Self-fulfilling Business Cycles with Production Network ∗

Feng DongTsinghua University

Fei ZhouHong Kong Baptist University

May, 2021

Abstract

What is the role of production networks in inducing self-fulfilling business cycles? We builda continuous-time multisector business cycle model with input-output linkages and credit con-straints to study this. Credit constraints faced by productive firms endogenously create self-fulfilling business cycles: an expected decline in firm value tightens constraints and further de-presses equity value, generating a financial multiplier and thus self-fulfilling business cycles. The-oretically, we derive that the financial multiplier nests the input-output multiplier. We illustratethat the likelihood of self-fulfilling business cycles depends on intermediate input share througha “size effect” and a “diluting effect”: the combination of two effects has a U-shaped relation withthe financial multiplier. We also illustrate that the network structure has an important but ambigu-ous impact on self-fulfilling business cycles. Quantitatively, we demonstrate that tightening creditconstraints in sectors with higher Domar weights in the production network is more likely to leadto a self-fulfilling equilibrium.

Keywords: Production Networks, Credit Constraints, Financial Contagion, Endogenous Cy-cles, Indeterminacy.

JEL Classification: D24, E23, E32, E44.

∗We are thankful to Zhen Huo, Pengfei Wang, Yi Wen for their invaluable advice, and Jess Benhabib, David Cook,Yan Ji, Ernest Liu, Yang Lu, Micheal Song, along with seminar participants at Tsinghua University for useful discussionsand comments. Feng Dong acknowledges the financial support from the National Natural Science Foundation of China(#71903126). The usual disclaim applies.

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1. Introduction

Since the Great Recession, the role of financial frictions in shaping business cycle fluctuations hasbecome a central theme in economics. It is well known that financial shocks themselves can beimportant driving forces (see, among others, Jermann and Quadrini (2012), Eggertsson and Krugman(2012), and Guerrieri and Lorenzoni (2017)). More surprisingly, severer financial frictions may alsogive rise to self-fulfilling business cycles (Liu and Wang (2014)). On the other hand, a rapidly growingliterature on production networks (Gabaix (2011), Acemoglu, Akcigit, and Kerr (2016), Baqaee andFarhi (2020)) has pointed out that sectoral heterogeneities and input-output linkages are important inpropagating local shocks. The goal of this paper is to understand the effects of the joint presence ofcredit constraints and production networks in inducing self-fulfilling business cycles.

We build a continuous-time multisector business cycle model with input-output linkages andcredit constraints. In each production sector, a continuum of firms has access to a constant-return-to-scale technology that uses other sectors’ outputs as intermediate goods, but they possess hetero-geneous productivities and are subject to working capital constraints and fixed operational costs. Inthe absence of credit constraints, only the most productive firms operate. When such constraints arepresent, all firms with productivities higher than a threshold operate. Crucially, the credit constraintdepends on the aggregate economic conditions in an endogenous way: an increase in economy-wideTFP not only directly improves production efficiency but also increases firms’ equity value, whichin turn relaxes their credit constraints. What follows is a reduction of misallocation: the cutoff pro-ductivity increases, resources shift towards more productive firms, and misallocation is mitigated.This indirect channel can be viewed as a particular financial multiplier. In addition, when firms areinterconnected across sectors, an expansion in one sector boosts the demand for goods from othersectors, which in turn relaxes the credit constraints in other sectors. This interaction between thegeneral equilibrium feedback effects due to trade linkages and the financial multiplier caused bycredit constraints can induce sufficient amplification that makes the aggregate production functionappear to display increasing returns to scale.

Our first result is an exact analytical characterization of the financial multiplier with dynamiclinkage across sectors. We show that the rise of self-fulfilling business cycles hinges on the size ofaggregate financial multiplier. Holding primary input factors fixed, the direct effect on aggregateoutput of a TFP shock in one sector equals its cost-based Domar weight, which is generically largerthan the revenue-based Domar weight. These two Domar weights coincide only if the financial frictionvanishes, in which case the familiar Hulten’s theorem applies. Indirectly, the allocation efficiency inall sectors is improved. The indirect effects are governed by the magnitude of the fixed costs andproductivity dispersion in each sector. The financial multiplier therefore also nests the input-outputmultiplier, which is greater than one.

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Our second result involves the weight and the architecture of the production network and revealstheir impact on self-fulfilling equilibrium. We show that the financial multiplier is non-monotonicin both weight and the architecture of the production network. First, consider a case in which allsectors are symmetric. The financial multiplier displays a U-shape in intermediate input share. Therise of intermediate input share has two effects: a “size effect” and a “diluting effect”. The formereffect indicates that the sales of intermediate firms increase, and this tends to amplify the financialmultiplier as intermediate firms that are subject to the financial constraint expand. The latter effectoccurs because as sales expand, fixed operating costs become relatively less important, which dampensthe financial multiplier. On the other hand, we can vary the structure of the production network.The financial multiplier again displays a U-shape in interconnection between sectors. We find thatthe network structure has an important but ambiguous effect in inducing a self-fulfilling equilibrium,in the sense that making sectors more or less interconnected can either increase or decrease thelikelihood of economic indeterminacy.

Our third result quantitatively demonstrates that tightening sector-specific credit constraints hasheterogeneous effects on inducing self-fulfilling business cycles depending on a sector’s position inthe network. In general, tightening a sector’s credit constraint leads to a larger financial multiplier andhence a higher chance of self-fulfilling business cycles. However, the strength of this channel dependson the relative importance of this sector in the economy. An upstream sector is more “central”, and thecorresponding increase in the financial multiplier is more dramatic there. It follows that tighteningthe financial constraints in these more central sectors would more easily induce self-fulfilling businesscycles.

Overall, our results highlight how the nature of interactions determines the sectors’ relativeimportance in shaping aggregate self-fulfilling fluctuations.

Related Literature. To the best of our knowledge, our paper is the first to shed light on the roleof production networks in shaping self-fulfilling business cycles. Correspondingly, our paper mostdirectly draws from and contributes to the literature on self-fulfilling equilibria in real business cycles.Benhabib and Farmer (1994) first notes that increasing returns to scale can generate indeterminateequilibria. Farmer and Guo (1994), Basu and Fernald (1995), and Basu and Fernald (1997) examinethe quantitative importance of such indeterminacy. To generate empirically plausible increasingreturns, Galí (1993), Schmitt-Grohé (1997) and Wang and Wen (2008) resort to countercyclical markup,Benhabib and Farmer (1996) and Benhabib and Nishimura (2012) turn to two-sector model with mildwithin-sector externality, while Wen (1998) and Benhabib and Wen (2004) introduce variable capacityinto the discussion. We contribute to this research agenda by introducing a production networkthat can endogenously amplify increasing returns. Our paper also complements to the literature onexpectation-driven self-fulfilling business cycles. A partial list includes Benhabib, Wang, and Wen

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(2015), Chahrour and Gaballo (2017), and Acharya, Benhabib, and Huo (2021).Our paper also belongs to the active recent research agenda on production networks. Following

the pioneering contribution of Long and Plosser (1983) on multisector real business cycles, Foerster,Sarte, and Watson (2011), Gabaix (2011), Acemoglu, Carvalho, Ozdaglar, and Tahbaz-Salehi (2012),Acemoglu, Ozdaglar, and Tahbaz-Salehi (2015), Acemoglu, Akcigit, and Kerr (2016), Atalay (2017),Oberfield (2018), Baqaee (2018), Baqaee and Farhi (2018), Baqaee and Farhi (2019), Liu (2019), Al-tinoglu (2020), and Luo (2020), Bigio and La’o (2020), among others, enrich the theory and provideeconometric evidence. Additionally, see Carvalho and Tahbaz-Salehi (2019) for a comprehensive sur-vey on production networks. We apply the insights and tools developed by this body of work whichmainly focuses on static models to the possibility of indeterminate equilibria in RBC models.

The most related paper to ours is Liu and Wang (2014), which shows that financial frictions notonly amplify the cycles but can also generate self-fulfilling business cycles due to an endogenousaggregation with increasing returns to scale. There are mainly three key differences between Liu andWang (2014) and our paper. First, the model in Liu and Wang (2014) considers production economywithout intermediate goods input, while our paper considers a more complicated production network,treating Liu and Wang (2014) as a special case. That being said, the second most innovative part ofour paper is that we investigate the implications of the network structure in a multisector model andthe input-output structure is shown to shift the indeterminate region, i.e., the possibility region ofendogenous cycles. Lastly, since our model builds a richer production structure which allows forheterogeneous sectoral financial constraints, we are able to identify the key sector whose financialcondition is crucial for the stability of equilibrium.

The remainder of our paper is organized as follows. Section 2 introduces our basic theoreticalmodel. Section 3 studies effect on aggregate output of sectoral misallocation. Section 4 presentsour investigation of financial multiplier and self-fulfilling business cycles when altering weight andarchitecture of the production network. Section 5 evaluates the importance of sectors at differentpositions in the network in shaping self-fulfilling business cycles. Section 6 concludes the paper,while the Appendix contains all the proofs.

2. Model

Time is continuous and indefinite. The economy is populated by a continuum of homogeneoushouseholds, a continuum of final-goods producers and N continua of intermediate goods producers.

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2.1 Households

There is no intertemporal borrowing and lending for households or entrepreneurs. The only way totransfer wealth over time is by accumulating physical capital. 1

We model the worker side via a representative household with preferences given by

maxCh ,t ,Lt ,Ih ,t

E0

∫ ∞

0e−ρh t [

u(Ch ,t) − h(Lt)]

dt , (2.1)

where ρh is the discount factor, Ch ,t is the consumption rate, Lt is the labor supply, and Ih ,t is thecapital investment rate. We also impose that u(X) � log X and h(X) � ψX1+γ

1+γ , where ψ is the disutilityfrom working, and γ is the inverse Frisch elasticity. The budget constraint of the representativehousehold is

Ch ,t + Ih ,t ≤ Rt Kh ,t + Wt Lt , (2.2)

where Rt and Wt denote the capital rental price and wage, respectively. Kh ,t is capital owned bythe household. Households also have access to a linear technology to transform final goods intoinvestment goods Ih ,t . The law of motion of capital Kh ,t is

ÛKh ,t � −δKh ,t + Ih ,t .

The households take Rt and Wt as given and choose a path of consumption rate, working intensityand investment rate, Ch ,t , Lt and Ih ,t , respectively, to maximize their utility function (2.1). These implythat household Euler equations are given by

ÛCh

Ch� R − δ − ρh , (2.3)

WCh

� ψLγ . (2.4)

Next, we will turn to the problems for entrepreneurs. There is a continuum of final goodsproducers and N continua of intermediate goods producers, and each intermediate goods produceris labeled by i.

2.2 Final Goods Producers

The final goods producers simply assemble intermediate goods and have no access to any savingtechnology.2 The price of final goods is normalized to be 1. A representative final goods producertakes intermediate goods prices and the final goods (numéraire) price as given and chooses the

1This assumption is introduced to emphasize the impact of financial frictions on capital misallocation.2There is no intertemporal choice for the final goods producers whatsoever.

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amount of intermediate goods input to maximize its per period profit,

max{Xit }

{Yt −

N∑i�1

Pit Xit

}. (2.5)

Xit denotes the input of sector i’s intermediate goods, and Pit is the price for such intermediategoods. The production function for final goods Yt is given by

Yt �

N∏i�1

Xφiit .

