 )(s 1,0 + s 2,0 ). W 2,t = A 2 k 2,t-1 -R 1 b 2,t-1 = (1 -γ 2 )A 2 k 2,t-1 = (1 -γ 2 )A t 2 β t-1 1 γ 2 + β t-1 2 (1 -γ 2 ) (s 1,0 + s 2,0 )
These are decreasing in the credit limit γ 2 . The reason behind is that when γ 2 goes up, the interest rate R 1 increases which makes the repayment R 1 b 2,0 increases. However, the capital k 2,0 of agent 2 is already the sum of aggregate saving s 1,0 + s 2,0 which can no longer increase. By consequence, the net worth

W 2,1 = A 2 k 2,0 -R 1 b 2,0 decreases. 14
This makes the saving of agent 2 go down, and, hence, the output decreases. The mechanism can be summarized by the following schema:

Credit limit γ 2 ↑ ⇒ Interest rate ↑ ⇒ Agent 2's net worth ↓ ⇒ ⇒ Saving ↓ ⇒ Production ↓ ⇒ • • • (57)
However, this mechanism does not happen when the credit limit γ 2 of agent 2 is high enough.

Remark 4. In equilibrium without borrowing constraints, we have R t = A 2 , ∀t, and

s i,0 = β i w i,0 , ∀i s i,t = β i A 2 s i,t-1 = (β i A 2 ) t s i,0 , ∀t ≥ 1, ∀i = 1, 2 k 1,t = 0, b 1,t = -s 1,t k 2,t = s 1,t + s 2,t = A t 2 (β t 1 s 1,0 + β t 2 s 2,0 ) Y perf ect t = A 2 k 2,t-1 = A t 2 (β t-1 1 s 1,0 + β t-1 2 s 2,0 ) Since γ 2 (1 + s 2,0 s 1,0 ) < 1, we can verify that, for all t ≥ 2, Y t = A t 2 β t-1 1 γ 2 + β t-1 2 (1 -γ 2 ) (s 1,0 + s 2,0 ) < A t 2 (β t-1 1 s 1,0 + β t-1 2 s 2,0 ) = Y perf ect t .

A three-agent model

Lemma 6. Consider an economy with three agents having linear production functions

F i (k) = A i k and collateral constraint R t b i,t-1 ≤ γ i A i k i,t-1 . Assume that, β 3 ≥ max(β 1 , β 2 ), γ 2 A 2 < γ 3 A 3 < A 1 < A 2 < A 3 , and γ 2 1 -γ 2 s 2,0 + γ 3 A 3 A 2 -γ 3 A 3 s 3,0 < s 1,0 < γ 2 A 2 A 1 -γ 2 A 2 s 2,0 + γ 3 A 3 A 1 -γ 3 A 3 s 3,0 (β 1 γ 2 + β 2 (1 -γ 2 ))A 2 s 2,0 ≤ (β 3 -β 1 )γ 3 A 3 s 3,0
where s i,0 ≡ β i w i,0 and w i,0 is the endowment of agent i at the initial date.

Then there exists an equilibrium with the interest rates

R t = A 3 , ∀t ≥ 2, (58a) R 1 ∈ (A 1 , A 2 ) is determined by s 1,0 = γ 2 A 2 R 1 -γ 2 A 2 s 2,0 + γ 3 A 3 R 1 -γ 3 A 3 s 3,0 . (58b)
In such an equilibrium, the individual saving s i,t ≡ k i,t -b i,t is given by Agent 1:

s 1,0 = β 1 w 1,0 , s 1,t = β 1 R t s 1,t-1 ∀t ≥ 1, s 1,t = β t 1 R t • • • R 1 s 1,0
Agents 2 and 3:

s i,0 = β i w i,0 , ∀i; s i,1 = β i (1 -γ i )A i R 1 R 1 -γ i A i s i,0 , ∀i = 2, 3 s i,t = β i A 3 s i,t-1 = (β i A 3 ) t-1 s i,1 , ∀t ≥ 2, ∀i = 1, 2, 3.
The output is given by

Y 1 = i A i k i,0 = A 2 R 1 R 1 -γ 2 A 2 s 2,0 + A 3 R 1 R 1 -γ 3 A 3 s 3,0 Y 2 = A t-1 3 β t-1 1 R 1 s 1,0 + β t-1 2 (1 -γ 2 ) A 2 R 1 R 1 -γ 2 A 2 s 2,0 + β t-1 3 (1 -γ 3 ) A 3 R 1 R 1 -γ 3 A 3 s 3,0 .
Proof. See Appendix D.

In such an equilibrium, the most productive agent (agent 2) produces at any period while the least productive one never produces. At the period 1, there are two producers (agent 2 and agent 3) thanks to condition (58a). However, from period 2 on, only agent 3 is producer.

Let us focus on the effects of the credit limits on the aggregate production. According to the formula of the output and equation (58b), we can compute the derivative of the output and obtain the following result.

Proposition 15. Let conditions in Lemma 6 be satisfied. Then, we have that

∂Y 1 ∂γ 3 = ∂R 1 ∂γ 3 (A 3 -A 2 ) γ 2 A 2 (R 1 -γ 2 A 2 ) 2 s 3,0 > 0 (59a) ∂Y 1 ∂γ 2 = ∂R 1 ∂γ 2 (A 2 -A 3 ) γ 3 A 3 (R 1 -γ 3 A 3 ) 2 s 3,0 < 0 (59b)
For any period t ≥ 2, we have that

∂Y t ∂γ 3 = ∂R 1 ∂γ 3 A t-1 3 β t-1 1 s 1,0 + β t-1 3 (1 -γ 3 )A 3 -β t-1 2 (1 -γ 2 )A 2 γ 2 A 2 s 2,0 (R 1 -γ 2 A 2 ) 2 -A t-1 3 β t-1 3 A 3 R 1 R 1 -γ 3 A 3 s 3,0 (60a) ∂Y t ∂γ 2 = ∂R 1 ∂γ 2 A t-1 3 β t-1 1 s 1,0 + β t-1 2 (1 -γ 2 )A 2 -β t-1 3 (1 -γ 3 )A 3 γ 3 A 3 s 3,0 (R 1 -γ 3 A 3 ) 2 -A t-1 3 β t-1 2 A 2 R 1 R 1 -γ 2 A 2 s 2,0 (60b)
Since γ 2 < γ 3 , the output at period 1 is increasing in γ 3 (the credit limit of the most productive producer) but decreasing in γ 2 (the credit limit γ 2 of the least productive producer). This is consistent with Corollary 4 or point 1 in Proposition 9.

In the long run (when t is high enough), we have

β t-1 3 (1-γ 3 )A 3 -β t-1 2 (1-γ 2 )A 2 > 0 because β 3 > β 2 .
In such an equilibrium, when the credit limit of agent 2 increases, the aggregate output may decrease in any period.

Example 3. We illustrate our insight by the following simulation. In this simulation, we set that

β 1 = 0.2, β 2 = 0.2, β 3 = 0.95, s 1,0 = 4 = β 1 w 1,0 , s 2,0 = 4 = β 2 w 2,0 , s 3,0 = 3 = β 3 w 3,0 , γ 1 = 0.2, γ 3 = 0.3. Productivity: A 1 = 1, A 2 = 1.2, A 3 = 1.5.
We draw the output path for two cases: γ 2 = 0.3 and γ 2 = 0.35. We observe that

Y t (γ 2 = 0.35) < Y t (γ 2 = 0.30), ∀t ≥ 1.

Conclusion

We have build general equilibrium models with borrowing constraints to explain why the aggregate output may be decreasing (increasing, respectively) when the productivity or credit limit of producers increases (decreases, respectively). A positive homogeneous (productivity or financial) shock has a positive impact on the aggregate output. However, positive, asymmetric shocks may reduce the aggregate production. We have pointed out that both financial frictions and the productivity gap (or dispersion of productivity distribution) matter for this phenomenon.

A Proofs for Sections 2

Proof of Lemma 3. Since F i (0) = ∞, we have k i > 0 at optimum. The Lagrange function is

L = F i (k i ) -Rb i + λ i (S i + b i -k i ) + µ i (γ i F i (k i ) -Rb i )
It is easy to see that (k i , b i ) is a solution if and only if there exists (λ i , µ i ) such that

[k] : (1 + µ i γ i )F i (k i ) = λ i [a] : (1 + µ i )R = λ i , µ i ≥ 0, and µ i (γ i F i (k i ) -R i b i ) = 0.
These equations imply that:

A i f i (k i ) = F i (k i ) = R 1 + µ i 1 + γ i µ i ≥ R. (A.1)
Since F i is decreasing, we have k i ≤ k n i (R/A i ). We consider two cases. Case 1: The credit constraint is binding: γF i (k i ) = Rb i . In this case, (k i , b i ) is the solutions of the following equations:

b i = k i -S i (A.2) γF i (k i ) = R(k i -S i ), i.e., γ i R = k i F i (k i ) - S i F i (k i ) . (A.3)
Consider the function k/F i (k). Its derivative equals

F i (k)-kF i (k) (F i (k)) 2 which is non-negative because F is concave. So, the function G i (k) ≡ k-S i F i (k) is strictly increasing in k. More- over, lim k→0 G i (k) < γ i /R and G i (∞) > γ i /R (because F i (∞) < 1)
. Therefore, there exists a unique solution k i of equation (A.3), and this is positive. It is actually

k b i . We now investigate condition k i ≤ k n i . Since G i (k i ) = γ i /R, condition k i ≤ k n i is equivalent to G i (k n i ) ≥ γ i /R (because G i (k i ) = γ i /R) or, equivalently, R k n i (R/A i )-S i F i (k n i (R/A i )) ≥ γ i .
Conversely, assume that R

k n i (R/A i )-S i F i (k n i (R/A i )) ≥ γ i . We choose k i = k b i . Then, by definition of k b i , we have k i ∈ (S i , ∞). Therefore, we have R > R(1 - S i k i ) = γ i F i (k i ) k i ≥ γ i F i (k i )
where the last inequality follows the fact that

F i is concave. It means that R > γ i F i (k i ). So, we can define µ i , λ i by 1 - F i (k i ) R = µ i F i (k i ) R -γ i , λ i = R(1 + µ i ).
Therefore, (λ i , µ i ) and

(k i , b i ) satisfy conditions [k] and [b] above. It means that (k i , b i ) is a solution. Case 2: γ i F i (k i ) > Rb i .
In this case, we have µ i = 0, and hence

F i (k i ) = R, i.e, k i = k n i .
It remains to check that this value of k i satisfies the condition:

γ i F i (k i ) > Rb i = R(S i -k i ), i.e., γ i /R > G i (k n i ). Observe that if RG i (k n i ) < (≥)γ i , then G i (k n i ) < (≥)γ i /R = G i (k b i ), which implies that k n i < (≥)k b i .
The converse is easy. Notice that, in this case, agent borrows (i.e., b i > 0) if and only if k i > S or equivalently k n i > S. This means that her wealth is low and/or interest rate is low and/or her productivity is high.

