Abstract:
Using industry-level panel data and plausibly exogenous variation in supply conditions, we estimate the elasticity of retail price margins with respect to inventories along the retailer's optimal pricing curve. We find that this elasticity is negative and statistically significant, implying that lower finished-good inventories leads to higher price margins. We assess the implications of this channel for inflation dynamics within a New Keynesian Phillips curve (NKPC) framework that links inventories to retailers' markup behavior. Incorporating the inventory-sales ratio into the NKPC markedly improves the model's empirical fit and helps account for two notable recent inflation episodes: the missing disinflation of 2009-2011 and the COVID-era surge.
Keywords: Inflation; inventories; supply disruptions; Phillips curve.
JEL Classification Numbers: E31, E32, E22.
"Retail markups in a number of sectors have seen material increases in what could be described as a price-price spiral, whereby final prices have risen by more than the increases in input prices. The compression of these markups as supply constraints ease, inventories rise, and demand cools could contribute to disinflationary pressures."
The Phillips curve has been a key pillar of the literature on inflation and the business cycle. Nevertheless, its empirical versions to this date leave a lot of room for improvement in accounting for inflation dynamics. In particular, the COVID-era inflation surge that originated in the goods sector was not well captured by the standard Phillips curve models.
In this paper, we argue that inventories--specifically, retail inventories of finished goods--play a critical role in shaping inflation dynamics through their impact on retail prices.2 While the standard New Keynesian Phillips curve (NKPC) abstracts from inventories, we show that retail inventory fluctuations offer a key missing link: they help explain why prices rose sharply during periods of supply disruption and low inventory availability.
Our approach begins with a stylized model of a retail firm that jointly chooses prices and inventory stocks in the face of supply disruptions. We show that transitory shocks shift the firm's stocking policy without directly affecting its pricing decision, thereby enabling identification of the optimal pricing curve using instruments for inventory disruptions.
Empirically, we estimate the elasticity of retail price margins with respect to inventories using industry-level panel data spanning 2008q1 to 2025q1. Because pricing and stocking decisions are jointly determined, accurately estimating this elasticity requires an identification strategy. Motivated by our stylized model, we construct two shift-share variables that capture plausibly exogenous transitory supply shocks. We use these shift-share variables to instrument for changes in retailers' inventory-sales ratios. The first instrument reflects material supply shortages among upstream manufacturers from the Census Bureau's Quarterly Survey of Plant Capacity (QSPC). The second instrument reflects global shipping delays based on the Average Congestion Rate (ACR) of Bai et al. (2024). In both cases we regard supply shifts as occurring upstream, and map them to downstream retailers using exposure shares based on a retail industry's supply use from a given manufacturing industry.
Identification with shift-share instruments can be achieved by assuming exogeneity of either shifts or shares (Borusyak et al., 2022; Goldsmith-Pinkham et al., 2020). Our identification rests on exogenous shifts. In our setting, the ideal experiment would be to randomly assign observed transitory supply shifts across manufacturing industries. These shifts would affect downstream retail industries differently given the degree to which retailers rely on each manufacturer for their respective supply of finished goods. In practice, supply shifts are unobservable, so we adopt a quasi-experimental framework by constructing proxies for supply shifters.
The estimated elasticity is negative and economically significant: tighter inventories are systematically associated with higher price margins, even after controlling for cost variables. When instrumenting inventory changes with observed supply disruptions, to identify the elasticity along the retailer's optimal pricing curve, we estimate an even stronger negative relationship, supporting the hypothesis that inventory scarcity contributes directly to retail price increases and providing evidence of state-dependent retail pricing.
To assess the macroeconomic implications of this inventory-pricing channel, we extend the stylized retail firm model, embedding the negative estimated elasticity into a New Keynesian framework. Retailers face both Rotemberg-style nominal rigidities and time-varying markups arising from two sources: a Kimball (1995) demand structure and deep habit formation a la Ravn et al. (2006). This setup yields two NKPC representations. The sales-based NKPC highlights how inflation is tied to future marginal costs and marginal sales benefits, but is difficult to estimate directly except in the special case without time-varying markups. The stock-based NKPC, in contrast, relates inflation to observable inventory-sales ratios and expected marginal cost growth, offering a tractable and empirically robust formulation. In particular, the inventory-sales ratio serves as a sufficient statistic for understanding inflationary pressures arising from the model's endogenous retail markup behavior.
Estimating the stock-based NKPC using macroeconomic data, we find that incorporating the retail inventory-sales ratio markedly improves the model's empirical fit compared to standard NKPC specifications. In particular, the stock-based NKPC helps account for two prominent recent inflation episodes--capturing both the missing disinflation of 2009-2011 and the COVID-era surge. The underlying intuition is that detrended inventory levels were tight in both periods, generating additional inflationary pressures that bring model-implied inflation closer to the observed data. Because changes in the inventory-sales ratio orthogonal to movements in the expected marginal cost growth should reflect movements in retail sector markups within our model, the results highlight the important role that retail price markups may play in driving inflation dynamics.
Related literature.
The COVID-era surge in inflation has sparked renewed interest in the role of supply-side disruptions in driving price dynamics. A growing literature documents how production and transportation frictions contributed to rising costs across sectors 2022; Comin et al., 2023; Ferrante et al., 2023; Ortiz, 2022). These studies emphasize upstream channels|such as global supply chain These studies emphasize upstream channels--such as global supply chain bottlenecks, convex production technologies, and sectoral reallocation--as key drivers of inflationary pressures. We build on this line of research by focusing on a downstream margin that has received comparatively less attention: the pricing behavior of retailers facing inventory shortages. While prior work highlights how supply disruptions raise production costs, our contribution is to show that they can also amplify retail price margins when firms are constrained by limited inventory availability.
A paper closely related to our topic is Cavallo and Kryvtsov (2023), which uses online micro-level stockout data to study the inflationary effects of consumer product shortages during the pandemic. We complement their analysis by leveraging industry-level panel data, which offers two distinct advantages. First, our data encompasses a broader set of retailers, including the motor vehicle dealer industry--a key contributor to goods inflation in the early stages of the pandemic. Second, we use retail gross margin data--the difference between the sales price and the acquisition cost--rather than retail prices, and we control for several industry-level cost measures, such as average hourly earnings. This allows us to analyze retailers' pricing behavior beyond what can be attributed to underlying cost pressures during this period.