We normalize the final goods production to be constant returns to scale, i.e.,∑N

i�1 φi � 1. Conse-quently, the marginal cost of intermediate goods input i must be equal to its marginal benefit in finalgoods production,

Xit �φiYt

Pit. (2.6)

2.3 Intermediate Goods Producers

Each entrepreneur owns a sector, and each sector is denoted by i ∈ {1, ...,N}. There exists a continuumof firms in each sector, and each firm is indexed by ι ∈ [0, 1]. Entrepreneurs have preferences

maxCe ,i ,Ie ,i

E0

∫ ∞

0e−ρe t [u(Ce ,it)] dt , (2.7)

where ρe is the discount rate for the entrepreneurs and Ce ,it is their consumption. The budgetconstraints for the entrepreneurs are

Ce ,it + Ie ,it ≤ De ,it + Rt Ke ,it . (2.8)

In the spirit of Kiyotaki and Moore (1997) and Liu and Wang (2014), we assume that ρhρe

issmall enough that entrepreneurs have no incentive to accumulate any capital in equilibrium, i.e.,Ie ,it � 0, Ke ,it � 0. Moreover, De ,it denotes the dividends received by the entrepreneurs, and Πit(ι) isthe profit of firm ι operating in sector i. Since all firms in sector i are solely owned by one entrepreneur,the dividends come from the total profits of all firms,

De ,it �

∫ 1

0Πit(ι)dι. (2.9)

Timing. Time is continuous. However, to provide a better illustration, we examine the results“within a given moment”. Entrepreneurs first have to decide for each firm whether to stay in businessor exit. If they stay, a fixed operating cost has to be paid. Only after the cost is paid are entrepreneurs

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able to observe each firm’s individual draw of productivity. Seeing the productivity shock, theentrepreneurs then decide whether a firm will produce or not based on the expected profit. If afirm operates, working capital is loaned to the firm. Once production is completed, the entrepreneurchooses to repay or default on the working capital loan.

Production Technology. Each firm ι in sector i has access to a constant returns to scale technologythat transforms capital, labor and intermediate goods into sector i goods,

Oit(ι) � Ait Zit(ι)[Kit(ι)<i Lit(ι)1−<i

]1−αM,i

N∏

j�1Si jt(ι)ωi j

αM,i

,

where Oit(ι) is the firm’s output and Pit is the price of this product. Ait is the sector-specificproductivity shock. An operating firm rents capital Kit(ι) from households at rental rate Rt , hires

labor Lit(ι) at the competitive real wage Wt , and employs a bundle of intermediate goodsN∏

j�1Si jt(ι)ωi j .

Specifically, Si jt(ι) denotes the intermediate input, which is produced by sector j and used by sectori at time t, and its price is P jt . Finally, Zit(ι) is the firm-specific productivity shock, assumed to bei.i.d. both over time and across firms. Fi (·) is the cumulative density function of Zit(ι). We normalizeEi [Zit(ι)] � 1 for all sectors. To obtain a sharp result, throughout the paper, we let Zi(ι) conform to aPareto distribution to obtain a closed-form solution, i.e., Fi(Zit) � 1 −

(ZitZi

)−ηiwith Z i � 1 − 1

ηi< 1.3

We can also write αK,i � (1 − αM,i)<i , αL,i � (1 − αM,i) (1 − <i) , αS,i j � αM,iωi j ; then, αK,i , αL,i andαS,i j represent the elasticities of output with respect to the capital input, labor input and intermediategoods input, respectively. Since we assume constant returns to scale for firm production, αK,i + αL,i +∑N

j�1 αS,i j � 1. As a result, the production function is equivalent to

Oit(ι) � Ait Zit(ι)Kit(ι)αK,i Lit(ι)αL,i

N∏j�1

Si jt(ι)αS,i j .

Profit. Define the profit of a firm ι in sector i asΠit(ι), which depends on the entrepreneur’s decisionon the firm being operative or inactive after observing the idiosyncratic shock Zit(ι),

Πit(ι) � max{O ,I}

{ΠO ,it(ι),ΠI ,it(ι)},

{O ,I} denotes the choice between operating and staying idle. ΠO ,it(ι) is the momentary profit of anoperating firm,

ΠO ,it(ι) � maxKit (ι),Lit (ι),Si jt (ι)

Pit Oit(ι) − Rt Kit(ι) − Wt Lit(ι) −N∑

j�1P jtSi jt(ι) −Φi . (2.10)

3Mathematical requirement: ηi > 2. Consequently, E [Zit (ι)] � 1, Var [Zit (ι)] � 1ηi (ηi−2)

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Φi is the fixed operating cost that firms have to pay to stay in business, and this fixed cost is prepaidto draw an individual productivity shock.

On the other hand, if the firm stays inactive for some period (because of a low productivity draw)but remains in the business in the hope of becoming profitable later, the fixed costΦi must neverthelessbe paid. The fixed cost is financed by issuing equity to the entrepreneur who owns this sector.4 Themomentary “profit” (actually loss from overhead costs) is ΠI ,it(ι),

ΠI ,it(ι) � −Φi .

Continuation Value. Define Vit as the continuation value for the firm to stay in business, which isalso the ex ante value of a firm before paying the fixed cost. Let Vit(ι) be an individual firm’s value;then, Vit �

∫Vit(ι)dι �

∫V(Zit(ι))dF(Zit(ι)). The continuation value takes the following recursive

form:Vit � −Φi + max{VI ,it(ι),VO ,it(ι)}.

VI ,it(ι) is the value of an inactive firm, and VO ,it(ι) is the value of an operative firm. As there isno current output, the value of an inactive firm VI ,it(ι) only comes from the future discounted value,

VI ,it(ι) � max{

e−ρ dtE

[Λe ,it+dt

Λe ,itVit+dt

], 0

}.

The stochastic discount factor is given by Λe ,it+dtΛe ,it

, where Λe ,it � u′(Ce ,it) is the marginal utility ofthe entrepreneur owning sector i. The value of an operative firm VO ,it(ι) comes from both currentprofit and future value,

VO ,it(ι) � max {VD ,it(ι),VC ,it(ι)} .

{D , C} denotes the choice between defaulting and committing. The value of a firm committing to itsworking capital loan is VC ,it(ι) and that of a defaulting firm is VD ,it(ι).

Working Capital Constraint. Firms do not make upfront payments to households and to otherfirms. Instead, all the working capital is loaned to the firms as credit. Since firms have limitedliability, the credit offered is tailored to provide a sufficient repayment incentive. If the firm choosesto commit to the loan, its value is given by

VC ,it(ι) �©­«Pit Oit(ι) − Rt Kit(ι) − Wt Lit(ι) −

N∑j�1

P jtSi jt(ι)ª®¬ dt + max{

e−ρe dtΛe ,it+dt

Λe ,itVit+dt , 0

}.

If the firm chooses to default, then it can seize all the revenue at the risk of being caught. The

4Note here that by law of large numbers, the firms making money still outweigh the dormant firms. Therefore,entrepreneurs always enjoy a positive amount of consumption.

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event of being caught is a Poisson process that arrives with intensity Θi . Should a defaulting firm becaught, it is deprived of all future access to credit; thus, the value of the defaulting firm is given by

VD ,it(ι) � Pit Oit(ι)dt + (1 −Θi dt)max{

e−ρe dtΛe ,it+dt

Λe ,itVit+dt , 0

}.

To guarantee there is no default on the equilibrium path, the incentive compatibility (IC) conditionrequires that VC ,it(ι) ≥ VD ,it(ι), i.e.,

Θi e−ρe dt max{Λe ,it+dt

Λe ,itVit+dt , 0

}dt ≥ ©­«Rt kit + Wt lit +

N∑j�1

P jt si jtª®¬ dt .

In the limit, the IC condition takes a very simple form:

Rt Kit(ι) + Wt Lit(ι) +N∑

j�1P jtSi jt(ι) ≤ ΘiVit ≡ B it . (2.11)

Effectively, Θi ∈ [0, 1] represents the degree of efficiency of the credit markets: Θi � 1 correspondsto a perfect credit market, and Θi � 0 corresponds to the case where the credit market is completelyshut down. We use B it to denote the endogenous borrowing limit in sector i.

Cutoff Productivity. Now, we can examine the allocation of resources across firms. We will show inLemma 1 that for each sector, there exists a cutoff productivity Z∗

it , above which the firm participatesin production; otherwise, the firm stays inactive.

Lemma 1. (Individual behaviors) For each sector i, there exists a cutoff value Z∗it , such that firms’ optimal

capital input, labor input and intermediate goods input are given by

Kit(ι) �αK,iB it

R1{Zit (ι)≥Z∗

it } , Lit(ι) �αL,iB it

W1{Zit (ι)≥Z∗

it } , Si jt(ι) �αS,i jB it

P j1{Zit (ι)≥Z∗

it } ,

and firms’ optimal borrowing amount and output are as follows:

Bit(ι) � B it1{Zit (ι)≥Z∗it } , Oit(ι) �

Zit(ι)Z∗

it

B it

Pit1{Zit (ι)≥Z∗

it } .

The cutoff productivity Z∗it is determined by

Z∗it ≡

1Ait

(Rt/Pit

αK,i

)αK,i (Wt/Pit

αL,i

)αL,i N∏j�1

(P jt/Pit

αS,i j

)αS,i j

.

This lemma establishes that only firms with sufficiently high productivities choose to operate, andall operating firms will borrow to the limit. Because firms within a sector produce homogeneous goods

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and the marginal revenues from producing such goods are the same for all firms, the same amountsof capital, labor and intermediate goods are hired by heterogeneous operating firms. However, firmswith higher idiosyncratic productivities effectively enjoy a lower marginal cost of producing. As aresult, firms will borrow to their limit whenever their marginal costs are lower than their marginalrevenues, and such limit is regulated by a common sector-level credit constraint. However, since theproductivities are different, firms with higher productivities will produce more even with the sameamount of input.

Define the sector-level aggregate inputs as Kit , Lit , Si jt and sector-level output as Oit . By the lawof large numbers, the sector-level variables are given by

Kit � (1 − Fi(Z∗it))

αL,iB it

R, Lit � (1 − Fi(Z∗

it))αL,iB it

W, Si jt � (1 − Fi(Z∗

it))αS,i jB it

P j,

Oit �

∫Zit(ι)

Z∗it

B it

Pitdι �

B it

Pit Z∗it

∫Z∗

it

Zit(ι)dF(Zit(ι)). (2.12)

Given the sector input{Kit , Lit , Si jt

}, the sector output can also be written as

Oit �

∫ 1

0Ait Zit(ι)

Kit(ι)αK,i Lit(ι)αL,i

N∏j�1

SαS,i j

i jt

dι � Ait · KαK,iit LαL,i

it

N∏j�1

SαS,i j

i jt , (2.13)

where we denote Ait � AitEi

(Zit |Zit ≥ Z∗

it

)> Ait as the endogenous sector-specific productivity.

The production function is observationally equivalent to an economy where firms in each sector arehomogeneous and operate at decreasing returns to scale (DRS); the details are shown in Appendix C.

2.4 Input-Output Linkages

In this section, we introduce input-output linkages between sectors. We define input-output matrices,Leontief inverse matrices and Domar weights in this economy. This section builds on concepts widelyused in the literature on production networks (see, for example, Baqaee and Farhi (2018) and Baqaeeand Farhi (2020)).

Sectoral Expenditure Shares. To properly define the input-output matrices, we first examine theexpenditure share for each sector. Sector i’s expenditures on inputs from sector j are P jtSi jt . Sectori’s expenditure from sector j is a fraction of sector i’s revenue by (2.12), and accordingly, one can seethat sector i’s demand for intermediate goods, labor and capital are given by

Si jt � αS,i jtPit Oit

P jt, Lit � αL,it

Pit Oit

Wt, Kit � αK,it

Pit Oit

Rt, (2.14)

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where αS,i jt ≡ αS,i jZ∗

itEi(Zit (ι)|Zit (ι)≥Z∗

it), αL,it ≡ αL,i

Z∗it

Ei(Zit (ι)|Zit (ι)≥Z∗it), αK,it ≡ αK,i

Z∗it

Ei(Zit (ι)|Zit (ι)≥Z∗it)

are de-fined as endogenous expenditure shares.

Assumption. Eit

(Zit (ι)

Z∗it

|Zit ≥ Z∗it

)≥ 1 weakly decreases with Z∗

it . 5

Equation (2.14) shows that Ei

(Zit (ι)

Z∗it

|Zit(ι) ≥ Z∗it

)creates a wedge between the sector-level endoge-

nous expenditure shares and the primitive firm-level elasticities of output with respect to variableinputs. Note that Ei

(Zit (ι)

Z∗it

|Zit(ι) ≥ Z∗it

)measures the relative gap between the average and the lower

bound of an operating firm’s productivity. Assumption 2.4 basically states that such a gap weakly nar-rows when the productivity floor is raised. This is a very mild assumption because intuitively when theproductivity floor increases, the dispersion of productivities decreases. Since we assume the Paretodistribution for idiosyncratic productivity shocks, we immediately have Ei

(Zit (ι)

Z∗it

|Zit(ι) ≥ Z∗it

)�

1Zi

,which is a constant and independent of Z∗

it , i.e., assumption 2.4 is satisfied.

