Online Appendix for Finance-thy-Neighbor. radeT Credit ... · by banking institutions de ned as the...

Online Appendix for "Finance-thy-Neighbor.

Trade Credit Origins of Aggregate Fluctuations."

Not for Publication.

Margit Reischer, November 2020

Appendix B provides an overview of the matrix notation and operations applied in this

paper. Appendix C describes the data sources and variables used to derive the results in

Section 2 and to calibrate the model. Appendix D discusses the equilibrium conditions

of the model introduced in Section 3 and contains the proofs of the respective lemmata.

Appendix E derives the partial equilibrium similiar to Bigio and La'O (2019) and Ap-

pendix F provides detailed proofs of the lemmata and propositions in Section 3.2. The

calibration and the results of the quantitative application are discussed in Appendix G.

B Matrix and Variable Notation 2

C Data 3

C.1 Data Sources and Sample Description . . . . . . . . . . . . . . . . . . . . . 3

C.2 Bank versus Trade Credit . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

D Model-Derivations 8

D.1 The Household's Optimization Problem . . . . . . . . . . . . . . . . . . . . 8

D.2 The Firms' Optimization Problem . . . . . . . . . . . . . . . . . . . . . . . 8

D.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

E Equilibrium 16

E.1 National Accounting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

E.2 Partial Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

F Log-Linearization 24

F.1 Log-Linearized Equilibrium and Wedges . . . . . . . . . . . . . . . . . . . 24

F.2 Credit Costs, Links and Interest Rates . . . . . . . . . . . . . . . . . . . . 32

F.3 Partial Equilibrum Structural Output Response . . . . . . . . . . . . . . . 34

G Quantitative Application 36

G.1 Model Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

G.2 Business Cycle Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

G.3 Model Fit and Additional Simulations . . . . . . . . . . . . . . . . . . . . . 44

Georgetown University, email: [email protected].

Online Appendix - Not for Publication

B: Matrix and Variable Notation

Matrix and Variable Notation. Matrices are denoted as bold capital letters (e.g. X)

and vectors as bold small letters (e.g. x). Since the economy consists of M sectors,

matrices in the following are of size [M ×M ] and vectors are of size [M × 1], unless they

are associated with the vectorized matrices, vec(Φ) or vec(Θ).

Auxiliary Matrix Operations

inv(.) ... inverse of a matrix

diag(.) ... extracts diagonal entries of matrix X; generatesdiagonal matrix using vector x

vec(.) ... vectorizes matrix X by stacking each column

... denotes the Hadamard product

⊗ ... denotes the Kronecker product

ι ... vector of ones

J ... = ιι′.

Production and Credit Network. The matrices ΩX and Θ denote the intermediate

production structure and credit network, respectively. To simplify notation let Ω = ΩX

below. Both the production and credit network can be mapped into standard graph

theoretical notation. Following Carvalho (2010), dene the set of M sectors as the vertex

set V = v1, ..., vM and let E(Ω) and E(Θ) be subsets of all ordered pairs of vertices

vk, vs, with vk, vs ∈ V dened as

E(Ω) = vk, vs ∈ V2 : vk, vs ∈ E(Ω) if Sector s supplies inputs to Sector k (B.3a)

E(Θ) = vk, vs ∈ V2 : vk, vs ∈ E(Θ) if Sector s extends credit to Sector k (B.3b)

Therefore, the Cobb-Douglas technology implies that the entry [Ω]ks = ωks denotes the

share of good s in the total intermediate input use of sector k and it is assumed that∑Ms=1 ω

Xks = 1 ∀k ∈ 1, ...,M. Similarly, [Θ]ks = θks denotes the share of intermediate

good expenditures of sector k obtained on trade credit from supplier s, which is endoge-

nously determined in the model. Due to the complementary of the production and the

credit network, it follows that E(Θ) ⊆ E(Ω). In other words, a credit link between sec-

tor k and s only exists if both sectors also engage in input trade. The production as

well as the credit network can be described as directed graphs G(Ω) and G(Θ) applying

Denition 1 in Carvalho (2010):

Denition B.1. G(Ω) = G(V , E(Ω)) is a directed sectoral trade linkages graph with

vertex set V and edge set E(Ω), where each element of E(Ω) is a directed arc from element

k to s. Similarly, G(Θ) = G(V , E(Θ)) denes the credit linkages graph.

2


C: Data

C.1. Data Sources and Sample Description

GDP and Sectoral Output. Data on nominal and real aggregate US GDP, aggregate Im-

ports and Exports, Gross Output and the corresponding GDP-Deator are obtained from

the Bureau of Economic Analysis (BEA). Sectoral data are obtained from the summary

tables on "Use of Commodities by Industries After Redenitions" provided by the BEA.

Prices and Wages. Data on total hours worked and sectoral prices are obtained from the

Bureau of Labor Statistics (BLS). In particular, I combine the respective variables from

the MFP- and the LPC-Database in order to deal with missing data when required.

Lending Standards. The Senior Loan Ocer Opinion Survey on Bank Lending Practices

- conducted by the Federal Reserve - reports the tightening in lending standards (LS)

by banking institutions dened as the net percentage of domestic respondents tightening

their standards for commercial and industrial (C&I) loans.

Sectoral Credit Spreads and Federal Funds Rate. The sectoral credit spreads (rZkt) are

derived in Gilchrist and Zakraj²ek (2012) and provided to me by the authors. The federal

funds rate (rB0t) is obtained from the FRED, Federal Reserve Bank of St. Louis database.

Balance Sheet and Income Statement Data of US Firms. The Compustat database is

used to infer sectoral trade credit (shares) based on the balance sheet data on accounts

receivable and payable of US rms. A rm is included in the sample if

(1) non-missing NAICS-classication

(2) headquarter in the US

(3) non-missing and non-negative data on balance sheet and income statement items(a)

(4) accounts receivable do not exceed sales

(5) the sum of accounts payable and cash do not exceed total production costs

(6) non-missing consecutive observations over the time period 2005-2010

In addition, rms are excluded who either enter or exit the Compustat database during

the period 2005-2010. In total, 2,686 rms are included in the initial sample per year

on average. The average number of rms in each sector and their representativeness of

each industry are presented in Table C.3 across all years 1997-2016. The reduced sample

used to construct Figure 2.1 and 2.2 only contains rms with non-missing values over the

sample period 1997-2016. Furthermore, a rm's observation is assigned to the previous

calendar year if its scal year ends in the months January through May and assigned to

the current calendar year if its scal year ends in the months June through December.

3

https://www.bea.gov/

https://www.bls.gov/

https://www.federalreserve.gov/data/sloos.htm

https://fred.stlouisfed.org/series/fedfunds

https://wrds-web.wharton.upenn.edu/wrds/


(a) The following balance sheet and income statement items(a) are obtained to construct

the variables discussed in Section 2 and to calibrate the model as shown in Section 4. A

short description based on the denition given in the Compustat database is included.

Accounts Payable (ap) and Receivable (ar). AP are trade obligations due within

one year and are included in current liabilities (lc). AR represent amounts on open

account owed by customers for goods and services sold.

Cost of Goods Sold (cogs) are all expenses directly related to the production of goods

and services sold to customers.

Total Assets (at) and Sales (R) realised during the scal period.

Total (lt) and Current Liabilities (lc). Total Liabilities (lt) are the sum of current

liabilities (lc), long-term debt and other non-current liabilities. Current Liabilities

include debt in current liabilities (dlc), accounts payable (AP ) and other liabilities

due within one year.

Total Long-Term (dlt) and Current Debt (dlc). Long-Term Debt (dlt) is dened as all

debt obligations due in more than one year. Current Debt (dlc) denotes all interest-

bearing obligations due after the current year including long-term debt due in one

year and short-term borrowings/notes payable (np).

(b) In addition, the following variables are obtained:

Net Income (ni) dened as the scal period income or loss after subtracting expenses

and losses from all revenues and gains.

Dividends (dv) paid for capital and Cash (ch) holdings of a company.

Interest Expenditures (xint) of the company on short- and long-term debt, and In-

terest Income (idit) from interest-bearing obligations held by the company.

Notes Payable (np) denoting the total amount of short-term notes including i.a. bank

acceptances and overdraft, commercial paper.

Depreciation (dp) associated with spreading the actual cost of tangible capital assets

over their useful life and R&D Expenditures (xrd) on the development of new products.

Any missing values are assigned the value of zero and a rm-year observation is excluded

if the sum of net income (ni), dividends (dv) and interest expenditures (xint) is zero. The

composition and representativeness of both samples of US rms obtained from Compustat

- (a) restricted and (b) unrestricted with respect to the time coverage and missing values

of rms - is presented in Table C.3 at a sector level. The less restrictive sample (b) is

used to calibrate the sector-to-sector equilibrium trade-credit shares in Section 4.1.

4


C.2. Bank versus Trade Credit

The variables included in the regressions presented in Table C.2 are dened in the follow-

ing. Sectoral aggregates are obtained by summing the respective balance sheet item or

income variable over all Compustat rms assigned to the sector in a given year.

Trade-Credit Shares. The variable θPkt denes the share of accounts payable (ap) in

total cost of production (cogs) and θRkt denotes the share of accounts receivable (ar)

in total sales (sales).

Financial Shares. The variable θFkt is calculated as the ratio of current (dlc) to long

term debt (dlt) and θEkt denotes the share of cash (ch) in total production cost (cogs).

Size. The relative size of a sector is measured by the share of a sector's assets (at)

in total assets in the economy.

Leverage. The leverage ratio, LV Skt , is calculated as the share of total debt (dlt+ dlc)

in sales (R). Note that I use sales in the denominator rather than total assets (at) as

accounts receivable are also included as a share of total sales. The average leverage

is the average aggregate leverage over the entire sample period 1997-2016.

Detrended Cost of Production, CXkt , are obtained by detrending total sectoral cost of

production (cogs) using an hp-lter with a smoothing constant of 6.25 as suggested

for annual data by Ravn and Uhlig (2002).

Using a panel of 45 sectors from 2000 to 2014, Equation (2.1a) and (2.1b) in Section

2 are both estimated by OLS including sector and year xed eects. The estimated

coecients and corresponding clustered standard errors at the sector level are reported in

Table C.2. Panel (a) reports the estimation results for Equation (2.1a). Besides the share

of accounts receivable in total sales at a sector level, θRkt, additional regressors include

the leverage ratio, LV Skt , the ratio of current to long-term debt, θFkt, the share of cash in

total production cost, θEkt, as well as the relative size of a sector, Sizekt, measured by

the share of a sector's assets in total assets as dened above. The estimation results of

Equation (2.1a) suggest that a one percent increase in the share of revenues extended

on credit signicantly increases a sector's risk premium by approximately 0.3 percent.

These results are robust to the inclusion of additional variables controlling for the access

to external funds, the ability of rms to repay their debt and the size of the rm. e.g.

Petersen and Rajan (1997); Jacobson and von Schedvin (2015); Costello (2018).

Panel (b) reports the estimation results for Equation (2.1b). Since price data on the

cost of trade credit are not readily available due to the nature of the contract, the analysis

is limited to the interest rate on bank loans which are approximated by calculating the

sum of the federal funds rate and the sectoral credit spread, rZkt, derived in Gilchrist and

5


Zakraj²ek (2012). In addition to the cost of bank credit, rBkt, the detrended total cost of

inputs, CXkt , is included as a regressor which allows to control for the size of each sector in

terms of their production costs. The estimated coecients and corresponding clustered

standard errors imply that an increase in the cost of bank credit by 100 basis points

(rBkt + 0.01) increases the share of production costs nanced using trade credit by 0.12

percentage points.