The relationship between inventories and inflation has been explored in several studies, particularly in the context of business cycle models. Jung and Yun (2005), Lubik and Teo (2012), and Kryvtsov and Midrigan (2010, 2013) examine how inventory dynamics interact with pricing, often within frameworks that assume constant elasticity of substitution (CES) in demand. These papers typically find either a limited role for inventories in explaining inflation dynamics or a need for additional mechanisms--such as time-varying, countercyclical markups--to reconcile theory with the data, especially in response to monetary policy shocks. Our empirical analysis using industry-level data confirms that retail price margins respond in ways that are inconsistent with CES demand, supporting the need for richer pricing structures. In contrast to the prior literature, our model explicitly incorporates standard features that generate time-varying price markups, including habit formation and Kimball-style demand. We derive a representation of the NKPC in which the inventory-sales ratio serves as a sufficient statistic for retail markups. This approach allows us to empirically assess the role of time-varying markups in inflation dynamics without relying on direct and often noisy measures of price margins.
Our paper also contributes to the empirical Phillips curve literature, which has grappled with the declining explanatory power of traditional NKPC specifications in recent decades (Mavroeidis et al., 2014; Stock and Watson, 2020; Coibion and Gorodnichenko, 2025). A central debate in this literature concerns the inclusion of forward-looking variables that capture the information and beliefs shaping firms' pricing decisions. For example, Coibion and Gorodnichenko (2015) show that replacing professional inflation forecasts with household survey expectations can help resolve the 2009-2011 missing disinflation puzzle, highlighting the value of expectation measures that better reflect firms' information set. Reis (2023) similarly argues for combining multiple indicators of inflation expectations to obtain early and more accurate signals of inflation dynamics. In this spirit, we propose the inventory-sales ratio as a complementary forward-looking variable that reflects retailers' near-term beliefs and strategic pricing behavior. Incorporating this measure directly into the NKPC significantly improves its empirical fit, allowing it to account for key episodes such as the 2009-2011 missing disinflation and the surge in goods inflation following the COVID-era supply disruptions.
The remainder of the paper is organized as follows. Section 2 presents a stylized retail firm model that motivates our identification strategy for estimating the inventory stock elasticity of the retail price margin. Section 3 implements thisstrategy using industry-level panel data. Section 4 extends the stylized model and derives a NKPC with inventories that is consistent with the empirically observed negative inventory elasticity of retail markups. Section 5 estimates the inventory-augmented Phillips curve to quantify the macroeconomic importance of retail pricing behavior, as reflected in the inventory-sales ratio. Section 6 concludes.
In this section, we present a stylized model of a retail firm to illustrate how temporary supply disruptions can be used to identify the firm's optimal pricing curve as a function of its inventory stock. The key insight is that while the firm's optimal stocking decision depends on both current and expected future costs of acquiring goods, its pricing decision depends only on the expected future cost. Therefore, changes in the current cost of acquiring goods--so long as they do not affect expectations of future costs--will shift the firm's stocking curve but leave its pricing curve unchanged. In particular, if supply disruptions are perceived to be transitory relative to other cost shocks, they can serve as valid instruments to trace out the firm's pricing curve. We first show this identification logic in a static one-period model, and then extend the insight to a dynamic two-period setting that makes explicit the role of intertemporal inventory decisions in separating pricing from stocking behavior.
To build intuition about the retail firms' joint decisions over stocking and pricing, we consider a one-period model in which the firm purchases a stock of finished goods and sets a price at which to sell them. The firm simultaneously chooses the stock level and the selling price, taking as given the demand function and the unit cost of stocking. Demand for the good, denoted $$s$$, is given as $$s={s}(a,p)$$ where $$a$$ is the stock and $$p$$ is the price.3 We assume the demand function satisfies $$\partial {s}/ \partial a>0$$, $$\partial^{2}{s}/\partial a^{2}<0$$, and $$\partial {s}/\partial p<0$$. The first two conditions are standard in the inventory literature. For example, in stockout avoidance models, an additional unit of stock raises sales up to the point where stockouts are eliminated; in stock-elastic demand models, sales increase with an additional unit of stock but at a diminishing rate.
After sales, the firm may liquidate unsold inventory, $$a-s$$, incurring a cost. The total cost of stocking, net of liquidation, is given by:
| $$\displaystyle (c+\xi)a - (1-\gamma) c (a-s),$$ |
where $$c$$ is the unit cost of stocking, $$\xi$$ captures an additional shadow cost faced by the retailer (e.g., supply disruptions not embedded in $$c$$), and $$\gamma$$ governs the liquidation cost. The parameter $$\xi$$ is a stand-in for the difficulty in stocking by the retail firm not captured in the unit cost of stocking $$c$$.
The firm's profit is given by:
| $$\displaystyle \phi = p {s}(a,p) - (c+\xi)a +(1-\gamma)c(a-{s}(a,p)).$$ |
The firm chooses $$a$$ and $$p$$ to maximize this profit.
The first order condition with respect to $$a$$ (stocking) is:
| $$\displaystyle p\frac{\partial {s}}{\partial a} - (c+\xi) + (1-\gamma)c \left(1-\frac{\partial {s}}{\partial a} \right) = 0,$$ |
which simplifies to:
| $$\displaystyle (p- (1-\gamma)c)\frac{\partial {s}}{\partial a} = \gamma c + \xi.$$ | (1) |
Absent both a shadow cost of stocking ($$\xi = 0$$) and a liquidation cost ( $$\gamma = 0$$), the right-hand side of (1) becomes zero. In that case, as long as $$p > (1 - \gamma)c$$, the firm would optimally stock an infinite quantity ( $$a \rightarrow \infty$$), driving $$\partial {s} / \partial a \rightarrow 0$$. Therefore, at least one of $$\xi$$ or $$\gamma$$ must be strictly positive to yield a finite interior solution. The resulting stocking curve determines the optimal level of the stock as a function of the price, conditional on the cost parameters $$c$$, $$\xi$$, and $$\gamma$$.