Input-output Matrices. We define the revenue-based input-output matrix to be αSt , the i j-th elementof which is sector i’s expenditure on the intermediate goods from sector j as a share of i’s total revenue

αS,i jt �P jtSi jt

Pit Oit.

Analogously, define the cost-based input-output matrix to be αSt , the i j-th element of which is

αS,i jt �P jtSi jt

Wt Lit + Rt Kit +∑N

j�1 P jtSi jt,

which also corresponds to the elasticity of output with respect to intermediate goods input fromsector j.

Leontief Inverse Matrix. We define the revenue-based Leontief inverse matrix as

Ψt �(I − α′

S,t)−1

�

∞∑k�0

(α′

S,t) k.

Intuitively, the i j-th element of Ψt is a measure of i’s total reliance on j as a supplier. Furthermore,we define the cost-based Leontief inverse matrix as

Ψt �

(I −α′

S,t

)−1�

∞∑k�0

(α′

S,t

) k.

5This assumption is satisfied for a large family of distributions, including Pareto, uniform, log-normal, power, Weibull,Frechet, Logistic, etc. In particular, we can easily verify that Ei

(ZiZ∗

i|Zi ≥ Z∗

i

)is a constant for the Pareto distribution, and

strictly decreases with Z∗i for the other aforementioned distributions.

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The i j-th element of Ψt records the direct and indirect exposures of the cost of i to the price of j

through the production network. Note that this is still a partial-equilibrium elasticity where factorprices are considered fixed.

Domar Weights. We define the revenue-based Domar weight λit of the intermediate goods produceri as its sales share as a fraction of aggregate output

λit �Pit Oit

Yt.

In general,∑N

i�1 λi > 1, since there are not only final sales but also intermediate sales. Domar weightsare a measure of sector weights when we integrate or aggregate all the sectors.

Lemma 2. (Monotonicity of Domar weights) We can decompose the Domar weights 6

λt �(I − α′

S,t)−1

φ � Ψtφ,

where

λ �

λ1...

λN

, φ �

φ1...

φN

, αS,t �

αS,11t . . . αS,1Nt...

...

αS,N1t . . . αS,NNt

.Under Assumption 2.4, λit weakly increases with Z∗

it .

Lemma 2 shows that the more efficient a sector is, the more important this sector will be in theproduction network. Moreover, from (2.6), sector i’s sales to final goods producers are a fraction ofits total sales, Xit �

φi

λitOit .

Analogously, define the cost-based Domar weight as λi to measure the importance of i as a supplierfor final goods producers,

λt �

(I −α′

S,t

)−1φ �Ψtφ.

Because of the endogenous wedge created by the credit constraints (Eit

[Zit (ι)

Z∗it

|Zit ≥ Z∗it

]≥ 1), the

revenue-based and cost-based input-output matrices always satisfy αS ≤ αS. It is then easy to showthat the revenue-based Domar weight for any given sector is never larger than the cost-based Domar

6Note that if there were no within-sector heterogeneity or if there were no financial frictions, then Z∗it

Ei

(Zit (ι)|Zit (ι)≥Z∗

it

) → 1,

and thus αS,i jt → αS,i j , αL,it → αL,i , αK,it → αK,i . Moreover, αS,t → αS and λt → λ are all constants, which immediatelyimplies that the Hulten theorem is satisfied in the polar case where economy is frictionless. Additionally, we can compareour microfounded results with the reduced-form setup by Bigio and La’O (2016), whose model is static, and the wedge isexogenously postulated.

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weight,

λ �

∞∑n�0

(α′

S)n

φ ≤∞∑

n�0

(α′

S

)nφ � λ.

Final Expenditure Shares. The economy-wide aggregate demands for labor and capital are givenby

Lt �

N∑i�1

Lit �

N∑i�1

αL,itPit Oit

Yt

Yt

Wt� αL,t

Yt

Wt, (2.15)

Kt �

N∑i�1

Kit �

N∑i�1

αK,itPit Oit

Yt

Yt

Rt� αK,t

Yt

Rt. (2.16)

Let αL,t , αK,t be the total expenditure shares of labor and capital,

αL,t ≡ λ′tαL,t , αK,t ≡ λ

′tαK,t ,

Aggregate Output and Aggregate Capital. When we aggregate the whole economy, final goodsproduction can be expressed as a Cobb-Douglas production function using aggregate capital andaggregate labor.

Proposition 2.1. (Aggregate output and endogenous TFP) The aggregate output is given by

Yt � At KαK,tt L

αL,tt , (2.17)

where the aggregate elasticities of final goods output with respect to capital and labor are given by αK,t �

λ′tαK , αL,t � λ′

tαL, with αK,t + αL,t � 1. We define endogenous TFP as

At � ζt

N∏i�1

Aλitit ,

where Ait � AitEi

(Zit |Zit ≥ Z∗

it

)> Ait can be interpreted as endogenous sector-specific productivity. More-

over, we define ζt as the TFP component that is affected by the allocation of factors and intermediate goods,

ζt �

N∏i�1

kαK,iit lαL,i

it

N∏j�1

(αS,i jt

λit

λ jt

)αS,i j λit

·N∏

i�1

(φi

λit

)φi

,

with kit � Kit/Kt and lit � Lit/Lt being the capital and the labor share employed by each sector.

The endogenous TFP is the cost-based Domar-weighted average product of all sectors’ endogenousproductivities, corrected by the general equilibrium effect from the resource allocation.

13

GDP is calculated by the value-adding approach,

GDP � Yt −N∑

i�1Pit Xit +

N∑i�1

Pit Oit −N∑

j�1P jtSi jt −Φi

� Yt −

N∑i�1

Pit Xit +

N∑i�1

Pit©­«Xit +

N∑j�1

S jitª®¬ −

N∑i�1

N∑j�1

P jtSi jt −N∑

i�1Φi � Yt −

N∑i�1Φi .

Another observation is that the accumulation of aggregate capital is given by

ÛKt � −δKt + (αK,t + αL,t)Yt − Ch ,t . (2.18)

It follows that the accumulation of capital is related to aggregate output Yt (given by equation(2.17)) and household consumption Ch ,t (given by equation (2.3)). Note that the households’ decisionsregarding labor supply Lt and consumption Ch ,t are all nonlinear functions of Kt . In addition,(αK,t + αL,t) could also be highly nonlinear in principle since Z∗

it is a complicated function of Yt .Therefore, the law of motion of capital is highly nonlinear.

2.5 Equilibrium Definition and Characterization

Market Clearing Conditions. The markets for capital, labor and intermediate goods all clear

Kh ,t �

N∑i�1

Kit , Lt �

N∑i�1

Lit , Oit � Xit +

N∑j�1

S jit . (2.19)

In addition, summing up entrepreneurs’ and workers’ budget constraints and by the aforementionedmarket clearing conditions, we can obtain the aggregate resource constraints in the economy,

Yt −N∑

i�1Φi � Ch ,t + Ce ,t + Ih ,t + Ie ,t . (2.20)

Competitive Equilibrium. Given sequences of sector-level productivities Ait and distributions offirm-specific productivities F(Zit), a competitive equilibrium consists of a set of prices {Pit , Rit ,Wit},the consumption, labor supply and investment choices of the households

{Ch ,t , Lt , Ih ,t

}, the con-

sumption and production choices of the entrepreneurs {Ce ,it}, the intermediate goods input chosenby final goods producers {Xit}, the capital, labor and intermediate goods input, and the borrowingamount chosen by the intermediate goods producers

{Kit(ι), Lit(ι), Si jt(ι)

}such that:

i. households maximize the utility (2.1) subject to their budget constraints (2.2);

ii. entrepreneurs maximize their utility (2.7) subject to their budget constraints (2.8);

14

iii. final goods producers choose intermediate goods inputs to maximize their profits (2.5);

iv. intermediate goods producers choose capital, labor and intermediate goods inputs to maximizetheir profits (2.10) subject to the collateral constraint (2.11); and

v. the markets for all goods and factors clear by (2.19) and (2.20).

3. Sectoral Misallocation

Due to imperfect contract enforcement, productive firms cannot operate at their full capacities, andtheir production is restricted by working capital constraints. This leads to misallocation and the sup-pression of aggregate productivity. In this section, we will consider the impact of credit constraints onaggregate productivity through the channel of misallocation. Apparently, without credit constraints,only firms with the highest productivities operate. However, when credit constraints take effect, lessproductive firms can also participate, and such misallocation lowers average productivity. We nowdraw a connection between credit constraints and aggregate productivity. The ultimate goal is to fullycharacterize the aggregate output.

Debt Limits. With the presence of working capital constraints, an increase in sectoral output willhave an additional effect of relaxing the borrowing limits, which in turn drives up the averageproductivity in this sector and results in yet higher sectoral output. Credit constraints create apositive feedback loop that amplifies productivity shocks.

Proposition 3.1. Given the vector of cutoff productivities Z∗t �

[Z∗

1t , ..., Z∗Nt

] ′ and aggregate output Yt , theloan-to-output ratio depends on the tightness of the credit constraints and the profitability of the average firm,

B it

Yt�Θi

ρe

[ξ

(Z∗

it

)− ϕit

]≡ f (Z∗

it), (3.1)

where ϕit � Φi/Yt . ξ(Z∗it) − ϕit is the expected marginal profit for a firm,

ξ(Z∗it) � λit

©­«1 − αK,it − αL,it −N∑

j�1αS,i jt

ª®¬On the other hand, from the credit demand side (2.12), there is a monotonic mapping between Z∗

t and the sectorloan-to-output ratio,

B it

Yt�

λit Z∗it∫

Z∗it

Zit(ι)dF(Zit(ι))≡ g(Z∗

it). (3.2)

In addition, f (Z∗it) is a weakly decreasing function of Z∗

it , while g(Z∗it) is weakly increasing in Z∗

it .

15

Proposition 3.1 pins down the supply of credit (3.1) and the demand of credit (3.2).

Cutoff Productivities. Given the aggregate output Yt , the cutoff productivities can be uniquelydetermined if we combine (3.1) and (3.2),

Θi

ρe

[ξ

(Z∗

it

)− ϕi

]�

λit Z∗it∫

Z∗it

Zit(ι)dF(Zit(ι)).

Note that since the idiosyncratic productivities conform to the Pareto distribution, αS,i jt , αL,it , αK,it ,

λit , λit , ξit , kit , lit , αK , αL are all constants such that

αS,i j � αS,i jZ i , αL,i � αL,iZ i , αK,i � αK,iZ i ,

ki �Kit

Kt�

αK,i λi∑Ni�1 αK,i λi

, li �Lit

Lt�

αL,i λi∑Ni�1 αL,i λi

.

Similarly, the firms’ expected profit not excluding fixed costs is also a constant,

ξi �

1 − ©­«αK,i + αL,i +

N∑j�1

αS,i jª®¬ Z i

λi �(1 − Z i

)λi � λi/ηi .

The cutoff productivities can immediately be connected to aggregate output,

Z∗it �

[Θi

ρe(ηi − 1)

(1 − ηiΦi

λiYt

)]1/ηi

Z i . (3.3)

This result shows that Z∗it is increasing in Θi . If contract enforcement is stronger (i.e., Θi is higher),

more credit will be made available for the more productive firms, and the resources will shift towardsthose firms. In accordance, the cutoff productivity Z∗

it will be increased. By the same token, Z∗it also

strictly increases in Yt , provided that the fixed cost is positive (Φi > 0). Higher aggregate outputinflates the value of productive firms, which in turn relaxes the credit constraints for more productivefirms and facilitates resource reallocation to the more productive firms.

