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Network Origins of the Money-Output Relation

Vipin P Veetil

arXiv:2609.38503v1econ.THcs.SI

TL;DR

The paper asks how monetary shocks can affect short-run output despite money’s long-run neutrality. It models firms exchanging intermediate inputs with heterogeneous delivery delays and uneven shock incidence, and finds that these mechanisms generate deadweight loss and asymmetric output responses.

  • Problem

    The paper asks how money can affect short-run output when it appears to have little bearing on long-run output.

    Source: §1 · p. 2

  • Method

    The paper models firms exchanging delayed intermediate inputs and introduces monetary shocks as uneven changes in firms’ nominal balances.

    Source: §1 · pp. 2, 4

  • Results

    Positive and negative monetary shocks generate deadweight loss, while reallocation can dominate locally and allow GDP to rise after a positive shock.

    Source: §1 · p. 4

  • Takeaways & Limitations

    Longer, more roundabout input chains delay a disturbance’s arrival at usable production but enlarge its cumulative loss.

    Source: §1 · p. 5

    The assumed size pattern constrains firms closer to final demand, but does not restrict sizes elsewhere in the supply chain or require any single retailer to be small.

    Source: §4.1 · p. 29

Abstract

from arXiv · show

This paper develops a dynamic production-network model in which monetary non-neutrality emerges under fully flexible prices. Firms finance intermediate-input purchases from current nominal balances, and intermediate goods arrive with heterogeneous lead times across suppliers. A monetary shock then has two distinct effects. The first is miscoordination: because production uses dated input bundles assembled under past nominal conditions, the shock disturbs the composition of usable inputs (within and across firms) and thereby generates a deadweight loss. The second is reallocation: because the incidence of the shock is tilted toward smaller firms, which are disproportionately downstream, the shock redistributes resources along the supply chain. More specifically, positive shocks push resources downstream, negative shocks pull them upstream. This means that under a negative shock, both the miscoordination and the reallocation effect reduce downstream output and therefore GDP. Under a positive shock, reallocation raises GDP while miscoordination reduces it. The money-output relation is therefore sign-asymmetric: negative shocks generate sharp contractions, while positive shocks generate at most mild expansions, and sometimes none. Computational experiments on a reconstructed US production network generate output declines after a monetary contraction that are about three times as large as the increases after a monetary expansion.

1 Introduction

The paper explains how uneven monetary shocks disrupt dated input bundles and reallocate resources along production networks, creating real effects despite flexible prices. It develops this account through a dynamic network model, relates it to prior work, and previews its analysis.

1 Introduction

  • Uneven shock incidence creates both miscoordination in dated input bundles and supply-chain reallocation: positive shocks push activity downstream, while negative shocks pull it upstream.Smaller firms are more exposed and tend to be closer to final demand, so their purchasing-power changes shift resource demand along the chain.Source: §1 · pp. 2–4
  • Both positive and negative monetary shocks generate deadweight loss, though reallocation can dominate locally and allow GDP to rise after a positive shock.The possibility arises for an output aggregator sufficiently biased toward one end of the supply chain, even though downstream firms also face miscoordination.Source: §1 · p. 4
  • Longer, more roundabout input chains delay a disturbance’s arrival at usable production but enlarge the cumulative loss as fast and slow adjustments separate.The economy absorbs the disturbance more slowly, not more cheaply.Source: §1 · p. 5
  • A positive monetary shock can raise current GDP by drawing resources away from upstream production, effectively borrowing intermediate goods from future output.The paper describes this as society borrowing from itself because upstream goods today become downstream goods tomorrow.Source: §1 · p. 5

1.1 Related literature

  • Unlike work that treats networks mainly as amplifiers of price stickiness, this paper makes the network itself the source of monetary non-neutrality through disrupted intermediate-input flows.Heterogeneous input delays layer production across past nominal conditions, allowing shocks to disrupt transition dynamics while long-run real quantities remain unchanged.Source: §1.1 · p. 6
  • The paper extends a transient production-network model by adding dated inputs, whose placement in the network shapes how monetary and real shocks affect aggregate variables over time.The introduction then previews the model, inefficiency and reallocation analyses, joint GDP results, network experiments, and conclusion.Source: §1.1 · p. 6, §1.2 · p. 7

2 The Model

The section develops a model of monetary adjustment in which firms connected through a fixed supplier network purchase intermediate inputs with heterogeneous lead times, leaving dated orders in a production pipeline. It examines how an uneven one-time monetary shock misaligns firms’ balances and prolongs real adjustment, with sufficiently small shocks converging to a new stationary equilibrium under flexible-price market clearing.