Table C.2: Bank versus Trade Credit

(a) Cost of External Finance

VAR (1) (2) (3) (4)

ln(θRkt) 0.33* 0.33* 0.34* 0.32*

(0.161) (0.157) (0.153) (0.151)

ln(LV Skt) 0.03 0.03 0.04 0.03

(0.122) (0.122) (0.121) (0.120)

θFkt 0.33+ 0.32+ 0.31+

(0.167) (0.166) (0.163)

θEkt 0.57 0.64

(0.457) (0.452)

Sizekt 2.98*

(1.331)

NObs 579 579 579 579

R2 0.76 0.77 0.77 0.77

Constant Yes Yes Yes Yes

Sector FE Yes Yes Yes Yes

Year FE Yes Yes Yes Yes

(b) Trade-Credit Shares

VAR (1) (2) (3)

rBkt 0.12+ 0.13+ 0.12+

(0.064) (0.063) (0.062)

θFkt -0.03 -0.03 -0.03

(0.022) (0.021) (0.021)

θEkt -0.02 -0.02

(0.055) (0.056)

CXkt -0.02*

(0.010)

NObs 579 579 579

R2 0.79 0.79 0.80

Constant Yes Yes Yes

Sector FE Yes Yes Yes

Year FE Yes Yes Yes

Note: Panel (a) and (b) report the estimated coecients of Equation (2.1a) and (2.1b), where the dependentvariables are: (a) sectoral credit spreads (ln(rZkt)) derived in Gilchrist and Zakraj²ek (2012) and (b), the share of

accounts payable in total cost of production (θPkt). The set of control variables at a sector level derived from a panel of

Compustat rms includes: the share of accounts receivable in total sales (θRkt), the leverage ratio (LV Skt), the ratio of

current to long-term debt (θFkt), the share of cash in total production cost (θEkt), the relative size of a sector (Sizekt)

measured by the share of a sector's assets in total assets and detrended total production costs (CXkt). The estimatedcoecients are obtained by estimating Equation (2.1a) and (2.1b) by OLS including both sector and year xed eects.Clustered standard errors at the sector level are reported in parentheses, ** p<0.01, * p<0.05, + p<0.1

6


Table C.3: Sample Description

(a) Stylised Facts (b) Calibration

ID Sector Description #Firms RP(Y) RP(R) #Firms RP(Y) RP(R) NL

1 11 Agriculture 4 0.03 0.01 13 0.15 0.06 0.17

2 211 Oil and Gas 6 0.88 0.55 57 1.88 1.13 0.57

3 212 Mining, except 211 7 0.54 0.27 26 1.03 0.52 0.10

4 213 Support for 212 6 0.68 0.51 20 0.82 0.60 0.03

5 22 Utilities 55 0.81 0.48 175 2.47 1.47 0.11

6 23 Construction 11 0.06 0.04 30 0.08 0.04 0.01

7 311T2 Food, Beverages and Tobacco 38 1.04 0.29 92 1.63 0.45 0.04

8 313T6 Textile, Apparel and Leather 24 1.94 0.61 69 3.04 0.95 0.08

9 321 Wood Products 6 0.22 0.06 14 0.48 0.14 0.07

10 322T3 Paper Products and Printing 18 0.85 0.31 46 1.30 0.48 0.11

11 324 Petroleum and Coal Products 8 4.52 1.17 21 6.22 1.63 0.06

12 325 Chemical Products 52 1.10 0.45 185 1.72 0.70 0.13

13 326 Plastics and Rubber Products 12 0.50 0.17 33 0.73 0.24 0.14

14 327 Nonmetallic Mineral Products 9 0.43 0.18 21 0.61 0.25 0.15

15 331 Primary Metals 19 1.15 0.28 40 1.66 0.42 0.16

16 332 Fabricated Metal Products 27 0.42 0.17 59 0.61 0.25 0.15

17 333 Machinery 51 1.05 0.38 131 1.50 0.55 0.07

18 334 Computer and electronic Products 65 1.13 0.62 274 2.42 1.34 0.10

19 335 Electrical Equipment and Components 19 0.41 0.17 49 1.16 0.48 0.13

20 3361MV Motor Vehicles, Bodies and Parts 23 0.46 0.10 52 0.98 0.23 0.06

21 3364OT Other Transportation Equipment 18 2.42 0.97 34 2.99 1.20 0.04

22 337 Furniture and Related Products 12 0.47 0.18 24 0.82 0.31 0.04

23 339 Misc Manufacturing 17 0.48 0.23 81 0.79 0.38 0.09

24 42 Wholesale Trade 56 0.41 0.29 145 0.76 0.53 0.06

25 441 Motor Vehicle and Parts Dealers 7 0.30 0.21 17 0.76 0.53 0.01

26 445 Food and Beverage Stores 6 0.93 0.64 16 1.53 1.06 0.00

27 452 General Merchandise Stores 11 4.63 3.15 22 5.11 3.49 0.00

28 4A0 Other Retail 6 1.63 0.76 19 2.24 1.04 0.01

29 481 Air Transport 6 2.09 1.13 8 2.31 1.25 0.02

30 482 Rail Transport 14 0.23 0.10 25 0.30 0.14 0.07

31 484 Truck Transport 4 1.28 0.83 26 6.13 3.67 0.07

32 486 Pipeline Transport 5 0.83 0.41 26 0.88 0.46 0.19

33 48A9 Other Transport and Warehousing 41 0.85 0.54 135 1.31 0.84 0.10

34 511 Publishing Industries 5 0.03 0.02 71 0.23 0.13 0.04

35 512 Motion Picture and Sound 1 0.37 0.22 13 0.52 0.31 0.13

36 513 Broadcasting & Telecommunications 13 0.53 0.27 93 1.55 0.79 0.10

37 514 Information Services 4 1.77 0.99 59 2.33 1.28 0.22

38 54 Professional & Technical Services 20 0.04 0.02 135 0.10 0.06 0.17

39 55 Management of Companies - - - - - - 0.00

40 56 Administrative & Waste services 31 0.17 0.11 85 0.27 0.17 0.10

41 62 Health Care & Social Assistance 21 0.10 0.08 83 0.17 0.13 0.00

42 71 Arts, Entertainment, Recreation 8 0.04 0.02 32 0.13 0.08 0.02

43 72 Accommodation & Food Services 20 0.19 0.10 79 0.30 0.17 0.01

44 81 Other Services except GOV 5 0.01 0.01 14 0.02 0.01 0.04

45 GOV Government and Education 5 0.01 0.04 37 0.01 0.09 0.00

Mean 18 0.86 0.41 61 1.41 0.68 0.09

Stdv 16 1.01 0.53 56 1.47 0.78 0.09

Min 1 0.01 0.01 8 0.01 0.01 0.00

ID (35) (44) (44) (29) (45) (44) (39)

Max 65 4.63 3.15 274 6.22 3.67 0.57

ID (18) (27) (27) (18) (11) (31) (2)

Note: This table presents the NAICS (2007) IDs and descriptions of the sectors included in thecalibration of the model. In addition, the table reports the average number of rms (#Firms) ineach industry included in the sample from Compustat over the entire sample period 1997-2016 used(a) to calculate aggregate statistics for the respective nancial variables presented in Figure 2.1Section 2 and (b) in the calibration of the model discussed in Section 4.1. The representativeness ofthe sample for each sector is calculated as the share of total sales of rms in industry value added(RP(Y)) and in gross industry output (RP(R)) as reported by the BEA. The last column reports theaverage net-lending position of each sector based on Denition 2.1.

7


D: Model-Derivations

Appendix D.1 presents the household's problem and Appendix D.2 discusses the prot

maximization problem of the nal and intermediate good producing rm. Appendix D.3

comments on the system of equations representing the equilibrium of the model.

D.1. The Household's Optimization Problem

A representative household maximises expected life-time utility

maxC,L

E0

∞∑t=0

βtU (C,L) (t) subject to P (t)C(t) = w(t)L(t) + Π(t) + T (t) (D.1)

In addition to labor income, wL, the household receives total prots (dividends) of rms,

Π =∑

m πm. If the household also owns the nancial sector, she will receive the interest

payments on bank credit in the form of additional transfers, T (t), which will be further

discussed in Section E.1. As there is no uncertainty in the model and the household does

not face an intertemporal decision, I can drop both the expectations operator and the

time subscript for notational convenience. Let the utility function be given by Equation

(D.2) such that the FOCs imply the optimal consumption-labor choice in Equation (D.3).

U (C,L) =C1−εC

1− εC− L1+εL

1 + εL. (D.2) and

LεL

C−εC=

w

P(D.3)

D.2. The Firms' Optimization Problem

Both intermediate and nal goods are produced by a representative, price-taking rm in

the respective sector. While the nal good producing rm does not face a working capital

constraint and thus only solves the prot maximization problem, the intermediate good

producing rm faces two maximisation problems each period: (1) Prot Maximization

Problem, and (2) Credit Decision Problem. Appendix D.2.1 and D.2.2 discuss the prot

maximization problem of the nal and intermediate good rm, respectively. The credit

decision problem of an intermediate good producing rm is discussed Appendix D.2.3.

D.2.1. Final Good Producer's Optimal Input Decisions

The representative rm's dual problem is to (1) choose the optimal amount of the sectoral

inputs V0 = x0mm to minimize the cost of producing F = 1 and (2) maximize prots by

choosing the optimal amount F produced. (1) First, let PF denote the cost-minimizing

8


expenditures on inputs of the nal good producing rm solving

PF ≡ minV0

∑m

pmx0m subject to F = A0

∏m

xωFm0m

where P is the price index of the input composite and A0 = exp(zq0)A0, as outlined in the

main text. The normalization constant is dened as [A0]−1 =∏

m

(ωFm)ωFm and the pro-

ductivity shock zq0 is normally distributed as N (0, σ2q,0) such that A0 = A0 in equilibrium.

The FOCs of the cost-minimization problem imply that P =∏

m pωFmm . (2) The FOCs of

the nal good producing rm's prot maximization problem yield P = P . The optimalnal demand for sector m's output is therefore given by Equation (D.4) and the optimal

price charged for the nal good is shown in Equation (D.5).

x0m = ωFm

(pmP

)−1

F (D.4) and P =∏m

pωFmm . (D.5)

D.2.2. Intermediate Good Producer's Optimal Input Decisions

Production Technology. Using Equation (3.2) to substitute for the composite of

(productive) labor and intermediate inputs, Vk, and the composite of intermediate inputs,

Xk, in Equation (3.1), implies that the production function (3.1) can be written as

qk =(

exp(zqk)Ak

(kαkk `Q,1−αkk

)ηk (∏s

xωXksks

)1−ηk )χk. (D.6)

The normalization constants, AQk , AVk and AXk , are dened as follows and imply that

Ak = AQk

(AVk(AXk)1−υk

)1−αkηk

[AQk

]−1

= χk (αkηk)αkηk (1− αkηk)(1−αkηk)[

AVk

]−1

= (υk)υk (1− υk)(1−υk)[

AXk

]−1

=∏

s

(ωXks)ωXks .

Consequently, the normalization constant of the sector-specic productivity level, Ak =

exp(zqk)Ak, is given by

[Ak]−1 = χk

[ηkα

αkk (1− αk)(1−αk)

]ηk [(1− ηk)

∏s

(ωXks)ωXks

]1−ηk.

The productivity shocks zqk are normally distributed as N (0, σ2q,k) such that Ak = Ak in

equilibrium.

9


Derivation of Optimization Problem. Total revenues of rm k from selling its

output and providing trade credit to its customers are given by

Rk =M∑c=0

(1− θck) pkxck + (1 + rTk )θckpkxck =M∑c=0

(1 + rTk θck

)pkxck = φRk pkqk (D.7)

where the last line substitutes the constraints of sector k for production (3.12) and trade

credit extended (3.13), which are binding in equilibrium. Consequently, the revenue

wedge, φRk , and the average trade-credit share extended to rm k's customers, θCk , are

φRk = 1 + rTk θCk (D.8) , where θCk =

M∑c=0

θckxckqk

(D.9)

with θ0k = 0. The binding working capital constraint implies that total costs of production

including interest payments are

(1 + rBk )BCk +∑s

(1 + rTs )θkspsxks = φLkw(`Qk + `Tk

)+∑s

φXkspsxks (D.10)

where the respective credit wedges are

φLk = 1 + rBk (D.11) and φXks = 1 + (1− θks)rBk + rTs θks. (D.12)

Note that the intermediate goods credit wedge equals a weighted average of the interest

rates on bank and trade credit. Substituting for the binding working capital constraint

implies that prots can be written as

πk = φRk pkqk − φLkw(`Qk + `Tk

)−∑s

φXkspsxks (D.13)

such that the rm's prot maximization problem is given by 3.8 in the main text.

Prot Maximization Problem. The rm's prot maximization problem is solved

as a dual problem in two steps: (1) Given interest rates, r, trade-credit shares, Θ, the

input composite, Vk, and the share of sales extended on trade credit to customers, θCk , rm

k rst chooses production inputs to minimize total costs of production. Having derived the

cost-minimizing input expenditures, rm k then solves for the (2a) optimal level of output

choosing, Vk, the optimal trade-credit shares (2b) demanded from suppliers, θksMs=1 and

(2c) extended to customers, θCk . Each step will be discussed in detail in the following.

Proof of Lemma 3.1. (1) Cost-Minimzation. For given credit links, the optimal input

demand is derived in two steps: given total input expenditures, the rm minimizes ex-

penditures on (a) composite and (b) individual inputs.

10


(a) Let pVk Vk be the cost-minimizing total expenditures on inputs solving

pVk Vk ≡ min`Qk ,Xk

φLkw`Qk + pXk Xk subject to Vk = AVk

(`Qk

)υk (Xk

)1−υk

where pVk is a composite of input costs and pXk is the price of the intermediate composite,

Xk. The FOCs yield that the minimum expenditures pVk Vk to produce one unit of Vk are

pVk =(φLkw

)υk (pXk )(1−υk). (D.14)

The optimal demand for labor `Qk and the composite intermediate input Xk is given by

`Qk =(1− αk)ηk1− αkηk

(φLkw

pVk

)−1

Vk (D.15) and Xk =(1− ηk)1− αkηk

(pXkpVk

)−1

Vk. (D.16)

(b) Similarly, let pXk Xk be the cost-minimizing total expenditures on intermediate goods:

pXk Xk ≡ minxkss

∑s

φXkspsxks subject to Xk = AXk∏s

(xks)ωXks .

The FOCs imply that the aggregate price index of the composite intermediate good Xk is

pXk = φXk∏s

(ps)ωXks (D.17) where φXk =

∏s

(φXks)ωXks . (D.18)

The optimal demand for the intermediate input from supplier s, xks, is given by

xks = ωXks

(φXkspspXk

)−1

Xk = ωXks(1− ηk)1− αkηk

(φXkspspVk

)−1

Vk. (D.19)

Exploiting the result of Lemma 3.2 implies that optimal demand for productive labor, `Qk ,

and intermediate inputs, xks, can be written in terms of total revenues, Rk = φRk pkqk,

`Qk = (1−αk)ηkχk(φLkw

φRk pk

)−1

qk, (D.20) xks = ωXks(1− ηk)χk(φXkspsφRk pk

)−1

qk. (D.21)

This completes the proof of Lemma 3.1.