Corollary 1. (Aggregate output) The equilibrium outcome of total output Yt is given by 7

Yt � ζN∏

i�1

[Ait

Z∗it

Z i

]λi

KαKt L

αLt � ζ

N∏i�1

[Aλi

it

[Θi

ρe(ηi − 1)

(1 −

ηiϕi

λi

)]λi/ηi]

KαKt L

αLt , (3.4)

7If N � 1, i.e., for the one-sector model, then immediately ζ � 1, and the aggregate output in equation (2.17) is simplyreduced to

Yt � AtE(Zt |Zt ≥ Z∗

t)· KαK

t LαLt ,

This coincides with the one-sector model in Moll (2014), Liu and Wang (2014), where At is the aggregate TFP shock andαK , αL are the capital share and labor share.

16

where

ζ �

N∏i�1

kαK,ii lαL,i

i

N∏j�1

(αS,i j

λi

λ j

)αS,i j λi

·N∏

i�1

(φi

λi

)φi

, αK � λ′αK , αL � λ′αL .

Amplification Effect. Our intuition regarding the self-fulfilling business cycles hinges on an en-dogenous amplification mechanism. To understand how credit constraints amplify business cyclefluctuations, let us consider a hypothetical increase in sector-specific productivity Ai and see how thelabor market responds.

On the labor supply side, there exist three competing forces. First, the marginal productivity oflabor increases, as does the wage. The wage effect will boost labor supply. Second, a higher marginalproductivity of capital drives up the interest rate. The interest rate effect makes households morewilling to work today. Third, the production expansion allows households to earn a higher income.The wealth effect makes that households desire more leisure and supply less labor. The overall changein supply of labor is determined by the relative forces of these three effects. Under usual configuration,the labor supply curve moves up. On the labor demand side, there are also three forces taking effect,each reinforcing the other. First, when the marginal productivity of labor increases, firms tend tohire more workers. Second, with the presence of credit constraints, production expansion leads toan increase in firm value and thus enables productive firms to borrow more. This reallocation effectdrives up the sector-level endogenous productivity Ait and allows firms to hire more workers. Third,an increase in sector i’s productivity Ai will also positively affect other sectors’ production throughthe input-output linkages, which creates positive reallocation effects throughout the economy. Wecall this effect as “reallocation through network effects”. All three forces will shift the labor demandcurve out.

The equilibrium employment is jointly determined by the supply and demand side of the labormarket as discussed above. But no matter what direction the equilibrium employment eventuallymoves, the financial constraints and input-output linkages are more likely to amplify such movement.

With credit constraints, the loan-to-output ratio responds more than proportionately to changesin total output, because higher output reduces the average fixed cost and thus alleviates the effectivecredit constraint. The loan-to-output ratio is therefore procyclical. On the other hand, the cutoffproductivity is also procyclical. This is because with higher output, the credit and labor shift towardsmore efficient firms. This reallocation effect further raises endogenous sector productivity and therebyreinforces the initial increase in sector level productivity shock.

Moreover, in the presence of the production network, the general equilibrium effect of reallocationis further amplified because a higher aggregate output simultaneously relaxes credit constraints forall sectors. But of course, a sector level productivity shock may have mixed effect on the total outputdepending on the architecture of the network and the position of the sector in the network.

17

4. Multisector Business Cycles

We have established that the reallocation effect and production network are essential for amplifyingproductivity shocks. We now turn to studying self-fulfilling business cycles. We will show that creditconstraints together with the production network can make the aggregate production observationallyequivalent to an economy that operates at increasing returns to scale, thus making it more likely tobe exposed to self-fulfilling fluctuations.

4.1 Steady State

This section studies the steady states of the model. Suppose all sectors share the same level of with-insector heterogeneity: ηi � η, then in the steady state, the aggregate output solves following nonlinearequation

Yη+∑λi−ηαK

(YK

)ηαK

�

(ζ

N∏i�1

Aλii LαL

)η N∏i�1

[Θi

ρe(η − 1)

(Y − ηΦi

λi

)]λi

, (4.1)

where

ζ �

N∏i�1

kαK,ii lαL,i

i

N∏j�1

(αS,i j

λi

λ j

)αS,i j λi

·N∏

i�1

(φi

λi

)φi

, αK � λ′αK , αL � λ′αL .

Figure 1 shows how steady-state output is determined.8 The blue solid curve plots the left-hand sideof equation (4.1), while the red dashed curve plots the right-hand side. The intersection of two curvesyields two steady states YL ,YH . However, here we will mainly focus on the high steady state YH .

8Parameters: Φ � 0.005,φ �

[0.50.5

], αM � 0.5, η � 6,αS � αM ·

[a 1 − a

1 − b b

], a � 0.3, b � 0.3, γ � 0, ψ � 1, δ � 0.025.

18

Figure 1: Steady-state aggregate output

4.2 Local Dynamics

Let us now establish the possibility of a continuum of equilibria around the steady state in this section.In order to characterize the local dynamics around steady state, we henceforth work with the log-linearized solution around a steady state. We denote the steady state of Xt as X and the percentagedeviation of Xt from the steady-state as Xt .

Lemma 3. Log-linearizing equations (3.3) and (3.4) yields

Z∗it � υiYt , (4.2)

Yt �

N∑i�1

λiAit +

N∑i�1

λi Z∗it + αKKt + αL Lt , (4.3)

where υi �Φi

Yλi−ηiΦi�

ϕi

λi−ηiϕi�

(λi YΦi

− ηi

)−1.

Lemma 3 highlights a positive feedback loop in local dynamics. Let us still consider a hypotheticalimprovement in sector-specific productivity Ai . From equation (4.3), this will lead to an increase intotal output Yt . How does an increase in Yt affect the sectoral productivity cutoff Z∗

it? From equation(4.1) we know that in the steady state, the admissible parameter space is λi Y

Φi> ηi , i.e. sectoral fixed cost

Φi is relatively unimportant compared with sectoral sales λiY and/or within-sector heterogeneity issubstantial. Therefore, υi > 0 and equation (4.2) underscores that a higher aggregate output improvesallocation efficiency in sector i. We call this reallocation effect: a higher total output helps alleviatethe effective sectoral financial constraint and improves the sectoral productivity by reallocating creditto more productive firms. Such reallocation effect is more manifest with even lower fixed cost andhigher degree of heterogeneity. Note that as ηi → ∞, i.e., as firm heterogeneity vanishes, the effect of

19

aggregate output Yt on cutoff productivity Z∗it wanes. Equation (4.3), on the other hand, states that,

given (Kt , Lt), a positive sector-specific TFP shock Ait and an improvement in allocation efficiency Z∗it

contribute to boosting aggregate output. 9

Financial Multiplier. The joint work of production network and credit constraints create a financialmultiplier in local dynamics. Combining equations (4.2) and (4.3) yields

Yt � µ ·[

N∑i�1

λiAit + αKKt + αL Lt

], (4.4)

where µ is defined as the financial multiplier, which is given by

µ ≡ 11 −∑N

i�1 υiλi�

11 − ϱ .

Here, we write ϱ �

N∑i�1υiλi . Obviously, ϱ �

N∑i�1υiλi is sufficient to represent the financial multiplier µ.

Figure 2: Financial Multiplier

We illustrate in Figure 2 the mechanism through which credit constraints together with productionnetwork can generate an endogenous financial multiplier. An increase in output enables high-productivity firms to borrow more and produce more. As a result, it leads to reallocation that impliesa higher endogenous sector-specific productivity Ait by raising the sector-specific productivity cutoffZ∗

i . This reallocation effect generates the sector-specific financial multiplier υi (shown in equation(4.2)). On the other hand, the increase in sector-level productivity results in higher aggregate outputthrough the production network, and this sector-level network effect is measured by Domar weight λi .Thus, increased output Y leads to a ϱ-fold additional increase in output Y. The one-round financialmultiplier ϱ is a cost-based, Domar-weighted average of all sector-specific financial multiplier υi . Thisone-round financial multiplier will take effect in infinite rounds, as such an additional increase inoutput Y will lead to a new round of multiplication and so forth. Therefore, the total effect of the

9At the extensive margin, υi � 0 if Φi � 0, i.e., there is no fixed cost in sector i. The fixed cost is not essential forour results. Instead, it is obvious that what matters is the endogenously procyclical leverage ratio Bit/Yt , as indicated byequation (3.1). If Φi � 0, but instead borrowing constraintΘit is endogenous, we can still have a positive effect of Yt on Z∗

it ,which is qualitatively similar to the findings in equation (4.2).

20

financial multiplier would be µ �1

1−ϱ . For the convenience of exposition in the following sections, ϱand µ are interchangeably referred to as economy-wide financial multipliers, as they are one-to-onemapped.

Generalized Hulten Theorem. The aggregate effects of productivity shocks on total output varyacross sectors. There are two layers of amplification for uniform sector-level technology shocks:amplification through the production network and amplification through credit constraints.

Proposition 4.1. (Response of output to sector productivity shocks) As shown by equations (4.3) and(4.4),

∂ ln Yt

∂ ln Ait� λi ,

N∑i�1

∂ ln Yt

∂ ln Ait�

N∑i�1

λi ≡ χ > 1 (network multiplier),

d ln Yt

d ln Ait�

λi

1 −∑Ni�1 υiλi

� λiµ > λi > λi (network multiplier + financial multiplier).

Therefore the Hulten theorem fails to hold here since the first-order impact on output of a TFPshock to an industry λiµ is larger than that industry’s sales as a share of output λi . The Hultentheorem holds if and only if either of three scenarios occurs: (a) Var(Zit) � 0, i.e., ηi → ∞; thus, thereexists no firm heterogeneity. (b) Φi � 0; thus, the leverage ratio B it/Yt is constant, and there is nomisallocation due to financial frictions. (c) Θi → ∞; only the most productive firm produces.

On the other hand, χ captures the percentage change in output in response to a uniform one-percent increase in all sectors’ technology. It captures a notion of returns to scale at the aggregatelevel. The amplification of this uniform technology shock arises because goods are reproducible.

Aggregate Increasing Returns. It is direct to see that endogenous increasing returns to scale (IRS)emerges in reduced-form aggregate production function,

∂Yt

∂Kt+∂Yt

∂Lt� µ > 1.

In particular, We do not need Φi > 0 for all i ∈ N to obtain IRS,

∃ i ∈ N such that Φi > 0 �⇒ µ �1

1 −∑Ni�1

λiΦi

λi Y−ηiΦi

> 1.

This condition describes that as long as the fixed cost is present in at least one sector, the endoge-nous IRS would always emerge in this economy. Next we will show that the equilibrium indeterminacystems from endogenous IRS.

Indeterminate Region. The reduced-form aggregate production in our model exhibits increasingreturns to scale, which makes the economy prone to self-fulfilling sunspot-driven business cycles.

21

Now we examine when sunspot-driven fluctuations are likely to occur. We define the range ofparameters that can induce multiple equilibria as “indeterminate region”.

We focus on local dynamics around the steady state and abstract from fundamental shocks. Theequilibrium can be summarized as the following log-linearized system of equations:

Yt � µ(αKKt + αL Lt

),

Lt �1

1 + γ

(Yt − Ch ,t

),

ÛKt � (αK + αL)YK

(Yt − Kt

)− Ch

K

(Ch ,t − Kt

),

ÛCh ,t � αKYK

(Yt − Kt

).

Lemma 4. Without fundamental shocks, the perfect foresight equilibrium can be equivalently summarized bythe following log-linearized system of equations in Yt and Kt :[ ÛYt

ÛKt

]� J

[Yt

Kt

].

The Jacobian matrix is given by J �

[x11 x12

x21 x22

]. Each element in the Jacobian matrix is

x11 �YKαK − ϵCK(αK + αL)

ϵCY+

Ch

KϵCK , x12 �

Ch

KϵCK (ϵCK − 1)

ϵCY+

YKϵCK (αK + αL) − αK

ϵCY,

x21 �YK(αK + αL) −

Ch

KϵCY , x22 �

Ch

K(1 − ϵCK) −

YK(αK + αL) ,

where

ϵLK �−αK

αL, ϵLY �

1µαL

, ϵCK �αK(1 + γ)αLµ

, ϵCY �αLµ − 1 − γ

αLµ,

YK

�δ + ρh

αK,

Ch

K�

(δ + ρh

) αK + αL

αK− δ.

Following the seminal works by Benhabib and Farmer (1994) and the important extensions byWen (1998) and Benhabib and Wang (2013), among others, we know that the necessary and sufficientconditions for the existence of sunspot-driven fluctuations are det (J) > 0 and tr (J) < 0.