2 The Model

  • Firms produce using labor and intermediate goods ordered from other firms, so each date’s input bundle reflects orders placed at different past dates.A representative household supplies labor and buys from some firms; prices clear markets at every date during the decentralized adjustment.Source: §2 · p. 7

2.1.1 Production technology

  • Firms combine labor with vintage-layered intermediate inputs, with nested CES aggregation allowing substitution within and across delivery vintages.Both substitution curvatures are negative, and inputs of different vintages are poorer substitutes than inputs of the same vintage.Source: §2.1.1 · pp. 8–9

2.1.2 Production network and supplier shares

  • The production network links each firm to its suppliers, and cost minimization determines how its intermediate-input spending is divided across suppliers and vintages.The stationary supplier-share matrix has columns summing to one.Source: §2.1.2 · pp. 9–10

2.1.3 Household preferences

  • The representative household supplies labor and spends fixed wage-income shares on firms’ differentiated goods, with every firm required to reach final demand through a finite sales chain.This reachability condition makes every firm’s stationary sales strictly positive.Source: §2.1.3 · p. 10

2.2 The stationary competitive equilibrium

  • A stationary competitive equilibrium exists and is unique up to the nominal wage under the stated supplier, technology, and final-demand reachability conditions.Lead times affect this equilibrium through the vintage nesting, not through the dating of orders; the stationary equilibrium is efficient among stationary allocations.Source: §2.2 · p. 11

2.3 Monetary shock

  • The only exogenous disturbance is a one-time monetary expansion or contraction, and the analysis studies shocks small enough to remain near the stationary equilibrium.The shock changes the distribution of firm balances, while the aggregate money stock is conserved after the initial change.Source: §2.3 · p. 12, §2.4 · p. 15
  • At the proportional-incidence benchmark, money is neutral; the model’s non-neutrality arises because shocks land unevenly across firms.Smaller firms receive a larger transfer relative to their balances than under proportional incidence.Source: §2.3 · pp. 12–13

2.4 Decentralized adjustment after a monetary shock

  • After the shock, firms spend their balances under fixed supplier shares, so balance misalignments propagate through the network while total money is conserved.Each firm’s wage bill is fixed, leaving changes in its balance to affect intermediate-input spending; the nominal wage is indexed to the post-shock money stock.Source: §2.4 · pp. 14–15
  • Assumption 3 bounds the nominal redistribution rate and allows for misalignment to travel upstream tier by tier without its bound depending on the number of firms.In a tiered supply chain, a retail-end misalignment can travel upstream without decaying until it reaches the top.Source: §2.4 · p. 16
  • Dated inputs make the pipeline of outstanding orders a second state variable alongside balances, and its due orders form the dated input bundle used for production.Orders with different lead times remain in transit across periods; higher pipeline layers contain orders that have not yet matured.Source: §2.4 · pp. 16–17
  • Real feedback is damped but can prolong adjustment beyond money redistribution, with each round taking up to the longest lead time.Out-of-proportion inputs alter output and clearing prices, which affect customers’ later orders; fixed labor limits output responses to input changes. [s11.p29#d2ef23] [s11.p30#607afe]Source: §2.4 · p. 17
  • For sufficiently small shocks, the post-shock trajectory exists and converges locally to a new stationary equilibrium, with outputs, prices, and orders sharing the convergence bound.In the new equilibrium, prices and balances change in proportion to the money stock while real quantities return to their stationary values; money matters along the transition. [s11.p38#1a4f97]Source: §2.4 · p. 18

3 Miscoordination within Firms

Dated input bundles can become compositionally misaligned after a monetary shock, reducing output during the transition. The cumulative loss depends on shock size and on how intermediate-input lead times shape the mismatch.