A.1 Proof of Theorem 1 (economy with strictly concave technologies)

To simplify notations, we write

k n i (R) and k b i (R) instead of k n i ( R A i ) and k b i ( R γ i A i , S i ) (see Definition 2).
We also introduce the so-called aggregate capital demand function:

B n (R) ≡ n i=1 k n i (R) + m i=n+1 k b i (R) if n ≤ m -1 m i=1 k n i (R) if n = m. Lemma 7. B n (R n ) > B n+1 (R n+1 ) = B n (R n+1 ).
Proof. Indeed, since R n < R n+1 , we notice that

B n (R n ) ≡ n i=1 k n i (R n ) + m i=n+1 k b i (R n ) > B n (R n+1 ) = n i=1 k n i (R n+1 ) + m i=n+1 k b i (R n+1 ) = n+1 i=1 k n i (R n+1 ) + m i=n+2 k b i (R n+1 )
where the last equality follows

k b n+1 (R n+1 ) = k n n+1 (R n+1 ). Therefore, B n (R n ) > B n+1 (R n+1 ) = B n (R n+1 ) ∀n.
We state an intermediate step whose proof is based on Lemma 3 and Corollary 2.

Lemma 8. Let assumptions in Theorem 1 be satisfied. Consider an equilibrium ((k i , b i ) i , R) and an index n ∈ {1, . . . , m -1}.

1. If R > R m , Lemma 3 implies that credit constraint of any agent is not binding.

So, the equilibrium coincides to that of the economy without credit constraints. Therefore, we have

R = R * > R m . 2. If R > R n , then credit constraint of any agent i ≤ n is not binding. Hence k i = k n i (R) < k n i (R n ) ∀i ≤ n. Condition R > R n also implies that k b i (R) < k b i (R n ). Therefore, we have i S i < B n (R n ).
3. If R ≤ R n+1 , then credit constraint of any agent i ≥ n + 1 is binding, and hence

k i = k b i (R) ≥ k b i (R n+1 ) ∀i ≥ n + 1. Moreover, we have k i ≥ k n i (R) ≥ k n i (R n+1 ). Therefore, we have i S i ≥ B n (R n+1 ).
We now prove Theorem 1. Let us consider an equilibrium. Since there is at least one agent whose credit constraint is not binding, we have R > R 1 .

Step 1. Suppose that R ∈ (R n , R n+1 ]. So, credit constraint of any agent i ≥ n + 1 is binding and that of any agent i ≤ n is not binding. Hence, the capital demand is

i k i = n i=1 k n i (R) + m i=n+1 k b i (R). (A.4)
Therefore, the equilibrium interest rate is determined by

n i=1 k n i (R) + m i=n+1 k b i (R) = S ≡ i S i . (A.5)
The left-hand side is decreasing in r, and hence this equation has a unique solution.

Since R ∈ (R n , R n+1 ], we have

n i=1 k n i (R n ) + m i=n+1 k b i (R n ) > i S i ≥ n i=1 k n i (R n+1 ) + m i=n+1 k b i (R n+1 ).
Conversely, if this condition holds, by using properties of functions

k b i , k n i , we can easily prove that R ∈ (R n , R n+1 ]. Indeed, if R > R n+1 , then point 2 of Lemma 8 implies that S < B n+1 (R n+1 ). This contradicts to S ≥ B n+1 (R n+1 ). If R ≤ R n , then point 3 of Lemma 8 implies that S ≥ B n-1 (R n ) = B n (R n ). This contradicts to S < B n (R n ). Therefore, we obtain R ∈ (R n , R n+1 ].
Step 2. We now suppose that R * > R m . We will prove that credit constraint of any agent is not binding. Suppose that

B = {i ∈ {1, . . . , m} : agent i's borrowing constraint is binding} = ∅. Let n : 1 ≤ n ≤ m -1 be the highest element in B, i.e., credit constraint of any agent i ≥ n + 1 is binding while that of any agent i ≤ n is not. We have R ∈ (R n , R n+1 ]. So, k b i (R) ≥ k b i (R n + 1) > k(R m ) and k n i (R) ≥ k n i (R n+1 ) ≥ k n i (R m ). Hence, we get that i S i = n i=1 k n i (R) + m i=n+1 k b i (R) ≥ m i=1 k n i (R m ). (A.6)
However, by definition of R * , we have

i S i = m i=1 k n i (R * ) < m i=1 k n i (R m ). (A.7)
This is a contradiction.

Step 3. We now prove that R n ≤ R * ∀n ≤ m -1. Indeed, in the regime R n , for any i ≥ n + 1, agent i's credit constraint is binding. Hence, Lemma 3 follows that

k b i (R n ) ≤ k n i (R n ) ∀i ≥ n + 1. Consequently, we get that m i=1 k n i (R * ) = S = n i=1 k n i (R n ) + m i=n+1 k b i (R n ) ≤ m i=1 k n i (R n ) which implies that R * ≥ R n .

A.2 Characterization of general equilibrium with linear technologies

Denote

D n ≡ m i=n A n S i A n -γ i A i ∀n ≥ 1, B n ≡ m i=n+1 A n S i A n -γ i A i ∀n ≥ 1. (A.8)
where by convention, m i=n x i = 0 if n > m. By definition (A.8), we observe that

S m 1 -γ m = D m < • • • < D n+1 < B n < D n < B n-1 < • • • < B 1 = m i=2 A 1 S i A 1 -γ i A i . (A.9)
Denote R L n the greatest solution of the following equation:

15 m i=n+1 γ i A i R -γ i A i S i Asset demand = n i=1 S i Asset supply or equivalently m i=n+1 RS i R -γ i A i Capital demand = S Capital supply (A.10)
Theorem 2 (characterization of general equilibrium with linear technologies). Assume that

F i (K) = A i K ∀i and A 1 < • • • < A m .
Then, there exists a unique equilibrium.

The equilibrium interest rate is determined by the following:

R = A i in the regime A i . R L i in the regime R i . (A.11)
where 1. the regime A n (with n ∈ {1, . . . , m}) is the set of all economies satisfying

A n > max i (γ i A i ) and B n ≤ S ≤ D n 2. the regime R n (with n ∈ {1, . . . , m -1}) is the set of all economies satisfying (a) either max i (γ i A i ) < A n < R L n < A n+1 (or equivalently max i (γ i A i ) < A n and D n+1 < S < B n )
15 It should be noticed that the function f (x) ≡ m i=n+1 xSi x-γiAi is not continuous at point γ i A i with i ≥ n + 1. However, it is continuous and decreasing in the interval (max i≥n+1 (γ i A i ), ∞). Then, the equation f (x) = S has a unique solution, r n , in such interval.

(b) or

A n ≤ max i (γ i A i ) < R L n < A n+1 (or equivalently A n ≤ max i (γ i A i ) < R L n and D n+1 < S).
Remark 5. In Theorem 2, we assume that A 1 < • • • < A m . However, we can characterize the set of equilibria in the general case where some agents have the same productivity. Indeed, without lost of generality, we can (1) rank that A i ≤ A i+1 , ∀i, and assume that (2) the set {A i : i ∈ {1, . . . , m}} has the cardinal p, p ≤ m and its distinct values are (A it ) p t=1 , where

A 1 = A i 1 < A i 2 < • • • < A ip = A m . We can decompose that A 1 , A 2 , . . . , A m = A 1 , . . . , A 1 i 1 times , A i 1 +1 , . . . , A i 1 +i 2 i 2 times , . . . , A i 1 +•••+i p-1 , . . . , A m im times Let us denote A t ≡ A it , S t ≡ i:A i =A i t S i .
Then, we can use the same argument in Theorem 2 (but we replace m by p, A i by A i , S i by S i ) to determine the unique equilibrium interest rate. However, there may be multiple equilibrium allocations when one of the sets {i :

A i = A i 1 }, . . . , {i : A i = A ip } has multiple elements. A.2.1 Proof of Theorem 2
Theorem 2 is a direct consequence of the existence of equilibrium and Lemmas 9-13 below. First, the following result is a direct consequence of Lemma 2.

Lemma 9. Assume that A 1 < A 2 < • • • < A m . If max i (γ i A i ) ≥ A n and there exists an equilibrium, then R > A n .
By comparing B n , D n with the aggregate capital supply S ≡ m i=1 S i , we obtain the following result.

Lemma 10. Assume that

A 1 < A 2 < • • • < A m . Denote S ≡ m i=1 S i the aggregate capital. Consider an equilibrium. 1. If A n > max i (γ i A i ) and R > A n , then B n > S. Consequently, if A n > max i (γ i A i ) and B n ≤ S, then R ≤ A n . 2. If A n > max i (γ i A i ) and R < A n , then S > D n . Consequently, if A n > max i (γ i A i ) and S ≤ D n , then R ≥ A n .
Proof.

1. Since R > A i for any i = 1, . . . , n, Lemma 2 implies that k i = 0, a i = -S i ∀i = 1, . . . , n. Hence, we have, by using market clearing condition,

n i=1 S i = - n i=1 a i = m i=n+1 a i ≤ m i=n+1 γ i A i R -γ i A i S i < m i=n+1 γ i A i A n -γ i A i S i (A.12)
where the first inequality follows b

i ≤ γ i A i S i R-γ i A i while the last inequality follows R > A n > max i (γ i A i ) and the fact that the function F unc(R) m i=n+1 γ i A i R-γ i A i S i is decreasing in (max i (γ i A i ), +∞).
Notice that this function is not decreasing in the interval (0, ∞).