Proposition 4.2. (Equilibrium indeterminacy) The necessary and sufficient condition for equilibrium inde-terminacy is given by

max{µ∗

1 , µ∗2}< µ < µ∗

3 ,

22

where

µ∗1 �

1 + γ

αL, µ∗

2 �δ(1 + γ

)αK+αLαK

(δ + ρh

) (1 + γ

)αK − αLρh

, µ∗3 �

1αK.

Proposition 4.2 implies that equilibrium indeterminacy arises if the size of financial multiplierµ falls within certain range. µ∗

1 and µ∗2 jointly determine the lower bound of the indeterminacy

range. Notice that indeterminacy will only arises when a hypothetical increase in expected futureequity value can be validated in the equilibrium. With a reasonably high financial multiplier µ,the reallocation effect can generate sufficiently large increment in aggregate TFP. This makes thelabor demand shift sufficiently up to offset the wealth effect on labor supply side. As a result, theequilibrium employment and the expected future equity value indeed increase as expected. On theother hand, µ∗

3 defines the upper bound. If µ > µ∗3, the amplification effect is too strong to the

extend that it generates endogenous growth rather than endogenous cycle. Notice that both the sizeof financial multiplier µ and its range that creates indeterminacy are influenced by the productionnetwork.

Next, we will discuss some characteristics of the local dynamics. Since we are particularly inter-ested in the role of the production network, we will sketch how the network weight and architecturealter the likelihood of sunspot-driven fluctuations. In the first exercise, we set the network structureto be symmetric and demonstrate how the intermediate input share can impact the formation ofself-fulfilling business cycles. The second exercise turns to the asymmetric production network andinspects which network structure is more likely to drive the economy into multiple equilibria.

4.2.1 Altering the Intermediate Input Share

We now address how the intermediate input share alters the economy-wide financial multiplier. Forcomputational convenience, we assume the following symmetric input-output table:

αS � αMω �

[αM 00 αM

] [ω 1 − ω

1 − ω ω

].

We fix the input-output linkage ωwhile altering the intermediate input share αM . We assume thatthe ratio of total fixed costs to final goods output is constant to shut down the pro-cyclical leveragechannel emphasized in Liu and Wang (2014) and only focus on the role played by the productionnetwork in driving indeterminacy. We further assume that the fixed cost Φi and financial constraintΘi are symmetric across sectors. Recall that the economy-wide financial multiplier is the weighted

23

average of all sector-level financial multipliers,

µ ≡ 11 −∑N

i�1 υiλi�

11 − ϱ .

where ϱ �

N∑i�1υiλi , and the sector-level financial multiplier υi �

(λi/ϕi − ηi

)−1hinges on the relative

importance of fixed cost in this sector.

Figure 3: Financial multiplier when altering the intermediate input share

Figure 3 plots the response of the economy-wide financial multiplier to the change in the interme-diate input share.10 The key observation is that the impact of an increase in the intermediate inputshare αM on the financial multiplier µ exhibits a U-shape. Since all sectors are symmetric, we canconsider one sector as an example. The upward-sloping blue curve corresponds to the “size effect”:an increase in the intermediate input share boosts the sales of intermediate firms, which is reflected byan increasing Domar weight. This tends to amplify the sector-level financial multiplier since interme-diate firms that are subject to financial constraints expand. On the other hand, the downward-slopingred curve corresponds to the “diluting effect” because higher sales make the fixed cost play a lessimportant role and thus dampen the procyclicality of the loan-to-output ratio. To be more precise,

higher sales will dampen the sector-level financial multiplier υi �

(λi/ϕi − ηi

)−1.

These two forces are always pushing in opposite directions. However, the combination of thetwo determines the trend of economy-wide financial multiplier µ (represented by the black dashedcurve). The curve is U-shaped, which indicates that the diluting effect dominates the size effect whenthe intermediate input share is not too large, and the relationship reverses when the intermediate

10Parameters: ϕ1 � ϕ2 � 0.065,φ �

[0.50.5

], η � 6, a � 0.2, γ � 0, ψ � 1, δ � 0.025. To calibrate Θ, match with Liu and

Wang (2014), BY−Φ �

BY(1−ϕ) � 2.08. Then Θ �

2.08(1−ϕ)ρe

ξ(Z∗1)+ξ(Z∗

2)−ϕ.

24

input share reaches a sufficiently high level.Furthermore, two blue shaded areas correspond to intermediate input shares that can lead the

economy into multiple equilibria and are thus referred to as the “indeterminate region”. Multipleequilibria can emerge either when the intermediate input share is too small or too large. Whenintermediate input share is too small, due to a relatively high fixed cost burden, intermediate firmsare bound by too tight financial constraints. When intermediate input share is too large, intermediatefirms that are subject to financial constraints are too large, and the size of financial multiplier is alsolarge.

4.2.2 Altering the Network Architecture

Another key target of this paper is to investigate the relevance of the production network structureto economic instability. Consider the effect when we modify the network structure but hold otheraspects of the economy fixed.

For computational convenience, we assume the following input-output table:

αS � αMω �

[αM,1 0

0 αM,2

] [ω1 1 − ω1

1 − ω2 ω2

].

where the intermediate input shares of a sector’s own production are ω1 and ω2. We fix ω2 and alterω1. A higher ω1 means that sector 1 is a more “self-reliant” than “dependent” sector. A higher ω1

also means that sectors are less interconnected. To isolate the effect of altering the network structureon the economy-wide financial multiplier, we again fix ϕi so that the relative fixed cost size does notaffect the financial multiplier in this economy.

Figure 4: Financial multiplier when altering the network structure, ω2 � 0.5

We illustrate how the financial multiplier changes when altering the network structure in Figure

25

4.11 On the horizontal axis, we vary the value of ω1. On the vertical axis, we report the economy-widefinancial multiplier and all its decomposed effects. We address this question in three steps.

First, when ω1 rises, sector 1 becomes more “self-reliant” and reduces its dependence on sector2. As a result, the relative importance of sector 1 (reflected by Domar weight λ1) increases, while therelative importance of sector 2 (reflected by Domar weight λ2) is undermined. This is in line with the“size effect”: the increase in ω1 boosts the sales of sector 1 and reduces the sales of sector 2.

Second, as ω1 increases, sector 1’s financial multiplier υ1 decreases. This is due to the “dilutingeffect”: the increase in sector 1’s sales makes the fixed cost in sector 1 become relatively less importantand effectively relaxes the financial constraint in sector 1. For sector 2, the reversal property holds, sothe financial multiplier υ2 is increasing in ω1.

Third, the economy-wide financial multiplier ϱ is the Domar-weighted average of the sector-levelfinancial multiplier. The blue curve corresponds to the joint effects of the “size effect” and “dilutingeffect” in sector 1. The blue curve is downward sloping, indicating that the “diluting effect” dominatesthe “size effect” in sector 1. The reverse is true for sector 2; see the red, upward-sloping curve. Thus,the combination of these two sectors’ weighted average financial multiplier, or the black dashed curve,exhibits a U-shape. This helps us to see that when the network becomes more interconnected, it doesnot necessarily lead to a higher level of financial multiplier ϱ. The trend of the financial multiplier isdetermined by the relative forces of two sectors’ weighted financial multipliers.

5. Critical Sector in Driving Sunspot Fluctuations

We now use a parameterized version of our model to illustrate the importance of sectors at differentpositions in the network in shaping self-fulfilling business cycles. In particular, we tighten thefinancial constraints on sectors with different Domar weights and see which operation first triggerseconomy-wide multiple equilibria.

The parameters used in this numerical exercise are summarized in Table 1. The calibration matchesquarterly data. To simplify our discussion, we will focus on a two-sector economy.

11Parameters: ϕ1 � ϕ2 � 0.095, φ �

[0.50.5

], η � 6, αM � 0.5, b � 0.5, γ � 0, ψ � 1, δ � 0.025.

26

Table 1: Parameters used for steady state calibration

Parameter Description Value

ρh household discount rate 0.01ρe entrepreneur discount rate 0.02<i capital income share 0.3αM,i intermediate goods income share 0.5δ capital depreciation rate 0.025γ inverse of Frisch elasticity 0ψ effort disutility 1ηi shape of Pareto distribution 6φi final goods expenditure share [0.5 0.5]

Heterogeneous Financial Shocks. We are curious about whether some sectors’ financial constraintsare more influential than others on the financial stability of the entire economy. This questionis particularly relevant when the government has limited credit resources to allocate and has toprioritize which sector to subsidize. We consider an asymmetric two-sector economy to demonstratethe relative financial importance of different sectors in inducing self-fulfilling business cycles. Forcomputational convenience, we assume the following input-output table:

αS � αM,i ·[0.9 0.10.9 0.1

]Sector 1 mainly uses its own product as the intermediate input, while sector 2 mainly uses the othersector’s product as the intermediate input. Thus, in essence, sector 1 is in a more influential position inthis production network. We refer to sector 1 as the upstream industry and sector 2 as the downstreamindustry. Now, given this network structure, we ask whether multiple equilibria are more likely tooccur when the financial shock affects more influential sectors in the economy. We want to see theinteraction between the asymmetric financial constraint and asymmetric production network. Otherthan the contract enforcement level Θi , both sectors are identical in fixed cost Φi � Φ. For the sectornot being treated, we set Θi � 1.

27

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

(a) Shock Θ1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

(b) Shock Θ2

Figure 5: Financial multiplier when altering contract enforcement

We conduct two experiments by shocking sector 1 and sector 2’s credit market efficiency valuesΘ1 and Θ2, respectively. Figure 5 shows the results of these financial shock treatments. On thehorizontal axis, we vary the financial constraint of a chosen sector. On the vertical axis, we report theeconomy-wide financial multiplier and its decomposition.

We begin with Figure 5a, which corresponds to the response of the economy-wide financialmultiplier when tightening the financial constraint in the upstream sector 1. As the level of Θ1

moves from 0.9 to 0.1, the working capital constraint is exogenously tightened. Consequently, theDomar-weighted financial multiplier of the upstream sector increases; see the blue downward-slopingcurve. However, this is not the whole story, and the downstream sector 2 is also affected throughthe input-output linkages. In fact, the Domar-weighted financial multiplier of the downstream sector(λ2υ2) is not only shifted but also shifted faster than that in sector 1; see the red, steeper downward-sloping curve in the same graph. In this sense, treatment in the upstream sector can easily “alter” thefinancial landscape in the downstream sector. The joint forces of the two sectors shape the economy-wide financial multiplier represented by the black dashed line, which is quite steep, suggesting avery sensitive response of the economy-wide financial multiplier to the upstream sector’s financialconstraint.

Instead, if we tighten the financial constraint in the downstream sector 2, which is shown in Figure5b, the Domar-weighted financial multipliers λ1υ1 and λ2υ2 in the two sectors co-move – they bothincrease in response to a negative sector 2 financial shock. However, since the downstream sector has alow Domar weight in this economy, an exogenous tightening of its financial constraint leads to a minorchange in its own sector; see the red, relatively flat line. Moreover, since sector 2 occupies a relativelyinferior node in the production network, it is less powerful in “altering” the financial landscape ofsector 1; see the blue, even flatter line. Consequently, the combination of two sectors’ effects leadsto a highly insensitive response by the economy-wide financial multiplier, reflected by a much flatter

28

black dashed line. We conclude that the upstream sector has a larger effect on the economy-widefinancial multiplier. This also translates into a larger impact in inducing economy-wide instability.

Let us visualize how self-fulfilling fluctuations are connected with sector-level financial constraintsthrough Figure 6. The horizontal axis corresponds to different levels of fixed costs, and the verticalaxis corresponds to different levels of financial constraints in the treatment sector, holding the othersector’s financial constraint fixed at 1; a lowerΘi means a tighter financial constraint. The red shadedarea documents (Θ1 ,Φ) pairs that can lead the economy to multiple equilibria if the financial shockaffects sector 1. On the other hand, the blue shaded area collects (Θ2 ,Φ) pairs that drive the economyto multiple equilibria. We call these shaded areas “indeterminate regions”. The areas to the left ofthese two indeterminate regions, including the indeterminate regions, are “admissible regions”.