The first order condition with respect to price is:
| $$\displaystyle s = -(p-(1-\gamma)c)\frac{\partial {s}}{\partial p}.$$ | (2) |
This pricing equation pins down the optimal price as a function of the level of the stock, taking $$c$$ and $$\gamma$$ as given. For example, suppose the demand function has constant price elasticity:
| $$\displaystyle {s}(a,p) = u(a)\times p^{-\eta},$$ |
where $$u(a)$$ is increasing in $$a$$ and $$\eta>1$$. Then the price markup is constant and given by:
| $$\displaystyle \mu \equiv \frac{p}{(1-\gamma)c} = \frac{\eta}{\eta-1},$$ |
which is the standard result under CES demand. In this case, the pricing curve is flat, and changes in the stock have no effect on the markup.
Figure 1: Illustration of a Retailer's Equilibrium
Taken together, the optimal stocking and pricing curves jointly determine the firm's equilibrium stock and price (or markup). A key feature of this simple model is that the shadow cost $$\xi$$ appears only in the stocking condition--not in the pricing condition. As a result, shifts in $$\xi$$ move the stocking curve without affecting the pricing curve, allowing us to trace out the firm's pricing behavior. Figure 1 illustrates this identification strategy. When $$\xi = \xi^{*}$$, the firm's equilibrium stock and markup are $$(a^{*}, \mu^{*})$$. When $$\xi$$ increases to $$\xi^{**} > \xi^{*}$$, the equilibrium shifts to $$(a^{**}, \mu^{**})$$. Supply disruptions that affect $$\xi$$ not captured by $$c$$ therefore serve as valid instruments to recover the pricing curve. If the pricing curve is flat--as in the case of constant price elasticity--then changes in $$\xi$$ would shift the stock without altering the markup. In contrast, if the pricing curve is downward sloping, as shown in the figure, then a higher $$\xi$$ leads to lower stock and a higher markup. Verifying this inverse relationship between inventory and price margins is the focus of the empirical analysis that follows.
We now extend the logic of the one-period model to a dynamic, two-period setting in which inventories can be carried across periods. This extension verifies that the identification strategy remains valid when the firm makes intertemporal stocking decisions.
Assume the retail firm lives for two periods, facing the same demand function in each period as in the one-period model, $$s={s}(a,p)$$. In the first period, the firm purchases a stock of goods $$a_1$$, sets a price $$p_1$$, and sells $$s_1 = {s}(a_1, p_1)$$. The remaining inventory $$a_1 - s_1$$ is carried over to the second period, subject to a depreciation rate $$\delta$$. In the second period, the firm purchases $$n$$ additional units, bringing the total available stock to:
| $$\displaystyle a_{2} = (1-\delta)(a_{1}-s_{1}) + n.$$ |
It then sets price $$p_2$$ and sells $$s_2 = {s}(a_2, p_2)$$ units. As in the one-period model, leftover inventory at the end of the second period is liquidated at cost, governed by the parameter $$\gamma$$.
The firm maximizes the sum of discounted profits:
| $$\displaystyle \pi = p_1 s_1 - (c_1 + \xi_1) a_1 + \beta \mathbb{E}_1 \left[ p_2 s_2 - c_2(a_2 - (1 - \delta)(a_1 - s_1)) + (1 - \gamma)c_2(a_2 - s_2) \right],$$ |
where $$\beta$$ is the discount factor, and $$\mathbb{E}_1$$ denotes expectations conditional on information available in period 1. We assume that the second-period supply disruption $$\xi_2$$ is unanticipated, so $$\mathbb{E}_1 \xi_2 = 0$$.
The first-order condition with respect to $$a_1$$ (initial stocking) is:
| $$\displaystyle \left(p_1 - \beta(1 - \delta)\mathbb{E}_1 c_2\right) \frac{\partial {s}(a_1, p_1)}{\partial a_1} = c_1 + \xi_1 - \beta(1 - \delta)\mathbb{E}_1 c_2,$$ |
which mirrors the one-period stocking condition in equation (1).
Similarly, the first-order condition with respect to $$p_1$$ is:
| $$\displaystyle s_1 = -\left(p_1 - \beta(1 - \delta)\mathbb{E}_1 c_2\right) \frac{\partial {s}(a_1, p_1)}{\partial p_1},$$ |
which corresponds to the pricing condition in equation (2).
As in the one-period case, the shadow cost $$\xi_1$$ enters only the stocking equation and not the pricing equation. This separation confirms that temporary supply disruptions--captured by changes in $$\xi_1$$--remain valid instruments for identifying the slope of the pricing curve in a dynamic environment.
In the next section, we turn to the data and implement this identification strategy using industry-level panel data to estimate the elasticity of retail price margins with respect to the stock of goods. In empirical work, we proxy the stock using the end-of-period inventories. Indeed, our stylized model implies a tight relationship between the stock of goods, $$a$$, and end-of-period inventories, $$inv$$, as given by:
| $$\displaystyle inv = a-s.$$ |
Letting $$is=inv/s$$ denote the inventory-sales ratio and $$\overline{IS}$$ its long-run average, we obtain the following log-linear approximation:
| $$\displaystyle \hat{a} = \hat{s} + \frac{\overline{IS}}{\overline{IS}+1}\widehat{is},$$ |
where hatted variables denote log-deviations from their respective averages. This expression implies that changes in the stock-sales ratio and the inventory-sales ratio are proportionally related. As such, conditional on sales, fluctuations in the stock of goods are directly reflected in movements of the inventory-sales ratio. Accordingly, we use the inventory-sales ratio as an empirical proxy for the stock of goods in our model--controlling for sales--as it conveys equivalent information about stock dynamics and is both directly observable and widely used in the empirical literature.
In this section, we use industry-level panel data to estimate how the retail and wholesale margin price reacts to changes in the inventory stock.4
We assemble quarterly data for eight industries across the retail and wholesale trade sectors, which we collectively refer to as the retail sector for simplicity, spanning 2008q1 to 2025q1. From the Bureau of Labor Statistics (BLS), we collect industry-specific Producer Price Index (PPI) data, average hourly earnings, and total hours worked. Note that in the trade sectors, the PPI reflects gross margins--that is, the difference between the sales price and the acquisition cost of goods sold by retailers. From the Bureau of Economic Analysis (BEA), we incorporate information on real inventories and real sales, which are sourced from the National Income and Product Accounts tables. We merge the BLS and BEA data at the three-digit North American Industry Classification System (NAICS) level, which represents the most granular classification available across these data sets. Further details on data construction are provided in the Appendix.