3 Miscoordination within Firms

  • Within firms, dated inputs moving in different proportions create a composition mismatch that lowers output relative to inputs moving by their average proportional change.The loss is zero exactly when all dated inputs at the firm move in the same proportion.Source: §3 · pp. 18, 20

3.1 Cumulative loss from production inefficiency

  • Cumulative production-inefficiency loss is finite and bounded by a fixed multiple of (1−θ)^2π^2; for small shocks, equal-sized expansions and contractions destroy the same output within firms.The leading coefficient is positive when some firm’s first-order input responses differ across suppliers.Source: §3.1 · p. 21

3.2 Lead times and the loss from production inefficiency

  • A wider spread of lead times around an unchanged mean generally raises the date-specific loss, weighted by input responses to lead time, unless reduced routing mismatch offsets the increase.The increase is tied to the outer curvature 1−ρ_c and the rise in lead-time variance.Source: §3.2 · pp. 21–23
  • Whether longer mean lead times raise or lower loss depends on whether monetary and real waves dampen or amplify one another as their timing separates.Their alignment depends on whether intermediate-input demand and money travel in the same direction through the network.Source: §3.2 · pp. 23–24
  • Lengthening every lead time changes cumulative loss through altered comovements among the monetary wave and rounds of the real wave, not through changes in each wave’s own dispersion.Loss rises when separated waves dampen one another and falls when they amplify one another.Source: §3.2 · pp. 24–25

4 Monetary Reallocation

The section characterizes how monetary shocks reallocate activity across the supply chain, defining firms’ positions by the time their output takes to reach final demand. It also examines how upstream output contractions traveling downstream can offset the initial reallocation pattern.

4 Monetary Reallocation

  • Because shocks land more heavily on smaller downstream firms, positive shocks raise downstream output initially but can later turn that expansion into contraction as upstream output falls.Negative shocks produce the mirror image, pulling resources upstream; dated orders carry these movements to later dates.Source: §4 · p. 25

4.1 Supply-chain structure

  • The supply-chain framework measures firms’ household-linked sales through downstreamness and assumes retailers are smaller than other buyers in the input markets they share.Lead times are assumed to lengthen away from final demand, with retail firms mostly using shorter-lead-time inputs and top-tier firms mostly using longer-lead-time inputs.Source: §4 · p. 25, §4.1 · pp. 28–29

4.2 Impact reallocation

  • Impact reallocation vanishes when no supplier sells to households and all firm customers of each supplier share the same expenditure impulse on every active shortest-lead-time link.Under those conditions, money is neutral at impact.Source: §4.2 · p. 31
  • A positive shock shifts activity toward final demand at impact, as more heavily affected buyers outbid others for fixed short-lead-time output; negative shocks reverse these effects.The retail tier’s gains in shared markets come at the expense of firms further upstream and the household.Source: §4.2 · p. 32

4.3 Persistence of the impact reallocation

  • Upstream output contractions can offset downstream impact expansions when they reach retailers while their impact-date orders are still maturing.Reduced output travels down the supply chain one lead time per link; what matters is how long the output takes to reach final demand.Source: §4.3 · p. 33
  • F_i(T) is the share of firm i’s output that reaches final demand within T periods, rising with T to one.It is the distribution of delivery time to final demand, measuring a firm’s supply-chain position in periods.Source: §4.3 · p. 33
  • Following purchase chains, household expenditure pays firm i its Domar weight, m*_i/w*, in total.The passage defines the portion of that total spent on output firm i delivers to final demand within T periods.Source: §4.3 · p. 33

Appendix B.4).

Dated input orders sustain the downstream advantage of a positive shock on retail lead-time dates, but can also carry upstream shortages downstream and reverse that expansion later.

Appendix B.4).

  • The position-tilted output response combines dated buyer-supplier contrasts along supply paths, with pass-through discounts limiting the total tilt mass at each date.The contrasts are evaluated at the order dates of the input bundle.Source: Appendix B.4 · p. 34
  • The impact advantage persists on retail lead-time dates because vintages of orders placed when retailers outbid other buyers mature through the dated-input pipeline.At each retail lead-time date, a vintage of the impact-date orders matures.Source: Appendix B.4 · p. 35