Proof of Corollary 3.1. For given credit costs, r, and shares, Θ, using Equation (D.15)

and (D.19), total costs of productive inputs including interest payments are

CQk = φLkw`

Qk +

∑s

φXkspsxks = pVk Vk. (D.22)

11


Then marginal cost of production including interest-cost are given by

pVk =(φLk

)υk ( ∏s

(φXks)ωXks )(1−υk)

︸︷︷︸= φVk

(w)υk (∏

s

(ps)ωXks

)(1−υk)

︸︷︷︸= mcVk

(D.23)

where φVk denotes the composite credit wedge which is a function of the credit links

between sectors. The properties of φVk with respect to v ∈ rBk , rTk , θks are given by

∂φVk∂v

1

φVk=

υkφLk

∂φLk∂v

+ (1− υk)∑s

ωXksφXks

∂φXks∂v

Straight forward calculations imply that

∂φVk∂v

1

φVk=

0 < υk

φLk+ (1− υk)

∑s ω

Xks

(1−θks)φXks

for v = rBk

0 < +(1− υk)ωXksθksφXks

for v = rTs

0 ≶ −(1− υk)ωXks(rBk −r

Ts )

φXksfor v = θks

Corollary 3.1 therefore follows directly from Lemma 3.1.

Proof of Lemma 3.2. (2a) Prot Maximization. Using the results of Lemma 3.1 and

Corollary 3.1, prots of rm k in Equation (D.13) can also be written as

πk = φRk pkqk − pVk Vk − (1 + rBk )w`Tk . (D.24)

The FOC with respect to Vk is given by

∂πk∂Vk

: pVk Vk = (1− αkηk)χkφRk pkqk. (D.25)

Equation (D.25) implies that total input expenditures (including interest rate costs) are

a fraction of total revenues. The optimal goods price equals

pk =MCV

k

MP Vk

=φVkφRk

mcVk(1− αkηk)χkqkV −1

k

(D.26)

and the eective mark-up over marginal costs of production, φVk (φRk )−1, is a combination

of credit and revenue wedges. This completes the proof of Lemma 3.2.

Properties of MC. In the following, Assumption D.1 derives a lower bound for the

convexity of the risk premium, rZk , in the measure of default probability, θZk , such that the

marginal cost of production, pVk , are convex in the share of revenues extended on trade

credit to customers, θCk . To this end, two auxiliary measures are rst dened below.

12


Denition D.1 (Denition of MC-Wedge Elasticities). Dene εBV,k as the elasticity of the

marginal cost wedge wrt the interest rate on bank credit, and εBε,k as the elasticity of εBV,k,

wrt the interest rate on bank credit:

εBV,k =∂φVk∂rBk

rBkφVk

and εVε,k =∂εBV,k∂rBk

rBkεBV,k

, with E0k =εBV,krBk

and E1k = −∂E0k

∂rBk

such that E0k, E1k < 1 and

E0k =υkφLk

+ (1− υk)∑s

ωXks(1− θks)φXks

and E1k =υk

(φLk )2+ (1− υk)

∑s

ωXks(1− θks)2

(φXks)2

.

Assumption D.5 (Credit Cost Parameter µ). Let the degree of convexity of the risk

premium, rZk , dened in Equation (3.7) in the joint default probability measure, θZk , be

µ > µ = maxµLBm

Mm=1

and

[µLBm ]−1 = 1− (E1m − E20m)

E0m

rZm = 1−(1− (εVε,k + εBV,k)

) rZmrBk.

Derivation of Assumption D.5. Using Equation (3.7) and Denition D.1

∂rBk∂θCk

= µrZkθZk,

∂2rBk∂(θCk )2

=(µ− 1)

θZk

∂rBk∂θCk

and∂E0k

∂θCk= −E1k

∂rBk∂θCk

,

the rst and second order derivative of marginal costs, pVk , wrt the average trade-credit

share extended to customers are

∂pVk∂θCk

=∂pVk∂φVk

∂φVk∂rBk

∂rBk∂θCk

= E0kpVk

∂rBk∂θCk

∂2pVk∂(θCk )2

=∂

∂θCk

(∂pVk∂θCk

)=

∂E0k

∂θCk

∂rBk∂θCk

pVk + E0k∂2rBk∂(θCk )2

pVk + E0k∂rBk∂θCk

∂pVk∂θCk

.

The second order derivative can be alternatively written as

∂2pVk∂(θCk )2

=

[−(E1,k − E2

0,k)∂rBk∂θCk

+ E0,k(µ− 1)

θZk

]∂rBk∂θCk

pVk .

It follows that E1,k > E20,k and 1 > εVε,k + εBV,k, due to Jensen's Inequality for f(.) being a

convex function

f(∑

s

wvsφ

vks

)≤∑s

wvsf(φvks

)where wv

s ∈ υk, (1− υk)ωXksMs=1 and φvks ∈ (φLk )−1, (1− θks)(φXks)−1Ms=1.

13


D.2.3. Optimal Credit Decisions

Proof of Lemma 3.4. Having derived the cost-minimizing input expenditures for given

trade-credit shares, Θ, the input composite, Vk, and the share of sales extended on trade

credit to customers, θCk , rm k now chooses Vk = θksMs=1, θCk , Vk to maximise prots

while taking the demand for trade credit from their customers, θckMc=1, as given.

maxVk

φRk pkqk − pVk Vk − (1 + rBk )w`Tk subject to (3.1), (3.6), (3.7),

and the feasibility constraint θCk , θksMs=1 ∈ [0, 1] .

Optimal Portfolio Choice. (a) The FOC wrt θksMs=1 is

∂πk∂θks

: −[∂pVk∂θks

Vk + (1 + rBk )∂w`Tk∂θks

]− λ1 = 0 (D.27)

where λ1 is the Kuhn-Tucker Lagrange multiplier associated with the feasibility con-

straints. Applying the results of Lemma 3.1 and Corollary 3.1 and assuming that the

optimal θks ∈ (0, 1)∀k, s, such that λ1 = 0, the FOC implies that

∆kspsxks1 + rBk

= κT0,ks +κT1,ks

(θSk )2(θks − θSk )

and Equation (3.22) in the main text follows. Note that if rm k can adjust its credit

portfolio frictionless such that κT1,ks = 0∀ks, then the optimal demand for trade credit is

θks =

1 if psxks∆ks − (1 + rBk )κT0,ks > 0 for ∆ks > 0

0 if psxks∆ks − (1 + rBk )κT0,ks < 0 for ∆ks < 0

Thus, the introduction of non-linear management costs of credit lines and respective

parameter choices ensures that the demand for trade credit will have an interior solution.

(b) The 2nd order derivative of πk with respect to θks is given by

∂2πk∂(θks)2

= −[∂2pVk∂(θks)2

Vk + (1 + rBk )∂2w`Tk∂(θks)2

](D.28)

and - using Equation (3.22) - is negative at the optimum if

∆ks

(θks − θκks

)φXks

<1

1− (1− υk)ωXks

holds. Note that ∆ks

(θks − θκks

)can be interpreted as the net-interest rate costs associ-

ated with choosing an optimal credit-portfolio share above or below the trade-credit share

14


determined solely by the parameters of the model, θκks. The condition for a maximum

implies an upper bound for the share of net-interest cost of optimal credit portfolio devi-

ations, ∆ks

(θks − θκks

), in gross credit costs, φXks, related to nancing k's expenditures on

input s. While the LHS is ∈ [0, 1), the RHS of condition is greater than one such that

the optimal trade-credit share θks maximizes prots.

Optimal Share of Revenues Extended on Trade Credit. (a) The FOC wrt θCk is

∂πk∂θCk

:∂φRk∂θCk

pkqk =∂pVk∂θCk

Vk +∂rBk∂θCk

w`Tk (D.29)

where the feasibility constraint is non-binding. The interest rate on trade credit charged

is the solution to Equation (D.29). Using the results of Lemma 3.1 and Corollary 3.1,

the FOC implies that

rTk pkqk =∂rBk∂θCk

BCk where BCk = w(`Qk + `Tk ) +∑s

(1− θks)psxks (D.30)

and Equation (3.23) in the main text follows.

(b) The second order derivative with respect to θCk is

∂2πk∂(θCk )2

= −[∂2pVk∂(θCk )2

Vk +∂2rBk∂(θCk )2

w`Tk

]< 0 (D.31)

Let µ be given by Assumption D.5, then rTs implied by the optimal trade-credit share

extended to customers, θCk , maximizes rm k's prots.

D.3. Summary

The system of equations consists of (5+12M+2M2) equations (minus one after accounting

for the numeraire) in the same number of unknowns:

(a) Aggregate Quantities (3): C,F, L

(b) Sectoral Quantities (6M +M2): `Qk , `Tk , qk, Vk, Xk, xksMs=1, x0kMk=1

(c) Prices (2 + 3M): w,P, pVk , pXk , pkMk=1

(d) Credit Costs and Shares (3M +M2): rBk , rTk , θCk , θksMs=1Mk=1

Accounting for the numeraire (w) and substituting reduces the number of equations to

(3 + 4M) with the following set of unknowns: C,F, Vk, θCk Mk=1, P, pk, rTk Mk=1. The

reduced set of equations is used to simulate and directly solve for the equilibrium of the

model in Section 4 applying the iterative procedure described in Appendix G.1. Total

nominal (Mk) and real (xMk) imports as well as capital (kk) are included to ensure a

consistent mapping of the model to the data and are treated as constants.

15


E: Equilibrium

This appendix derives the partial equilibrium of the model economy introduced in Section

3 taking the cost, r, and composition of credit, Θ, as given. Since the model economy

introduced in Section 3 nests the economy in BL(2019), the partial equilibrium of this

economy maps into similar expressions as derived in BL(2019) masking that trade-credit

costs and linkages are in fact endogenous and will distort the propagation of shocks as

discussed in more detail in Section 3.2. Taking into account the calibration exercise,

Appendix E.1 rst comments on the national accounting and the equilibrium level of

capital in this economy. Appendix E.2 derives the partial equilibrium expressions of sales,

prices, output, aggregate GDP, labor, the aggregate eciency and labor wedge following

the same steps as in BL(2019).

E.1. National Accounting

Capital and Investment. The model presented in Section 3 is static, where

capital is treated as a constant in the production function (3.1) and is equal to its steady

state level. In order to derive the equilibrium level of capital used in the calibration of

the model in Section 4.1, I now discuss the optimization problem of rm k for the case

in which rms own and invest in their capital stock by purchasing the nal good for

investment, ik. Dene zk = [zqk, zbk] and let the law of motion for capital be given by

k′k = ik + (1− δ)kk. (E.1)

The intermediate good rm k's prot maximization problem is formulated recursively as

V(zk, k) = maxV,k′

(πk − Pik) + Etm′V(z′k, k′) subject to (E.1),(D.13),(3.1), (3.6), (3.7),

and feasibility constraints on trade-credit shares θCk , θks ∈ [0, 1] and non-negativity con-

straints for all variables. In addition to the set of choice variables from the static opti-

mization problem, V , rm k now also chooses its capital stock for next period, k′k. Since

the rm applies the stochastic discount factor of the household: m = βtλ = βt(Ct)−εC

and m′ = βt+1(Ct+1)−εC , the capital euler equation is given by

P = βE[(

C

C ′

)εC (αkηkχkR

′k(k′k)−1 + P ′(1− δ)

)](E.2)

In equilibrium, all variables are constant such that equilibrium nominal capital expendi-

tures are a fraction of sectoral revenues, PrKk kk = αkηkχkRk, where 1 = β(1 + rKk − δ).The law of motion of capital implies that equilibrium investment is given by ik = δkk ∀k.

16


Due to the static nature of the model, it is assumed that capital never depreciates, δ = 0,

such that investment equals, ik = 0∀k, and I =∑

m im = 0.

Revenue and Expenditure Decomposition. In equilibrium, the intermediate

good rm spends its revenues on the respective factors including credit costs and prots:

Rk ·

χk(1− ηk) = φXk pXk Xk ... intermediate inputs

χkηk(1− αk) = φLkw`Qk ... productive labor

χkηkαk = PrKk kk ... capital

(1− χk) = Πk ... non-productive labor, φLkw`Tk , and dividends, dk.

Dene sπk = (1 − χk) and denote sdk as the share of Πk spent on net dividend payments

and (1 − sdk) as the share spent on non-productive labor including interest rate costs,

respectively. Total cost of production including interest rate costs are then

Ck = φLkw`k +∑s

φXkspsxks = pVk Vk + (1− sdk)sπkRk = sCk Rk (E.3)

where sCk = (1− αkηk)χk + (1− sdk)sπk denotes the total cost share in revenues. Similarly,

the share of revenues spent on dividends payments, is simply given by (1−sCk ). Total cost

of production, Ck, can be decomposed into a share, (1−θBk ), that is spent on actual input

expenditures, and a share, θBk , that is spent on bank interest payments. The equilibrium

expenditures of sector k on bank interest payments, rBk∑

s(1− θks)psxks + rBk w`k, equal

θBk Ck =[

(1− ηk)χk∑s

ωXkssXks + sLk

((1− αk)ηkχk + (1− sdk)sπk

) ]Rk = sBk Rk

where sXks = rBk (1 − θks)/φXks, sLk = rBk /φLk denote the shares of interest rate payments in

the respective input expenditures. Total cost of production and credit management costs

excluding bank interest rate payments are given by (1− θBk )Ck = (sCk − sBk )Rk.