To formally examine the relationship between inventory holdings and changes in the industry gross margin, we estimate the following regression,
| $$\displaystyle \Delta_k\log(PPI_{i,t}) = \beta\Delta_k\log(IS_{i,t}) + X_{i,t}\gamma + \varepsilon_{i,t},$$ | (3) |
where $$\Delta_k$$ denotes the $$k$$-th difference operator. The outcome variable is the percentage change in the gross margin, measured as the log-difference in the PPI from period $$t-k$$ to $$t$$. The coefficient of interest, $$\beta$$, captures the relationship between changes in the inventory-sales ratio and changes in the gross margin. All control variables are contained in $$X_{i,t}$$.
Because stocking and pricing decisions are jointly determined, estimating equation (3) via ordinary least squares (OLS) would yield a biased estimated of $$\beta$$. In particular, as shown in the stylized model, unobserved persistent or anticipated shocks can shift both the pricing and stocking curves. Shifts in the pricing curve induce a positive correlation between stocks and margins. As a result, we expect the OLS estimate of $$\beta$$ to be biased upward.
To estimate equation (3), we therefore adopt an identification strategy which exploits plausibly exogenous and transitory supply-side variation which we discuss in turn.
Production delays in manufacturing typically show up in capacity utilization data. Indeed, capacity utilization have taken center stage in recent work on inflationary pressures arising from production bottlenecks (e.g., Boehm and Pandalai-Nayar, 2022; Comin et al., 2023). Yet utilization is endogenous, reflecting both demand and supply forces. Unlike Boehm and Pandalai-Nayar (2022), who emphasize demand-driven movements in utilization and their interaction with supply conditions, our goal is to isolate upstream supply disturbances that shift utilization orthogonally to contemporaneous retail demand (conditional on controls).
Our first instrument exploits the establishment-level reason codes in the QSPC--the survey underlying official capacity utilization statistics. A key advantage of the QSPC is that it records why establishments operate below capacity, allowing us to isolate supply-driven variation in utilization. For each three-digit NAICS manufacturing industry, we collect the share of underutilizing establishments that cite "insufficient supply of materials."5
Firm-reported material shortages have been shown to track supply-driven disturbances in production (Balleer and Noeller, 2024; Braun et al., 2024). Building on this evidence, we take the raw shortage shares and, industry-by-industry, residualize them on their own lag, measures of contemporaneous aggregate demand and industry-specific demand, and quarter fixed effects for seasonality. The aggregate demand proxy follows Shapiro (Forthcoming). The industry-specific demand proxy is the common component of gross output quantities and prices (extracted separately by industry) that loads positively on both. This procedure removes predictable demand movements and serial correlation, leaving an innovation that reflects unexpected, supply-driven disruptions.
Formally, our QSPC shift-share instrument is defined as:
| $$\displaystyle Z_{i,t}^{QSPC} = \sum_j s_{ij,t_{0}}\times \Delta_k g^{QSPC}_{j,t},$$ | (4) |
where the shift variable is,
| $$\displaystyle \Delta_k g^{QSPC}_{j,t} = \sum_{\ell=0}^{k} \varepsilon^{QSPC}_{j,t-\ell}.$$ | (5) |
The shift variable is defined as the residualized share of plants in upstream industry $$j$$ reporting material shortages (as described above). Meanwhile, the share variable, $$s_{ij,t_0}\equiv{\omega_{ij,t_0}}/{\omega_{i,t_0}}$$, is defined as the retailer $$i$$'s exposure to a given manufacturing industry $$j$$ in period $$t_{0}$$. The numerator, $$\omega_{ij,t_{0}}$$, is the value of inputs purchased by retail industry $$i$$ from manufacturing industry $$j$$ in the base year $$t_{0}$$.6 The denominator, $$\omega_{i,t_{0}}=\sum_{j}\omega_{ij,t_{0}}$$, represents the total value of inputs (manufacturing and non-manufacturing) used by retailer $$i$$. The term $$s_{ij,t_{0}}$$ thus reflects the input share sourced from manufacturing industry $$j$$ within retailer $$i$$'s total input bundle.
We next consider an instrument for global supply disruptions based on the ACR index of Bai et al. (2024), available from January 2017 to June 2024. The ACR index measures global port congestion and is regarded as an exogenous indicator of shipping disruptions.
Our ACR shift-share instrument is defined as:
| $$\displaystyle Z_{i,t}^{ACR} = \sum_j \bigg{(}s_{ij,t_{0}} \times \frac{m_{j,t_0}}{m_{t_0}} \bigg{)}\times \Delta_k g^{ACR}_{t},$$ | (6) |
where the shift variable is,
| $$\displaystyle \Delta_k g^{ACR}_{t} = \sum_{\ell=0}^{k}\varepsilon_{t-\ell}^{ACR},$$ | (7) |
and $$\varepsilon_t^{ACR}$$ is the residual from a regression of the ACR index on its own lag. Because the ACR shift variable is a time series, we define our share variable as a composite measure of exposure where $$s_{ij,t_{0}}$$ denotes the retailers' exposure from the BEA 2007 Use Table, as before, and $${m_{j,t_0}}/{m_{t_0}}$$ denotes the manufacturers' import share in 2007. Thus, the share variable is an exposure weight mapping global supply disruptions to manufacturers according to import exposure, and to retailers based on the use shares.
We estimate equation (3) via OLS as well as with our instrumental variables, using the aforementioned industry-level data across retailers. The control variables specified in $$X_{i,t}$$ consist of the $$k$$-period log-differences of the following industry-level variables: real sales, average hourly earnings, total hours worked, and an exposure-weighted sum of upstream manufacturing suppliers' PPI. The upstream PPI for each retail industry is defined as:
| $$\displaystyle PPI_{i,t}^{\text{upstream}} = \sum_{j}s_{ij,t_{0}} \times PPI_{j,t},$$ | (8) |
where $$PPI_{j,t}$$ denotes the PPI of manufacturing industry $$j$$ in period $$t$$. We also specify a range of fixed effects in $$X_{i,t}$$. First, we specify industry fixed effects to control for time-invariant unobserved retailer characteristics as well as the time-invariant component of the shifts. In addition, we specify industry-specific linear time trends to absorb gradual, deterministic retail industry-specific trends over time. Finally, we specify time fixed effects. For the QSPC instrumental variable regression, however, we specify an interaction of the sum of exposure shares with time fixed effects.