Figure 6: Indeterminacy: financial shock to sectors with different levels of influence

Given one particular level of fixed costΦ∗, if we tighten the financial constraint, i.e., decreaseΘ from1 towards 0, then tightening the upstream sector’s financial constraint more easily drives the economyinto the “indeterminate region” than tightening the downstream sector’s financial constraint; the linewith an arrow first reaches red shaded indeterminate region when Θ1 declines to approximately 0.5and then reaches the blue shaded “indeterminate region” whenΘ2 declines to approximately 0.2. Thisreveals the importance of financial abundance for the upstream sector in stabilizing the economy. Weconclude that if the government has limited resources and attention, it should focus on improvingfinancial market efficiency for sectors with higher Domar weights.

29

6. Conclusion

Are certain production network weights and structures more likely to drive the economy into a self-fulling equilibrium? Are certain sectors’ levels of financial market efficiency more important for theeconomy’s stability? This paper makes a contribution in answering these questions.

In this paper, we develop a flexible and tractable framework that studies the interactions offinancial frictions and input-output linkages and their joint impact on sunspot-driven fluctuations.In the benchmark model, we incorporate the production network into a self-fulfilling business cyclemodel. We show that the economy permits a simple but rich aggregation, which allows an analyticalanalysis of the financial multiplier and network multiplier.

By introducing the production network, we investigate the endogenous financial multiplier andfind that it is linked with the weight and structure of the production network. The impact of theintermediate input share on the economy-wide financial multiplier is U-shaped. In addition, multipleself-fulfilling equilibria can arise when we alter the network structure.

Quantitatively, we also show that when tightening the credit constraint for a particular sector,its effect in terms of inducing self-fulfilling business cycles hinges on the relative importance of thissector: a sector with higher Domar weight is more likely to trigger sunspot-driven fluctuations aftera financial depression.

30

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33

Appendices

A. Household Euler Equation

The current-value Hamiltonian for households is given by

H (Kh , Ch , L, λ) � log (Ch) − ψL1+γ

1 + γ+ λ [−δKh + RKh + WL − Ch] .

To generate the sufficient conditions for the optimum,

H (Kh , Ch , L, λ)C

�1

Ch− λ � 0,

H (Kh , Ch , L, λ)L

� −ψLγ + λW � 0,

Ûλ � ρhλ − H (Kh , Ch , L, λ)K

� ρhλ − λ [R − δ] � 0.

These imply that household Euler equations are given by

ÛCh

Ch� R − δ − ρh ,

WCh

� ψLγ .

B. Individual Firm Policy

Lemma 5. (Individual Firm Policy Functions) For each sector i, there exists a cutoff value Z∗it , such that the output

Oit(ι) and debt Bit(ι) raised by an individual firm that draws firm-specific productivity Zit(ι) are given by

Oit(ι) �

{Zit (ι)

Z∗it

BitPit, if Zit(ι) ≥ Z∗

it

0, if Zit(ι) < Z∗it

, (B.1)

Bit(ι) �

{B it , if Zit(ι) ≥ Z∗

it

0, if Zit(ι) < Z∗it

, (B.2)

where the sector cutoff Z∗it is determined by

Z∗it ≡

1Ait

(Rt/Pit

αK,i

)αK,i (Wt/Pit

αL,i

)αL,i N∏j�1

(P jt/Pit

αS,i j

)αS,i j

.

Proof. Denote Πit(ι) as the profit of firm ι in sector i

Πit(ι) � maxKit (ι),Lit (ι),Si jt (ι)

Pit Oit(ι) − Rt Kit(ι) − Wt Lit(ι) −N∑

j�1P jtSi jt(ι).

Given the capital rental rate Rt , the wage rate Wt , and the vector of intermediate goods prices {Pit}, if the

34

optimal choice is an interior solution, then the first order conditions lead to

Kit(ι) �ηiαiPit Oit(ι)

Rt,

Lit(ι) �ηi(1 − αi)Pit Oit(ι)

Wt,

Si jt(ι) �(1 − ηi)ϖi jPit Oit(ι)

P jt.

The necessary condition for interior solutions is

Oit(ι) � Ait Zit(ι)[(ηiαiPit Oit(ι)

Rt

)αi ( ηi(1 − αi)Pit Oit(ι)Wt

)1−αi] ηi

N∏j�1

( (1 − ηi)ϖi jPit Oit(ι)P jt

)ϖi j

1−ηi

.

This necessary condition for interior solution implies that there exists an unique cutoff value for firm-specificproductivity shock Z∗

it ,

Z∗it �

1Ait

(Rt/Pit

ηiαi

)ηiαi ( Wt/Pit

ηi(1 − αi)

)ηi (1−αi ) N∏j�1

(P jt/Pit

(1 − ηi)ϖi j

) (1−ηi )ϖi j

.

If a firm has unlimited access to the credit market, then it will produce infinite amount if its draw of firmspecific productivity Zit(ι) > Z∗

it , while stay out of production if Zit(ι) < Z∗it . And Any amount of production

can be supported if Zit(ι) � Z∗it .

The idea behind is very simple. The marginal revenue of producing one unit of intermediate goods is Pit ,while the corresponding marginal cost is denoted byΨit(ι)

Ψit(ι) �1

Ait Zit(ι)

(Rt

αK,i

)αk ,i(

Wt

αL,i

)αL,i N∏j�1

(P jt

αS,i j

)αS,i j

,

where αK,i � ηiαi , αL,i � ηi(1 − αi), αS,i j � (1 − ηi)ϖi j . The individual firm will only participate in productiononce their marginal producing cost is no larger than marginal benefit.

With the presence of credit constraint, the gross production expense is capped by the borrowing limitB it . The borrowing limit effectively controls the scale of production for firm whose productivity exceeds thethreshold Z∗

i . Given a borrowing limit B i , the total expenses are fixed. Then the individual firm output is givenby

Oit(ι) � Ait Zit(ι)(αK,iB it

Rt

)αK,i (αL,iB it

Wt

)αL,i

N∏j�1

(αS,i jB it

P jt

)αS,i j �Zit(ι)

Z∗it

B it

Pit.

C. Frictionless Economy: Decreasing Return to Scale Production

In this section we will show that the aforementioned financially frictional economy is observationally equivalentto an economy where firms in each sector are homogeneous and operate at decreasing return to scale (DRS).

35

We label the latter economy as the DRS economy. Taking the allocation and prices from the original economyas given, construct the DRS economy in the following way.

Since all firm in sector i are homogeneous, the production function for a representative firm is

Oit(ι) � Ait Kit(ι)αK,it Lit(ι)αL,it

N∏j�1

Si jt(ι)αS,i jt ,

where the factor share is given by

αK,it ≡ αK,iZ∗

it

Ei

(Zit(ι)|Zit(ι) ≥ Z∗

it

) ,αL,it ≡ αL,i

Z∗it

Ei

(Zit(ι)|Zit(ι) ≥ Z∗

it

) ,αS,i jt ≡ αS,i j

Z∗it

Ei

(Zit(ι)|Zit(ι) ≥ Z∗

it

) .

Notice that we borrow the value of Z∗it and Ei

(Zit(ι)|Zit(ι) ≥ Z∗

it

)from original economy, they are just numbers

and does not represent firm productivity heterogeneity any more in this section. Also the production isdecreasing return to scale since αK,it + αL,it + αS,i jt �

Z∗it

Ei(Zit (ι)|Zit (ι)≥Z∗it)< 1. The sector level productivity is given

by

Ait � AitEi(Zit |Zit ≥ Z∗

it

)· KαK,i−αK,it

it LαL,i−αL,itit

N∏j�1

SαS,i j−αS,i jt

i jt ,

where Kit , Lit , Si jt are sector level production input. We claim that the sector level allocation and prices inoriginal economy are also the equilibrium sector allocation and prices in this DRS economy. We need to verifythat the optimal condition of firms are still satisfied. The problem of a representative firm in sector i is

maxKit (ι),Lit (ι),Si jt (ι)

Pit Oit(ι) −©­«Rt Kit(ι) + Wt Lit(ι) +

N∑j�1

P jtSi jt(ι)ª®¬ .The first order conditions are

Rt Kit(ι) � αK,itPit Oit(ι),Wt Lit(ι) � αL,itPit Oit(ι),P jtSi jt(ι) � αS,i jtPit Oit(ι).

which are satisfied in the original equilibrium.

36

D. Proof of Lemma 2

Proof. The sales of the sector i can be decomposed into final sales as well as intermediate sales according tointermediate goods market clearing condition (2.19)

Pit Oit

Yt�

Pit Xit

Yt+

∑j

PitS jit

Yt.

Thus the revenue-based Domar weight for sector i has to follow

λit � φi +∑

j

αS, ji

Z∗jt

E[Z jt(ι)|Z jt(ι) ≥ Z∗jt]λ jt � φi +

∑j

αS, jit λ jt .

More compactly in matrix formation,λt � φ + α′

S,tλt ,

λt �(I − α′

S,t)−1

φ �

∞∑k�0

(α′

S,t) kφ,

where

λt �

λ1t...

λNt

, φ �

φ1...

φN

, αS,t �

αS,11t . . . αS,1Nt...

...

αS,N1t . . . αS,NNt

.Now we prove that λit is weakly increasing in Z∗

i . Suppose Z∗i > Z∗

i , we want to show λt(Z∗i ) > λt(Z∗

i ),where both Domar weights are given by definition,

λt(Z∗i ) � φ + αS,t(Z∗

i )′φ +

(αS,t(Z∗

i )′)2

φ + ...,

λt(Z∗i ) � φ + αS,t(Z∗

i )′φ +(αS,t(Z∗

i )′)2φ + ...

Then in full matrix term, taking the productivity cutoff change in sector 1 as an example,

λ1t...

λNt

�

φ1...

φN

+αS,11t(Z∗

1) . . . αS,1Nt(Z∗1)

......

αS,N1t . . . αS,NNt

φ1...

φN

+αS,11t(Z∗

1) . . . αS,1Nt(Z∗1)

......

αS,N1t . . . αS,NNt

2 φ1...

φN

+ ...Then the Domar weight for sector 1 was previously,

λ1t(Z∗1) � φ1 +

N∑j�1

αS,1 jt(Z∗1)φ j + A(Z∗

1), (D.1)

where A(Z∗1) represents the i-th entry in the summation of matrices from the third term on. But when Z∗

1 rises

37

to Z∗1, Domar weight for sector 1 changes to,

λit(Z∗1) � φ1 +

N∑j�1

αS,1 jt(Z∗1)φ j + A(Z∗

1). (D.2)

Comparing the second term in RHS of (D.1) and (D.2), apparently by assumption 2.4,

N∑j�1

αS,1 jt

Ei

(Z1tZ∗

1|Z1t ≥ Z∗

1t

) φ j ≤N∑

j�1

αS,1 jt

Ei

(Z1t

Z∗1|Z1t ≥ Z∗

1t

) φ j .

By similar logic A(Z∗1) ≤ A(Z∗

1). Thus λ1t(Z∗1) ≤ λit(Z∗

1).

E. Proof of Proposition 2.1

Proposition 2.1. (Aggregate Output with Endogenous TFP) The Aggregate output is given by

Yt � At Kλ′αKt Lλ′αL

t ,

where At , i.e., the endogenous aggregate TFP, is given by

At ≡ Zt ·N∏

i�1

(AitEi

(Zit |Zit ≥ Z∗

it

) )λi ,

with aggregate productivity

At �

N∏i�1

(AitEi

(Zit |Zit ≥ Z∗

it

) )λiN∏

i�1

kαK,iit lαL,i

it

N∏j�1

(αS,i jt

λit

λ jt

)αS,i j λi

·N∏

i�1

(φi

λit

)φi

,

and λ �Ψtφ, λ � Ψtφ, kit � Kit/Kt and lit � Lit/Lt .