Since

R < A n , again Lemma 2 implies that k i = R R-γ i A i S i and a i = γ i A i R-γ i A i S i ∀i ≥ n. We have m i=n A n S i A n -γ i A i S < m i=n RS i R -γ i A i = m i=n k i ≤ m i=1 S i = S (A.13)
where the first inequality follows

A n > R > max i (γ i A i ). Lemma 11. R = A n if and only if A n > max i (γ i A i ) and B n ≤ S ≤ D n . 16 Proof. If R = A n , we have k i = 0 ∀i ≤ n -1 and k i = RS i R-γ i A i ∀i ≥ n + 1. This implies that A n = R > max i (γ i A i ). Since 0 ≤ k n ≤ RS i R-fnAn , we have m i=n+1 RS i R -γ i A i ≤ i k i = m i=n k i ≤ m i=n RS i R -γ i A i = i=n A n S i A n -γ i A i (A.14) By converse, suppose that A n > max i (γ i A i ) and m i=n+1 AnS i An-γ i A i ≤ S ≤ m i=n AnS i An-γ i A i . Applying points 1 and 2 of Lemma 10, we have R ≥ A n and R ≤ A n . Hence R = A n .
By combining Lemma 10 and the fact that R > max i (γ i A i ), we obtain the following result.

Lemma 12. Assume that A 1 < A 2 < • • • < A m . Consider an equilibrium. If R ∈ (A n , A n+1 ), then A n+1 > max i (γ i A i ) and R = R L n (hence R L n ∈ (A n , A n+1 )).
We now identify the necessary and sufficient conditions under which R = R L n .

Lemma 13. R = R L n = A n if and only if one of the following conditions is satisfied:

1. max i (γ i A i ) < A n < r L n < A n+1 , or equivalently max i (γ i A i ) < A n and D n+1 < S < B n 2. A n ≤ max i (γ i A i ) < R L n < A n+1 , or equivalently A n ≤ max i (γ i A i ) < R L n and D n+1 < S.
In any case, we have that

R L n ∈ [A n , A n+1 ). Proof. Part 1. Assume that R = R L n = A n . By definition of R and R L n , we have m i=n+1 RS i R-γ i A i = S, and R L n > max i (γ i A i ). We will prove that R = R L n ∈ (A n , A n+1 ). If R ≤ A n , then R < A n+1 , and hence k i = RS i R-γ i A i ∀i ≥ n+1. Since m i=n+1 RS i R-γ i A i = S = i k i . We have k i = 0 ∀i ≤ n, and hence k n = 0. This implies that R ≥ A n . Therefore, we have R = A n , a contradiction. Thus, we have R > A n . If R ≥ A n+1 , we have k i = 0 ∀i ≤ n. Hence S = i k i ≤ m i=n+1 RS i R-γ i A i . Since m i=n+1 RS i R-γ i A i = S, we have k i = RS i R-γ i A i ∀i ≥ n + 1. Hence A n+1 ≥ R. So, R = A n+1 . We have just proved that R ≤ A n+1 . By definition of R, we get that A n+1 > 16 We need condition A n > M ≡ max i (γ i A i ) because that R > max i (γ i A i ). Condition m i=n+1 AnSi An-γiAi ≤ S ensures that R ≤ A n while condition S ≤ m i=n AnSi An-γiAi ensures that R ≥ A n . max i (γ i A i ). If R L n = A n+1 , then applying Lemma 11, we have m i=n+2 A n+1 S i A n+1 -γ i A i = B n+1 ≤ S. However, by definition of R L n , we have m i=n+1 A n+1 S i A n+1 -γ i A i = S, contradiction. Therefore, we obtain R L n < A n+1 . We have just proved that R L n ∈ (A n , A n+1 ).
Applying point 2 of Lemma 10, we have S > D n+1 . There are two cases:

1. max i (γ i A i ) ≥ A n . In this case, we have A n ≤ max i (γ i A i ) < R L n < A n+1 . 2. max i (γ i A i ) < A n . We get max i (γ i A i ) < A n < R L n < A n+1 . Notice that, in this case, R L n ∈ (A n , A n+1 ) is equivalent to D n+1 < S < B n .
Part 2. Conversely, assume that (i)

A n ≤ max i (γ i A i ) < R L n < A n+1 or (ii) max i (γ i A i ) < A n < R L n < A n+1 . 1. If A n ≤ max i (γ i A i ) < R L n < A n+1 . Condition A n ≤ max i (γ i A i ) implies that R > A n . Then k i = 0 ∀i ≤ n, and hence S = m i=n+1 k i ≤ m i=n+1 RS i R-γ i A i By definition R L n , we have S = m i=n+1 R L n S i R L n -γ i A i . Since the function f (X) ≡ m i=n+1 XS i X-γ i A i is decreasing in the interval (max i≥n+1 (γ i A i ), ∞) and R, R L n > max i (γ i A i ), we have R ≤ R L n . This implies that R ∈ (A n , A n+1 ). Therefore, Lemma 12 implies that R = R L n . 2. If max i (γ i A i ) < A n and D n+1 < S < B n . We have S < D n because D n > B n .
According to point 2 of Lemma 10, we have R ≥ A n .

Condition S > D n+1 implies that S > B n+1 because D n+1 > B n+1 . According to point 1 of Lemma 10, we have R ≤ A n+1 .

If R = A n+1 , then Lemma 11 implies that S ≤ D n+1 . This is a contradiction because S > D d+1 .

If R = A n , Lemma 11 implies that S ∈ [B n , D n ]. However, S ≤ B n . Thus, we have

S = B n = m i=n+1 AnS i An-γ i A i . Since A n > max i (γ i A i ), then A n = R L n , a contradiction.
Summing up, we have R ∈ (A n , A n+1 ). By applying point 3 of Lemma 12, we have R = R L n .

Remark 6. We can check that the regimes in Theorem 2 are not overlap, and the union of these regimes is equal to the set of economies satisfying

A 1 < • • • < A m , or, formally, E = ∪ m i=1 A i ∪ ∪ m-1 i=1 R i (A.15a) X ∩ Y = ∅ ∀X, Y ∈ {A 1 , . . . , A m , R 1 , . . . , R m-1 } and X = Y. (A.15b) Denote M ≡ max i (γ i A i ).
By definition, we see that:

1. The economy E ≡ (F i , γ i , S i ) i=1,...,m ∈ A 1 if and only if A 1 > max i (γ i A i ) and S > B 1 .

1. Suppose that there exists

E ∈ R n,1 ∩ R h,2 . It means that A n > max i (γ i A i ); D n+1 < S < B n ; A h+1 > max i (γ i A i ) ≥ A h ; D h+1 < S. Since h > n, then A h > A n > max i (γ i A i ). This is a contradiction because max i (γ i A i ) ≥ A h . So, we have R n,1 ∩ R h,2 = ∅.
2. Suppose that there exists

E ∈ R n,2 ∩ R h,1 . It means that A n+1 > max i (γ i A i ) ≥ A n ; D n+1 < S;A h > max i (γ i A i ); D h+1 < S < B h . Since h ≥ n + 1, we have B h ≤ B n+1 < D n+1 < S < B h , a contradiction.
B Proofs for Section 3.1

Proof of Proposition 1. In the case

γ 2 < A 1 λA 2 S 1 S 1 +S 2 = A 1 A 2 S 1 S 1 +S 2 and γ 2 < A 1 A 2 S 1 S 1 +S 2 , we have that Y (A 1 , A 2 ) -Y (A 1 , A 2 ) = A 1 S 1 + A 2 S 2 A 1 (1 -γ 2 ) A 1 -γ 2 A 2 -A 1 S 1 -A 2 S 2 A 1 (1 -γ 2 ) A 1 -γ 2 A 2 = (A 1 -A 1 )S 1 + A 2 S 2 (1 -γ 2 ) A 1 A 2 -A 1 A 2 (A 1 -γ 2 A 2 )(A 1 -γ 2 A 2 ) Point 1. When A 2 A 2 ≥ A 1 A 1 ≥ 1, we have (A 1 -A 1 )S 1 > 0 and A 1 A 2 -A 1 A 2 ≥ 0. By consequence, we get that Y (A 1 , A 2 ) -Y (A 1 , A 2 ) > 0.
Point 2. We can compute that By Taylor's theorem, we get point 2.

Proof of Corollary 3. Point 1.a is a direct consequence of Corollary 1. Point 1.b is a direct consequence of Theorem 1.

We now prove point 2. We make use of Theorem 2. We firstly consider the regime R n with n ≤ m -1. In this regime, we have .17) Notice that Y = A m S if and only if n + 1 = m.

Y = Y n = m i=n+1 rA i S i r -f i A i ≤ m i=n+1 A m rS i r -f i A i = A m S. (A
We now consider the regime A n with n ≤ m. In this regime, we have

Y = A n n i=1 S i + m i=n+1 A n (1 -f i )A i S i A n -f i A i = A n m i=1 S i + A n m i=n+1 A n (A i -A n ) A n -f i A i ≤ A n S + (A m -A n ) m i=n+1 A n A n -f i A i ≤ A n S + (A m -A n )S = A m S.
where the last inequality is from the condition

m i=n+1 AnS i An-f i A i in the regime A n . It is easy to see that Y = A m S if and only if either (i) n + 1 > m or (ii) n + 1 = m and A m-1 A m-1 -fmAm S m = S.
Combining these two cases, we obtain our result.

Proof of Proposition 3. Since the production function has Cobb-Douglas form, all agents produce in equilibrium. According to (24), we have

∂Y ∂A 1 = f 1 (k 1 )
Productivity effect .18) According to FOCs, we have

+ i =1 A 1 f 1 (k 1 ) -A i f i (k i ) -∂k i ∂R ≥ 0 ∂R ∂A 1 ≥ 0 Allocation effect . ( A
[k] : (1 + µ i γ i )F i (k) = λ i [a] : (1 + µ i )R = λ i , µ i ≥ 0, and µ i (γ i F i (k i ) -R i b i ) = 0.
These equations imply that: .19) This implies that R ≥ max j γ j F j (k j ) ≥ max j γ j F j (S)). Thus, R ≥ max j γ j F j (S)) > 0, ∀A 1 .