7Table 1: Industry Regression Results -- Panel A: One-quarter Difference ($$k=1$$)
| Estimation method | Dependent variable: Gross price margin (OLS) | Dependent variable: Gross price margin (IV) | Dependent variable: Gross price margin (IV) | Dependent variable: Gross price margin (IV) |
|---|---|---|---|---|
| Instrument | - | QSPC | ACR | QSPC, ACR |
| IS ratio | -0.143***, (0.038) | -0.579***, (0.215) | -0.583***, (0.179) | -0.648***, (0.166) |
| First-stage F-stat | 5.62 | 9.50 | 12.30 | |
| AR Wald F (p-value) | 0.080 | 0.001 | <0.001 | |
| Hansen J (p-value) | 0.800 | |||
| Observations | 536 | 536 | 232 | 232 |
| Sample | 2008q1-2025q1 | 2008q1–2025q1 | 2017q2–2024q2 | 2017q2–2024q2 |
Notes: Driscoll–Kraay standard errors are reported in parentheses. Instruments for the IV columns are: reported material supply shortages from the Quarterly Survey of Plant Capacity (QSPC) and Average Congestion Rate (ACR). * p<0.10, ** p<0.05, *** p<0.01.
Table 1: Industry Regression Results -- Panel B: Two-quarter Difference ($$k=2$$)
| Estimation method | Dependent variable: Gross price margin (OLS) | Dependent variable: Gross price margin (IV) | Dependent variable: Gross price margin (IV) | Dependent variable: Gross price margin (IV) |
|---|---|---|---|---|
| Instrument | - | QSPC | ACR | QSPC, ACR |
| IS ratio | -0.121**, (0.048) | -0.786***, (0.233) | -0.446***, (0.064) | -0.570***, (0.066) |
| First-stage F-stat | 11.09 | 31.32 | 15.51 | |
| AR Wald F (p-value) | 0.012 | <0.001 | <0.001 | |
| Hansen J (p-value) | 0.180 | |||
| Observations | 536 | 528 | 224 | 224 |
| Sample | 2008q1–2025q1 | 2008q2-2025q1 | 2017q3–2024q2 | 2017q3–2024q2 |
Notes: Driscoll–Kraay standard errors are reported in parentheses. Instruments for the IV columns are: reported material supply shortages from the Quarterly Survey of Plant Capacity (QSPC) and Average Congestion Rate (ACR). * p<0.10, ** p<0.05, *** p<0.01.
We first estimate equation (3) using OLS. The results are reported in the OLS column of Table 1. Panels A and B present estimates for the one-quarter difference ($$k=1$$) and the two-quarter difference ($$k=2$$), respectively. The OLS estimate in Panel A suggests that changes in the inventory-sales ratio are negatively related to changes in the retailer's gross price margin. We find a similar negative relation at the two-quarter horizon (Panel B). Specifically, a one percentage point decline in the inventory-sales ratio over two quarters is associated with a roughly 0.12-0.14 percentage point increase in the gross margin over the same period.
We next implement instrumental variables (IV) estimation using 2SLS, introducing the instruments first individually and then jointly. The corresponding results are reported in the final three columns of Table 1. At both horizons ($$k=1$$ and $$k=2$$), a one percentage point decline in the inventory-sales ratio leads to a statistically significant increase in retail gross margins. The estimated elasticities range from approximately -0.45 to -0.79. These estimates are economically significant, implying that a one standard deviation decrease in the inventory-sales ratios raises gross margins by about one standard deviation.
The first-stage F-statistics suggest that the instruments are generally strong, particularly at the two-quarter horizon and when using ACR or both instruments jointly. For the one-quarter specification with QSPC alone, the F-statistic is somewhat lower--possibly reflecting retail pricing frictions at higher frequencies--but inference remains supported by the Anderson-Rubin Wald test, which is robust to weak instruments.
In the final column of Table 1, we use both instruments jointly and continue to estimate a statistically significant and negative elasticity. The first stage F-statistics remain above conventional thresholds, and we can also implement a test of overidentifying restrictions, which provides additional support for the validity of our estimates.8
Relative to OLS, the IV estimates are more negative, indicating that the OLS estimates are biased upward. This pattern is consistent with the intuition described above and with the mechanism illustrated in the stylized model. In particular, an upward shift in the pricing curve results in higher margins and greater inventory stocks. Because OLS does not account for these unobserved shifts in the pricing curve, it induces a positive correlation between the inventory-sales ratio and gross margins, thereby biasing the OLS estimates upward.
Table 2: Industry Regression Results (Including Other Input Costs) -- Panel A: One-quarter Difference ($$k=1$$)
| Estimation method | Dependent variable: Gross price margin (OLS) | Dependent variable: Gross price margin (IV) | Dependent variable: Gross price margin (IV) | Dependent variable: Gross price margin (IV) |
|---|---|---|---|---|
| Instrument | - | QSPC | ACR | QSPC, ACR |
| IS ratio | -0.342*, (0.195) | -1.089***, (0.367) | -1.356, (0.544) | -0.986***, (1.439) |
| First-stage F-stat | 9.38 | 1.86 | 12.17 | |
| AR Wald F (p-value) | 0.064 | 0.520 | 0.039 | |
| Hansen J (p-value) | 0.81 | |||
| Observations | 288 | 96 | 86 | 86 |
| Sample | 2019q1–2025q1 | 2019q1–2025q1 | 2019q1–2024q2 | 2019q2–2024q2 |
Notes: Driscoll–Kraay standard errors are reported in parentheses. Instruments for the IV columns are: material shortages reported in the Quarterly Survey of Plant Capacity (QSPC) and Average Congestion Rate (ACR). * p<0.10, ** p<0.05, *** p<0.01.