Proof. Recall from (2.13) that the sector output is,

Oit � AitEi(Zit(ι)|Zit(ι) ≥ Z∗

it

)· KαK,i

it LαL,iit

N∏j�1

SαS,i j

i jt ,

and from (2.14) that demand for intermediate goods j is,

Si jt � αS,i jtPit Oit

P jt� αS,i jt

Pit Oit/Yt

P jt O jt/YtO jt � αS,i jt

λit

λ jtO jt ,

38

we know that

ln Oit � lnEi(Zit(ι)|Zit(ι) ≥ Z∗

it

)+ ln Ait

+ αK,i ln Kit + αL,i ln Lit +

N∑j�1

αS,i j ln Si jt

� lnEi(Zit(ι)|Zit(ι) ≥ Z∗

it

)+ ln Ait

+ αK,i ln kit + αL,i ln lit +

N∑j�1

αS,i j ln

(αS,i jt

λit

λ jt

)+ αK,i ln Kt + αL,i ln Lt +

N∑j�1

αS,i j ln O jt ,

Then we know thatlnOt � at +αK ln Kt +αL ln Lt +αS lnOt + co

t ,

where an entry in at isait ≡ lnEi

(Zit |Zit ≥ Z∗

it

)+ ln Ait , (E.1)

and an entry in cot is

coit ≡ αK,i ln kit + αL,i ln lit +

N∑j�1

αS,i j ln

(αS,i jt

λit

λ jt

).

In turn, we know that

lnOt � (I −αS)−1 (at +αK ln Kt +αL ln Lt + co

t)

�

[ (I −α′

S

)−1] ′ (

at +αK ln Kt +αL ln Lt + cot)

�Ψ′t(at +αK ln Kt +αL ln Lt + co

t).

Since λit �Pit Oit

Yt�

φ j OitXit

, we know that

ln Xit � ln Oit + ln(φi

λit

),

more compactly,lnXt � lnOt + cx

t ,

where cxit � ln

(φi

λit

).

Also by final goods production function Yt �N∏

i�1Xφi

it ,

39

ln Yt �

N∑j�1

φ j ln X jt

� φ′ lnXt

� φ′ lnOt +φ′cxt

� φ′Ψ′t(at +αK ln Kt +αL ln Lt + co

t)+φ′cx

t

� λ′t(at +αK ln Kt +αL ln Lt + co

t)+φ′cx

t ,

and by (E.1),

λ′tat �

N∑i�1

λi ait �

N∑i�1

ln(AitEi

(Zit |Zit ≥ Z∗

it

) )λi� ln

[N∏

i�1

(AitEi

(Zit |Zit ≥ Z∗

it

) )λi

].

In sum,Yt � At K

λ′αKt Lλ′αL

t ,

where

At � exp[λ′

tat(Z∗

t)+ λ′co

t(Z∗

t)+φ′cx

t(Z∗

t) ]

�

N∏i�1

AitEi(Zit(ι)|Zit(ι) ≥ Z∗

it

)kαK,i

it lαL,iit

N∏j�1

(αS,i jt

λit

λ jt

)αS,i j λi

·N∏

i�1

(φi

λit

)φi

.

Notice that the factor intensities add up to 1,

λ′αK + λ′αL � λ′ (αK +αL)� φ′ (I −αS)−1 (αK +αL)

� φ′∞∑

n�0αn

S (1 −αS1)

� φ′1 +φ′∞∑

n�1αn

S1 −φ′∞∑

n�0αn

SαS1

� φ′1 � 1.

F. The Law of Motion for Aggregate Capital

Proof. Note that, the aggregate law of motion of Kt is given by

ÛKt � −δKt + It ,

where Kt � Kh ,t +∑N

i�1 Ke ,it , and It � Ih ,t +∑N

i�1 Ie ,it and It can be obtained from the aggregate resource

40

constraint

It � Yt −N∑

i�1Φi − Ch ,t − Ce ,t ,

and thus aggregate entrepreneur’s consumption is given by aggregate budget constraint, and by finalexpenditure share (2.15) and (2.16),

Ce ,t �

N∑i�1

Ce ,it

�

N∑i�1

Pit Oit − Rt Kit − Wt Lit −N∑

j�1P jtSi jt −Φi

�

N∑i�1

Pit©­«Xit +

N∑j�1

S jitª®¬ − Rt Kit − Wt Lit −

N∑j�1

P jtSi jt −Φi

�

N∑i�1

Pit Xit − Wt Lt − Rt Kt −N∑

i�1Φi

�

N∑i�1

PitφiYt

Pit− Wt Lt − Rt Kt −

N∑i�1Φi

� Yt − Wt Lt − Rt Kt −N∑

i�1Φi

�

(1 − λ

′tαK,t − λ

′tαL,t

)Yt −

N∑i�1Φi

� (1 − αK,t − αL,t)Yt −N∑

i�1Φi .

Consequently, the mapping between ÛKt and Kt can be obtained as

ÛKt � −δKt + (αK,t + αL,t)Yt − Ch ,t .

G. Proof of Proposition 3.1

Proposition 3.1. Given vector of cutoff productivity Z∗t �

[Z∗

1t , ..., Z∗Nt

] ′ and Yt , the sector loan-to-GDP ratiodepends on the tightness of the credit constraint,

B it

Yt� Θi

ξ

(Z∗

it

)ρe

− Φi

ρe Yt

.Meanwhile, there is a monotonic mapping between Z∗

t and sector loan-to-GDP ratio by (2.12),

41

B it

Yt�

λit Z∗it∫

Z∗it

Zit(ι)dF(Zit(ι)).

Proof. Recall from collateral constraint (2.11),

B it � ΘiVit .

Note that from Section 2, Vit �∫

Vit(ι)dι �∫

V(Zit(ι))dF(Zit(ι)) denotes the ex ante value of the firm.Obviously Vit is also the continuation value of sector i which is the present value of the firm. From (2.8) and(2.9), Ce ,it � De ,it �

∫ 10 Πit(ι)dι, Vit is given by

Vit �

∫ 1

0

(∫ ∞

te−ρe (s−t) u′(Ce ,is)

u′(Ce ,it)Πe ,is(ι)ds

)dι �

∫ ∞

te−ρe (s−t) Ce ,it

Ce ,is

(∫ 1

0Πe ,is(ι)dι

)ds �

Ce ,it

ρe.

In the equilibrium, Ce ,it is the aggregate dividends payment from all firms. And by sectoral demandderived earlier (2.14) as well as the definition of Domar Weight, the consumption of sector i entrepreneur isrelated to the final output in the following way,

Ce ,it �

∫ 1

0Πit(ι)dι �

∫ 1

0

©­«Pit Oit(ι) − Rt Kit(ι) − Wt Lit(ι) −N∑

j�1P jtSi jt(ι) −Φi

ª®¬ dι

� Pit Oit − Rt Kit − Wt Lit −N∑

j�1P jtSi jt −Φi

� Pit Oit©­«1 − αK,it − αL,it −

N∑j�1

αS,i jtª®¬ −Φi

� ξ(Z∗it)Yt −Φi .

Note that since αK,it , αL,it , αS,i jt , λit are all functions of Z∗t , so is ξ.c

ξ(Z∗it) � λit

©­«1 − αK,it − αL,it −N∑

j�1αS,i jt

ª®¬ .Therefore, the sectoral debt limit is related to the final output as such,

B it

Yt�Θ

ρe i

[ξ

(Z∗

it

)− Φi

Yt

]≡ f (Z∗

it)

We will show that with Yt given, ξ(Z∗

it

)is a weakly decreasing function in Z∗

it and so is f (Z∗it). For now, denote

νi � Ei

(Zit

Z∗i|Zit ≥ Z∗

it

).

ξi �

∞∑k�0

(α′

S,t) kφ

(1 − 1

νi

)

42

On the other hand, if we look from credit demand side, (2.12) shows that loan-to-GDP ratio is related tocutoff productivity Z∗

t ,B it

Yt�

λit Z∗it∫

Z∗it

Zit(ι)dF(Zit(ι))≡ g(Z∗

it).

Notice that g(Z∗it) is an increasing function is Z∗

i . As λit is weakly increasing in Z∗it . Whereas Z∗

it∫Z∗

itZit (ι)dF(Zit (ι))

�

1

Ei

(ZitZ∗

i|Zit≥Z∗

it

)(1−F(Z∗

it )), where Ei

(Zit

Z∗i|Zit ≥ Z∗

it

)is weakly decreasing in Z∗

it , and(1 − F(Z∗

it))

is decreasing in Z∗it

too.

H. Derivation of Steady States Output

We use X to denote the steady state of variable Xt . From Corollary 1, the equilibrium aggregate productionfunction is given by

Yt � ζN∏

i�1

[Aλi

it

[Θi

ρe(ηi − 1)

(1 − ηiΦi

λiYt

)]λi/ηi]

KαKt L

αLt .

Suppose all the sectors share the same level of productivity heterogeneity, i.e. ηi � η for all i. In the steadystate, the aggregate output solves the following nonlinear equation

Yη+∑λi−ηαK

(YK

)ηαK

�

(ζ

N∏i�1

Aλii LαL

)η·

N∏i�1

[Θi

ρe(η − 1)

(Y − ηΦi

λi

)]λi

,

where

ζ �

N∏i�1

kαK,ii lαL,i

i

N∏j�1

(αS,i j

λi

λ j

)αS,i j λi

·N∏

i�1

(φi

λi

)φi

, αK � λ′αK , αL � λ′αL .

The Euler equation of the households (2.3) together with the aggregate demand for capital (2.16) yields thesteady-state capital rental rate, which is inversely related to the output-capital ratio,

R � αKYK

� δ + ρh ,YK

�δ + ρh

αK.

Moreover, from the law of motion for aggregate capital (2.18), we can obtain the aggregate investment rate,which yields the steady-state consumption-capital ratio,

It � (αK + αL)Yt − Ch ,t � δKt ,Ch

K� (αk + αl)

YK

− δ.

Moreover, the aggregate labor demand (2.15) together with the labor supply Euler equation (2.4) yields thesteady-state wage and steady-state labor,

W � αLYL

� ψLγCh , L �

(αL

ψ

Y/KCh/K

) 11+γ

�©­«αL

ψ1

(αK + αL) − δαKδ+ρh

ª®¬1

1+γ

.

43

The steady-state aggregate output has the following representation:

Yη+∑λi−ηαK

(δ + ρh

αK

)ηαK

�

ζ(

N∏i�1

Aλii

) ©­«αL

ψ1

(αK + αL) − δαKδ+ρh

ª®¬αL1+γ

ηN∏

i�1

[Θi

ρe(η − 1)

(Y − ηΦi

λi

)]λi

.

On the other hand, Y is bounded from below, i.e., Y ≥ Ymin , because the cutoff productivity is boundedfrom below, i.e., Z∗

i ≥ Z i (when Θi � 0, all firms produce, and this includes the most unproductive firms). Thelower bound of output Ymin is given by

Y � ζN∏

i�1Aλi

i LαL

N∏i�1

[Z∗i

Z i

]λi

KαK ≥[ζ

N∏i�1

Aλi

i LαL

]KαK ≡ Ymin .

I. Proof of Lemma 3

In this appendix we will show how we log linearize the Z∗i − Y system.

Denote xt � log Xt − log X, so that Xt � X(1 + xt). Also, when X → 0, we have exp X � 1 + X.

Z∗it �

[Θit

ρe(ηi − 1)

(1 − ηiΦi

λiYt

)]1/ηi

Z i ,

Linearizing above equation around the steady state, denote Xt �Θit

ρe (ηi−1) −ΘitYt

ηiΦi

λiρe (ηi−1),

Z∗i (1 + Z∗

it) � X1/ηi

(1 + Xt

) 1ηi Z i

� X1/ηi

(exp Xt

) 1ηi Z i

� X1/ηi exp(

1ηi

Xt

)Z i

� X1/ηi

(1 +

1ηi

Xt

)Z i .

Since in the steady state, the equation degenerate into

Z∗i � X1/ηi Z i .

Thus we haveZ∗

it �1ηi

Xt .

44

On the other hand,

X(1 + Xt) �Θ

ρe(ηi − 1)(1 + Θit

)− Θ

YηiΦi

λiρe(ηi − 1)1 + Θit

1 + Yt

�Θ

ρe(ηi − 1)(1 + Θit

)− Θ

YηiΦi

λiρe(ηi − 1)exp Θit

exp Yt

�Θ

ρe(ηi − 1)(1 + Θit

)− Θ

YηiΦi

λiρe(ηi − 1)exp

(Θit − Yt

)�

Θ

ρe(ηi − 1)(1 + Θit

)− Θ

YηiΦi

λiρe(ηi − 1)

(1 + Θit − Yt

).