Total Value Added and Foreign Trade Adjustments. In anticipation of the is-

sues involved in mapping the equilibrium of the model to the data, the following paragraph

discusses the adjustment of the equilibrium expressions while accounting for imports. In

particular, the adjustments involve that (1) total nal demand faced by intermediate good

producing rms also includes foreign demand (MQk ) for some sectors and (2) that some

capital services (MKk ) and the income of the nancial sector from interest payments on

bank loans, Tk, are imported and therefore not part of total domestic income. A detailed

description of the adjustments made to the IO-tables is provided in Appendix G.1.

17


In order to obtain analytical expressions of the partial equilibrium of this economy

described in Appendix E.2, sector specic imported demand,MQm, as well as total imported

demand,MQ0 , are expressed as a share of total labor income, while sector specic imported

capital services, MKk , are expressed as a share of sectoral domestic revenues

MQ0 =

∑m

MQm =

∑m

pkxMk =∑m

ωFk sMk wL = sM0 wL and MK

k = sKk Rk.

Total nominal imports,M, thus equal the sum of foreign demand, expenditures on foreign

capital and nancial services. Total nominal value added, PY = PF −M, is derived by

consolidating the household and rm budget constraints and net-exports

PY = wL+∑k

(1− sKk )Rk − φLkw`k −∑s

φXkspsxks = wL+∑k

(1− sCk − sKk )Rk (E.4)

where the last equality uses the total cost share derived in the previous paragraphs.

Similarly, total nominal nal demand, PF = PY +M, can be written as

PF = (1 + sM0 )wL+∑k

Rk −∑s

(1 + rTs θks)psxks − w`k = (1 + sM0 )wL+∑k

sRkRk (E.5)

where sRk = 1 − (sCk − sBk ). Note that the market clearing condition of the nal good

sector is given by PF = PC. In other words, the nal good is consumed by the domestic

household only, who spends total domestic income (GDP) and imports on consumption

expenditures. Note that the total income of households includes imports and that the

equilibrium total consumption expenditures can also be expressed as

PC = M+∑k

pkqk −∑k

∑s

psxks (E.6)

where the market clearing condition for aggregate trade credit has been applied. While

nominal imports are included as a constant in the model-simulations in order to be con-

sistent with IO-tables, imports for nal demand are abstracted from in the following.

E.2. Partial Equilibrium

Lemma E.1 (Revenues in PE). Let the wage rate be the numeraire and dene sRk as in

Appendix E.1. Then (a) the vector of revenues of intermediate good producers is

R = CRΦRΩF ιL = κRφL (E.7)

where

CR =[I −ΦR

(inv(ΦX) ΩX

)′diag(χ (ι− η))−ΦRΩF

(ι(sR)′

)]−1

, (E.8)

18


the entries of the wedge-matrices ΦR and ΦX are given by

[ΦR]kk = 1 + rTk∑c

xckqkθck and [ΦX ]ks = 1 + rBk (1− θks) + rTs θks (E.9)

(b) and the revenues of the nal good producer are

R0 =(

1 + (sR)′κRφ

)L. (E.10)

Proof of Lemma E.1. (a) The revenues of the Intermediate Goods Firm k are

Rk = φRk pkqk = φRk pk

M∑c=0

xck. (E.11)

The left-hand side of Equation (E.11) represents the revenues generated by selling the

domestically produced output, qk, and the right hand side denotes revenues generated by

total demand (net of quantity-imports). Using the optimal demand for sector k's output

derived in Appendix D and applying the denitions of Appendix E.1, the revenues of rm

k are

Rk =∑c

(φRkφXck

)ωXck(1− ηc)χcRc + φRk ω

Fk

(wL+

∑m

sRmRm

)Dene the vector of revenues of the intermediate goods rm as R = [R1 · · · RM ]′. Then

stacking yields the following system of equations

R = ΦR[(inv(ΦX) ΩX

)′diag(χ (ι− η)) + ΩF

(ι(sR)′

)]R+ ΦRΩF ι (wL)

where the entries of matrices ΦR and ΦX are dened in Lemma E.1. Equation (E.7)

follows where the coecient matrix CR is dened in Equation (E.8) with typical entry

[(CR)−1]

kc= 1|c=k − (1 + rTk θ

Ck )

[ωXck(1− ηc)χc

1 + rBc (1− θck) + rTk θck+ ωFk s

Rc

]

(b) The revenues of the Final Good Firm equal nal consumption sales, R0 = PF = PC.

Using the denitions of Appendix E.1 and Equation (E.7), Equation (E.10) follows from

direct calculations.

Lemma E.2 (Prices in PE). (a) Let the wage rate w be the numeraire and dene α =

(χ α η). Then the vector of partial equilibrium sectoral prices is

log (p) = CP[−χ zq + log(φP ) + (ι− χ) (log(κRφ ) + ι log(L)) + α log(rK)

](E.12)

19


where the coecient matrix is dened as

CP =[I − diag

(χ (ι− η)

)ΩX − α (ιω′F )

]−1

(E.13)

and the vector of price wedges is given by

log(φP ) = χ (ι−α η) log(φV )− log(φR). (E.14)

(b) The aggregate price level is

log(P ) = ω′F log(p). (E.15)

Proof of Lemma E.2. (a) Substituting for output of sector k in Rk = φRk pkqk using

sector k's production function as given in Equation (D.6) as well as the prot maximizing

intermediate input composite (D.25) and capital (E.2), yields

Rk = φRk pk

[exp(zqk)Rk

(PrKk

)−αkηk (pVk )−(1−αkηk)]χk

.

Applying the results of Corollary 3.1, taking logs and rewriting yields

log(pk)− (1− ηk)χk∑m

ωXkm log(pm)− αkηkχk∑m

ωFm log(pm) = ...

log(φPk ) + (1− χk) log(Rk)− χk[zqk − αkηk log(rKk )− (1− αk)ηk log(w)

](E.16)

where log(φPk ) = (1−αkηk)χk log(φVk )− log(φRk ). Stacking Equation (E.16) for all sectors,

using R = κRφ (wL) and taking the wage rate as the numeraire yields Equation (E.12).

(b) The aggregate price level is given by Equation (E.24) and follows from Section D.2.1.

Lemma E.3 (Sectoral Output in PE). Let the wage rate w be the numeraire and dene

α = (χ α η). The vector of sectoral output is given by

log(q) = CP[χ zq − α log(rK)

]− φQ +

[I − CPdiag (ι− χ)

][log(κRφ ) + ι log(L)

](E.17)

where the output wedge is dened as

φQ = CP log(φP ) + log(φR). (E.18)

Proof of Lemma E.3. The vector of sectoral output is derived by substituting R and

p using the results of Lemma (E.1) and (E.2) in log(q) = log(R)− log(p)− log(φR) and

taking the nominal wage rate w as the numeraire. Lemma E.3 follows.

20


Denition E.1 (Eciency Wedge). Dene CY = ω′FCP and let the aggregate labor share

and productivity be given as

λ = CY (ι− χ) (E.19) and log(Zz) = CY diag(χ)zq. (E.20)

The eciency wedge is given by

log(ΦZφ ) = log(ΦZ

Φ) + log(ΦZK) (E.21)

where the components attributed to credit costs and capital are dened as follows

(a) credit costs

log(ΦZΦ) = −CY log(φP )− CY diag(ι− χ) log(κRφ ) + log(κYφ ) (E.22a)

(b) internal rental rate of equilibrium capital

log(ΦZK) = −CY diag(χ α η) log(rK). (E.22b)

The aggregate share of total value added in aggregate labor supply is given by

κYφ = ι′κYφ = 1 +∑m

[1− sCm − sKm

][κRφ ]m (E.23)

where the respective cost-shares are dened in Appendix E.1 and sY denotes the vector of

sectoral shares of value added in revenues.

Derivation of Denition E.1. Dene CY = ω′FCP . Substituting the vector of prices,

p, in log(P ) = ω′F log(p) with Equation (E.12) and collecting terms yields

log(P ) = −[log(ΦZ

φ )− log(κYφ )]− log(Zz) + λ log(L) (E.24)

and Equation (E.19) to (E.23) follow. The vector of sectoral shares of total value added

in aggregate labor supply, κYφ , as well as the sectoral share of value added in revenues,

sY , are dened in Lemma E.4. Since

1 =∑m

(1− sLm)[(1− αm)ηmχm + (1− sdm)sπm][κRφ ]m

holds in equilibrium due to labor market clearing, Equation (E.23) follows.

21


Lemma E.4 (Value Added). Let the wage rate w be the numeraire and let the corre-

sponding expenditures shares, sLk , sXks, s

dk, s

πk and sKk be dened in Appendix E.1.

(a) Real sectoral value added is given by

log (y) =[ι log(ΦZ

φ )−(ι log(κYφ )− log(κYφ )

) ]+ ι[log(Zz) + (1− λ) log(L)

](E.25)

where κYφ is the vector of sectoral shares of value added in aggregate labor, κYφ = sY κRφ ,and sY , equals the share of sector k's value added in revenues with typical entry

[sY ]k =[(1− sLk )

((1− αk)ηkχk + (1− sdk)sπk

) ]+[1− sCk − sKk

]. (E.26)

(b) Real aggregate GDP equals the sum of sectoral value added such that

Y = ZzΦZφL

(1−λ) (E.27)

where Zz denotes aggregate productivity and ΦZφ equals the aggregate eciency wedge

dened in Denition E.1.

Proof of Lemma E.4. (a) Value added of sector k equals total labor expenditures and

prots net of capital imports and interest payments on bank loans:

Pyk = w`k + πk − sKk Rk = sYk Rk = [κYφ ]kL.

Using the accounting results of Appendix E.1 as well as Lemma 3.1, the share of sectoral

value added in revenues, sYk , is dened in Equation (E.26). Substituting for revenues using

Equation (E.7) allows to rewrite sectoral value added as a share of aggregate labor supply,

[κYφ ]k. Stacking, taking logs and applying Equation (E.24), yields Equation (E.25).

(b) Real GDP equals the sum of sectoral value added and is given by

PY = P∑k

yk = wL+∑k

(1− sCk − sKk )Rk =∑k

sYk Rk = ι′κYφL.

Taking logs, dening the wage rate as the numeraire and applying Equation (E.24) yields

log (Y ) = − log(P ) + log(κYφ)

+ log (L) = log(Zz) + log(ΦZφ ) + (1− λ) log(L)

such that the aggregate production function is given in Equation (E.27).

22


Lemma E.5 (Aggregate Labor and Output in GE). Let the wage rate w be the numeraire.

(a) Assuming that aggregate consumption is a share γ of GDP, Y , aggregate labor supply

is given by

log(L) = κZL[log(Zz) + log(ΦZ

φ )]− κLL

[log(κYφ ) + εC log(γ)

](E.28)

where

κZL =1− εC

1 + εL − (1− εC)(1− λ), κLL =

1

1 + εL − (1− εC)(1− λ). (E.29)

(b) Aggregate GDP is then dened as

log (Y ) = κZY[log(Zz) + log(ΦZ

φ )]− κLY

[log(κYφ)

+ εC log(γ)]

(E.30)

where

κZY =1 + εL1− εC

κZL , κLY = (1− λ)κLL. (E.31)

Proof of Lemma E.5. From the household's problem in Section D.1, the log of labor

supply is

εL log(L) = log(w)− log(P )− εC log(C).

Taking the nominal wage rate as the numeraire, assuming that aggregate consumption is

a share γ of GDP, Y , and using Equation (E.24) and (E.27), aggregate labor supply is

[εL + εC(1− λ)] log(L) = −λ log(L)− log(κYφ ) + (1− εC)[log(Zz) + log(ΦZ

φ )]− εC log(γ).

Rearranging and collecting terms yields Equation (E.28).

(b) Substituting for equilibrium aggregate labor supply, L, in Equation (E.27) yields

aggregate GDP as shown in Equation (E.30).

Denition E.2 (Eciency Wedge). Following BL(2019) and applying the concepts in-

troduced in Chari et al. (2007), the aggregate labor wedge, ΦLφ , is dened as

−ULUC

=w

P= ΦL

φ(1− λ)Y L−1.

It denotes the wedge between the household's marginal rate of substitution between con-

sumption and labor and the aggregate marginal product of labor. Using the results of

Lemma E.5 the aggregate labor wedges equals

log(ΦLφ) = − log(1− λ)− log(κYφ ).

23


F: Log-Linearization

In this appendix, I log-linearize the model around its equilibrium, using x = x exp(x) ≈x(1 + x) and x = dlog(x) = log x/x. The log-linearized equilibrium responses of revenues,

prices and sectoral output in terms of (1) productivity shocks, (2) general equilibrium

adjustments in the aggregate labor supply and (3) distortions introduced as credit wedges

are derived in Appendix F.1. In addition, credit wedges are decomposed into eects

attributed to changes in interest rates on bank and trade credit and changes in trade-credit

shares. The log-linearized response of credit costs and shares is discussed in Appendix F.2.