4.4 Reversal of the impact reallocation

  • The reversal combines a fading monetary advantage as excess balances move upstream with shortages propagated from upstream firms deprived of inputs at impact.Dated orders carry both forces through the supply chain over lead times.Source: §4.4 · pp. 36, 38
  • Under the stated assumptions, downstream-tilted shocks raise the position-tilted output index on retail lead-time dates, yet condition (R) can make it negative afterward.The reversal occurs beyond every retail lead-time date, at a bounded horizon.Source: §4.4 · pp. 37–38
  • The reversal condition is easiest to meet in short supply chains with broad sourcing, where fewer sourcing rounds separate retailers from the contracting region.Condition (R) compares the contracting region’s impact-date tilt mass with other impact-date orders and the post-impact response.Source: §4.4 · pp. 37–38

5 Monetary Non-neutrality

Section 5 combines dated-input misalignment and supply-chain reallocation to account for GDP’s response; it shows that the response differs by shock sign and over time.

5 Monetary Non-neutrality

  • GDP over a transition equals the value drawn from goods in transit minus deadweight losses from within-firm input inefficiency and across-firm misallocation.The losses also include welfare lost when consumption across goods and dates departs from stationary proportions.Source: §5 · pp. 39–40

5.1 Asymmetric monetary non-neutrality

  • At first order, GDP responses are equal and opposite and deadweight loss vanishes; second-order asymmetry combines losses under either shock sign with stronger reallocation after contraction.Contraction shifts balances across firms relative to a smaller post-shock money stock, moving more output along the supply chain.Source: §5.1 · p. 40
  • Under Theorem 3’s conditions, an expansion raises GDP over the first reporting interval, while an equal contraction lowers it more, by more than the sum of their deadweight losses.The theorem requires a positive GDP margin and a retail shortest lead time.Source: §5.1 · p. 42
  • A sufficiently large expansion can lower GDP when deadweight loss exceeds the reallocative gain; when expansion raises GDP, upstream production may still contract and inputs may fit less efficiently.The reallocative gain grows in proportion to the shock, while deadweight loss grows with its square.Source: §5.1 · p. 44

5.2 The intertemporal effects of monetary shocks

  • Across the transition, GDP gains are offset by losses, leaving cumulative GDP equal to minus transition deadweight loss; an early expansion gain is repaid with interest.The implicit interest is the transition’s production-inefficiency and misallocation losses; the early gain comes from running down goods in transit.Source: §5.2 · pp. 45–46

6 Computational Experiments

Computational experiments find sign-asymmetric GDP responses in both synthetic and reconstructed US supply chains: contractions are sharper than expansions. The experiments also locate the asymmetry along the supply chain and show how it varies with shock size, input curvature, and network size.

6 Computational Experiments

  • Experiments compare tiered synthetic networks with reconstructed US networks, impose shorter lead times downstream, and shock stationary economies with labor fixed.The reconstructed US network originally has no intermediate-input lead times, so the experiments impose them.Source: §6 · pp. 46–47

6.1 Experiments on synthetic supply chains and the robustness of analytical results

  • The synthetic-network contraction trough is 0.3%, versus a 0.2% expansion peak; the peak ratio is 1.3 across 10^3–10^5 firms.The GDP path has an early rise followed by a delayed fall below its pre-shock level.Source: Figure 1 · p. 48, §6.1 · p. 48
  • At π=0.10, peak ratios reach 7.3 on the US chain and 1.7 on the synthetic chain, versus 1.1 and 1.0 at π=0.002.The ratio also rises as inputs become more complementary, and asymmetry persists when inputs are substitutes.Source: §6.1 · p. 49, Figure 2 · p. 50

6.2 The size of monetary non-neutrality in the US economy

  • At the US full census, contraction lowers GDP 1.5% versus a 0.5% expansion rise; the peak ratio is 3.2.Both responses peak in the second quarter.Source: §6.2 · pp. 49–50, Figure 3 · p. 51
  • Synthetic peak responses are the same at 10^3 and 10^5 firms, so monetary non-neutrality does not shrink as the economy grows.The paper attributes this to the aggregate shock’s cross-sectional balance misalignment, which is not diluted by a law of large numbers.Source: §6.2 · p. 51, Figure 4 · p. 52
  • At 105 US firms, the antisymmetric response rises from about zero upstream to 3.0% in the most downstream decile, while the symmetric response is negative in every decile.The antisymmetric component is the first-order reallocation effect; the symmetric component offsets downstream gains under expansion and deepens contraction.Source: §6.2 · pp. 52–53, Figure 5 · p. 53