Again in the steady state,

X �Θi

ρe(ηi − 1) −Θi

YηiΦi

λiρe(ηi − 1).

Then we have,XXt �

Θi

ρe(ηi − 1)Θit −Θi

YηiΦi

λiρe(ηi − 1)

(Θit − Yt

).

Xt � Θit +Θi

XYηiΦi

λiρe(ηi − 1)In turn we have,

Z∗it �

1ηi

[Θit +

Θi

XYηiΦi

λiρe(ηi − 1)Yt

]

�1ηi

Θit +

ΘiY

ηiΦi

λiρe (ηi−1)Θi

ρe (ηi−1) −ΘiY

ηiΦi

λiρe (ηi−1)

Yt

�

1ηi

[Θit +

ηiΦi

Yλi − ηiΦiYt

]�

1ηiΘit + υiYt ,

where

υi �Φi

Yλi − ηiΦi�

ϕi

λi − ηiϕi�

(λiYΦi

− ηi

)−1

.

Since we assume that the credit tightness is fixed at Θi , the log-linearized cutoff productivity degeneratesinto

Z∗it � υiYt .

Now we log-linearize the aggregate production function

Yt � ζN∏

i�1

[Ait

Z∗it

Z i

]λi

KαKt L

αLt .

45

Take log of both sides,

log Yt � log ζ +N∑

i�1λi

[log Ait + log Z∗

it − log Z i

]+ αK log Kt + αL log Lt .

At the steady state,

log Y � log ζ +N∑

i�1λi

[log Ai + log Z∗

i − log Z i

]+ αK log K + αL log L.

Find the difference of the two equations,

Yt �

N∑i�1

λiAit +

N∑i�1

λi Z∗it + αK Kt + αL Lt .

J. Proof of Lemma 4

Proof. The evolvement of Ch ,t and Kt are respectively,

ÛCh ,t

Ch ,t� αK

Yt

Kt− δ − ρh ,

ÛKt

Kt� −δ + (αK + αL)

Yt

Kt− Ch ,t

Kt.

The local dynamics around steady state can be summarized by the following equation system

Yt � µ(αK Kt + αL Lt

),

Lt �1

1 + γ

(Yt − Ch ,t

),

ÛKt � (αK + αL)YK

(Yt − Kt

)− Ch

K

(Ch ,t − Kt

),

ÛCh ,t � αKYK

(Yt − Kt

).

This system can be reduced into law of motion of (Yt , Kt) pair,

Ch ,t �αK(1 + γ)

αLKt −

1 + γ − αLµ

αLµYt � ϵCK Kt + ϵCYYt

Lt �−αK

αLKt +

1µαL

Yt � ϵLK Kt + ϵLYYt

ÛKt �

[(αK + αL)

YK

− Ch

KϵCY

]Yt −

[(αK + αL)

YK

+Ch

K(ϵCK − 1)

]Kt ,

ÛCh ,t � ϵCKÛKt + ϵCY

ÛYt

ÛYt �

[YKαK − ϵCK(αK + αL)

ϵCY+

Ch

KϵCK

]Yt +

[Ch

KϵCK (ϵCK − 1)

ϵCY+

YKϵCK (αK + αL) − αK

ϵCY

]Kt

46

where

ϵLK �−αK

αL, ϵLY �

1µαL

, ϵCK �αK(1 + γ)

αL, ϵCY � −

1 + γ − αLµ

αLµ,

Y/K �δ + ρh

αK, Ch/K � (αK + αL)

YK

− δ �

(αK + αL

αK

) (δ + ρh

)− δ.

We can vectorize the system, [ ÛYtÛKt

]� J

[Yt

Kt

].

Each element in the Jacobian matrix J �

[x11 x12

x21 x22

]is given by

x11 �YKαK − ϵCK(αK + αL)

ϵCY+

Ch

KϵCK , x12 �

Ch

KϵCK (ϵCK − 1)

ϵCY+

YKϵCK (αK + αL) − αK

ϵCY,

x21 �YK(αK + αL) −

Ch

KϵCY , x22 �

Ch

K(1 − ϵCK) −

YK(αK + αL) .

K. Proof of Proposition 4.2

Proof. The necessary and sufficient condition for indeterminate equilibrium to exist is given by

det(J) > 0, tr(J) < 0

Notice that

det(J) � x11x22 − x12x21 �

(1 + γ

) (δ + ρh

) [δ −

(δ + ρh

) αK+αLαK

] (1 − µαK

)1 + γ − µαL

,

tr(J) � x11 + x22 �

(1 + γ

) (δ + ρh

) αK+αLαK

µαK − δ(1 + γ

)− ρhµαL

1 + γ − µαL.

Obviously, (δ + ρh

) αK + αL

αK> δ + ρh > δ,

thus det(J) > 0 is equivalent to1 − µαK > 0, 1 + γ − µαL < 0,

and tr(J) < 0 is equivalent to[ (1 + γ

) (δ + ρh

) αK + αL

αKαK − ρhαL

]µ > δ

(1 + γ

).

47

Therefore, we can obtain from the above three inequalities the range of µ that is both necessary and sufficientfor the indeterminate equilibrium to exist.

max{µ∗

1 , µ∗2}< µ < µ∗

3

where

µ∗1 �

1 + γ

αL

µ∗2 �

δ(1 + γ

)αK+αLαK

(δ + ρh

) (1 + γ

)αK − ρhαL

µ∗3 �

1αK

L. Admissible Parameters

In this section we find endogenous restrictions on (Φ,Θ) combination such that interior solution of steadystate output exists. To make our analysis precise and simple, we assume symmetry in fixed cost Φ acrosssectors, i.e. Φi � Φ. For simplicity, we also assume all sectors share same productivity distribution, i.e. ηi � η.Consequently Z i � Z.

In the steady state, the aggregate production function is

Y � ζN∏

i�1

[Ai

Z∗i

Z i

]λi

KαK LαL � ζN∏

i�1

[Aλi

i

[Θi

ρe(η − 1)

(1 − ηΦi

λiY

)]λi/η]

KαK LαL ,

which is equivalent to

Y1+∑N

i�1 λi/η � ζ

[N∏

i�1Aλi

i

[Θi

ρe(η − 1)

(Y − ηΦi

λi

)]λi/η]

KαK LαL .

Denote κ � 1 +∑N

i�1 λi/η. Then the system of steady state can be expressed as the following

Y �

{ζ

[N∏

i�1Aλi

i

[Θi

ρe(η − 1)

(Y − ηΦi

λi

)]λi/η]

KαK LαL

} 1κ

, (L.1)

αLY/KCh/K

� ψLγ+1 ,

0 � αKYK

− δ − ρh , (L.2)

0 � −δ + (αK + αL)YK

− Ch

K.

48

From (L.2), we know Y/K �δ+ρhαK

≡ YK . Then (L.1) can be simplified as

Y �

[ζ

N∏i�1

Aλii LαL

]KαK

N∏i�1

[Θi

ρe(η − 1)

(1 − ηΦi

λiY

)]λi/η,

Y1−αK YαKK �

[ζ

N∏i�1

Aλii LαL

]N∏

i�1

[Θi

ρe(η − 1)

(1 −

ηϕi

λi

)]λi/η,

Yη+∑λi−ηαK Y

ηαKK �

(ζ

N∏i�1

Aλii LαL

)η N∏i�1

[Θi

ρe(η − 1)

(Y − ηΦi

λi

)]λi

, (L.3)

where

ζ �

N∏i�1

kαK,ii lαL,i

i

N∏j�1

(αS,i j

λi

λ j

)αS,i j λi

·N∏

i�1

(φi

λi

)φi

, αK � λ′αK , αL � λ′αL .

On the other hand Y has a lower bound Ymin , because Z∗i ≥ Z i . (When Θ � 0, all firms have to produce,

this includes the most unproductive firms.) Ymin is then given by

Y � ζN∏

i�1

[Ai

Z∗i

Z i

]λi

KαK LαL � ζN∏

i�1Aλi

i LαL

N∏i�1

[Z∗i

Z i

]λi

KαK

≥ ζN∏

i�1Aλi

i LαL

N∏i�1

[Z i

Z i

]λi

KαK �

[ζ

N∏i�1

Aλi

i LαL

]KαK ≡ Ymin ,

Ymin ≡[ζ

N∏i�1

Aλii LαL

]KαK ,

Ymin �

[(ζ

N∏i�1

Aλii LαL

)Y−αKK

] 11−αK

.

Another constraint is to make sure (L.3) is well defined,

Y >ηΦi

λi.

This places an upper bound on Φi ,

Φi <Ymin min(λi)

η.

49

M. Global Dynamics

This section moves from the characterization of the local dynamics around steady states to the global dynamics.Under Pareto distribution, the dynamic system on

{Yt , Kt , Lt , It , Ch ,t ,Wt , Rt

}is given by,

Yt � ζN∏

i�1

[Aλi

it

[Θit

ρe(ηi − 1)

(1 −

ηiϕi

λi

)]λi/ηi]

KαKt L

αLt (M.1)

Wt � αLYt

Lt,

Wt

Ch ,t� ψLγt , Rt � αK

Yt

Kt,

ÛCh ,t

Ch ,t� Rt − δ − ρh , (M.2)

ÛKt � −δKt + (αK + αL)Yt − Ch ,t . (M.3)

Finally, the clearing condition of the labor markets implies that

Lt �

(αL

ψ1

Ch ,t/Yt

) 11+γ

. (M.4)

In sum, we have four variables{Kt ,Yt , Ch ,t , Lt

}with four equations (M.1), (M.2), (M.3), and (M.4). Substi-

tuting (M.4) into (M.1) yields an analytic formulation of Ch ,t as a function of (Kt ,Yt):

Ch �

ζ ©­«N∏

i�1Aλi

it

(Θi

ρe(ηi − 1

) (1 − ηiΦi

λiYt

)) λiηi ª®¬ K

αKt

(αL

ψYt

) αL1+γ

Y−1t

1+γαL

.

Taking log on both sides yields

ln Ch ,t �1 + γ

αL

[ln ζ +

N∑i�1

(λi ln Ait +

λi

ηln Θi

ρe(ηi − 1)

)+

αL

1 + γln αL

ψ

+

N∑i�1

λi

ηiln

(1 − ηiΦi

λiYt

)+ αK ln Kt +

(αL

1 + γ− 1

)ln Yt

].

We set Ait � Ai to consider deterministic case. Taking derivative of both sides with respect to t yields

ÛCh ,t

Ch ,t�

1 + γ

αL

N∑

i�1

λi

ηi

©­«ηiΦi

λi Yt

1 − ηiΦi

λi Yt

ª®¬ÛYt

Yt+ αK

ÛKt

Kt+

(αL

1 + γ− 1

) ÛYt

Yt

�

1 + γ

αL

N∑

i�1

λi

ηi

©­«ηiΦi

λi Yt

1 − ηiΦi

λi Yt

ª®¬ +αL

1 + γ− 1

ÛYt

Yt+αK

(1 + γ

)αL

ÛKt

Kt. (M.5)

Substituting (M.5) into (M.2), we have

1 + γ

αL

N∑

i�1

λi

ηi

©­«ηiΦi

λi Yt

1 − ηiΦi

λi Yt

ª®¬ +αL

1 + γ− 1

ÛYt

Yt+αK

(1 + γ

)αL

ÛKt

Kt� αk

Yt

Kt− δ − ρh .

50

Consequently, these yield two differential equations on (Kt ,Yt). That is, we can simplify the dynamicsystem into a 2-dimensional autonomous dynamical system on (Kt ,Yt).

ÛKt � −δKt + (αk + αl)Yt − Ch (Kt ,Yt) ,

ÛYt �

[αK − αK(1+γ)

αL(αK + αL)

]Yt −

[δ + ρh − δ αK(1+γ)

αL

]Kt +

αK(1+γ)αL

Ch (Yt , Kt)

1+γαL

[N∑

i�1

λiηi

(ηiΦiλi Yt

1− ηiΦiλi Yt

)+

αL1+γ − 1

] · Yt

Kt.

51