Appendix F.3 derives the rst order approximation of the (partial equilibrium) structural

output responses discussed in the main text.

To introduce additional notation, the eects of log-changes of the variables of interest

are determined by the entries of the corresponding elasticity matrices E, which are non-

linear functions of the steady state of the economy. In the subsequent derivations, the

following simplifying assumptions are made: (1) I abstract from productivity shocks and

consider the partial equilibrium case dened in 3.4 only. (2) The wage rate is taken as

the numeraire and capital is constant such that kk = 0.

F.1. Log-Linearized Equilibrium and Wedges

Proof of Lemma 3.5 . (a) Revenues of the intermediate and nal good producing

sector are

Rk = φRk pk

M∑c=0

xck, and R0 = wL+M∑m=1

πm + rBmBCm

where sectoral prots and bank loans are given by

πk = Rk − (1 + rBk )BCk −∑s

(1 + rTs )θkspsxks, and BCk = w`k +∑s

(1− θks)psxks

Using the results of Lemma 3.1, taking the wage rate as the numeraire and log-linearizing:

Rk = −φSX,k +∑c

[WXR ]ckRc + [W F

R]kkR0, and R0 = ELF L+

∑m

[WRF ]mmRm − φSF .

Note that R0ELF = L and the entries of the weight matrices equal the sales and income

shares and are dened in the main text. The intermediate, φSX,k, and nal, φSF , sales

wedges are given by

φSX,k =∑c

[WXR ]ckφ

Xck − φRk and φSF =

∑m

φFm (F.1)

24


with

φFm =∑s

[W TF ]msr

Ts + [W θ

F ]msθms − [W ΦF ]msφ

Xms − [W φ

F ]mφLm. (F.2)

The entries of the respective elasticity matrices are

R0[W ΦF ]ms = (1 + rTs θms)psxms

R0[W θF ]ms = (1 + rTs )rBmAPms(φ

Lm)−1

R0[W φF ]m = w`Qm

R0[W TF ]ms = rTs APms

(F.3)

Stacking Rk for each sector, substituting for nal revenues, rearranging and collecting

terms implies that the response of the vector of intermediate revenues is

R = −φRκ +WRWFR ιE

LF L and φRκ = WRφ

S. (F.4)

The matrix WR is dened in Equation (3.24) in the main text and captures all direct

and indirect demand interactions in the economy. The intermediate sales credit wedge,

φS, is obtained by stacking Equation (F.1) and summarizes the eect of changes in credit

costs and the composition of credit on intermediate sales. Equation (3.26) follows. The

log-change in the response of nal sales and the nal sales wedges are

R0 =(1 + ι′WR

FWRWFR ι)ELF L− φFκ and φFκ = ι′WR

FWRφS + φSF (F.5)

The nal sales credit wedge, φFκ , summarizes the eect of changes in credit costs and the

composition of credit on the income of households.

(b) Prices. The log-linearization of Equation (D.23) yields

pVk = φVk + (1− υk)∑s

ωXksps, where φVk = υkφLk + (1− υk)

∑s

ωXksφXks.

Using the results of Lemma 3.2 implies that the log-linearisation of

Rk = φRk pk [ exp(zqk)AQk

(kk)αkηk

(Vk)(1−αkηk) ]χk

yields

pk = (1− ηk)χk∑m

ωXkmpm + φPk + (1− χk)Rk − χkzqk , where φPk = χkφVk − φRk

and χk = (1− αkηk)χk, such that an entry of Equation (3.25) dened in the main text is

given by

φPk = (1− αk)ηkχkφLk + (1− ηk)χk∑s

ωXksφXks − φRk . (F.6)

Stacking sectoral prices, substituting for R, dening WRP = WPdiag(ι − χ)WR and

25


rearranging yields

p = φPκ +WRPW

FRιE

LF L−WPdiag(χ)zq , where φPκ = WP φ

P −WRP φ

S (F.7)

and the matrix WP is dened in Equation (3.24) in the main text. The composite price

credit wedge, φPκ , summarizes the composite eect of credit costs and trade-credit shares

on marginal cost of production and sales due to DRS. The log-linearisation of the aggregate

price level yields P =∑

m ωFmpm.

The auxiliary Lemmata F.1 to F.3 derived in the following show that changes in both

the combines sales and price wedge are a linear combination of the changes in credit costs

and the credit composition, τ . An increase in sales wedge reduces intermediate revenues

and captures the eect of changes in interest rates and trade-credit shares on intermediate

demand and households' income. Similarly, an increase in the price wedge increases prices

and summarizes the eect of changes in credit costs and the credit portfolio on the cost

of production as well as demand, due to the presence of decreasing returns. The lemmata

follow from tedious but straightforward log-linearization of the respective wedges. The

proofs presented in the following are a condensed version of the derivation steps. Further

details on the algebra are available upon request.

Lemma F.1 (Credit Wedges and Management Costs). (a) Credit Wedges. The log-

linearized revenue wedge of sector k is

φRk = [ETφ(R)]kkr

Tk +

∑c

[Eθφ(R)]ckθck +GE(W ). (F.8)

The labor and intermediate credit wedge deviations for each sector k are given by

φLk = [EBφ(L)]kkr

Bk and φXks = [EB

Φ ]ksrBk + [ET

Φ ]ksrTs − [Eθ

Φ]ksθks (F.9)

where the entries of the elasticity matrices are dened as follows

Rk[ETφ(R)]kk = rTkARk

Rk[Eθφ(R)]ck = rTkARck

φLk [EBφ(L)]kk = rBk

φXks[EBΦ ]ks = (1− θks)rBk

φXks[ETΦ ]ks = θksr

Ts

φXks[EθΦ]ks =

(rBk − rTs

)θks

(F.10)

The sign of the elasticity wrt to trade-credit shares depends on the sign of the interest-rate

dierential ∆ks = rBk − rTk . All remaining elasticities are positive.

26


(b) Credit Management Costs. The log-linearization of Equation (3.6) yields

CTk =

∑s

[EθC ]ksθks. (F.11)

Applying Equation (3.22), the typical entry of the elasticity matrix equals

CTk [Eθ

C ]ks = κT0,ksθκks + κT1,ks

(θks − θSkθSk

)θks

θSk=

∆ksAP ks

1 + rBk. (F.12)

The sign of the entries of the elasticity matrix depends on the interest dierential.

Proof. The proof of Lemma F.1 follows from direct calculations.

Lemma F.2 (Sales Wedge). The response of the sales wedge, φS, dened in Equation

(3.26) is given by

φSk = −∑m

[EBS ]kmr

Bm +

∑m

[ETS ]kmr

Tm +

∑m

∑s

[EθS]k,msθms (F.13)

Let vector of bank credit obtained for productive inputs be given by Equation (3.27). The

elasticity of sector k's sales wedge wrt changes in sector m's bank interest rate, rBm, is

[EBS ]km = rBm

∑i

[WBS ]k,miBC

Qmi, where [WB

S ]k,m: =[[W

B(L)S ]km , [W

B(X)S ]k,m:

],

wrt sector m's trade-credit interest rate, rTm, and sector m's trade-credit share obtained

from sector s, θms, is

[ETS ]km = rTm

∑c

[W TS ]k,cmARcm and [Eθ

S]k,ms = [W θS ]k,msAPms.

The entries of the respective weight matrices are given by

[WB(L)S ]km =

ωFkpqk

1

φLm, [W

B(X)S ]k,ms =

ωFkpqk

(1 + θmsrTs )− Is=k

pqk

1

φXms

[W TS ]k,nm =

ωFkpqk

(1− θnm)rBn −(φXnm − φRm

φRm

)Im=k

pqm

1

φXnm

[W θS ]k,ms =

ωFkpqk

[(1 + rTs )rBm + ∆ms

φXmsφLm

]−[rBm + rTs

(φXms − φRs

φRs

)]Is=kpqk

1

φXms

The sign of the respective entries are functions of the nal demand structure such that

[W .S]ki is positive if (ωFk ≥ ωFk ) and non-positive if (ωFk < ωFk ).

Proof. The results of Lemma F.1 are used to substitute for the log-changes in interme-

diate credit and revenue wedges. Consequently, both the nal, φSF , dened in Equation

27


(F.2) and the intermediate sales wedge, φSX , dened in Equation (F.1) are expressed as

a linear combination of changes in credit costs and the credit composition, τ . The log-

change in the sales wedge, φS, for sector k as shown in Equation (F.13) is then obtained

by combining the log-linearised intermediate and nal sales wedge. Lemma F.2 follows.

Corollary F.1 (Total Sales Wedge). The response of sector k's total sales wedge, φRκ ,

dened in Equation (F.4), is

[φRκ ]m = −∑m

[EBR ]kmr

Bm +

∑m

[ETR]kmr

Tm +

∑m

∑s

[EθR]k,msθms. (F.14)


elasticity of sector k's total sales wedge wrt changes in sector m's bank interest rate, rBm,

[EBR ]km = rBm

∑i

[WBR(R)]k,miBC

Qmi where [WB

R(R)]k: =[[W

B(L)R(R)]km , [W

B(X)R(R) ]k,m:

],



[ETR]km = rTm

∑c

[W TR(R)]k,cmARcm and [Eθ

R]k,ms = [W θR(R)]k,msAPms.

Dene Av = WR for v = R. Then the respective entries of the coecient matrices are

[WB(L)R(R)]km =

(∑n

ωFn[AR]knpqn

)1

φLm

[WB(X)R(R) ]k,ms =

(∑n

ωFn[AR]knpqn

)(1 + θmsr

Ts )− [AR]ks

pqs

1

φXms

[W TR(R)]k,cm =

(∑n

ωFn[AR]knpqn

)(1− θcm)rBc −

[AR]kmpqm

(φXcm − φRm

φRm

)1

φXcm

[W θR(R)]k,ms =

(∑n

ωFn[AR]knpqn

)[(1 + rTs )rBm + ∆ms

φXmsφLm

]− [AR]ks

pqs

[rBm + rTs

(φXms − φRs

φRs

)]1

φXms

Proof. Using Equation (F.4) and the results of Lemma F.2 yields the response of the

total sales wedge of sector k. Corollary F.1 follows.

Denition F.1 (Demand Eects). Let the matrices WBR(v),W

TR(v) and W

θR(v) be dened

in Corollary F.1. The subscript R(v) for v ∈ R,P,Q denotes the coecient matrix

summarizing the nal demand eects on (R) revenues AR = WR, (P ) prices AP =

WPdiag(ι− χ)WR, and (Q) output AQ = (I −WPdiag(ι− χ))WR.

28


Lemma F.3 (Total Price Wedge). The response of sector k's total price wedge, φPκ ,

dened in Equation (F.7) is

φPκ,k =∑m

[EBP ]kmr

Bm +

∑m

[ETP ]kmr

Tm −

∑m

∑s

[EθP ]k,msθms (F.15)


elasticity of sector k's total price wedge wrt changes in sector m's bank interest rate, rBm,

[EBP ]km = +rBm

∑i

[WBP ]k,miBC

Qmi and [WB

P ]k: =[[W

B(L)P ]km , [W

B(X)P ]k,m:

],



[ETP ]km = rTm

∑c

[W TP ]k,cmARcm and [Eθ

P ]k,ms = [W θP ]k,msAPms.

The respective entries of the coecient matrices are given by

[WB(L)P ]km =

[WP ]km

Rm+ [W

B(L)R(P )]km , [W

B(X)P ]k,ms =

[WP ]km

Rm+ [W

B(X)R(P ) ]k,ms

[W TP ]k,cm =

[WP ]kc

Rc− [WP ]km

Rm− [W T

R(P )]k,cm

[W θP ]k,ms =

[WP ]ks

RsrTs +

[WP ]km

Rm(rBm − rTs ) + [W θ

R(P )]k,ms

The weight matrices, W ..R(P ), are dened in Denition F.1.

Proof. The total price wedge φPκ is dened in Equation (F.7) and is a combination of the

price (F.6) and the sales wedge (F.1) derived in Lemma 3.5. The results of Lemma F.1

are then used to substitute for the log-changes in intermediate credit and revenue wedges

in Equation (F.6), (F.1) and (F.2). Using Corollary F.1, Lemma F.3 then follows from

tedious but straight forward calculations.

Proof of Proposition 1 . Using Equation (F.4) and (F.7) derived in Lemma 3.5 to

substitute for the vector of changes in intermediate revenues and prices in q = R− p−φR, yields the log-change in sectoral output and the sectoral output wedge presented in

Equation (3.28) and (3.29), respectively. The general equilibrium eects from changes in

aggregate labor supply are given by

GEQ(L) = (I −WPdiag(ι− χ))WRWFRιE

LF L

Using the results of Lemmata F.2 and F.3, and substituting for the price (F.6), sales (F.1),

and ultimately credit and revenue wedges using the results of Lemma F.1, implies that the

29


sectoral output wedge can be written as a linear combination of changes in credit costs

and the credit portfolio as shown in Equation (3.29). The respective elasticity matrices

follow from straight forward calculations, where

[WBQ]k: =

[[W

B(L)Q ]km , [W

B(X)Q ]k,m:

],

and the entries of the respective weight matrices are as follows

[WB(L)Q ]km =

[WP ]km

Rm− [W

B(L)R(Q)]km , [W

B(X)Q ]k,ms =

[WP ]km

Rm− [W

B(X)R(Q) ]k,ms

[W TQ ]k,cm =

Im=k

Rm+

[WP ]kc

Rc− [WP ]km

Rm+ [W T

R(Q)]k,cm

[W θQ]k,ms =

[WP ]km

Rm(rBm − rTs ) +

([WP ]ks − Is=k)

RsrTs − [W θ

R(Q)]k,ms

The weight matrices, W ..R(Q), are dened in Denition F.1.

