a general equilibrium model of supply chain...

A General Equilibrium Model of Supply ChainInteractions and Risk Propagation

John Birge and Jing Wu1

1University of Chicago Booth School of Business

SCF Symposium, Madrid

John Birge and Jing Wu (Chicago Booth) Equilibrium in Supply Chain Networks June 20, 2016 1 / 29

S&P 500 Supply Chain Network

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Figure: Who are my customers (left) and suppliers (right)

Green: Manufacturing, Blue: Transportation Warehousing, Red: Wholesale Retail


Outline

Empirical Observations: Propagation of risk on two levels (direct andindirect)

1st-order effects (direct propagation)

2nd-order effects (systematic risk)

Equilibrium Network Model

Implications of the Model

Conclusions and Future Directions


Empirical Observations

Pricing and Risk Basics

Model of share price at time t:

pt =∞∑s=0

e−(rs+δs )sds

Expected dividends ds

Depends on supply chain partners (first-order).Changes may be delayed due to inattention or invisibility.

Risk premium, δs

Depends on multiplicity of connections to transmit risk (second-order).Reliability issues may create nonlinear effects on the risk of network position.


Literature

Literature

1st-order effects

Industry level: Menzly & Ozbas (2007), Shahrur, Becker, & Rosenfeld (2010),Fruin, Osiol, & Wang (2012).

Firm level: Hendricks & Singhal (2003), Cohen & Frazzini (2008), Atalay,Hortacsu, & Syverson (2013, working).

2nd-order effects

Asset pricing: Sharpe (1964), Lintner (1965), Fama & French (1993).

Network risk: Acemoglu, Carvalho, Ozdaglar, & Tahbaz-Salehi (2012),Anupindi & Akella (1993), Cachon, Randall, & Schmidt (2007), Ahern (2012),Carvalho and Gabaix (2013), Kelly, Lustig, & Nieuwerburgh (2013, working),Herskovic (2014, working).


Data

Empirical Observations: Data

Scope is limited to U.S. public listed firms

Stock data: CRSP (monthly returns over July 2011 - June 2013)

Supply chain sales data (SPLC)

Compustat: SEC public filings (10% rule).

Bloomberg terminal (320k units): conference call transcripts, capital marketpresentations, firm press releases, product catalogs, firm websites.

Both are public information.

SEC’s Statement of Financial Accounting Standards No. 14 (SFAS 14)

“if 10% or more of the revenue of an enterprise is derived from sales to any singlecustomer, that fact and the amount of revenue from each such customer shall bedisclosed” in interim financial reports issued to shareholders


Data First-order Effects

Example Relationship

Customer to SupplierCalloway Golf/Coastcast (Cohen and Frazzini (2008))

Calloway misses earning forecase by half ($0.36 from $0.70).

Calloway’s stock price drops 30%.

Coastcast share price (50% of sales to Calloway) unchanged for one month.



Example Relationship

Supplier to Customer

Philips/Sony/Ericsson v. Nokia

Fire in Philips plant, key chip supplier for Nokia and Ericsson, in March 2000.

Philips states 1-week shutdown, then revises to 6 weeks.

Nokia (multi-sourcing) reacts quickly.

Ericsson (single sourcing) reacts slowly, lost $2.34B, acquired by Sony.



First-order Effects

w inij denotes the supplier weight of j as a fraction of i ’s procurement.

woutij denotes the customer weight of j as a fraction of i ’s sales.

w inij =

salesjiProcurement i

=salesji∑Nk=1 saleski

,woutij =

salesijSales i

=salesij∑Nk=1 salesik

.

ri,t is the return of firm i in month t.

The following specification is tested:

ri,t = α + β1ri,t−1 + β2

∑j

w inij rj,t−1 + β3

∑j

woutij rj,t−1

+β4

∑j

w inij rj,t + β5

∑j

woutij rj,t + εi,t (1)

Hypothesis:

Suppliers’ and customers’ concurrent performance relates to the firm.Supplier momentum (one-month lag) may be related to firm performance(following Cohen and Frazzini (2008)).



First-order Effects

Results

Table: Fama-Macbeth Regression of Concurrent Returns and Momentum.