7 Conclusion

  • The model combines downstream reallocation after positive shocks, upstream reallocation after negative shocks, and deadweight loss from inefficient input proportions to produce sharper contractions than expansions.The mechanism operates with flexible prices and makes welfare costs depend on supply-chain structure and firm-size distribution.Source: §7 · pp. 54–55

A.1.1 The within-period sequence

  • The within-period sequence turns current balances into intermediate orders at stationary supplier shares, clears prices against available output, and then produces using dated inputs.The analytical results require small shocks, fixed supplier shares, a spectral condition, and bounded firm-size ranges; the experiments relax these restrictions.Source: §6 · p. 46, §A.1.1 · pp. 59–60

A.1.2 Proof of existence and uniqueness of the stationary equilibrium

The stationary equilibrium is constructed from a unique positive price fixed point, and the appendix establishes local convergence after sufficiently small monetary shocks. The convergence envelope does not depend on the number of firms.

A.1.2 Proof of existence and uniqueness of the stationary equilibrium

  • The log-unit-cost map is a contraction, yielding a unique stationary price fixed point.The proof uses supplier shares as derivatives of log prices with respect to log supplier prices.Source: §A.1.2 · p. 61
  • A unique positive fixed point of the unit-cost map determines stationary prices, from which the proof recovers shares, sales, allocation, and the pipeline.With time-invariant prices and balances, dated production reduces to the static identity; the constructed allocation and prices form a stationary competitive equilibrium, unique up to the nominal wage.Source: §A.1.2 · pp. 61–62

A.1.3 Proof of local convergence after a monetary shock

  • After a monetary shock, log output is a delayed contraction of log inputs, so outputs and orders obey a geometric convergence envelope.The dated-input bound covers inherited pipeline entries as well as orders placed at the current date.Source: §A.1.3 · pp. 63–64
  • The convergence-bound constants depend on model parameters and lead-time bounds, but not on the number of firms.The argument also gives a positive expenditure and demand region under sufficiently small shocks.Source: §A.1.3 · pp. 62–64

A.2.1 Proof of the sign of the loss from production inefficiency

The proof shows that the loss from production inefficiency is nonnegative and vanishes only when dated inputs move in the same proportion.

A.2.1 Proof of the sign of the loss from production inefficiency

  • The loss from production inefficiency is nonnegative, with equality exactly when both variance components vanish.The aggregate loss inherits this sign because it sums firm-level losses with nonnegative weights.Source: §A.2.1 · p. 65
  • The aggregate loss is nonnegative because it is a weighted sum of firm-level losses with nonnegative weights.Source: §A.2.1 · p. 65
  • Both variance components vanish exactly when a firm’s dated inputs all move in the same proportion.Equivalently, the inputs have the same proportional change across suppliers within each supported vintage layer.Source: §A.2.1 · p. 65

A.2.2 Proof of the order of the cumulative loss from production inefficiency

The cumulative loss from production inefficiency is bounded and has a quadratic small-shock limit whose coefficient is determined by supplier-response variance. The adjacent reallocation proofs state thresholds and tolerances for downstream-concentrated tilts.

A.2.2 Proof of the order of the cumulative loss from production inefficiency

  • A geometric bound on dated-input deviations makes the cumulative production-inefficiency loss finite for sufficiently small shocks.The bound is uniform over normalized weight vectors and depends on the economy through the stated envelope parameters.Source: §A.2.2 · p. 65
  • The cumulative production-inefficiency loss divided by π^2 converges to C(ω), a sum of nonnegative curvature-weighted variances of first-order input responses.The coefficient vanishes exactly when those responses are constant across suppliers at every supported firm and date.Source: §A.2.2 · p. 66

A.3 Proofs of Section 4: Monetary Reallocation

  • The reallocation proofs specify a threshold for downstream-concentrated tilts and a tolerance for condition (N).Source: §A.3 · p. 66

A.3.1 Proof of impact reallocation

The proof derives impact-date expenditure contrasts from heterogeneous firm responses and uses path bounds to establish sign persistence over a finite horizon.