Similarly, Corollary F.2 highlights that the response of the aggregate GDP wedge, ΦYφ ,

can be decomposed into the total eect of credit distortions in prices, ΦPφ , and household's

prot and interest income, ΦΠφ . Similarly, matricesW

..Y summarize all direct and indirect

cost and demand interactions between sectors in this economy.

Corollary F.2 (Aggregate GDP). The response of aggregate GDP is given by

Y = Zz + ΦYφ +GEY (L) where Zz = ι′ΩFWPdiag(χ)zq. (F.16)

The change in the aggregate TFP-wedge

−ΦYφ = +ΦΠ

φ + ΦPφ = eτY τ +GE(L, W ) (F.17)

can be decomposed into changes capturing the total eect of changes in credit cost and the

credit portfolio on (a) aggregate prices

ΦPφ = ι′ΩF φPκ and φPκ = WP φ

P −WPdiag(ι− χ)WRφS (F.18a)

and (b) total prots and income from bank interest rate payments rebated to households:

ΦΠφ = ι′φΠ

κ and φΠκ = WR

Y WRφS + φF (F.18b)

The entries of vector eτY = [eBY , eTY , e

θY ] represent the elasticity of aggregate GDP wrt to

changes in credit costs and the credit portfolio

[eBY ]m = rBm∑i

[WBY ]miBC

Qmi, [eTY ]m = rTm

∑c

[W TY ]cmARcm and [eθY ]ms = [W θ

Y ]msAPms

30


The matrices, W ..Y , are dened in the proof and are functions of the equilibrium, E, and

the input-output price, WP , and demand, WR, relations dened in Equation (3.24).

Proof. (a) Let τ be the share of banks owned by foreign households such that only

δ = (1− τ) of the bank interest rate expenditures are redistributed to households. Thus,

household's labor, prot and transfer income from sector k is given by

Pyk = w`k + πTk , where πTk = Rk − φLτ,kBCk −∑s

(1 + rTs )APks (F.19)

with φLτ,k = (1 + τrBk ) and φXτ,ks = (1 + τrBk (1 − θks) + rTs θks). Using results of Lemmata

3.1 and 3.5 to substitute for bank loans, accounts payable, intermediate revenues and

the aggregate price index, the log-linearization of Equation (F.19) yields the response of

sector k's real value added and value added credit wedge

ωYk yk = ωYk Zz + φYκ,k +GEy(ˆk, L) where φYκ,k = −φΠ

κ,k − ωYk ΦPφ

and ωYk = PYk/PY and∑

k ωYk = 1. The prot and price credit wedges are derived for

τ = 0. Equation (F.18a) and (F.18b) follow. The household's income wedge, φTk , is

φTk = +[WBπ ]kkr

Bk +

∑s

[W Tπ ]ksr

Ts +

∑s

[W θπ ]ksθks − [W φ

π ]kφLk −

∑s

[W Φπ ]ksφ

Xks.

Note that φTk equals the nal sales wedge, φFk , in Equation (F.2) for τ = 0 such that

the matrices W ..π = W ..

F and WRY = WR

F . Using R0 = PY , the entries of the elasticity

matrices are given by

R0[WBπ ]kk = τrBk BCk

R0[W Tπ ]ks = rTs AP ks

R0[W φπ ]kk = φLτ,kw`

Qk

R0[W Φπ ]ks = φXτ,kspsxks

R0[W θπ ]ks = δ(1 + rTs )rBk AP ks(φ

Lk )−1

R0[WRY ]kk = πk + δT k

The general equilibrium eect from changes in labor are

GEy(ˆk, L) = +

w`k

PYˆk + e′k

[WR

Y −ΩY ΩFWPdiag(ι− χ)]WRW

FRιL(R0)−1L.

(b) The log-linearization of total real value added is given by

Y =∑m

ωYmym, where ΦYφ =

∑m

φYκ,m and GEY (L) =∑m

GEy(ˆm, L).

Equation (F.16) and (F.17) follow. Let τ = 0, substituting for the sales, φS, and price, φP ,

wedges in Equation (F.18a) and (F.18b) using the results of Lemma 3.5, collecting terms

31


and reindexing allows to express the aggregate TFP-wedge response in terms of changes

in credit, revenue and nal demand wedges dened in Lemma F.1. Using the results of

Lemma F.1 to F.3 implies that the aggregate TFP-wedge response can be written in terms

of changes in the cost of bank and trade credit and changes in the credit portfolio, τ .

Equation (F.17) follows, where

[WBY ]m: =

[[W

B(L)Y ]m , [W

B(L)Y ]m:

]and the entries of the corresponding weight matrices are given by

[WB(L)Y ]m = +

∑k

ωFk [WB(L)P ]km −

∑k

[WRY ]kk[W

B(L)R(R)]km −

1

φLm

1

R0

[WB(X)Y ]ms = +

∑k

ωFk [WB(X)P ]k,ms −

∑k

[WRY ]kk[W

B(X)R(R) ]k,ms −

1 + rTs θms

φXms

1

R0

[W TY ]cm = +

∑k

ωFk [W TP ]k,cm +

∑k

[WRY ]kk[W T

R(R)]k,cm +(1− θcm)rBm

φXcm

1

R0

[W θY ]ms = −

∑k

ωFk [W θP ]k,ms +

∑k

[WRY ]kk[W θ

R(R)]k,ms +

[(1 + rTs )rBm + ∆ms

φXmsφLm

]1

φXms

1

R0

.

F.2. Credit Costs, Links and Interest Rates

Lemma F.4. The vector of log-linearized responses of interest rates on bank and trade

credit, and trade-credit shares between sectors, τ , are

τ =

0 0 +Eθ

B

−EBT +ET

T +EθT

+EBθ −ET

θ −Eθθ

τ +

+EZb

B

+EZbT

0

εb +GEτ = Eττ τ +EZb

τ εb +GEτ . (F.20)

The entries of the diagonal elasticity matrices of the interest rate on bank and trade credit

with respect to shocks to the bank risk premium are given by

[EZbB ]kk = rZk (rBk )−1 and [EZb

T ]kk = φRk .

The entries of the remaining elasticity matrices in, Eττ , capture direct cost and indirect

demand eects of changes in credit costs and the credit composition on τ . The elasticity

of the interest rate on trade credit and trade-credit shares of sector k wrt changes in the

interest rate on bank credit, rBm,

[EBT ]km = rBm

∑i

[WBT ]k,miBC

Qmi, and [EB

θ ]ks,m = rBm∑i

[WBθ ]ks,miBCmi,

32


and in the interest rate on trade credit, rTm, and trade-credit shares, θms, is

[ETv ]em = rTm

∑c

[W Tv ]e,cmARcm , [Eθ

v]e,ms = [W θv ]e,msAPms

for v ∈ T, θ and e = k if v = T and e = ks otherwise. The direct eects of changes in

the credit composition on the bank interest rate are

[EθB]k,ck = [W θ

B]k,ckARck.

The matrices, W ..v, are dened in the proof and are functions of the equilibrium of the

economy, E, and the input-output price,WP , and demand,WR, relations given in (3.24).

Proof of Lemma F.4. The log-linearization of the interest rate on bank credit (3.7), on

trade credit (3.23) and of the trade-credit share between sector k and s (e = ks) (3.22)

and tedious algebra yields

rBk = +∑c

[EθB]ckθck + [EZb

B ]kkzbk +GEB

rTk = −∑m

[EBT ]kmr

Bm +

∑m

[ETT ]kmr

Tm +

∑m

∑n

[EθT ]k,mnθmn + [EZb

T ]kkzbk +GET

θe = +∑m

[EBθ ]emr

Bm −

∑m

[ETθ ]emr

Tm −

∑m

∑n

[Eθθ ]e,mnθmn +GEθ

Stacking all equations yields Equation (F.20) and the elasticity matrices are dened as in

Lemma F.4. Let Cs denote the cardinality of the customer set of supplier s and dene

[KRθ ]ks = sgn(∆ks)

∣∣∣∣ ∆ks

[Kθ]ks

∣∣∣∣ ,where 0 < [Kθ]ks = ∆ks

[θks

θks − θκks−∆ks

θks

φXks

]

The weight matrices wrt changes in the interest rate on bank credit are

[WBT ]k,m: =

[[W

B(L)T ]km , [W

B(X)T ]k,m:

]and [WB

θ ]e,m: =[[W

B(L)θ ]em , [W

B(X)θ ]e,m: , [W

B(θ)θ ]em

]where the respective entries are given by

[WB(L)T ]km =

Im=k

φLm+BCTk [W

B(L)R(R)]km

φRkBCk

[WB(X)T ]k,ms =

(1− θks)φXms

Im=k +BCTk [WB(X)R(R) ]k,ms

φRkBCk

,

[WB(L)θ ]ks,m = [KR

θ ]ks[WB(L)R(R)]km, [W

B(X)θ ]ks,ms = [KR

θ ]ks[WB(X)R(R) ]k,ms,

[WB(θ)θ ]ks,m = [Kθ]

−1ks

1−∆ks

[(1− θks)φXks

+1

φLk

]Im=k

w`Tk.

33


The entries of the weight matrices wrt changes in the interest rate on trade credit are

[W TT ]k,cm =

(1− θkm)

φXcmIc=k −BCTk [W

T (X)R(R) ]k,cm

φRkBCk

[W Tθ ]ks,cm = [Kθ]

−1ks

1 + ∆ks

θks

φXks

1

Cs1

ARcsIm=s + [KR

θ ]ks[WTR(R)]k,cm.

The entries of the weight matrices wrt to changes in trade-credit shares aecting the

interest rate on bank and trade credit and trade-credit shares capture direct (C) cost and

(S) demand and indirect demand eects.

[W θB ]k,cm =

µ

θZk

rZkrBk

1

pqk, [W θ

T ]k,ms = (C)− (S) + [W θR(R)]k,msφ

Rk

BCTkBCk

and [W θθ ]e,ms = [KR

θ ]e[WθR(R)]k,ms,

where

(C) =

[(µ− 1)

θZk+ rTk

θD0 + µθCkθZk

]Is=kpqk

and (S) =

[(1 + rTk )−∆ks

φXksφLk

]φRkφXks

Im=k

BCk.

The matrices capturing the eect of revenues are dened in Corollary F.1.

F.3. Partial Equilibrum Structural Output Response

Proof of Proposition 3. (1) In a rst step, the structural credit responses are derived

up to a rst-order approximation in partial equilibrium, dened in Denition 3.4. Ap-

plying the results of Lemma F.4 and using the rst order approximation of the credit

multiplier Ψτ in Proposition 2, implies that the rst order approximation of the responses

of credit-costs and shares to nancial shocks, zbk = εbk, are given by τ ≈ τ = Ψτεb. As

a result, the rst order approximation of the partial equilibrium structural response of

(a) the interest rate on bank credit is [rB]k =∑M

m=1[ΨBτ ]kmε

bm, where

[ΨBτ ]km = [EZb

B ]mmIm=k

(b) the interest rate on trade credit is [rT ]k =∑M

m=1[ΨTτ ]kmε

bm , where

[ΨTτ ]km = −[EB

T ]km[EZbB ]mm + (Im=k + [ET

T ]km)[EZbT ]mm

= −rZm∑i

[WBT ]k,miBC

Qmi + [EZb

T ]mmIm=k + rTm[EZbT ]mm

∑c

[W TT ]k,cmARcm

(c) the trade-credit share for e = ks is [θ]e =∑M

m=1[Ψθτ ]emε

bm , where

[Ψθτ ]em = [EB

θ ]em[EZbB ]mm − [ET

θ ]em[EZbT ]mm

= +rZm∑i

[WBθ ]e,miBCmi − rTm[EZb

T ]mm∑c

[W Tθ ]e,cmARcm.

(2) Note that the partial equilibrium structural output response for sector k to credit

34


shock follows from applying Denition 3.4 to Proposition 1

qk =∑m

[EZqQ ]kmz

qm − φ

Qk such that

∂qk∂εbi

= −∂φQk

∂εbi.

Substituting for the previously derived partial equilibrium structural credit responses, τ ,

implies that the response of the structural output wedge is

φQk =∑m

[EB

Q ]kmrBm + [ET

Q]kmrTm −

∑s

[EθQ]k,msθms

=∑m

[SQB ]km + [SQT ]km

εbm

where SQT = SQT (T )−SQT (θ). Substituting the response of credit costs and shares with the

previous results implies that entries of the matrix summarizing the eect of the nancial

shock on output via direct and indirect changes in the interest rate on bank credit is

[SQB ]km =∑n

[EBQ ]kn[ΨB

τ ]nm.