α ri,t−1∑

j winij rj,t−1

∑j w

outij rj,t−1

∑j w

inij rj,t

∑j w

outij rj,t

Ave. Coef -0.001 -0.088*** 0.036** 0.024 0.399*** 0.755***(T-Stat) (-0.96) (-11.06) (2.17) (0.95) (20.90) (3.12)

Ave. Coef 0.009*** -0.090*** 0.057*** 0.004(T-Stat) (10.38) (-9.08) (2.96) (0.09)

Ave. Coef 0.009*** -0.047***(T-Stat) (10.53) (-6.96)

Ave. Coef 0.008*** 0.022**(T-Stat) (11.09) (1.83)

Ave. Coef 0.008*** -0.040(T-Stat) (10.92) (-0.66)

Ave. Coef 0.003*** 0.619***(T-Stat) (3.61) (37.25)

Ave. Coef -0.002** 0.992***(T-Stat) (-2.26) (4.54)

Ave. Coef 0.004*** 0.018* 0.625***(T-Stat) (4.51) (1.57) (36.44)

Ave. Coef -0.002* 0.001 1.001***(T-Stat) (-1.92) (0.0274) (4.51)

Ave. Coef -0.001* 0.393*** 0.744***(T-Stat) (-1.80) (22.48) (3.20)

*p-value<10%, **p-value<5%, ***p-value<1%

Controls: MKT, SMB, HML, MOM, cross-firm effect, industry effect.


Data Second-order Effects

Second-order Effects

Assumption

Acemoglu, Carvalho, Ozdaglar, & Tahbaz-Salehi (2012) (and the extension later)finds that microeconomic idiosyncratic shocks lead to aggregate flucturations,which meansA firm’s systematic risk is formed from the aggregation of idiosyncratic shocks.

Effects of connections may be nonlinear due to interactions - riskdiversification or aggregation?

Firm level shocks may be exogenously correlated due to geographicalproximity and sector proximity.

A manufacturer (e.g., Nokia) may have diversification incentives to add anindependent supplier to increase reliability (reduce systematic risk exposurewith greater centrality).A distributor (e.g., a beverage distributor) may have concentration incentivesto add similar suppliers (e.g., French wineries) to build on existing capabilities(increase systematic risk exposure with greater centrality).



Second-Order Results: Manufacturing

Table: Factor Sensitivities by Eigenvector Centrality for Manufacturing Firms.

N3 Factor Loadings

Portfolio α (%) Rmt − Rft SMB HML MOM Adj. R2(%)

1(High) 0.235 0.888*** 90.85(1.50) (15.47)0.114 0.894*** -0.347* 0.018 0.084 90.01(0.49) (12.23) (-2.07) (0.119) (1.025)

2 0.295* 0.773*** 88.74(1.78) (13.79)0.277 0.938*** -0.184 -0.453*** -0.061 93.77(1.34) (14.28) (-1.22) (-3.29) (-0.83)

3 0.328 1.060*** 92.78(1.33) (17.60)0.482* 0.953*** 0.363* -0.005 -0.008 93.04(1.86) (11.63) (1.93) (-0.03) (-0.09)

4 0.356 1.256*** 87.45(0.89) (12.97)0.571 1.087*** 0.446 0.130 -0.142 87.82(1.36) (8.22) (1.47) (0.47) (-0.96)

5(Low) 0.507 1.410*** 85.54(1.55) (11.96)0.934* 1.157*** 0.780** -0.257 -0.132 87.53(1.95) (7.63) (2.24) (-0.80) (-0.78)

High-Low -0.272* -0.522(-1.72) (-3.92)-0.820* -0.263 -1.127** 0.275 0.216(-1.96) (-1.28) (-2.40) (0.64) (0.94)

*p-value¡10%, **p-value¡5%, ***p-value¡1%

Controls: MKT, SMB, HML, MOM, cross-industry concentration.



Second-Order Results: Logistics

Table: Factor Sensitivities by Eigenvector Centrality Centrality for Logistics Firms.