Table 2: Industry Regression Results (Including Other Input Costs) -- Panel B: Two-quarter Difference ($$k=2$$)
| Estimation method | Dependent variable: Gross price margin (OLS) | Dependent variable: Gross price margin (IV) | Dependent variable: Gross price margin (IV) | Dependent variable: Gross price margin (IV) |
|---|---|---|---|---|
| Instrument | - | QSPC | ACR | QSPC, ACR |
| IS ratio | -0.674***, (0.221) | -1.752***, (0.426) | -1.589**, (0.764) | -1.521***, (318) |
| First-stage F-stat | 13.59 | 3.69 | 17.49 | |
| AR Wald F (p-value) | 0.0005 | 0.181 | 0.006 | |
| Hansen J (p-value) | 0.91 | |||
| Observations | 273 | 91 | 81 | 81 |
| Sample | 2019q2-2025q1 | 2019q1–2025q1 | 2019q1–2024q2 | 2019q2–2025q1 |
Notes: Driscoll–Kraay standard errors are reported in parentheses. Instruments for the IV columns are: material shortages reported in the Quarterly Survey of Plant Capacity (QSPC) and Average Congestion Rate (ACR). * p<0.10, ** p<0.05, *** p<0.01.
Retailers may face additional costs beyond the labor and acquisition costs that we explicitly control for. If these costs are omitted, we cannot rule out that the estimated increase in gross margins reflects higher unobserved input costs. To address this concern, we estimate a version of our regressions that includes controls for other non-labor and non-capital costs, drawn from the BLS satellite series of net inputs to industry price indexes. These data are available only from December 2018 for NAICS 441, 445, and 452, and from December 2020 for the remaining retail and wholesale industries.
Because these series are available only over a shorter horizon, the sample size is reduced and the estimates are correspondingly noisier. Nonetheless, as reported in Table 2, the results remain broadly consistent with our baseline findings, indicating that our main conclusions are robust to the inclusion of these additional cost controls.
We report further robustness checks in the Appendix. These include first-stage regression results and a set of balance tests for the QSPC instrument at both at the manufacturing and retail levels. We also redefine the QSPC instrument to address potential measurement concerns.
Our findings suggest that retail prices tend to rise in response to exogenous supply disruptions, even after accounting for a broad set of controls, including measures of input costs. In the context of the NKPC literature, this evidence aligns with a strand of research emphasizing the role of time-varying retail price markups in shaping inflation dynamics. The next section builds on this perspective by further exploring its implications.
That said, we acknowledge that alternative interpretations of our results are possible and warrant further investigation. First, given our reliance on industry-level data, we are unable to observe the composition of retail goods. Retailers may respond to supply disruptions by shifting toward higher-margin products, which would mechanically raise measured price margins without requiring active price increases. Second, we cannot rule out unobserved cost pressures. In particular, absent data on retailers' expectations of future marginal costs, it remains possible that firms raise prices to maintain rather than expand markups. Third, we remain agnostic on manufacturer markups and do not study their interaction with retailer markups--a topic examined in detail by Alvarez-Blaser et al. (2025) using data from a large global manufacturer. These alternative mechanisms highlight the need for richer data to fully disentangle pricing responses, which we leave for future work.
Our empirical investigation reveals a significantly positive response in the retail price margin to supply disruptions that adversely affect the stocking behavior of the retail industry. Building on this relationship, we examine whether the systematic link between the dynamics of the retail inventory-sales ratio and the retail price markup can help improve the empirical performance of the NKPC.
To address this question, we extend the stylized model introduced in Section 2. Specifically, we incorporate inventory management into the retail sector and introduce two standard sources of state-dependent price markups: the demand specification of Kimball (1995) and the deep habits framework of Ravn et al. (2006). These mechanisms generate time-varying price margins that align with our empirical findings and are widely utilized in the literature on inflation and business cycles.
Subsequently, we derive two representations of the linearized NKPC with inventories. We demonstrate that incorporating the retail inventory-sales ratio as an observable variable in the NKPC effectively captures time-varying price markups, which are otherwise difficult to measure directly.
We first derive the retailers' demand function. To introduce time-varying price markups even without nominal rigidities, we specify the following aggregator where the aggregate habit-adjusted consumption, $$C_{t}$$, is implicitly defined as
| $$\displaystyle \int^{1}_{0}v_{j,t}\Upsilon\left(\frac{s_{j,t}- \theta s_{j,t-1}}{v_{j,t}C_{t}}\right) dj = 1.$$ |
The variables $$s_{j,t}$$ and $$v_{j,t}$$ denote the real sales and household preference of variety $$j\in[0,1]$$ in period $$t$$, respectively. The parameter $$\theta\in[0,1]$$ measures the degree of habit formation in consumption of each variety. This specification encompasses both the state-dependent demand elasticity a la Kimball (1995) and deep-habit formation in customer markets analyzed in Ravn et al. (2006). Following Klenow and Willis (2016), we assume the aggregator $$\Upsilon(x)$$ such that
| $$\displaystyle \Upsilon'(x) = \left(\frac{\eta-1}{\eta}\right)\exp{\left(\frac{1-x^{\frac{\psi}{\eta}}}{\psi}\right)},$$ |
This function is consistent with CES when $$\psi>0$$ approaches zero and $$\eta>1$$ is the price elasticity of demand under CES. More generally, the price elasticity of demand is expressed as
| $$$\displaystyle -\left(\frac{ds_{j,t}}{dp_{j,t}}\right)\left(\frac{p_{j,t}}{s_{j,t}}\right) = \frac{\Upsilon'(x_{j,t})}{x_{j,t}\Upsilon"(x_{j,t})}\frac{s_{j,t}-\theta s_{j,t-1}}{s_{j,t}} = \eta x_{j,t}^{-\frac{\psi}{\eta}}\frac{s_{j,t}-\theta s_{j,t-1}}{s_{j,t}},$$$ |
where $$x_{j,t} \equiv {(s_{j,t}-\theta s_{j,t-1})}/{(v_{j,t}C_{t})}$$. As discussed above, the price elasticity of demand is $$\eta$$ in the special case of CES ( $$\theta=\psi=0$$). Otherwise when $$\theta>0$$ or $$\psi>0$$, the price elasticity of demand varies due to its state and history dependence, allowing more flexibility for the model to be consistent with our empirical findings of a negative inventory stock elasticity of the price margin.