Similarly, the entries of the matrix summarizing the eect of direct and indirect changes

in the interest rate on trade credit and trade-credit shares

[SQT (T )]km = +∑n

[ETQ]kn[ΨT

τ ]nm and [SQT (θ)]km = −∑n

∑s

[EθQ]k,ns[Ψ

θτ ]ns,m.

Equation (3.33) and (3.34) follow by substituting Ψ..τ using the results of step (1), col-

lecting terms and simplifying expressions. The entries of the structural matrices are

SBQ,B = WBQ . Let W

B′T = [WB

T 0]. The structural weight matrices, S..Q,T , associated with

the eect of changes in (R) the interest rate on trade credit and (θ) the credit composition

on output are

[SBQ,T ]k,mi =

+∑n

∑c

rTnARcn[WTQ ]k,cn[W

B′T ]n,mi (R) indirect demand-eects

+∑n

∑s

APns[WθQ]k,ns[W

Bθ ]ns,mi (θ) direct cost and indirect demand-eects

[STQ,T ]k,cm =

+[W TQ ]k,cm (R) direct cost-eects

+∑n

∑i

rTnARin[WTQ ]k,in[W

TT ]n,cm (R) indirect demand-eects

+∑n

∑s

APns[WθQ]k,ns[W

Tθ ]ns,cm (θ) direct cost- and indirect demand-eects

35


G: Quantitative Application

Appendix G.1 rst outlines the algorithm used to calibrate the model introduced in Section

3 and any necessary adjustments made to the input-output tables to ensure a consistent

mapping of the equilibrium of the model to the data. Appendix G.2 presents business

cycle statistics on selected real and nancial variables. Appendix G.3 summarizes the

results of robustness exercises discussed in the main text.

G.1. Model Calibration

The iterative procedure outlined in Algorithm 1 is a rough sketch of the steps involved in

calculating the equilibrium of the model economy.

Algorithm 1 Calibration Steps

1: Adjustment of Nominal IO-Tables and Credit Network

2: Calculation of Risk Premium and Calibration of Production Parameters

3: Initial Guess of Intermediate Expenditure Shares ΩX

4: while |ΩXt − ΩX

t−1| > εΩ do

5: Initial Guess of Quantity Shares (wck = xck/qk)

6: while do|wck,t − wck,t−1| > εw

7: Calculate Equilibrium Financial Variables

8: Calculate Equilibrium Nominal Value Added

9: Calculate Equilibrium Prices and Quantities

10: Calculate Implied Productivity and Financial Shocks

11: Update Quantity Shares

12: end while

13: Update Intermediate Expenditure Shares

14: end while

15: Calculation of Parameters of Credit Management Cost Function

G.1.1. Adjustment of IO- and Financial Data

A. Trade Credit. Shares. The balance sheet data of a panel of US rms from Compustat

are rst used to calculate the share of accounts payable in total input expenditures, θPkt,

and the share of accounts receivable in total revenues, θRst, at an industry level.

36


Dealing with Missing Data and Domestic Non-Market Clearing. Since some industries

are not or under-represented in the Compustat sample, it is possible that observations on

industry trade-credit shares, θPkt and θRkt, are missing. I account for missing observations

as follows: (a) If a sector is missing all trade-credit data, all trade-credit shares are set

to zero which implies that this sector is neither extending nor receiving trade credit.

(b) If the time series contains some missing observations, I rst identify the period with

the highest number of consecutive non-missing observations. Using the rst and last

observation of this period, I use the median growth rate of trade-credit shares in the

sample to extrapolate the level of the respective share for the remaining observations.

As the model assumes a closed economy, all trade credit relations are between domestic

rms. Therefore, market clearing for domestic trade credit

APt =∑

mAPmt =∑

mARmt = ARt (G.21)

is implemented by rst calculating the implied level of total sectoral accounts payable,

APt, and receivables, ARt, using the sectoral trade-credit shares, θPkt and θ

Rkt, and the total

intermediate expenditures and sales recorded in the IO-tables. If Equation (G.21) does

not hold, sectoral accounts receivable (shares) are adjusted by the share of exports (Xkt)

in total sales (Rkt) for each sector.

Inter-Sectoral Trade-Credit Linkages. The inter-industry trade-credit share from supplier

s to customer k is constructed as a (sales) weighted average of the trade-credit shares, as

suggested in Altinoglu (2018)

θks =Rs

Rk +Rs

θPk +Rk

Rk +Rs

θRs (G.22)

and is non-zero only if both sectors also engage in trade in intermediate inputs.

B. Prot Decomposition. Bank interest rate expenditures, are recorded as part of

the gross operating surplus in the IO-tables net of interest-income (idit). (see Horowitz

and Planting, 2009). The gross operating surplus - GOP - (π) is decomposed into capital

expenditures (dp), dividend payments (ni+dv) and bank interest rate expenditures (xint)

by exploiting information on the composition of gross operating prots reported in the

income statements of the panel of US rms from Compustat: From the income statements

it follows that π+idit = dp+ni+dv+xint. Thus, total gross prots - GRP - (Σ = π+idit)

are a multiple of the observed prots π. The dividend, sDV , interest rate expenditure,

sIR, and capital, sK , shares for decomposing the GOP recorded in IO-Tables are(1− idit

Σ

)−1

π = Σ such that sDVkt =ni+ dv

Σ, sIRkt =

xint

Σ, sKkt =

dp

Σ.

37


In addition, the share of gross prots in total sales, sGPRkt , and the share of interest

income in gross prots, sIIRkt , are calculated for each industry. The level of dividends,

interest payments and capital expenditures then follows directly from the GOP recorded

in the IO-table as discussed below.

C. Adjustments of IO-Tables. The model is calibrated using the summary tables on

"Use of Commodities by Industries After Redenitions" provided by the BEA. To ensure

an appropriate mapping of the model to the data, adjustments are made as outlined below.

Treatment of Used and Non-Comparable Imports. The dollar value of the row entries on

expenditures on "Used Goods" and "Non-Comparable Imports" are redistributed propor-

tionally across sector k's suppliers using the expenditure shares on each sector in k's total

intermediate sales. Any negative intermediate expenditures entries are set to zero.

Treatment of FIRE. I follow BL(2019) and interpret the production function (3.1) as the

technology at use related to the physical production inputs rather than interest rates,

insurance premia or rental rates. As in BL(2019), the expenditures on FIRE-services are

treated as part of capital gains and not as intermediate production expenditures. There-

fore, the corresponding rows of the IO-tables are reassigned to gross operating prots

and the purchases of FIRE-sectors are treated as part of nal demand. To avoid double

counting, the share of capital gains attributed to FIRE-expenditures is treated as income

accruing to foreign households and thus excluded from the calculation of GDP.

Inventories. Changes in inventories are recorded as part of nal uses. However, the

model is static and does not account for the accumulation of inventories. Therefore, as in

BL(2019), changes in inventories are subtracted from nal uses. The dollar value supplied

by sector i is then redistributed proportionally across i's intermediate customers using the

sales share of each sector in i's total intermediate sales. Following the adjustment of in-

termediate sales for changes in inventories, total intermediate expenditures and industry

output is recalculated for each sector.

Final Demand, Imports and Exports. While the model is a closed economy without invest-

ment, sectors in the US economy invest and engage in foreign trade. Two observations can

be made: (1) The majority of commodities in the US are (a) both produced domestically

and imported and (b) both used as intermediate inputs in production and consumed by

nal demand. (2) Total nal uses (consumption, investment and exports) of most sectors

exceed imports, implying that the majority of commodities in the US are also produced

domestically. Note that total domestic demand can exceed domestic production in some

sectors (e.g. Oil-Sector). In order to take the data to the model, investments and exports

are treated as part of domestic demand of the nal good producer. In the calibration, I

account for foreign trade (imports) in the form a intermediate sales residual in order to

38


ensure market-clearing. Note that by simply ignoring imports in the calibration of the

model or assigning imports to nal demand directly implies that good markets do not

clear in equilibrium. The calibration ensures that the national accounting identity holds.

Interest Income, Taxes and Prots. Gross operating prots - GOP - recorded in the IO-

tables include proprietor's and rental income, corporate prots, interest expenditures net

of interest income, capital expenditures, etc. A separate measure of interest expenditures

is obtained as follows.

(a) Negative Gross Operating Surplus. Only a few sectors over the period 1997-2016

record negative prots at a few selected points in time (six observations). Since the

model does not allow for negative prots, the gross operating surplus is set to zero

if a negative value was recorded. (GOPkt)

(b) Total Interest Income and Gross Prots. A sector's interest income is derived as

IIRkt = sIIRkt ·(sGPRkt ·Rkt) using industry sales recorded in the IO-tables and applying

the measures obtained in (A). Gross prots, GPRkt, are then calculated as the sum

of the gross operating surplus adjusted for negative prots, GOPkt, and the imputed

interest income IIRkt. To avoid overstating gross prots and the interest income

from combining two data sources, both measures are updated in two additional steps

by winsorizing (90th-quantile)

(1) the ratio of gross, GPRkt, to gross operating prots, GOPkt, and

(2) the ratio of operating costs, wt`t +∑

s pxks,t, to gross prots, GPRkt.

(c) Adjustment of Taxes and Dividends. The imputed interest income is added to the

gross operating surplus of a sector and deducted from taxes. Since the model does

not account for taxes, taxes are treated as part of dividend payments to households.

As a result, a sector's total value added is left unchanged while tax-payments net of

interest income and thus also total dividends can be negative.

Total Industry and Commodity Output. The dierence between total industry and com-

modity output is added to nal uses such that nominal output produced equals total sales

and markets clear. The sales residual is distributed between nal demand (consumption,

investment and exports) and imports using the respective share in total nal demand.

D. Labor Costs and Prices. Data on total hours worked and sectoral prices are

provided by the Bureau of Labor Statistics (BLS). Any missing values are dealt with by

combining the MFP- and the LPC-Database.

E. Interest Rates on Bank Credit. To ensure that the interest rate on bank credit

is consistent with the imputed interest rate expenditures for the extreme case that all

intermediate-input expenditures and labor costs need to be nanced using bank credit,

39

https://www.bls.gov/


the following adjustments are made:

(a) In a rst step, three dierent measures of the bank interest rate and the maximal

bank interest expenditures are calculated:

(1) The bank interest rate, rB0,kt, implied by the IO-expenditure data is calculated

using the interest expenditure share in gross prots derived in (A), sIRkt , the

imputed gross prots, GPRkt, and total operating costs, such that rB0,kt = (sIRkt ·GPRkt) : (w`kt +

∑spxks,t).

(2) The bank interest rate, rB1,kt, is calculated using the GZ-spread and used to

calculate the maximal possible interest rate expenditures based on the total

operating costs recorded in the IO-tables.

(3) As a third measure, the bank interest rate, rB2,kt, is calculated by combining the

level of the bank interest rate inferred from the IO-tables at the beginning of

the observation period, rB0,k1, with the growth rate of the interest rate calculated

using the GZ-spreads, rB1,kt.

The second measure is applied, unless the share of interest rate expenditures in gross

prots implied by (2) exceeds one, in which case the combined measure (3) is used.

(b) Both, the share of interest rate expenditures in gross prots as well as the growth

rate of the interest rates are winsorized in order to reduce the size of outliers caused

by combining dierent data sources.

G.1.2. Parameters of Bank Risk Permium

The parameter governing the convexity of the risk premium with respect to the combined

default risk, µ, is calibrated by rst estimating the following equation for each sector

log(rZkt) = µ0 + µ1 log(θZkt) + εkt where θZkt = θD0t + θCkt. (G.23)

The variable θCkt denotes the share of sectoral accounts receivables in total sales and

θD0t denotes the aggregate leverage - the ratio of long-term debt and debt in current

liabilities to total assets. The data-counterpart for the risk premium, rZkt, is the sectoral

credit spread calculated in Gilchrist and Zakraj²ek (2012) and the aggregate leverage is

calculated directly from the corresponding balance sheet items in Compustat. Table G.1

reports the OLS-regressions results of Equation (G.23) for the sample period 2000-2013

and the corresponding standard errors. The convexity parameter µ is then calibrated by

calculating the sales-weighted (RW) average of the estimated coecients of log(θZkt), such

that µ = 1.2. As a result µ equals 1.2 such that a one percent increase in θZkt increases

the bank risk premium by 1.2 percent.

40


Table G.1: Calibrated Parameter µ of Risk Premiumlog(rZ) 11 21 22 23 31T33 42 44A5 48A9 51 54A6 62 71A2 81

RW (0.3) (4.6) (7.8) (0.6) (39.1) (7.7) (18.5) (6.0) (9.0) (2.7) (1.7) (1.8) (0.1)

log(θZ) 0.57 0.44 0.80 -2.07∗ 0.92 0.89 2.39∗∗ 3.31∗∗ 0.60 0.55 0.10 0.96 -1.05

(0.70) (0.26) (0.80) (0.68) (0.98) (1.11) (0.73) (0.95) (1.13) (1.07) (0.40) (1.15) (1.19)

R2 0.07 0.19 0.08 0.44 0.07 0.05 0.47 0.50 0.02 0.02 0.01 0.05 0.06

NObs 11 14 14 14 14 14 14 14 14 14 14 14 14

Note: This table presents the results of an OLS regression of Equation (G.23) for selected industries. The sales shares

in total sales of each industry in percent is reported in row (RW ). All regressions include a constant; Std.Errors are

recorded in parentheses. ** p<0.01, * p<0.05, + p<0.1

G.1.3. Parameters of Credit Managements Costs

The parameters of the credit management cost function (3.6) are calibrated in three steps:

(1) First, a sector's share of total accounts payable in total intermediate cost of production

excluding interest rate payments, θPk , is calculated using Equation (3.22) and imposing

the assumption of common parameters: κT0,ks = κT0 , κT1,ks = κT1 and θSk = θS0 ∀k, s.