N4 Factor Loadings

Portfolio Alpha(%) Rmt − Rft SMB HML MOM Adj. R2(%)

1(High) 1.314*** 0.747*** 84.93(3.26) (7.62)

1.428*** 0.768*** 0.006 -0.589 0.024 86.43(3.44) (5.85) (0.02) (-2.14) (-0.16)

2 0.894*** 0.671*** 70.41(3.78) (11.67)

0.916*** 0.976*** 0.034 -0.502 0.031 72.32(2.41) (8.13) (0.13) (-1.99) (0.23)

3 0.812** 0.964*** 83.05(2.23) (10.89)

0.801** 0.758*** -0.140 -0.152 0.164 83.75(3.36) (10.03) (-0.81) (-0.96) (1.93)

4 0.708** 0.857*** 86.41(2.50) (12.40)

0.669** 0.916*** -0.171 -0.190 0.019 85.49(2.14) (9.26) (-0.75) (-0.92) (0.17)

5(Low) 0.759 0.776*** 69.60(1.44) (6.03)0.485 0.942*** -0.548 0.141 0.048 67.70(0.84) (5.17) (-1.31) (0.37) (0.23)

High-Low 0.556 -0.029(1.53) (-0.20)0.975* -0.175 0.553 -0.730 -0.024(1.93) (-0.90) (1.24) (-1.69) (-0.11)


Controls: MKT, SMB, HML, MOM, cross-industry concentration.



Second-Order Results: Mining, Utilities, and Construction

Table: Factor Sensitivities by Eigenvector Centrality for NAICS 2 Industries.

N2 Factor Loadings

Portfolio α (%) Rmt − Rft SMB HML MOM Adj. R2(%)

1(High) -1.153* 1.399*** 79.54(-1.74) (9.09)-1.179 1.458*** -0.091 -0.330 0.114 76.83(-1.52) (5.80) (-0.15) (-0.68) (0.45)

2 -0.897 1.512*** 79.42(-1.25) (9.06)-1.023 1.583*** -0.329 0.092 -0.103 76.28(-1.21) (5.76) (-0.48) (0.17) (-0.37)

3 -0.346 0.762*** 61.90(-0.62) (5.93)-0.680 0.935*** -0.458 0.262 0.155 59.96(-1.09) (4.63) (-0.92) (0.67) (0.76)

4 -0.374 1.129*** 72.88(-0.58) (7.58)-0.598 1.213*** -0.071 -0.376 0.261 71.72(-0.83) (5.20) (-0.12) (-0.84) (1.11)

5(Low) -0.479 1.339*** 78.22(-0.72) (8.74)-0.626 1.456*** -0.201 -0.215 0.221 75.94(-0.82) (5.90) (-0.33) (-0.45) (0.89)

High-Low -0.674* 0.060(-1.95) (1.25)-0.553 0.002 0.110 -0.115 -0.107(-1.51) (0.02) (0.51) (-0.68) (-1.20)




Centrality Implications to Different Industries

Both shock correlation and network topology matter for systematic risk.

Methodology:

Expected returns should be explained by all systematic risk factors.Split firms into quintiles based on centrality measures.If ∆α 6= 0 for two extreme quintile portfolios, supply chain network leads to”anomalies” in systematic risk.

Positions in the supply chain affects a firm’s exposure to the systematic riskbesides the network topology.

Upstream firms in manufacturing have diversification incentive to formsupplier connections to operationally hedge risk.Downstream firms in logistics have concentration incentive to form supplierconnections to leverage economy of scale thus aggregate risk.



Simple Investment Model

Suppose an economy with 2 regions (A and B) and 3 potential future states belowwith equal probability (Prob (S = Si ) = 1

3 , ∀i ∈ 1, 2, 3):S1: both A and B function;S2: A cannot produce and B can;S3: B cannot produce and A can.Next, suppose 4 firms: 3 manufacturers and 1 distributor.Manufacturers: limited capacity and payoff of 1 as long as one input regionfunctions.Sources:

Firm 1 only from region AFirm 2 only from region BFirm 3 from both

Firm 4 is the distributor and connects to both A and B with a fixed cost of 1 in allstates.Payoffs:Π1 = 1, 0, 1, Π2 = 1, 1, 0, Π3 = 1, 1, 1, Π4 = 1, 0, 0.



Investment Model Solution

Let Ω denote the covariance matrix for the firms’ payoffs.