γ i A i f i (k i ) ≤ R = A i f i (k i ) 1 + γ i µ i 1 + µ i ≤ A i f i (k i ), ∀i. (A
1. When A 1 is high enough. Note that lim A 1 →∞ R 1 = ∞. Hence, for A 1 high enough, we have that R 1 > S. We prove that the equilibrium interest rate goes to infinity when A 1 goes to infinity. Indeed, if agent 1's borrowing constraint is not binding, we have

R = A 1 f 1 (k 1 ) > A 1 f 1 (S). If agent 1's borrowing constraint is binding, we have R(k 1 -S 1 ) = γ 1 A 1 f 1 (k 1 ) which implies that R = γ 1 A 1 f 1 (k 1 ) k 1 -S 1 ≥ γ 1 A 1 f 1 (S 1 ) S -S 1 Hence, R ≥ min A 1 f 1 (S), γ 1 A 1 f 1 (S 1 ) S-S 1
. From this, we obtain that lim

A 1 →∞ R = ∞. Now, condition lim A 1 →∞ R = ∞ implies that borrowing constraint of any agent i ≥ 2 is not binding for A 1 high enough. So, A 1 f 1 (k 1 ) ≥ R = A i f i (k i ), ∀i ≥ 1.
By combining this and condition (A.18), we get that ∂Y ∂A 1 > 0 for A 1 high enough. 2. We will prove that when A 1 is small enough, the productivity effect is smaller than the allocation effect. To show ∂Y ∂A 1 < 0 for A 1 small enough, we will prove that lim

A 1 →0 k 1 = 0, lim A 1 →0 A 1 f 1 (k 1 ) -A 2 f 2 (k 2 ) < 0, lim A 1 →0 -∂k 2 ∂R > 0, and lim A 1 →0 ∂R ∂A 1 > 0. Since A 1 f 1 (k 1 ) ≥ R ≥ max j γ j F j (S)) > 0, we have lim A 1 →0 f 1 (k 1 ) = ∞.
Therefore, we have lim

A 1 →0
k 1 = 0, and lim

A 1 →0 i =1 k i = S. (A.20) Since lim A 1 →0 k 1 = 0, we get that γ 1 A 1 f 1 (k 1 )-Rb 1 = γ 1 A 1 f 1 (k 1 )-Rk 1 +RS 1 > 0 for A 1 small enough.
It means that the borrowing constraint of agent 1 is not binding. To sum up, we have

R = A 1 f 1 (k 1 ) ≥ max j γ j F j (S)
) > 0, for A 1 small enough.

Denote

B 1 = B 1 (R 1 ) ≡ k n 1 (R 1 ) + m i=2 k b i (R 1 ), B 2 = B 2 (R 2 ) = 2 i=1 k n i (R 2 ) + m i=3 k b i (R 2 ) B m = B m (R m ) = m i=1 k n i (R m )
where, to simplify notations, we write k n i (R) and

k b i (R) instead of k n i ( R A i ) and k b i ( R γ i A i , S i ) (see Definition 2). We see that D i ≡ B i -k n i (R i ), ∀i. Notice also that B 1 , . . . , B m depend on A 1 but D 2 , D 3 , . . . , D m do not. Moreover, lim A 1 →0 (B i - D i ) = 0, ∀i ≥ 2 because lim A 1 →0 k n 1 (R i ) = 0, ∀i ≥ 2. Condition R 2 < R 3 < • • • < R m implies that D 2 > • • • > D m . Since lim A 1 →0 B 1 = +∞ and lim A 1 →0 (B i -D i ) = 0, ∀i ≥ 2, we have B 1 > B 2 > • • • > B m for A 1 small enough.
(a) S < D m . Then we have S < B m . According to Theorem 1, the equilibrium coincides to that of the economy without frictions. Therefore, the output is increasing in A 1 .

(b) Let D n > S > D n+1 . In this case, we have B n > S > B n+1 for any A 1 small enough. According to Theorem 1, the equilibrium interest rate R is in the interval (R n , R n+1 ] and determined by

n i=1 k n i (R) + m i=n+1 k b i (R) = S ≡ i S i (A.21) Denote Z 2 (R) = n i=2 k n i (R) + m i=n+1 k b i (R).
When A 1 tends to zero, we have lim A 1 →0 k n 1 (R) = 0 and lim A 1 →0 R = R(0) where R(0) > 0 is uniquely determined by Z 2 (R(0)) = S. For i ≥ n + 1, agent i's borrowing constraint is binding: R(k i -S i ) = γ i A i f i (k i ) for any A 1 small enough. Let A 1 tend to zero, we have k i tends to k i (0), R tends to R(0), and

γ i A i f i (k i (0)) = R(0)(k i (0) -S i ).
Let σ be such that

γ i f i (k) kf i (k) < σ < S 1 S 1 + S n+1 + • • • + S m , ∀i ≥ n + 1, ∀k ∈ (0, S). (A.22)
According to condition (26), we have

R(0) -A i f i (k i (0)) = γ i A i f i (k i (0)) k i (0) -S i -A i f i (k i (0)) (A.23) ≤ A i f i (k i (0)) k i (0) -S i (σk i (0) -(k i (0) -S i ) (A.24)
By market clearing condition, we have

m i=n+1 k i = m i=2 (S i -k i ) + S 1 -k 1 + m i=n+1 S i ≥ S 1 -k 1 + m i=n+1 S i
Let A 1 tend to zero, we get that m i=n+1 k i (0) ≥ S 1 + m i=n+1 S i . Thus,

m i=n+1 σk i (0) -(k i (0) -S i ) = m i=n+1 S i -(1 -σ)k i (0) ≤ m i=n+1 S i -(1 -σ) S 1 + m i=n+1 S i < 0
Therefore, there exists j ∈ {n+1, . . . , m} such that σk j (0)-(k j (0)-S j ) < 0, and hence .25) In this regime, R 1 < R ≤ R 2 , agent 2 borrows and her credit constraint is binding while agent 1 is lender. Now, we prove part 1 of Proof of Proposition 4. Since Inada condition holds, all agents produce in equilibrium. According to (24), we have

R(0) -A j f j (k j (0)) ≤ A j f j (k j (0)) k j (0) -S j (σk j (0) -(k j (0) -S j ) < 0. (A
∂Y ∂A 1 =f 1 (k 1 ) + A 1 f 1 (k 1 ) -A 2 f 2 (k 2 ) -∂k 2 ∂R > 0 ∂R ∂A 1 > 0 . (A.32)
To show ∂Y ∂A 1 < 0 for A 1 small enough, we will prove that lim

A 1 →0 k 1 = 0, lim A 1 →0 A 1 f 1 (k 1 )- A 2 f 2 (k 2 ) < 0, lim A 1 →0 -∂k 2 ∂R > 0, and lim A 1 →0 ∂R ∂A 1 > 0.
According to FOCs, we have

[k] : (1 + µ i γ i )F i (k) = λ i [a] : (1 + µ i )R = λ i , µ i ≥ 0, and µ i (γ i F i (k i ) -R i b i ) = 0.
These equations imply that:

γ i A i f i (k i ) ≤ R = A i f i (k i ) 1 + γ i µ i 1 + µ i ≤ A i f i (k i ), ∀i. (A.33) This implies that R ≥ γ 2 F 2 (k 2 ) ≥ γ 2 F 2 (S). Thus, R ≥ γ 2 F 2 (S), ∀A 1 . Since R ≤ A 1 f 1 (k 1 ). So, we have lim A 1 →0 f 1 (k 1 ) = ∞. Therefore, we have lim A 1 →0
k 1 = 0, and lim

A 1 →0 k 2 = S. (A.34)
By consequence, we get that γ 1 A 1 f 1 (k 1 ) -Rb 1 = γ 1 A 1 f 1 (k 1 ) -Rk 1 + RS 1 > 0 for A 1 small enough. It means that the borrowing constraint of agent 1 is not binding. To sum up, we have R = A 1 f 1 (k 1 ) ≥ γ 2 F 2 (S) for A 1 small enough. Since R 2 does not depend on A 1 , we observe that lim A 1 →0 k n 1 (R 2 /A 1 ) = 0. So, by combining with the assumption k n 2 ( R 2 A 2 ) < S, we have k n 1 ( R A 1 ) + k n 2 ( R A 2 ) < S for A 1 small enough. According to point 3 of Lemma 2, we have R 1 < R 2 for A 1 small enough. Hence, we can apply Corollary 6 to obtain that the borrowing constraint of agent 2 is binding in equilibrium. It means that γ

2 A 2 f 2 (k 2 ) -Rk 2 + RS 2 = 0.
Look at the market clearing condition:

k n 1 ( R A 1 ) + k b 2 ( R γ 2 A 2 , S 2 ) = S ≡ i S i . When A 1 converges to 0, we have k n 1 ( R A 1 ) converges to 0. So, R converges to R(0) satisfying k b 2 ( R(0) γ 2 A 2 , S 2 ) = S. So, we have lim A 1 →∞ A 1 f 1 (k 1 ) -A 2 f 2 (k 2 ) = lim A 1 →∞ R -A 2 f 2 (k 2 ) = R(0) -A 2 f 2 (S). (A.35)
Since agent 2's borrowing constraint is binding: R(k 2 -S 2 ) = γ 2 A 2 f 2 (k 2 ) for any A 1 small enough. Let A 1 tend to zero, we have .36) check that the interest rate is uniquely determined by