θPk =

[θS0 − (θS0 )2κ

T0

κT1

]+

[(θS0 )2

κT1

]pqEk = β0 + β1pq

Ek (G.24)

where

θPk =

∑s θkspsxks∑s psxks

and pqEk =

[∑s(psxks)

2∆ks

(∑

s psxks)2

](b)

[∑s psxks

(1 + rBk )

](a)

. (G.25)

The variable pqEk denotes the eective net-interest expenditures and equals the (a) dis-

counted intermediate cost of production excluding interest rate payments (b) weighted by

the sector-specic Credit-Hirschman-Herndahl Index (HHI). Similar to the HHI-index

measuring the degree of monopoly power in an industry (Shepherd, 1987), the Credit-HHI

captures a sector's concentration of net-interest rate costs of production and ranges from

[min(∆k),max(∆k)] for ∆k = ∆ksMs=1. While the sign of the index depends on the rel-

ative cost of bank and trade credit, a higher absolute value implies a higher dependency

on a particular supplier. Since actual data on the cost of trade credit is not available, the

data-counterparts of θPk and pqEk are obtained by mapping the model to the data. Note

that, while θPk is stationary, pqEk is a non-stationary variable such that the detrended25

eective net-interest expenditures, pqE,hkt , are used in the estimation of Equation (G.24).

(2) In a second step, Equation (G.24) is estimated by OLS using a panel of 45 sectors from

2000-2014, while controlling for time and sector xed eects. The estimated coecients

and corresponding standard errors are reported below in parentheses in Equation (G.26)

θPkt = β0 + β1pqE,hkt + FE + εkt and θPkt = 0.11

(0.005)

∗∗ + 0.24(0.115)

∗pqE,hkt + FE. (G.26)

25The variable, pqEk , is rst detrended using an hp-lter with a smoothing constant of 6.25 (Ravn andUhlig, 2002) and then normalized by adding the cyclical component to the sectoral time-mean of pqEk .

41


(3) In the third and last step, Equation (G.24) is rst multiplied by (θks/θPk ). The link-

specic cost parameters, κT0,ks and κT1,ks, are then obtained by matching the adjusted

coecients of the net-interest expenditures on intermediate production expenditures in

Equation (G.24) and Equation (3.22) using the estimation results of Equation (G.26):

(θSk )2

κT1,ks≈ β1 ·

θksθPk

=⇒ κT1,ks = (θSk )2 · θPk

θksβ1

(G.27)

The parameters, κT0,ks, and κBk , are set to ensure that Equation (3.22) and (3.6) hold.

G.2. Business Cycle Statistics

Following the period-by-period mapping of the equilibrium of the model to the US econ-

omy in Section 4.1, Figure G.1 and Table G.2 document business cycle statistics for

selected real and nancial variables. Panel (a) plots the sectoral mean of the implied

banking shock and of the log-change of selected production inputs across time. Panel

(b) depicts the average log-changes in the interest rates on bank and trade credit as well

as in the trade-credit shares: The implied shock to risk premia rose by 26.9% and lead

to an increase of bank and trade-credit interest rates by 17.8% and 23.1%, respectively.

Average sectoral output declined by approximately 16.0% caused by a drop in labor and

the composite intermediate input by 9.5% and 27.8%, respectively. At the same time,

the average trade-credit share extended to customers declined by 17.2% and the average

share of intermediate expenditures obtained on trade credit dropped by 14.3%.

Figure G.1: Data - Mean Changes

(a) Real Variables and Financial Shocks (b) Financial Variables

Note: This gure plots the log-change in percent of the time series of selected real and nancial variables calculated

based on the data discussed in Section 4.1 from 1997-2016. Panel (a) plots the sectoral mean of the implied nancial

shock (zBk ), the log-change of real output (qk), labor (`k) and the intermediate input composite (qXk ). Panel (b) plots

the average log-change in the interest rate on bank (rBk ) and trade credit (rTk ) as well as in the trade-credit shares (θCk ,

θSk =∑Ms=1 θks/M).

42


Table G.2 reports the cross-sectional mean of the standard deviation of log-changes

in the sectoral variables of interest as well as the within sector correlation between (a)

log-changes in output and (b) log-changes in the cost of bank credit and the remaining

real and nancial variables. In addition, the sample is split based on their net-lending

position dened in 2.1 and obtained from the data presented in Section 2: a sector is

counted as a net-lender if its net-lending position is above the median net-lending share.

Table G.2: Data - Time-Series Correlation

(a) Real Variables

Total (97-16)

VAR All NB NL p-Value

STDV

q 0.069 0.068 0.069 0.921

` 0.065 0.052 0.078 0.282

qX 0.140 0.133 0.147 0.663

zB 0.154 0.152 0.155 0.893

CORR (q,`) 0.557 0.601 0.512 0.440

(q,qX) 0.809 0.784 0.834 0.384

(q,zB) -0.418 -0.385 -0.453 0.300

#OBS 45 23 22

(b) Financial Variables

Total (97-16)

VAR All NB NL p-Value

STDV

rB 0.114 0.114 0.114 0.978

rT 0.168 0.169 0.167 0.952

θC 0.134 0.130 0.138 0.658

θS 0.092 0.078 0.107 0.081CORR (rB,rT ) 0.922 0.953 0.890 0.065

(rB,θC) -0.239 -0.258 -0.220 0.624

(rB,θS) -0.341 -0.401 -0.277 0.104

#OBS 45 23 22

Note: This table reports the time-mean of the standard deviation and the correlation with output and the bank interest

rate of the log-change of the following variables: output (qk), labor (`k), the intermediate input composite (qXk ), the

interest rate on bank (rBk ) and trade credit (rTk ), the trade-credit shares (θCk , θSk =

∑Ms=1 θks/M). The rst column

reports the business cycle statistics for the entire sample. The second and third column report the same statistics for a

subgroup of sectors based on the net-lending position Denition 2.1. The p-values for the dierences in means between

the two groups are reported in the last column.

The business-cycle statistics of sectoral output, labor and the intermediate composite

are similar to those reported in BL(2019): Over the sample period, 1997-2016, labor

demand is on average less volatile whereas the demand for the composite intermediate

input is more volatile than output. Unsurprisingly, log-changes in output are positively

correlated with changes in production inputs. The dierence in means test suggests

that there is no signicant dierence in the volatility or output-correlation between net-

borrowing and net-lending sectors. The average trade-credit share extended to customers,

θC , and obtained from suppliers, θS, are negatively correlated with the cost of bank

nance: While the interest rates on bank and trade credit comove strongly, the latter

exhibits a higher standard deviation. This relates to the relative volatility of accounts

payable and liabilities and the observation that rms shifted their borrowing portfolio

towards bank nance in response to an increase in credit market frictions documented

in Section 2. The correlation between the cost of bank and trade credit as well as the

average trade-credit share obtained from suppliers seems to be signicantly higher for the

group of sectors classied as net-borrowers at a 10% signicance level: As net-borrowers

face a higher cost of bank nance, they are more likely to increase their lending rates and

shift more towards bank nance. This suggests that the smoothing eect of trade credit

may be more pronounced for net-borrowers as predicted by the model.

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G.3. Model Fit and Additional Simulations

Model Fit. Figure G.3 replicates Figure 2.1b and 2.1c in the main text using the

model-generated series only. Qualitatively, the model captures that (M3) trade credit is

more volatile than rms' liabilities and that (M4) the share of accounts payable and receiv-

able in total costs of production and sales are negatively correlated with aggregate credit

spreads in the economy. Quantitatively, the model accounts for 7.6(5.0)% of the variation

in total (current) liabilities, 25.6% of the variation in supplier credit and approximately

18.9(23.3)% of the variation in the share of trade credit in total costs(sales).

Figure G.3: Model-Implied Business Cycle Properties of Trade Credit

(a) Model Prediction 3 (b) Model Prediction 4

Note: The panels in this gure replicate the graphs presented in Figure 2.1 and plot the evolution of the log-changein percent of the simulated time series of Current (LC) Liabilities, Accounts Payable (AP ), the aggregate GZ-spread(RB), the share of AP in Total Costs of Goods Sold (θP ) and the share of AR in Total Sales (θR). All variables arereported in real terms using the aggregate price-index. The model-simulations are based on nancial shocks only. Thegures also report the standard deviation of the respective series in percent.

Figure G.4 depicts the predicted percentage changes in the aggregate labor wedge and

labor in addition to the log-changes in aggregate output and labor observed in the data.

Figure G.4: Model-Fit - Aggregate Outcomes

(a) Aggregate Labor Φ(L) (b) Labor

Note: Panel (a) in this gure plots the model-implied log-changes in the aggregate labor wedge in response to shocksto the cost of bank credit only. The log-changes of observed aggregate labor are measured against the right axis. Panel(b) plots both the log-changes of aggregate labor as implied by the model simulations in response to shocks to the costof bank credit on the left axis against those observed in the data. All log-changes are reported in percent.

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Robustness. Table G.3 summarizes the results of two simulation exercises evaluat-

ing the importance of asymmetries in (d) credit linkages and (e) liquidity shocks on the

results derived in the main text. Table G.4 presents the trade-credit multipliers associated

with the simulation exercises (a) and (c) in the main text, while comparing the benchmark

economy with both bank and supplier credit, to an equivalent economy with bank nance

only, E(0), where rms do not face any credit management costs (CTk = 0∀k).

Table G.3: Trade-Credit Multipliers of Counterfactual Simulations

Aggregate Sectors

(1) (2) (3) (4) (5) (6) (7)

CF EN ∆%(Y )09 ∆%(L)09 ∆%(ΦZ)09 ∆%(ΦL)09 ∆%(q)09 ∆%(`)09 ∆%(φV )09

(d)

TCA

E(θ) -0.880 -0.548 -0.525 -0.037 -0.718 -0.786 0.556

E(θ) -0.877 -0.551 -0.521 -0.043 -0.705 -0.803 0.542

M 1.003 0.995 1.008 0.854 1.019 0.979 1.027

(e)

TCS

E(θ) -0.880 -0.548 -0.525 -0.037 -0.718 -0.786 0.556

E(z) -0.883 -0.532 -0.539 -0.008 -0.736 -0.726 0.514

M 0.996 1.030 0.974 4.635 0.976 1.083 1.083

Note: This table documents the model simulated log-change of the following variables to aggregate shocks to sector-specic bank risk premia in an economy with bank and supplier credit, E(θ): aggregate output (Y ), labor (L),the aggregate eciency (ΦZ) and labor wedge (ΦL), average sectoral output (q), labor (`) and credit cost wedge(φV ). The trade-credit multipliers (M) are calculated as the ratio of responses of the variable in E(θ) to (d) theircounterparts in an equivalent economy with constant and symmetric trade-credit shares, E(θ) and (e) to the sameeconomy featuring symmetric shocks, E(z). All log-changes are reported in percent.

Table G.4: Trade-Credit Multipliers - No Credit Management Costs

Aggregate Sectors

(1) (2) (3) (4) (5) (6) (7)

CF EN ∆%(Y )09 ∆%(L)09 ∆%(ΦZ)09 ∆%(ΦL)09 ∆%(q)09 ∆%(`)09 ∆%(φV )09

(a)

TC0

E(θ) -0.880 -0.548 -0.525 -0.037 -0.718 -0.786 0.556

E(0) -0.480 -0.376 -0.207 -0.148 -0.191 -0.405 0.183

M 1.834 1.459 2.533 0.250 3.769 1.942 3.037

(c) NL

E(θ) -0.045 -0.022 -0.031 0.008 -0.096 -0.093 0.068

E(0) -0.010 -0.008 -0.004 -0.003 -0.014 -0.048 0.012

M 4.538 2.894 7.008 -2.622 6.905 1.927 5.560

(c) NB

E(θ) -0.236 -0.162 -0.131 -0.034 -0.046 -0.077 0.049

E(0) -0.177 -0.139 -0.076 -0.055 -0.017 -0.034 0.031

M 1.333 1.168 1.718 0.624 2.706 2.253 1.565

Note: This table documents the model simulated log-change of the following variables to shocks to sector-specicbank risk premia in an economy with bank and supplier credit, E(θ), and in an equivalent economy without creditmanagements costs and bank credit only, E(0): aggregate output (Y ), labor (L), the aggregate eciency (ΦZ) andlabor wedge (ΦL), average sectoral output (q), labor (`) and credit cost wedge (φV ). The trade-credit multipliers(M) are calculated as the ratio of responses of the variable in E(θ) to their counterparts in E(0). The counterfactualexercises feature an economy with bank nance only (TC0); (NL/NB) an economy in which only net-lenders (net-borrowers) experience an increase in their bank interest rates using Denition 2.1. All log-changes are reported inpercent.

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