As shown in the Appendix, the variety $$j$$ demand function is derived as follows:
| $$\displaystyle s_{j,t} = v_{j,t}C_{t}\Upsilon'^{-1}\left(\frac{p_{j,t}}{P_{t}}D_{t}\right) + \theta s_{j,t-1},$$ | (9) |
where the demand index $$D_{t}$$ is defined as
| $$\displaystyle D_{t} = \int^{1}_{0}\Upsilon'\left(\frac{s_{j,t}-\theta s_{j,t-1}}{v_{j,t}C_{t}}\right)\frac{s_{j,t}-\theta s_{j,t-1}}{C_{t}}dj,$$ |
and the price index $$P_{t}$$ is implicitly defined in
| $$\displaystyle 1 = \int^{1}_{0}v_{j,t}\Upsilon\left(\Upsilon'^{-1}\left(\frac{p_{j,t}}{P_{t}}D_{t}\right)\right)dj.$$ |
The monopolistically competitive retail firm $$j$$ in period $$t$$ purchases $$y_{j,t}$$ quantity of manufacturing goods at a unit price $$Q_{t}$$. It differentiates the good which is then stocked and sold to households at a price $$p_{j,t}$$. The stock of retail goods, $$a_{j,t}$$, is the sum of the undepreciated inventories from the previous period and the goods purchased in the current period:
| $$\displaystyle a_{j,t} = (1-\delta) inv_{j,t-1} + y_{j,t},$$ | (10) |
where $$inv_{j,t-1}$$ is the end-of-period inventories in period $$t-1$$ and $$\delta$$ is the inventory depreciation rate. In turn, the end-of-period inventories evolve as follows:
| $$\displaystyle inv_{j,t} = a_{j,t} - s_{j,t}.$$ | (11) |
Dividing each side by $$s_{j,t}$$, we obtain a relationship between the inventory-sales ratio and the stock-sales ratio. Since they differ only by a constant, the two ratios move together. The inventory-sales ratio is defined as:
| $$\displaystyle is_{j,t} = \frac{inv_{j,t}}{s_{j,t}}.$$ |
Demand for the retail good is given in (9). Following Bils and Kahn (2000) and Jung and Yun ( 2005), we assume that demand is also elastic to the stock of goods for sale:
| $$\displaystyle v_{j,t} = \left(\frac{a_{j,t}}{A_{t}}\right)^{\zeta},$$ | (12) |
where $$\zeta$$ is the sales elasticity to the stock of goods.9 The aggregate stock is defined as
| $$\displaystyle A_{t} = \left(\int^{1}_{0}a_{j,t}^{\zeta}dj\right)^{\frac{1}{\zeta}}.$$ |
Nominal price rigidity is introduced a la Rotemberg (1982). In detail, the retail firm faces a quadratic price adjustment cost in the following form:
| $$\displaystyle \frac{\nu_{p}}{2}\left(\frac{p_{j,t}}{p_{j,t-1}}-1\right)^{2}P_{t}S_{t},$$ | (13) |
where $$\nu_{p}$$ represents the degree of nominal price rigidity and $$S_{t}=\int^{1}_{0}s_{j,t}dj$$ is aggregate sales.
The retail firm maximizes the expected discounted sum of profits:
| $$\displaystyle \mathbb{E}_{0}\sum^{\infty}_{t=0}\Lambda_{0,t}\Phi_{j,t},$$ |
where the period-$$t$$ profit of the retailer is
| $$\displaystyle \Phi_{j,t} = p_{j,t}s_{j,t} - Q_{t}y_{j,t}.$$ |
The retailer is subject to the above conditions (9)-(13). The full optimality conditions are presented in the Appendix.
In standard New Keynesian models without inventories, the NKPC is obtained by combining the firm's pricing decision with its sales condition, under the assumption that the firm satisfies all demand at its chosen price. Implicit in this setup is that the retail firm's stocking decision is equivalent to its production and sales decision.
With inventories, this equivalence breaks down: retailers can satisfy demand not only by purchasing new goods but also by drawing down existing stocks. This distinction introduces an additional margin of adjustment and implies that there are multiple routes to deriving the NKPC. Below, we present the key equilibrium conditions that give rise to alternative representations of the NKPC.
The log-linearized optimal pricing condition in symmetric equilibrium is
| $$\displaystyle \hat{\pi}_{t} = \beta \mathbb{E}_{t}\hat{\pi}_{t+1} - \frac{1}{\nu_{p}}(\hat{\kappa}_{t} + \hat{c}_{t} - \hat{s}_{t}),$$ | (14) |
where hatted values denote log-deviations from the noninflationary steady state, and $${\kappa}_{t}$$ is the real marginal benefit of an additional unit of sales in period $$t$$--the Lagrange multiplier on the demand constraint (9), capturing the inframarginal revenue effect. Current inflation therefore depends on expected future inflation, the real marginal benefit of sales, and the gap between habit-adjusted consumption and sales. This condition also applies in the model without inventories. The key difference between models with and without inventories lies in how demand is met, as summarized by $$\kappa_{t}$$. In the model with inventories, $$\kappa_{t}$$ is not directly linked to contemporaneous real marginal cost because retailers can satisfy demand by drawing down existing inventories rather than immediately purchasing additional goods to restock.
The log-linearized optimal sales condition is
| $$\displaystyle \hat{\kappa}_{t} = \theta\beta\mathbb{E}_{t}(\hat{r}_{t,t+1}+\hat{\kappa}_{t+1}) - \left[\eta(1-\theta)\left(1+\psi\ln{\left(\frac{\eta-1}{\eta}\right)}\right)^{-1} - 1+\theta\beta \right]\mathbb{E}_{t}(\hat{r}_{t,t+1} +\widehat{mc}_{t+1}),$$ | (15) |
where $$r_{t,t+1}$$ is the real stochastic discount factor between period $$t$$ and $$t+1$$ and $$mc_{t}$$ the real marginal cost of producing a retail good in period $$t$$, equivalently the Lagrange multiplier on the law of motion for the inventory stock (10).