Ω =

13 − 1

6 0 16

− 16

13 0 1

60 0 0 016

16 0 1

3

Suppose we have a representative mean-variance investor, and letµ = [µ1, µ2, µ3, µ4] denote firms expected return.Then for any feasible returns µ the investor targets, the investor find the portfolioweights w = [w1,w2,w3,w4] by solving

minww′Ωw |w

′µ = µ;w

′1 = 1′

,



Simple Investment Results

The result of the equilibrium of investment is:µ1

µ2

µ3

µ4

=1

λ1

16w1 + 1

6w416w1 + 1

6w4

013w1 + 1

3w4

+λ2

λ1

Therefore,µ3 < µ1 = µ2 < µ4

i.e. the manufacturers have lower risk than the distributor, and the dual sourcingmanufacturer is less risky than the single sourcing manufacturer.Questions:

Does this result generalize to a broader classs of networks and what are otherempirical implications?Are the output representations consistent with an equilibrium model ofproduction?



Equilibrium Network Model and Relationship to Literature

Previous literature focus on the sector level only.

Lucas (1977) argues that microeconomic shocks would average out at theaggregated level proportional to 1√

n.

Acemoglu et al. (2012) suggests Lucas (1977) only holds under symmetricnetwork structure, and microeconomic shocks may lead to aggregatedfluctuations in asymmetric networks.

The change in the density of firm level connections is not captured.

We build a supply chain network model using two-level nested productionfunction capturing both the firm-level and sector-level connections.



Model Setup

An extension of Acemoglu et al. 2012

n industry sectors (S1, S2, ..., and Sn).

Firms in the same sector have the same Cobb-Douglas CRS productionfunction to produce perfectly substitutable products.

Supply chain relationships are established ex-ante.

xklij : output from firm l in sector j that inputs to firm k in sector i .

xi =∑

k∈Sixki : output from firm k in sector i .

xij =∑

k∈Si

∑l∈Sj

xklij : the production from sector j to sector i .

xi =∑

k∈Sixki : sector i ’s total production.

A unit labor allocating to to each firm (lki ) in each sector (li ), i.e.li =

∑k∈Si

lki and∑n

i=1 li = 1.

Consumption from by firm k in sector i is cki , and ci =∑

k∈Sicki .

Total consumption / GDP / labor wage is h.



Competitive Equilibrium

A competitive equilibrium of economy

We define a competitive equilibrium of economy with n sectors consisting of prices(pi , i ∈ 1, ..., n), wage h, consumption bundle

(ci =

∑k∈Si

cki ,∀i , k ∈ Si), and

quantities(lki , x

ki , x

klij ,∀i , j , k, l

)such that

1 the representative consumer maximizes her utility;2 the firms in each sector maximizes their profits (0 in expectation);3 the labor and good markets clear at both levels, i.e. for any firm k in any

sector i , and for any sector i ,

xki = cki +n∑

j=1

∑l∈Sj

x lkji ,∑k∈Si

lki = li

xi = ci +n∑

j=1

xji ,n∑

i=1

li = 1



Household and Firm Problems

The customer has Cobb-Douglas preferences over distinct goods from nsectors subject to budget constraint, that is

max u (c1, c2, ..., cn) = AΠni=1 (ci )

1n , s.t.

n∑i=1

pici ≤ h

Firm problem solves the following maximization problem

maxΠki = pix

ki − hlki −

n∑j=1

pj∑l∈Sj

xklij

xki = zki(lki)α n∏

j=1

∑l∈Sj

xklij

(1−α)wij



From Firm Connections to Sector Connections

Since firms face the same input prices and own the same productiontechnology, they will choose the same proportions of inputs:∑

l∈Sj

xklij = γki xij , lki = γki li

where γki =

∑l∈Sj

xklij

xij=

lkili

is the firm’s sector share.

Firm-level networks determine the shape of the sector shock distribution.

The Origin of Sector Shock

In sector i ’s output, i.e. xi = zi (li )α∏n

j=1 (xij)(1−α)wij , the sector productivity

shock is a sum of firm level shocks, weighted by each firm’s sector share.

zi =∑k∈Si

γki zki



From Firm Connetions to Sector Connetions

Firm-level connections affect the sector shock through the distribution of thefirm’s sector share γki .