γ 2 A 2 f 2 (S) = R(0)(S -S 2 ) = R(0)S 1 . So, we have R(0) -A 2 f 2 (S) = γ 2 A 2 f 2 (S) S 1 -A 2 f 2 (S) < 0 (A
R =                A 1 if S 1 ≥ γ 3 A 3 A 1 -γ 3 A 3 S 3 + γ 2 A 2 A 1 -γ 2 A 2 S 2 R 1 if γ 3 A 3 A 2 -γ 3 A 3 S 3 + γ 2 1-γ 2 S 2 < S 1 < γ 3 A 3 A 1 -γ 3 A 3 S 3 + γ 2 A 2 A 1 -γ 2 A 2 S 2 A 2 if γ 3 A 3 A 2 -γ 3 A 3 S 3 -S 2 ≤ S 1 ≤ γ 3 A 3 A 2 -γ 3 A 3 S 3 + γ 2 1-γ 2 S 2 R 2 if γ 3 1-γ 3 S 3 -S 2 < S 1 < γ 3 A 3 A 2 -γ 3 A 3 -S 2 A 3 if S 1 ≤ γ 3 1-γ 3 S 3 -S 2 (A.48) where R 2 = γ 3 A 3 1 + S 3 S 1 +S 2
and R 1 is the highest solution of the equation:

γ 2 A 2 R -γ 2 A 2 S 2 + γ 3 A 3 R -γ 3 A 3 S 3 = S 1 . (A.49) This equation implies that R(S 2 (R -γ 3 A 3 ) + S 3 (R -γ 2 A 2 )) = S(R -γ 2 A 2 )(R -γ 3 A 3 ), or equivalently S 1 R 2 -R (S 1 + S 2 )γ 2 A 2 + (S 1 + S 3 )γ 3 A 3 + Sγ 2 A 2 γ 3 A 3 = 0. (A.50a) So, the rate R 1 is computed by R = (S 1 + S 2 )γ 2 A 2 + (S 1 + S 3 )γ 3 A 3 + √ ∆ 2S 1 (A.50b) where ∆ ≡ (S 1 + S 2 )γ 2 A 2 + (S 1 + S 3 )γ 3 A 3 2 -4S 1 Sγ 2 A 2 γ 3 A 3 (A.50c)
There are 5 different cases. In each case, we can explicitly compute equilibrium outcomes thanks to Lemma 2.

Proof of Proposition 10 (homogeneous credit limit). Since F i (k t ) ≥ R, there are two cases.

(1) If F i (k i ) = R, then we have hence ∂k i ∂γ < 0. (2) If F i (k i ) > R, then borrowing constraint of this agent is binding. The market clearing condition i k i = i S i implies that i:F i (k i )=R ∂k i ∂γ + i:F i (k i )>R ∂k i ∂γ = 0. So, we have i:F i (k i )>R ∂k i ∂γ > 0. We now claim that ∂k i ∂γ > 0 for any agent with F i (k i ) > R.
For such agents we have γF

i (k n i ) -R(k n i -S i ).
Taking the derivative with respect to γ of both sides of this equation, we have

F i (k i ) + γF i (k i ) ∂k i ∂γ = ∂R ∂γ (k i -S i ) + R ∂k i ∂γ (A.51) i.e., ∂k i ∂γ = ∂R ∂γ γ R -1 F i (k i ) R -γF i (k i ) . (A.52)
By summing with respect to i such that F i (k i ) > R and noticing that i:F i (k i )>R ∂k i ∂γ > 0 and R -γF i (k i ) > 0 ∀i, we get that ∂R ∂γ γ R -1 ≥ 0. From this and (A.52), we obtain

∂k i ∂γ > 0 ∀i such that F i (k i ) > R.
We now observe that

∂Y ∂γ = i:F i (k i )=R F j (k i ) ∂k i ∂γ + i:F i (k i )>R F i (k i ) ∂k i ∂γ ≥ R m i=1 ∂k i ∂γ = 0. (A.53)

D Proofs of Section 4

Firstly, we provide a sufficient condition to check whether a sequence is an intertemporal equilibrium.

Lemma 14. Let the borrowing constraint of agent i at date

t is B i (R t , b i,t-1 , k i,t-1 ) ≤ 0, where • The function B i is continuously differentiable, increasing in b i and R, but de- creasing in k i . Moreover, for R given, the function B i (R, •, •) is convex. • B i (R, k i , b i ) ≤ 0 implies that F i (k i ) -Rb i ≥ 0. • i B i (R, k i , b i ) < 0, ∀R > 0, k i ≥ 0, such that i k i > 0, i b i = 0.
If the sequences (R t , (c i,t , k i,t , b i,t ) i ) t and (λ i,t , µ i,t , η i,t ) i,t satisfy the following conditions:

1. c i,t , l i,t , λ i,t , η i,t , µ i,t+1 are non-negative and R t > 0 for any t.

c

i,t + k i,t + R t b i,t-1 = F i,t (k i,t-1 ) + b i,t , and B i (R t , b i,t-1 , k i,t-1 ) ≤ 0, ∀i, ∀t. 3. i b i,t = 0, ∀t. 4. ∞ t=0 λ i,t c i,t < ∞, ∞ t=0 β t i u i (c i,t ) < ∞.

TVCs: lim

T →∞ β t i u i (c i,t )(k i,t -b i,t ) = 0, ∀i.
6. FOCs: ∀i, ∀t,

β t i u i (c i,t ) = λ i,t λ i,t = F i,t+1 (k i,t )λ i,t+1 -µ i,t+1 ∂B i ∂k (R t+1 , b i,t , k i,t ) + η i,t η i,t k i,t = 0 λ i,t = R t+1 λ i,t+1 + µ i,t+1 ∂B i ∂b (R t+1 , b i,t , k i,t ) µ i,t+1 B i (R t+1 , b i,t , k i,t ) = 0, then the list (R t , (c i,t , k i,t , b i,t ) i ) is an intertemporal equilibrium.
Proof of Lemma 14. Before proving this part, we should notice that this result requires neither u i (0) = 0 nor u i (0) = ∞. Let us prove our result. It is sufficient to prove the optimality of (c i , k i , b i ) for all i. Let (c i , k i , b i ) be a plan satisfying all budget and borrowing constraints and b

i,-1 -b i,-1 = 0 = k i,-1 -k i,-1 . We have T t=0 β t i (u i (c i,t ) -u i (c i,t )) ≥ T t=0 β t i u i (c i,t )(c i,t -c i,t ) = T t=0 λ i,t (c i,t -c i,t ) c i,t + k i,t + R t b i,t-1 ≤ F i,t (k i,t-1 ) + b i,t (B.1) B i (R t , b i,t-1 , k i,t-1 ) ≤ 0 (B.2) Budget constraints imply that c i,t = F i,t (k i,t-1 ) + b i,t -k i,t -R t b i,t-1 , c i,t ≤ F i,t (k i,t-1 ) + b i,t -k i,t -R t b i,t-1 λ i,t (c i,t -c i,t ) ≥λ i,t (F i,t (k i,t-1 ) + b i,t -k i,t -R t b i,t-1 -F i,t (k i,t-1 ) -b i,t + k i,t + R t b i,t-1 ) =λ i,t F i,t (k i,t-1 ) -F i,t (k i,t-1 ) -λ i,t (k i,t -k i,t ) + λ i,t (b i,t -b i,t ) -λ i,t R t (b i,t-1 -b i,t-1 )
According to FOCs, we have

λ i,t k i,t = λ i,t+1 F i,t+1 (k i,t )k i,t -µ i,t+1 ∂B i ∂k (R t+1 , b i,t , k i,t )k i,t + η i,t k i,t λ i,t b i,t = R t+1 λ i,t+1 b i,t + µ i,t+1 ∂B i ∂b (R t+1 , b i,t , k i,t )b i,t
This implies that

λ i,t (k i,t -k i,t ) = λ i,t+1 F i,t+1 (k i,t )(k i,t -k i,t ) -µ i,t+1 ∂B i ∂k (R t+1 , b i,t , k i,t )(k i,t -k i,t ) + η i,t (k i,t -k i,t ) (B.3) λ i,t (b i,t -b i,t ) = R t+1 λ i,t+1 (b i,t -b i,t ) + µ i,t+1 ∂B i ∂b (R t+1 , b i,t , k i,t )(b i,t -b i,t ) (B.4)
Therefore, we have that

T t=0 λ i,t (c i,t -c i,t ) ≥ T t=0 λ i,t F i,t (k i,t-1 ) -F i,t (k i,t-1 ) -λ i,t (k i,t -k i,t ) + T t=0 λ i,t (b i,t -b i,t ) -λ i,t R t (b i,t-1 -b i,t-1 ) ≥ T -1 t=0 λ i,t+1 F i,t+1 (k i,t ) -λ i,t (k i,t -k i,t ) -λ i,T (k i,T -k i,T ) + T -1 t=0 λ i,t -λ i,t+1 R t+1 (b i,t -b i,t ) + λ i,T (b i,T -b i,T ) =λ i,T k i,T -b i,T -(k i,T -b i,T ) + T -1 t=0 µ i,t+1 ∂B i ∂k (R t+1 , b i,t , k i,t )(k i,t -k i,t ) -η i,t (k i,t -k i,t ) + T -1 t=0 µ i,t+1 ∂B i ∂b (R t+1 , b i,t , k i,t )(b i,t -b i,t ) =λ i,T k i,T -b i,T -(k i,T -b i,T ) + T -1 t=0 η i,t (k i,t -k i,t ) + T -1 t=0 µ i,t+1 ∂B i ∂k (R t+1 , b i,t , k i,t )(k i,t -k i,t ) + ∂B i ∂b (R t+1 , b i,t , k i,t )(b i,t -b i,t )
We consider µ i,t+1

∂B i ∂k (R t+1 , b i,t , k i,t )(k i,t -k i,t ) + ∂B i ∂b (R t+1 , b i,t , k i,t )(b i,t -b i,t ) . µ i,t+1 ∂B i ∂k (R t+1 , b i,t , k i,t )(k i,t -k i,t ) + ∂B i ∂b (R t+1 , b i,t , k i,t )(b i,t -b i,t ) =µ i,t+1 (B i (R t+1 , b i,t , k i,t ) -B i (R t+1 , b i,t , k i,t )) + µ i,t+1 B i (R t+1 , b i,t , k i,t ) -B i (R t+1 , b i,t , k i,t ) (B.5) - ∂B i ∂k (R t+1 , b i,t , k i,t )(k i,t -k i,t ) - ∂B i ∂b (R t+1 , b i,t , k i,t )(b i,t -b i,t ) ≥µ i,t+1 (B i (R t+1 , b i,t , k i,t ) -B i (R t+1 , b i,t , k i,t )) = -µ i,t+1 B i (R t+1 , b i,t , k i,t )) ≥ 0, (B.6) because the function B i (R t+1 , •, •) is convex. It remains to prove that lim inf T →∞ λ i,T k i,T -b i,T -(k i,T -b i,T ) ≥ 0.
Proof of Lemma 4. The maximization problem of agent i is max