A.3.1 Proof of impact reallocation

  • At impact, the expenditure contrast between buyer and supplier is σ_i − σ_j, and supplier-weighted contrasts give relative expenditure impulses.The proof bounds impact-date contrasts in absolute value by a constant proportional to 1 − θ.Source: §A.3.1 · pp. 67–68
  • At the impact date, output’s first-order response is (1 − β_i)ξ_i,k = φ_i, with no cascade because supplier output has not yet moved.Source: §A.3.1 · p. 68
  • Because the reallocation objects are linear in the per-unit shock, a negative shock reverses the impact response.Source: §A.3.1 · p. 69

A.3.2 Proof of sign persistence over a finite horizon

  • For sign persistence, retail-tier impulse terms have a positive lower bound while off-tier tilt mass and path contributions are bounded under the stated conditions.The post-impact path component is zero through date k and begins only at date k + 1.Source: §A.3.2 · pp. 70–71

A.3.3 Proof of the two-phase reallocative impulse response

The proof establishes a negative arriving-contraction expression under (R) and uses linearity to obtain the mirror image for a negative shock. It also identifies a date beyond the retail lead-time dates for the contraction argument.

A.3.3 Proof of the two-phase reallocative impulse response

  • On retail lead-time dates, positivity of Dπ𝒬t(b) follows from Proposition 4.Source: §A.3.3 · p. 71
  • The proof selects a date τ− beyond the retail lead-time dates at which μτ−(𝒯) ≥ μ.The construction uses the sourcing-round threshold and the share of impact-date orders that mature k* periods or more later.Source: §A.3.3 · p. 71
  • The arriving-contraction bound makes the displayed reallocative expression negative under (R).Linearity in the per-unit shock gives the mirror image for a negative shock.Source: §A.3.3 · p. 72

A.4.1 Proof of asymmetric monetary non-neutrality

The proof derives asymmetric signs for the first-interval GDP response from derivative signs and a positive quadratic term. It extends the result to a quarterly average of proportional changes.

A.4.1 Proof of asymmetric monetary non-neutrality

  • The two derivative signs of f+ yield the signs and asymmetry within the radius allowed by Proposition 2.The function f+ is a finite average of maps real-analytic near zero that vanish there.Source: §A.4.1 · p. 72
  • The quadratic term is at least cY(1 − θ)π² + O(π⁴), and is positive once π is sufficiently small.Source: §A.4.1 · p. 73
  • The quarterly average of proportional changes has the same first derivative at zero, and its Hessian adds squares of first-order responses of order (1 − θ)².The first part of Theorem 3 holds for this average as well, with K+ raised by the added term.Source: §A.4.1 · p. 73

A.4.2 Proof of intertemporal resource accounting

The proof bounds deviations in post-shock orders and shows that the aggregate output response series converges absolutely to zero. Its resource-accounting inequality is strict when consumption departs from its stationary value.

A.4.2 Proof of intertemporal resource accounting

  • Post-shock orders placed at date s deviate proportionally by at most C_S κ^s |π|(1 − θ).Orders outstanding at date 0 were placed before the shock and do not deviate.Source: §A.4.2 · p. 73
  • The nondecreasing partial sums of nonnegative losses converge to the limit in (72), with strict inequality when consumption departs from its stationary value.The strictness follows because the consumption loss is then positive.Source: §A.4.2 · p. 73
  • The output-response series converges absolutely to zero, using the bound |ḟt| ≤ C_c κ^t(1 − θ).Source: §A.4.2 · p. 73

A.4.3 Proof of implicit interest cost of a monetary expansion

The proof establishes a positive implicit interest cost and relates shifted consumption to a positive consumption loss and excess. It also gives a quadratic bound on dated-input losses.

A.4.3 Proof of implicit interest cost of a monetary expansion

  • When consumed quantity moves, the corresponding consumption loss H_t and the excess are positive.Source: §A.4.3 · p. 74
  • After T0, f_t(π) < 0 for the stated horizon condition C_c κ^{T+1}|π|(1 − θ) ≤ 1.Source: §A.4.3 · p. 74
  • For a fixed economy, the dated-input loss ℰ_i(bx) is bounded above by C_ℰ∥bx∥².The bound follows from nonnegativity and a zero gradient at zero.Source: §A.4.3 · p. 74
  • The implicit interest cost I(π) is positive and equals L_Y(π) − G_Y(π).Source: §A.4.3 · p. 74

B Analytical Results and Proofs

The appendix establishes local well-posedness and analytic behavior of the model’s transition, and develops proof results on within-firm input decomposition and lead-time variation.