Define the influence vector as v′

= αn 1′

[I − (1− α)W ]−1 satisfyingvi = pixi∑n

i=1 pixithus

∑ni=1 vi = 1.

Supply Chain Network Systematic Risk

The aggregate output is a influence vector weighted sum of sector-specificproductivity shocks below.

y = lnh = v′ε

where ε is a column vector with εi = lnzi = ln(∑

k∈Sizki γ

ki

). The volatility of the

aggregate output (the systematic risk) is

Var [y ] = Var[v′ε]



Sparse v.s. Dense Supply Chain Networks

Sector B

Sector A

Sector B

Sector A

The firm’s sector share γki in the left case would have higher variance on thedistribution than the right case.Similar to Proposition 4 in Acemoglu et al. (2015), the expected total output

E [y ] decreases when Var[v′ε]

increases, i.e.

1. Supply Chain Network and Sector Performance

For concave production functions, a sparse firm-level supply chain network resultsin less total sector output than a dense network.



Simulation

Step 1 (Relationship-formation): Each firm chooses a set of suppliers.Ex-ante the market is perfectly competitive.Step 2 (Input-acquisition): Each firm draws i.i.d. production shock. Inputquantity depends on the supplier actual production.The dense network has a low sector weight variance (std 0.0001 v.s. 0.0023).

2. Supply Chain Network and Firm Volatility

A sparse network results in more volatile firm production than a dense network.

8.8 9 9.2 9.4 9.6 9.8 10 10.2 10.4 10.6

x 10−3

0

200

400

600

800

1000

1200

0 0.005 0.01 0.015 0.02 0.025 0.03 0.0350

200

400

600

800

1000

1200

1400

Figure: Sector Weight γki Distribution (Left: 80% of Suppliers, Right: 2% of Suppliers).



Simulation (cont.)

Both cases exhibit sizable and systematic deviations from the normaldistribution (2%: heavy left tail, 80%: heavy right tail).Only the sector weight with modest connection density is normal distributed.

Sufficient Statistics for Firm Production Variation

With firm-level supply chain connection variation, there is no guarantee that thefirm-level production is normally distributed.

−4 −3 −2 −1 0 1 2 3 49

9.2

9.4

9.6

9.8

10

10.2

10.4

10.6

10.8x 10−3

Standard Normal Quantiles

Qua

ntile

s of

Inpu

t Sam

ple

QQ Plot of Sample Data versus Standard Normal

−4 −3 −2 −1 0 1 2 3 40

0.005

0.01

0.015

0.02

0.025

Standard Normal Quantiles

Qua

ntile

s of

Inpu

t Sam

ple

QQ Plot of Sample Data versus Standard Normal

Figure: Q-Q Plot of the Sector Weight γki Distribution (Left: 80% of Suppliers, Right:

2% of Suppliers).John Birge and Jing Wu (Chicago Booth) Equilibrium in Supply Chain Networks June 20, 2016 27 / 29


More Concentrated Economic Activities during Crisis.

Core: most (eigenvector) central firms; Periphery: least central firms.Force-directed layout algorithm (Fruchterman and Reingold 1991).Left: network in July 2007; Right: network in June 2009.Economic activities for June 2009 supply chain network are moreconcentrated than July 2007.

Figure: Network in July 2007

Figure: Network in June 2009


Conclusions & Future Directions

Conclusions and Future Directions

Evidence of concurrent supplier and customer effects plus suppliermomentum effects on returns.

Investors’ limited attention to supplier firms relative to customer firms.Gradual diffusion of supply chain information downstream as opposed toupstream.

Evidence of decreasing returns to centrality in manufacturing and increasingreturns to centrality in logistics.

Supply chain structure is an ex-ante determined and ex-post identifiable sourcefor systematic risk.Upper-stream utility, mining, and construction firms behave similarly asmanufacturing.

Equilibrium model of firms connecting across sectors

Natural hedging decisions from manufacturersLower volatility effect for manufacturers implies conditions for increasingconections to cause lower risk for upstream and higher risk for downstream

Future DirectionsAdditional empirical tests (including default propagation)Incorporate of investment into the model formulation


a general equilibrium model of supply chain...

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