(c i ,k i ,b i ) ∞ t=0 β t i u i (c i,t ) subject to: c i,t + k i,t + R t b i,t-1 ≤ A i,t k i,t-1 + b i,t R t b i,t-1 ≤ γ i A i,t (k i,t-1 ) Denote s i,t = k i,t -b i,t the net saving of agent i at date t. Let R t = A 1,t ∀t. For agent 1, we have c 1,t + (k 1,t -b 1,t ) ≤ A 1,t (k 1,t-1 -b 1,t-1
). We can compute that

s 1,0 = β 1 w 1,0 , s 1,t = β 1 A 1,t s 1,t-1 ∀t ≥ 1 s 1,t = β t 1 A 1,t • • • A 1,1 s 1,0
For agent 2, since A 2,t > R t = A 1,t ∀t, her borrowing constraint is always binding: R t b 2,t-1 = γ 2 A 2,t k 2,t-1 . Therefore, we have

s 2,t = k 2,t 1 - γ 2 A 2,t+1 R t+1 , A 2,t k 2,t-1 -R t b 2,t-1 = (1 -γ 2 )A 2,t k 2,t-1 ∀t ≥ 1.
From this, we can compute that

s 2,t = β 2 (1 -γ 2 )A 2,t R t R t -γ 2 A 2,t s 2,t-1 = β 2 (1 -γ 2 )A 2,t R t R t -γ 2 A 2,t • • • β 2 (1 -γ 2 )A 2,1 R 1 R 1 -γ 2 A 2,1 s 2,0 k 2,t = 1 1 -γ 2 A 2,t+1 R t+1 s 2,t = R t+1 R t+1 -γ 2 A 2,t+1 s 2,t b 2,t = γ 2 A 2,t+1 R t+1 k 2,t = γ 2 A 2,t+1 R t+1 -γ 2 A 2,t+1 s 2,t
Therefore, we can find the capital of the first agent

k 1,t =s 1,t + b 1,t = s 1,t -b 2,t =β t 1 A 1,t • • • A 1,1 s 1,0 - γ 2 A 2,t+1 R t+1 -γ 2 A 2,t+1 β 2 (1 -γ 2 )A 2,t R t R t -γ 2 A 2,t • • • β 2 (1 -γ 2 )A 2,1 R 1 R 1 -γ 2 A 2,1 s 2,0
In order to keep k 1,t > 0 ∀t, we need that

β t 1 A 1,t • • • A 1,1 s 1,0 - γ 2 A 2,t+1 R t+1 -γ 2 A 2,t+1 β 2 (1 -γ 2 )A 2,t R t R t -γ 2 A 2,t • • • β 2 (1 -γ 2 )A 2,1 R 1 R 1 -γ 2 A 2,1 s 2,0 > 0 ⇔ β t 1 s 1,0 -β t 2 γ 2 A 2,t+1 A 1,t+1 -γ 2 A 2,t+1 (1 -γ 2 )A 2,t A 1,t -γ 2 A 2,t • • • (1 -γ 2 )A 2,1 A 1,1 -γ 2 A 2,1 s 2,0 > 0 or equivalently s 2,0 s 1,0 β 1 β 2 γ 2 1 -γ 2 β 2 β 1 (1 -γ 2 )A 2,1 A 1,1 -γ 2 A 2,1 • • • β 2 β 1 (1 -γ 2 )A 2,t+1 A 1,t+1 -γ 2 A 2,t+1 < 1∀t. This happens if sup t β 2 β 1 1 -γ 2 A 1,t A 2,t -γ 2 < 1 and s 2,0 s 1,0 β 1 β 2 γ 2
1-γ 2 ≤ 1. By applying Lemma 14, we can check that the above list (R t , (c i,t , k i,t , b i,t ) i ) is an equilibrium. We now compute the aggregate production

Y t = A 1,t k 1,t-1 + A 2,t k 2,t-1 = A 1,t β t-1 1 A 1,t-1 • • • A 1,1 s 1,0 - γ 2 A 2,t R t -γ 2 A 2,t β 2 (1 -γ 2 )A 2,t R t-1 R t-1 -γ 2 A 2,t-1 • • • β 2 (1 -γ 2 )A 2,1 R 1 R 1 -γ 2 A 2,1 s 2,0 + + A 2,t R t R t -γ 2 A 2,t β 2 (1 -γ 2 )A 2,t-1 R t-1 R t-1 -γ 2 A 2,t-1 • • • β 2 (1 -γ 2 )A 2,1 R 1 R 1 -γ 2 A 2,1 s 2,0 = β t-1 1 A 1,t A 1,t-1 • • • A 1,1 s 1,0 + β t-1 2 (1 -γ 2 ) t A 2,t A 1,t A 1,t -γ 2 A 2,t A 2,t-1 A 1,t-1 A 1,t-1 -γ 2 A 2,t-1 • • • A 2,1 A 1,1 A 1,1 -γ 2 A 2,1 s 2,0 .
Proof of Proposition 13. According to (50), we have

Y t = β t-1 1 A t 1 s 1,0 + β t-1 2 (1 -γ 2 ) t A t 2 A 1 A 1 -γ 2 A 2 t s 2,0 .
So, we get that

1 tA t-1 1 ∂Y t ∂A 1 = β t-1 1 s 1,0 -(1 -γ 2 ) γ 2 A 2 2 (A 1 -γ 2 A 2 ) 2 β 2 (1 -γ 2 )A 2 A 1 -γ 2 A 2 t-1 s 2,0 (B.7)
The output is more likely to be increasing in the TFP A 1 of the least productive agent (i.e., ∂Yt ∂A 1 > 0) if (1) the productivity gap A 2 A 1 is low or (2) the initial income gap s 2,0 s 1,0 is low or (3) the time preference gap β 2 β 1 is low.

1. If s 1,0 s 2,0

(A 1 -γ 2 A 2 ) 2 (1-γ 2 )γ 2 A 2 2
> 1, then it is easy to see that ∂Yt ∂A 1 > 0 ∀t.

2.

If s 1,0 s 2,0

(A 1 -γ 2 A 2 ) 2 (1-γ 2 )γ 2 A 2 2
< 1, we have immediately that ∂Y 1 ∂A 1 < 0. According to equation (B.7) and the fact that lim t→∞ β 1 (A 1 -γ 2 A 2 ) (1-γ 2 )A 2 β 2 t = ∞, we get point 2 of Proposition 13.

Notice that the capital of the first agent is

k 1,t = s 1,t + b 1,t = s 1,t -b 2,t = (β 1 R) t s 1,0 - γ 2 A 2 R -γ 2 A 2 β 2 (1 -γ 2 )A 2 R R -γ 2 A 2 t s 2,0
Condition k 1,t > 0 ∀t is equivalent to

β t 1 s 1,0 - γ 2 A 2 A 1 -γ 2 A 2 β 2 (1 -γ 2 )A 2 A 1 -γ 2 A 2 t s 2,0 > 0 ∀t ≥ 0.
This happens if and only if β 1 ≥ (1-γ 2 )A 2

We now prove the last part of Proposition 13. The net worth of agent 2: W 2,t ≡ Output -Payment.

W 2,t ≡ Output -Payment = A 2 k 2,t-1 -R t b 2,t-1 = (1 -γ 2 )A 2 k 2,t-1 = (1 -γ 2 )A 2,t R t R t -γ 2 A 2,t β 2 (1 -γ 2 )A 2,t-1 R t-1 R t-1 -γ 2 A 2,t-1 • • • β 2 (1 -γ 2 )A 2,1 R 1 R 1 -γ 2 A 2,1 s 2,0 = (1 -γ 2 )A 2 A 1 A 1 -γ 2 A 2 β 2 (1 -γ 2 )A 2 A 1 A 1 -γ 2 A 2 t-1 s 2,0
First, we look at

Y 1,t + Y 2,t Y 2,t = Y t Y 2,t = β t-1 1 A t 1 s 1,0 + β t-1 2 (1 -γ 2 ) t A 2 A 1 A 1 -γ 2 A 2 t s 2,0 A 2 A 1 A 1 -γ 2 A 2 β 2 (1-γ 2 )A 2 A 1 A 1 -γ 2 A 2 t-1 s 2,0 = β t-1 1 A t 1 s 1,0 A 2 A 1 A 1 -γ 2 A 2 β 2 (1-γ 2 )A 2 A 1 A 1 -γ 2 A 2 t-1 s 2,0 + (1 -γ 2 ) = β t-1 1 (A 1 -γ 2 A 2 ) t s 1,0 β t-1 2 A t 2 (1 -γ 2 ) t-1 s 2,0 + (1 -γ 2 )
This is increasing in the ratio A 1 /A 2 . Since

β 1 (A 1 -γ 2 A 2 ) β 2 (1-γ 2 )A 2 > 1, we have Yt Y 2,t → ∞, or equivalently Y 1,t Y 2,t → ∞.
The reason is that the rate o time preference of the less productive agent β 1 is high enough. We can also check that the share of net worth of the productive agent W 2,t Yt → 0. We now look at the growth rate

G t = Y t+1 Y t = β t 1 A t+1 1 s 1,0 + β t 2 (1 -γ 2 ) t+1 A 2 A 1 A 1 -γ 2 A 2 t+1 s 2,0 β t-1 1 A t 1 s 1,0 + β t-1 2 (1 -γ 2 ) t A 2 A 1 A 1 -γ 2 A 2 t s 2,0 = β 2 β 1 β 1 β 2 A 1 -γ 2 A 2 A 2 (1-γ 2 )
t+1 + s 2,0