B Analytical Results and Proofs

  • The appendix’s standing Assumption 2 and the bound π of Section 2.3 apply throughout without restatement.These are the conventions specified for Appendix A.Source: B Analytical Results and Proofs · p. 75

B.1 Lemmas for Section 2: The Model

  • With positive balances, expenditures, and inherited orders near the stationary state, within-period clearing has a unique positive solution and the transition and dated-input observation maps are real-analytic.The impact state varies affinely with the monetary shock, so dated inputs have first-order expansions at each fixed finite date.Source: §B.1 · p. 75

B.2 Lemmas for Section 3: Miscoordination within Firms

  • The input-bundle expansion separates scale and composition effects, while the curvature-weighted dispersion form is positive semidefinite and reduces to (1 − ρ) Covℱᵢ when ρ_c = ρ.The scale–composition form follows from a second-order expansion of the production identity.Source: §B.2 · pp. 77–78

B.3.1 An increase in the variance of the lead-time profile

  • When the lead-time variance rises at every firm, the first sum in the comparison decomposition is non-negative.The comparison is between profiles k′ and k.Source: §B.3.1 · pp. 81–82

B.3.2 A uniform shift of the lead-time profile

This section formalizes how lead-time shifts and delivery-time paths transmit monetary and real waves, expenditure, and reallocation through input bundles and production networks to GDP, including response asymmetry, GDP accounting, and deadweight-loss components. It also characterizes input misalignment and the GDP margin, including examples in which neither term in (B.69) has a determined sign.

B.3.2 A uniform shift of the lead-time profile

  • The monetary and real waves have finite H-norms, and their cumulative cross-term equals CEQ minus CDQ.This decomposition separates the cumulative loss into the two wave norms and their interaction.Source: §B.3.2 · p. 82
  • A uniform increase in lead times translates the monetary wave and dated expenditure impulse, while preserving the H-norms of translated profiles and rounds.The comparison holds with the same shares, labor shares, and weighting under both lead-time profiles.Source: §B.3.2 · pp. 83–84
  • The monetary and real waves dampen each other when supplier-dispersion covariances are nonnegative at weighted firms and positive at one, under the lemma’s common-lead-time and eigenvector conditions.For lead time k=1, the condition holds when a weighted firm has suppliers with different incidence values; for k≥2, it holds under an additional restriction on how supplier averaging affects dispersion.Source: §B.3.2 · p. 85

B.4 Lemmas for Section 4: Monetary Reallocation

  • The exposure advantage is bounded by the log-size gap over the intermediate-input share, subject to a labor-share correction.The lemma also extends the bounds to weighted averages over sets of firms.Source: §B.4 · p. 86
  • Each post-shock intermediate-input order response decomposes into a buyer-side expenditure channel and a supplier-side price channel.The expenditure response is linear in the first-order balance perturbation.Source: §B.4 · p. 87, The expenditure channel admits the… · p. 88

The expenditure channel admits the explicit form

  • The dated relative expenditure impulse combines supplier contrasts, and input-bundle scales transmit these impulses through the network as a convergent path sum.The position-tilted log-output response is a nonnegative combination of dated contrasts.Source: The expenditure channel admits the… · pp. 88–89
  • The delivery-time path mass gives the share of sales reaching final demand by a given horizon, and the reallocation impulse has an analogous expansion using household shares.The impulse separates impact-date contrasts from the remaining component, which is zero through the shortest lead time k.Source: The expenditure channel admits the… · pp. 90–91
  • A downstream-concentrated tilt sends at least mass μ to the top tier at some date after all retail lead-time dates.The arrival date lies in [k + k*, τ], while every retail lead-time date is below k*.Source: The expenditure channel admits the… · p. 92
  • Without dated inputs, production inefficiency loss is proportional to within-firm input-scale variance, and the output response equals minus that nonnegative loss.In this case, the position term P_t is zero at every date.Source: The expenditure channel admits the… · p. 93

