Production Network Inflation Dynamics

• The paper demonstrates that production networks, via spectral gaps and heterogeneous price adjustments, create persistent inflation effects and misaligned relative prices.
• It employs a finite firm model with a column-stochastic input–output matrix to capture both direct and indirect propagation of monetary shocks.
• The analysis reveals that central bank policies should integrate network metrics to accurately assess CPI elasticity and inflation transmission in open economies.

A production network model of inflationary dynamics formalizes how inflation propagates through interlinked firms and sectors, incorporating heterogeneous microeconomic adjustment, input–output linkages, and network structure. Recent work establishes that even in economies where micro-level price adjustments appear “flexible,” aggregate inflation can generate persistent, nontrivial misalignments in relative prices—mechanisms governed not by price rigidity per se, but by the topology and spectral characteristics of the production network itself (Veetil, 26 Jan 2026). In open economies, production networks additionally shape the transmission of sectoral productivity, wage, and import shocks to consumer prices, with both direct and indirect network-driven pathways (&&&1&&&).

1. Production Network Structure and Spectral Properties

The production network is formally represented as a finite set of firms $N = \{1, \dots, n\}$, with directed edges specified by a column-stochastic adjacency matrix $\mathbf{A} = (a_{ij})_{i,j\in N}$, where $a_{ij}$ is the share of firm $j$’s expenditure on input from supplier $i$ and $\sum_i a_{ij} = 1$ for all $j$. Nodes are characterized by (out-)degree $d_i = \#\{j : a_{ij} > 0\}$ and empirical networks display mildly negative assortativity, parameterized by $\kappa_i \approx B\,d_i^{-\nu}, \; \nu \in (0,1)$, where $\kappa_i$ is the expected degree of $i$'s neighbors.

The dynamic and equilibrium properties of the network are governed by the spectral gap $\gamma = 1 - \lambda_2$, where $\lambda_2$ is the second-largest eigenvalue of $\mathbf{A}$. A small spectral gap implies slow mixing—monetary shocks or liquidity injections propagate sluggishly, leading to persistent disequilibria in relative prices.

2. Microeconomic Adjustment and Aggregate Inflation Shock

Price-setting operates against a “cash-in-advance” constraint, where firm $i$’s money balance at $t$, $m_{i,t}$, gives nominal demand $\mathcal{D}_{i,t} = \sum_j a_{ij} m_{j,t}$ for its product. The flexible-price solution is $p_{i,t}^{\rm flex} = \mathcal{D}_{i,t} / q_{i,t}$, with money balances propagating via $\mathbf{m}_{t+1} = \mathbf{A} \mathbf{m}_t$.

Aggregate inflation arises from central bank liquidity injections $\pi M_t$ at each period, so total nominal balances evolve as $M_{t+1} = (1+\pi) M_t$. Steady-state price levels grow at the rate $\pi$, but individual firms only reset prices stochastically, with a hazard $\eta_{i,t} = g(\pi u_{i,t}) f(\delta_i)$ that increases in cumulative inflation since last reset, $u_{i,t}$, and depends on an “excess-degree” index $\delta_i = d_i^{\nu^2} - \mathbb{E}[d^{\nu^2}]$. This structure captures both common monetary drift and heterogeneous sectoral propensities to adjust.

3. Relative-Price Distortions and Network Propagation

Relative price distortion is central to the welfare costs of inflation in networked economies. For any numéraire sector $k$ (with $\delta_k = 0$), the relative price of firm $i$ is $r_{i,t} = p_{i,t} / p_{k,t}$ (with steady state $r_i^* = p_i^* / p_k^*$). Two principal measures of distortion are:

1. The $\ell_2$ gap $$\omega_t = \sqrt{n^{-1}\sum_{i=1}^n (r_{i,t} - r_i^*)^2}$$;
2. The Kullback–Leibler divergence $\psi_t = \sum_i R_{i,t} \log [R_{i,t} / R_i^*]$, where $R_{i,t}$ is the normalized share.

Eigenvector and spectral decomposition of the shock propagation yields persistent “transitory” components that decay with the spectral gap, but whose magnitude is proportional to the excess-degree $\delta_i$. Analytical results show (for small $\pi$): $$\omega = \pi\,\mathscr{W}_\omega(\alpha,\nu^2) + o(\pi), \quad \psi = \pi^2\,\mathscr{W}_\psi(\alpha,\nu^2) + o(\pi^2),$$ where $\alpha$ is the Pareto exponent (degree tail), $\nu^2$ is the assortativity parameter, and $\mathscr{W}_\omega,\mathscr{W}_\psi$ are positive moment functions. A heavier-tailed degree distribution (smaller $\alpha$), stronger assortativity (larger $\nu^2$), lower injection heterogeneity, and slower mixing all amplify $\omega$ and $\psi$ even if average price changes remain small (Veetil, 26 Jan 2026).