β 2 β 1 β 1 β 2 A 1 -γ 2 A 2 A 2 (1-γ 2 ) t + s 2,0 β t 2 (1 -γ 2 ) t+1 A 2 A 1 A 1 -γ 2 A 2 t+1 β t-1 2 (1 -γ 2 ) t A 2 A 1 A 1 -γ 2 A 2 t = β 2 β 1 β 1 β 2 A 1 -γ 2 A 2 A 2 (1-γ 2 )
t+1 + s 2,0

β 2 β 1 β 1 β 2 A 1 -γ 2 A 2 A 2 (1-γ 2 ) t + s 2,0 β 2 (1 -γ 2 )A 2 A 1 A 1 -γ 2 A 2 Since β 1 β 2 A 1 -γ 2 A 2 A 2 (1-γ 2 ) > 1, we have that G t → β 1 β 2 A 1 -γ 2 A 2 A 2 (1-γ 2 ) β 2 (1-γ 2 )A 2 A 1 A 1 -γ 2 A 2 = β 1 A 1 .
Proof of Lemma 5. Let us consider such an equilibrium. Since A 1 < R 1 < A 2 , and R t = A 2 > A 1 , ∀t, agent 1 does not produce and only invests in the financial asset. Denote s i,t ≡ k i,t -b i,t . Agent 1 is lender, k 1,t = 0, s 1,t = -b 1,t , ∀t. We can compute that

s 1,0 = β 1 w 1,0 , s 1,t = β 1 R t s 1,t-1 ∀t ≥ 1 s 1,t = β t 1 R t • • • R 1 s 1,0
For agent 2, since A 2 > R 1 , her borrowing constraints at date 0 is binding: R 1 b 2,0 = γ 2 A 2 k 2,0 . Therefore, we have

A 2 k 2,0 -R 1 b 2,0 = (1 -γ 2 )A 2 k 2,0 s 2,0 = k 2,0 1 - γ 2 A 2 R 1 k 2,0 = R 1 R 1 -γ 2 A 2 s 2,0 , b 2,0 = γ 2 A 2 R 1 -γ 2 A 2 s 2,0
The budget constraints of agent 2 write c 2,0 + s 2,0 = w 2,0

c 2,1 + s 2,1 = (1 -γ 2 )A 2 k 2,0 = (1 -γ 2 )A 2 R 1 R 1 -γ 2 A 2 s 2,0
c 2,t + s 2,t = A 2 s 2,t , ∀t ≥ 2 s 2,t = k 2,t -b 2,t , ∀t ≥ 2.

From this and the FOCs, we can compute the individual saving s i,0 = β i w i,0 , ∀i

s 2,1 = β 2 (1 -γ 2 )A 2 R 1 R 1 -γ 2 A 2 s 2,0 s i,t = β i A 2 s i,t-1 , ∀t ≥ 2, ∀i = 1, 2.
We now look at equilibrium. From the market clearing condition i b i,t = 0, we have that

s 1,0 = b 2,0 = γ 2 A 2 R 1 -γ 2 A 2 s 2,0 ⇔ R 1 = γ 2 A 2 (1 + s 2,0 s 1,0 )
By consequence, we find the saving of all agents:

s i,0 = β i w i,0 , ∀i s 1,1 = β 1 R 1 s 1,0 = β 1 γ 2 A 2 (1 + s 2,0 s 1,0 )s 1,0 = β 1 γ 2 A 2 (s 1,0 + s 2,0 ) s 2,1 = β 2 (1 -γ 2 )A 2 R 1 R 1 -γ 2 A 2 s 2,0 = β 2 (1 -γ 2 )A 2 γ 2 A 2 (1 + s 2,0 s 1,0 ) γ 2 A 2 (1 + s 2,0 s 1,0 ) -γ 2 A 2 s 2,0 = β 2 (1 -γ 2 )A 2 (s 1,0 + s 2,0 ) s i,t = β i A 2 s i,t-1 = (β i A 2 ) t-1 s i,1 , ∀t ≥ 2, ∀i = 1, 2.
It remains to find the sequence of capital (k i,t ). We have, ∀t ≥ 1 k 2,0 = s 1,0 + s 2,0 k 2,t = s 2,t + b 2,t = s 2,t -b 1,t = s 2,t + s 1,t , ∀t ≥ 1

k 2,1 = β 1 R 1 s 1,0 + β 2 (1 -γ 2 )A 2 R 1 R 1 -γ 2 A 2 s 2,0 = β 1 s 1,0 γ 2 A 2 (1 + s 2,0 s 1,0 ) + β 2 (1 -γ 2 )A 2 γ 2 A 2 (1 + s 2,0 s 1,0 ) γ 2 A 2 (1 + s 2,0 s 1,0 ) -γ 2 A 2 s 2,0 = β 1 γ 2 A 2 (s 1,0 + s 2,0 ) + β 2 (1 -γ 2 )A 2 γ 2 A 2 s 1,0 +s 2,0 s 1,0 γ 2 A 2 s 2,0 s 1,0 s 2,0 = β 1 γ 2 A 2 (s 1,0 + s 2,0 ) + β 2 (1 -γ 2 )A 2 (s 1,0 + s 2,0 ) k 2,t = i s i,t = i (β i A 2 ) t-1 s i,1 , ∀t ≥ 1 = (β 1 A 2 ) t-1 β 1 γ 2 A 2 (s 1,0 + s 2,0 ) + (β 2 A 2 ) t-1 β 2 (1 -γ 2 )A 2 (s 1,0 + s 2,0 ) = A t 2 β t 1 γ 2 + β t 2 (1 -γ 2 ) (s 1,0 + s 2,0 )
We now check that the above list ((c i,t , k i,t , b i,t ) i , R t ) t is an equilibrium. We use Lemma 14. It is easy to verify the market clearing conditions and the FOCs.

Condition R 1 ∈ (A 1 , A 2 ) is ensured by the assumption that

A 1 < γ 2 A 2 (1 + s 2,0 s 1,0 ) < A 2
• We verify borrowing constraints: R t+1 b 2,t ≤ γ 2 A 2 k 2,t . This is satisfied for t = 0. Let us consider t ≥ 1. This means that k 2,t -s 2,t = b 2,t ≤ γ 2 k 2,t , or, equivalently, (1 -γ 2 )k 2,t ≤ s 2,t . So, we must prove, for any t ≥ 1,

s 1,t = k 2,t -s 2,t ≤ 1 1 -γ 2 -1 s 2,t = γ 2 1 -γ 2
s 2,t , ∀t ≥ 1.

These conditions are satisfied because β 2 > β 1 .

• Tranversality conditions: lim T →∞ β t i u i (c i,t )(k i,T -b i,T ) = 0. It is easy to verify these conditions because β i ∈ (0, 1) and u (c) = 1/c. Proof of Lemma 6. Let us consider such an equilibrium. Since A 1 < R 1 < A 2 < A 3 , and R t = A 3 , ∀t, agent 1 does not produce and only invests in the financial asset.

Denote s i,t ≡ k i,t -b i,t . Agent 1 is lender, k 1,t = 0, s 1,t = -b 1,t , ∀t. We can compute that

s 1,0 = β 1 w 1,0 , s 1,t = β 1 R t s 1,t-1 ∀t ≥ 1 s 1,t = β t 1 R t • • • R 1 s 1,0
For agent i=2, 3, since A 3 > A 2 > R 1 , their borrowing constraints at date 0 are binding: R 1 b i,0 = γ i A i k i,0 . Therefore, we have

A i k i,0 -R 1 b i,0 = (1 -γ i )A i k i,0 , ∀i = 2, 3 s i,0 = k i,0 1 - γ i A i R 1 k i,0 = R 1 R 1 -γ i A i s i,0 , b i,0 = γ i A i R 1 -γ i A i s i,0
The budget constraint writes c i,0 + s i,0 = w i,0

c i,1 + s i,1 = (1 -γ i )A i k i,0 = (1 -γ i )A i R 1 R 1 -γ i A i s i,0
c i,t + s i,t = A 3 s i,t , ∀t ≥ 2, ∀i = 2, 3 s 2,t = k 2,t -b 2,t = -b 2,t , k 2,t = 0, ∀t ≥ 1 s 3,t = k 3,t -b 3,t , ∀t ≥ 2.

From this and the FOCs, we can compute the individual saving

s i,0 = β i w i,0 , ∀i, s i,1 = β i (1 -γ i )A i R 1 R 1 -γ i A i s i,0 , ∀i = 2, 3
s i,t = β i A 3 s i,t-1 , ∀t ≥ 2, ∀i = 1, 2, 3.

We now look at equilibrium. From the market clearing condition i b i,t = 0, we have that

s 1,0 = b 2,0 + b 3,0 = γ 2 A 2 R 1 -γ 2 A 2 s 2,0 + γ 3 A 3 R 1 -γ 3 A 3 s 3,0 .
This is to determine the interest rate R 1 . We have known R t = A 3 , ∀t ≥ 2. It remains to find the sequence of capital (k i,t ). We have, ∀t ≥ 1 k 3,t = s 3,t + b 3,t = s 3,t -b 1,t -b 2,t = s 3,t + s 1,t + s 2,t , ∀t ≥ 1

k 3,1 = β 1 R 1 s 1,0 + i=2,3 β i (1 -γ i )A i R 1 R 1 -γ i A i s i,0 k 3,t+1 = i s i,t+1 = i β i A 3 s i,t , ∀t ≥ 1
To sum up, we find that s i,0 = β i w i,0 , ∀i, k 1,t = 0, ∀t ≥ 0, k 2,t = 0, ∀t ≥ 1

k i,0 = R 1 R 1 -γ i A i s i,0 , b i,0 = γ i A i R 1 -γ i A i s i,0 , ∀i = 2, 3.
From this, we can compute the aggregate output:

Y 1 = i A i k i,0 = A 2 R 1 R 1 -γ 2 A 2 s 2,0 + A 3 R 1 R 1 -γ 3 A 3 s 3,0 ∀t ≥ 2, Y t = A 3 i s i,t-1 = A 3 i (β i A 3 ) t-2 s i,1 = A 3 (β 1 A 3 ) t-2 β 1 R 1 s 1,0 + i=2,3 β i (β i A 3 ) t-2 (1 -γ i )A i R 1 R 1 -γ i A i s i,0 = A t-1 3 β t-1 1 R 1 s 1,0 + i=2,3 β t-1 i (1 -γ i )A i R 1 R 1 -γ i A i s i,0 = A t-1 3 β t-1 1 R 1 s 1,0 + β t-1 2 (1 -γ 2 ) A 2 R 1 R 1 -γ 2 A 2 s 2,0 + β t-1 3 (1 -γ 3 ) A 3 R 1 R 1 -γ 3 A 3 s 3,0
We now check that the above list ((c i,t , k i,t , b i,t ) i , R t ) t is an equilibrium. We use Lemma 14. It is easy to verify the market clearing conditions and the FOCs.