B.5 Lemmas for Section 5: Monetary Non-neutrality

  • For sufficiently small shocks, a concave response makes the absolute response to a negative shock larger than that to an equally sized positive shock.The result applies over a finite set of dates where the response has positive first derivative and negative second derivative.Source: §B.5 · pp. 93–94
  • GDP accounting gives an exact identity linking log GDP change to the consumption–output wedge, whose even part is nonnegative under the stated assumptions.The bound on the wedge is uniform in cross-sectional size.Source: §B.5 · p. 94

And with the GDP margin

  • The GDP response has a uniform second-order expansion over any finite date set, and the reported second-derivative term has the sign of the shock for sufficiently small nonzero shocks.The first-order GDP response over the initial reporting interval also has a positive lower bound when c_Y > 0.Source: And with the GDP margin · pp. 95–96
  • The GDP response is analytic in the normalized shock, with its curvature controlled under uniform Hessian and slack assumptions.The misalignment is bounded by the slack, so the response’s leading term is the first-order response applied to the normalized shock.Source: And with the GDP margin · pp. 97, 98, 100
  • Production inefficiency and scale deviations contribute separate nonnegative components to the deadweight loss.The mix loss reflects input proportions moving differently from stationary levels; the scale loss reflects the bundle’s overall size differing from its stationary level.Source: And with the GDP margin · p. 99
  • The drawdown, deadweight loss, and excess value of goods in transit depend on the shock only through the normalized shock.The even part of the drawdown response has a second-order expansion involving the first derivative and Hessian of its response map.Source: And with the GDP margin · pp. 100–101
  • Deadweight loss splits exactly into nonnegative losses on consumed output and goods in transit; the transit loss is also part of the drawdown of goods in transit.The transit drawdowns sum to zero over the transition.Source: And with the GDP margin · p. 103
  • At the shortest positive lead time, first-order dated-input deviations equal σ_i − σ_j for suppliers at that lead time and are zero for other suppliers.Source: And with the GDP margin · p. 104
  • Example 1 states that neither term on the right of the third line of (B.69) has a determined sign.The following example varies the first retailer’s input share and, in one case, the household’s spending share.Source: And with the GDP margin · p. 105
  • At a=1/2, the first term, sum of the two terms, and D_2𝒴Π are about −3.1 × 10^-3, −6.8 × 10^-3, and −2.2 × 10^-3, respectively.The example uses two retailers with θ=0.8, ρ=−1, one-period lead times, and equal labor shares; the household spends 2/5 on the first retailer.Source: And with the GDP margin · p. 105
  • The first-order response f_1 is positive at both a=1/20 and a=1/2, while the second term is negative at both values.At a=19/20, the second term is positive, about 1.7 × 10^-2, when the household spends 1/10 on the first retailer.Source: And with the GDP margin · p. 105

C.1 Parameter Calibration

This appendix section documents the calibration of model curvature and supply-chain lead times and provides reference material on notation and dependencies among formal results.

C.1 Parameter Calibration

  • The baseline sets within-vintage curvature to ρ = −0.8, implied by an estimated same-product supplier substitution elasticity of about 0.55, and sets cross-vintage curvature equal to it.The elasticity estimate is attributed to Fujiy, Ghose and Khanna (2024); this baseline therefore has no cross-vintage amplification.Source: §C.1 · p. 106
  • At baseline, supplier lead times range from three to eight months for the most upstream buyers and two to five months for the most downstream buyers.The appendix describes integer lead times and presents baseline configurations and sweep ranges for both supply chains in Table C.2.Source: §C.1 · p. 106, Table C.2 · p. 107

C.2 Notation

  • The notation appendix groups symbols by theme and gives the paper section, Appendix B, or appendix where each is introduced.Its entries cover firm and network indices, technology, lead times, monetary shocks, post-shock dynamics, and output losses and reallocation.Source: §C.2 · pp. 108, 109, 111
  • The notation covers GDP and consumption responses, loss and reallocation measures, and parameters used to report the simulations.It also defines household balance, incidence, and transfer-cap parameters, alongside lead-time-window and slow-link parameters.Source: §C.2 · pp. 112–114

C.3 Dependency map

  • Figure C.1 maps dependencies among the paper’s formal results and Appendix B, ordered by dependency depth; Assumption 2 is linked only when a proof invokes it by name.An arrow indicates that a result’s statement or proof invokes another result.Source: Figure C.1 · p. 115
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