4. Analytical Results on Inflation–Price-Change Relationship

The model yields several results that challenge standard interpretations of inflation dynamics:

• Bounded Average Price Change: There exists a threshold $\zeta^* > 0$ (close to $\min\{1, \alpha - \nu^2\}$) such that the degree-weighted average absolute log-change in prices satisfies $\phi_t \leq \pi$ for all $\zeta < \zeta^*$ and finite $t$, with $\phi_t \to \pi$ in steady state.
• Near-Zero Correlation: The state-dependent hazard structure enables $\mathrm{Cov}(\phi, \pi) \approx 0$ generically, reconciling empirical findings of minimal correlation between inflation rates and price-change magnitudes.
• Large Relative Price Distortion: Substantial misalignment in relative prices can occur even under “flexible” micro-pricing, governed by network moments and spectral gap: $\omega = O(\pi)$, $\psi = O(\pi^2)$.
• Spectral Gap and Distortion Persistence: Small spectral gap $\gamma$ leads to persistent, superposed transient shocks, so price distortions accumulate and persist even if the average price index tracks the monetary injection rate.

5. Production Networks and CPI Elasticities in Open Economies

For disaggregated small open economies, the production network structure determines the elasticity of the consumer price index (CPI) to sectoral shocks (Silva, 2024). Let $A = (a_{ij})$ be the domestic input–output matrix and $\Gamma = (\gamma_{im})$ the import matrix. The (domestic) Leontief inverse, $(I-A)^{-1}$, encodes both direct and indirect propagation pathways.

In log-differential form,

$$\hat P = -b^\top (I-A)^{-1} \hat Z + b^\top (I-A)^{-1} A \hat W + [b^M + b^\top (I-A)^{-1} \Gamma] \hat P_M,$$

where $b$ and $b^M$ are domestic and imported consumption shares, $\hat Z$ are sectoral technology shocks, $\hat W$ wage shocks, and $\hat P_M$ import-price shocks.

Network structure implies:

• Elasticities to sectoral technology (productivity) shocks: $\varepsilon^{CPI}_{Z_s} = - [b^\top (I-A)^{-1}]_s$
• To factor prices: $\varepsilon^{CPI}_{W_f} = [b^\top (I-A)^{-1}A]_{f}$
• To import prices: $$\varepsilon^{CPI}_{P^M_m} = b^M_m + \sum_i b_i [(I-A)^{-1}\Gamma]_{im}$$

Direct and indirect trade exposures adjust the impact of shocks: indirect exporting dampens domestic inflation pass-through, while indirect importing amplifies CPI sensitivity to foreign price shocks. Empirically, network-adjusted Domar weights, labor shares, and import shares can differ substantially from their “raw” statistics, with significant consequences for quantitative inflation transmission in open economies.

6. Policy Implications and Misconceptions

The production network model demonstrates that inflation costs in networked economies predominantly arise from the mis-timing and heterogeneity of price adjustments propagating through input–output linkages, not simply the size or frequency of price changes. Standard microeconomic statistics (e.g., average price-change size) are weak indicators of actual misalignment; omission of network topology underestimates the distortionary effects of inflation.

Key implications include:

• Network topology (degree distribution, assortativity, spectral gap) modulates relative-price wedges, with heavy-tailed, negatively assortative, or slow-mixing networks generating larger and more persistent distortions.
• Central banks and policymakers should supplement traditional inflation metrics with sectoral propagation diagnostics, such as input–output centralities and spectral gaps, to assess true inflationary costs and to tailor monetary interventions accordingly (Veetil, 26 Jan 2026).
• In small open economies, neglecting production networks leads to systematic bias in estimating CPI elasticity to both domestic (productivity/wage) and foreign (import price) shocks, overstating domestic effects and understating imported inflation (Silva, 2024).

7. Empirical Calibration and Applications

Calibration with world input–output tables and national accounts shows that network-adjusted responses of CPI to productivity, wage, and import shocks differ significantly from traditional metrics. For example, in small open economies, network-adjusted Domar weights fall relative to raw sales-to-GDP, while network-adjusted import shares nearly double compared to direct shares. Application to the COVID-19 inflation episodes in Chile and the UK demonstrates that models incorporating full production network structure are better able to replicate observed inflation levels and volatility than those omitting network or trade linkages (Silva, 2024).

These findings collectively highlight the centrality of the network architecture—encoded in both the adjacency matrix and its spectral features—in mediating inflation dynamics and shaping optimal policy interventions in both closed and open economies.