The financial market clearing condition at date 0 becomes s 1,0 = γ 2 A 2 R 1 -γ 2 A 2 s 2,0 +

γ 3 A 3 R 1 -γ 3 A 3 s 3,0 . So, condition R 1 ∈ (A 1 , A 2
) is ensured by the assumption that

γ 3 A 3 A 2 -γ 3 A 3 s 3,0 + γ 2 1 -γ 2 s 2,0 < s 1,0 < γ 3 A 3 A 1 -γ 3 A 3 s 3,0 + γ 2 A 2 A 1 -γ 2 A 2 s 2,0
• We verify borrowing constraints: R t+1 b 3,t ≤ γ 3 A 3 k 3,t . This is satisfied for t = 0. Let us consider t ≥ 1. since R t+1 = A 3 , ∀t ≥ 1, the borrowing constraint becomes k 3,t -s 3,t = b 3,t ≤ γ 3 k 3,t , or, equivalently, (1 -γ 3 )k 3,t ≤ s 3,t . So, we must prove, for any t ≥ 0, s 1,t + s 2,t = k 3,t -s 3,t ≤ 1 1 -γ 3 -1 s 3,t = γ 3 1 -γ 3 s 3,t .

Since s i,t = β i A 3 s i,t-1 , ∀t ≥ 2, ∀i = 1, 2, 3, and β 3 ≥ max(β 1 , β 2 ), it suffices to prove this condition at date 1. Condition s 1,1 + s 2,1 ≤ γ 3 1-γ 3 s 3,1 is equivalent to

β 1 R 1 s 1,0 + β 2 (1 -γ 2 )A 2 R 1 R 1 -γ 2 A 2 s 2,0 ≤ γ 3 1 -γ 3 β 3 (1 -γ 3 )A 3 R 1 R 1 -γ 3 A 3 s 3,0
This is satisfied. Indeed, since s 1,0 = γ 3 A 3 R 1 -γ 3 A 3 s 3,0 + γ 2 A 2 R 1 -γ 2 A 2 s 2,0 , we have

β 1 R 1 s 1,0 + β 2 (1 -γ 2 )A 2 R 1 R 1 -γ 2 A 2 s 2,0 =β 1 R 1 γ 3 A 3 R 1 -γ 3 A 3 s 3,0 + γ 2 A 2 R 1 -γ 2 A 2 s 2,0 + β 2 (1 -γ 2 )A 2 R 1 R 1 -γ 2 A 2 s 2,0 =β 1 R 1 γ 3 A 3 R 1 -γ 3 A 3 s 3,0 + (β 1 γ 2 + β 2 (1 -γ 2 )A 2 s 2,0 R 1 R 1 -γ 2 A 2 ≤β 1 R 1 γ 3 A 3 R 1 -γ 3 A 3 s 3,0 + (β 3 -β 1 )γ 3 A 3 s 3,0 R 1 R 1 -γ 2 A 2 = γ 3 1 -γ 3 β 3 (1 -γ 3 )A 3 R 1 R 1 -γ 3 A 3 s 3,0

Figure 4 :

 4 Figure 4: Effects of credit limits γ 2 on the aggregate output.

The reader is referred to[START_REF] Matsuyama | Aggregate Implications of Credit Market Imperfections[END_REF],Quadrini (2011), Brunnermeier, Eisenbach, and[START_REF] Brunnermeier | Macroeconomics with Financial Frictions: A Survey[END_REF] for more complete reviews on the macroeconomic effects of financial frictions and to[START_REF] Buera | Entrepreneurship and Financial Frictions: A Macro-development Perspective[END_REF] for the relationship between entrepreneurship and financial frictions.

While we focus on firm heterogeneity, there is a growing literature studying the roles of household heterogeneity in macroeconomics (the reader is refereed to[START_REF] Kaplan | Microeconomic Heterogeneity and Macroeconomic Shocks[END_REF] for an excellent review on this topic).

Moreover, if n + 1 = m (i.e., only agent m produces), we have ∂Y ∂γm = 0.

Here, we do not focus the equilibrium existence; see, for instance, Bosi, Le[START_REF] Bosi | Intertemporal equilibrium with heterogeneous agents, endogenous dividends and collateral constraints[END_REF].

This insight is consistent with that in a two-period model in[START_REF] Pham | Effects of credit limit on efficiency and welfare in a simple general equilibrium model[END_REF].

2. E ∈ A m if and only if S ≤ D m .

3. E ∈ A n with n ∈ {2, . . . , m-1} if and only if A n > max i (γ i A i ) and B n ≤ S ≤ D n . 4. R n ≡ R n,1 ∪ R n,2 with n ∈ {1, . . . , m -1} where (a) R n,1 is the set of economies such that A n > max i (γ i A i ) and D n+1 < S < B n .

(b) R n,2 is the set of economies such that A n+1 > max i (γ i A i ) ≥ A n and D n+1 < S.

We now prove (A.15a) which implies the existence of equilibrium. It suffices to verify that E ⊂ ∪ m i=1 A i ∪ ∪ m-1 i=1 R i . Let us consider an economy E. There are only two cases.

1. max i (γ i A i ) < A 1 . In this case, we have max i (γ i A i ) < A n ∀n. Therefore, it is easy to see that

2. There exists n ∈ {1, . . . , m -1} such that A n+1 > max i (γ i A i ) ≥ A n . There are two sub-cases.

(a) S > D n+1 . In this case, E ∈ R n+1,2 .

(b) S ≤ D n+1 . Recall that M < A n+1 . In this case, we will prove that

Proof of (A.15b). Observe that the equilibrium interest rate is unique if (A.15b) holds. We have to prove that:

Following (A.9), it is easy to see that the two first equalities hold. We now prove that A n ∩R h,2 = ∅ ∀n, h. Suppose that there exists

From these conditions we get A n > A h , and hence n ≥ h + 1. Thus, we obtain S > D h+1 ≥ D n ≥ S, a contradiction. Therefore, we have

So, we have that

Therefore, we have

Again, by the market clearing condition

we have that

and hence, lim

By combining this with (A.29), lim R→R(0) ∂k i ∂R < 0, ∀i, and lim

(A.30)

By combining (A.26), (A.25), (A.30), and lim R→R(0) ∂k j ∂R < 0, we get that ∂Y ∂A 1 < 0 for any A 1 > 0 small enough.

Proof of Proposition 4. First, we state a corollary of Theorem 1.

Corollary 6. Let Assumptions 2, 3 and 4 be satisfied. Consider a two-agent model and assume that R 1 < R 2 .

1. In the regime R 2 , i.e., when

17 Notice that we always have that

because we assume that γ

f 2 (S) . Again, by the market clearing condition

First, since

and hence, lim

By combining this with (A.37) and lim A 1 →0 R = R(0) > 0, we get that lim

It is easy to see that, when A 1 is small enough, the

By combining (A.34), (A.36), (A.38), (A.39) and (A.32), we conclude that ∂Y ∂A 1 < 0 for any A 1 > 0 small enough.

We now consider the Cobb-Douglas production functions. In such a case, condition

For the sake of simplicity, we write

C Proofs for Section 3.2

Proof of Proposition 9. Under assumptions in Proposition 9, we can prove that the equilibrium interest rate is in (A n , A n+1 ) if and only if

Agents 1, . . . , n are lenders while agents n + 1, . . . , m are borrowers. It is easy to see that ∂Y ∂γ i = 0, ∀i ≤ n. For i ≥ n + 1, by using condition m i=n+1

we can compute that

The first point is a direct consequence of this expression and the fact that

(A.44)

Combining this with the expression of ∂Y ∂γ i , we obtain (38a). We also have

(A.47)

Combining this with the expression of ∂Y ∂γ i , we obtain (38b). Proof of Example 1. We focus here on the case max(γ 2 A 2 , γ 3 A 3 ) < A 1 (in this case the interest rate R may take any value in [A 1 , A m ]). Applying Theorem 2, we can According to (B.3) and (B.4), we have

because of (B.6) and the fact that

The budget constraint at date t implies that λ

Proof of Remark 3. Steady state analysis. Let us focus on an interior equilibrium (i.e., k i,t > 0, ∀i, t), we can write the FOCs

where µ i,t ≥ 0 is the multiplier with respect to the constraint R t b i,t-1 -γ i F i,t (k i,t-1 ) ≤ 0.

According to FOCs and the fact that B i is increasing in b but decreasing in k, we have that 1 ≥ R t+1 max i β i u i (c i,t+1 ) u i (c i,t ) , ∀i. Since k i,t > 0, ∀i, ∀t, there exists an agent, say agent i, whose borrowing constraint at date t + 1 is not binding. It means that µ i,t+1 = 0. By consequence, we have 1 = R t+1 β i u i (c i,t+1 ) u i (c i,t )

= R t+1 max j β j u j (c j,t+1 ) u j (c j,t ) . Therefore, we have R = 1/ max i {β i } at steady state.

The first-order conditions imply that λ i,t R t+1 -γ i F i,t (k i,t ) R t+1

= F i,t (k i,t )λ i,t+1 (1 -γ i ). By consequence, we obtain point 2.

• Tranversality conditions: lim T →∞ β t i u i (c i,t )(k i,T -b i,T ) = 0. It is easy to verify these conditions because β i ∈ (0, 1) and u (c) = 1/c.