To interpret this condition, it is helpful to begin with the benchmark New Keynesian model without inventories and with CES preferences across varieties ( $$\theta=\psi=0$$). The sales condition then reduces to
| $$\displaystyle \hat{\kappa}_{t} = -(\eta-1)\widehat{mc}_{t}.$$ |
Because the markup without inventories is the inverse of real marginal cost, this relation shows that the real marginal benefit of sales is proportional to the markup. Substituting this expression into (14) yields the familiar Rotemberg-type NKPC:
10| $$\displaystyle \hat{\pi}_{t} = \beta \mathbb{E}_{t}\hat{\pi}_{t+1} + \left(\frac{\eta-1}{\nu_{p}}\right)\widehat{mc}_{t}.$$ |
In this benchmark, the sales and stocking conditions coincide, since all sales must be met by contemporaneous purchases of goods. This equivalence implies a one-for-one mapping between marginal cost and the marginal benefit of sales, which is why the standard NKPC can be written directly in terms of real marginal cost.
With inventories, however, this one-for-one mapping breaks down. Additional sales need not be met by contemporaneous purchases. Instead, the retailer recognizes that sales can be satisfied by drawing down inventories, so the optimal sales decision depends on the expected discounted future marginal cost rather than on the current marginal cost. When $$\theta>0$$, habit formation adds an additional intertemporal channel: the expected discounted marginal benefit of future sales also matters. Combining (14) and (15) yields the following sales-based NKPC.
Proposition 1 (Sales-based NKPC). The sales-based NKPC is derived by combining (14) and (15): \begin{align*} \hat{\pi}_{t} = \beta \mathbb{E}_{t}\hat{\pi}_{t+1} + B_{1}\mathbb{E}_{t}(\hat{r}_{t,t+1}+\widehat{mc}_{t+1}) - B_{2}\mathbb{E}_{t}(\hat{r}_{t,t+1} + \hat{\kappa}_{t+1}) + B_{3}(\hat{s}_{t} - \hat{c}_{t}), \end{align*}
where
\begin{align*} B_{1} = \frac{1}{\nu_{p}}\left[\eta(1-\theta)\left(1+\psi\ln{\left(\frac{\eta-1}{\eta}\right)}\right)^{-1} - 1+\theta\beta \right], \quad B_{2} = \frac{\theta\beta}{\nu_{p}}, \quad B_{3} = \frac{1}{\nu_{p}}. \end{align*}In the CES case ($\theta=\psi=0$), the sales-based NKPC simplifies to
\begin{align*} \hat{\pi}_{t} = \beta \mathbb{E}_{t}\hat{\pi}_{t+1} + \left(\frac{\eta-1}{\nu_{p}}\right)\mathbb{E}_{t}(\hat{r}_{t,t+1} + \widehat{mc}_{t+1}). \end{align*}There are two main takeaways. First, without habits, expected future inflation and expected discounted real marginal cost are sufficient statistics for current inflation. Second, with habits, the sales-based NKPC is generally difficult to estimate empirically because it requires separate measures of the real marginal benefit of sales and the marginal cost.
Inventories also give rise to the equilibrium stocking condition, which characterizes how retailers choose inventory holdings. The log-linearized condition is
| $$\displaystyle \hat{a}_{t} = \hat{c}_{t} + \frac{1}{1-\beta(1-\delta)}[\beta(1-\delta)\mathbb{E}_{t}(\hat{r}_{t,t+1} + \widehat{mc}_{t+1})-\widehat{mc}_{t}] + \hat{\kappa}_{t}.$$ | (16) |
This condition highlights three key determinants of the retailer's stocking decision. First, aggregate demand ( $$\hat{c}_{t}$$) matters, as retailers target a desired inventory-sales ratio; higher expected sales translate into higher desired stocks. Second, the intertemporal substitution channel in stocking captures the difference between the expected discounted future marginal cost and the current marginal cost: when costs are expected to rise, firms have an incentive to buy more today and carry inventories forward. This channel parallels the mechanism emphasized by Crouzet and Oh (2016). Third, the term $$\hat{\kappa}_{t}$$ reflects the real marginal benefit of sales: holding more stock increases sales capacity and boosts expected profits, reinforcing the incentive to accumulate inventories.
Substituting out $$\hat{\kappa}_{t}$$ and denoting $$\overline{IS}$$ as the steady-state inventory-sales ratio, we obtain the following stock-based NKPC.
Proposition 2 (Stock-based NKPC). The stock-based NKPC is derived by combining (14) and (16): \begin{align*} \hat{\pi}_{t} = \beta \mathbb{E}_{t}\hat{\pi}_{t+1} + \tilde{B}_{1}[\beta(1-\delta)\mathbb{E}_{t}(\hat{r}_{t,t+1} + \widehat{mc}_{t+1})-\widehat{mc}_{t}] -\tilde{B}_{2}\widehat{is}_{t}, \end{align*}
where
\begin{align*} \tilde{B}_{1} = [{\nu_{p}(1-\beta(1-\delta))}]^{-1}, \quad \tilde{B}_{2} = \overline{IS}(\nu_{p}(\overline{IS}+1))^{-1}. \end{align*}Two insights stand out. First, inflation is driven by three observable statistics: expected future inflation, the expected change in discounted real marginal cost, and the inventory-sales ratio. Second, this representation is agnostic to the sources of endogenous markup variation arising from the Kimball aggregator or deep habits, since these demand effects are absorbed by the inventory-sales ratio. The dynamics of the ratio thus serve as a sufficient statistic for markup behavior in a wide class of models, including CES, Kimball demand, and deep habits.
We find that the stock-based NKPC offers greater flexibility by remaining agnostic about the sources of time-varying markups. Its key observable is the inventory-sales ratio, which--after purging the intertemporal substitution channel--primarily captures the inflationary role of the marginal benefit of sales, i.e., markups. This provides an empirically tractable proxy for time-varying markups in inflation dynamics, which we analyze in the next section.
It is also worth noting that, while our industry-level IV results exploit exogenous variation from short-lived supply disruptions, the stock-based NKPC does not take a stand on the origin of shocks. Its relative improvement in fit should therefore not be interpreted as evidence of the importance of supply disruptions per se, since any inflationary disturbance can propagate through inventories. Put differently, while our IV strategy isolates shifts in the stocking curve, the fluctuations in the inventory-sales ratio that enter the stock-based NKPC reflect both shifts of the curve and endogenous movements along it.
In this section, we estimate the stock-based NKPC using macroeconomic data and show that it outperforms both the canonical specification and the sales-based NKPC under CES demand. The improvement comes from including the inventory-sales ratio as an observable, which captures the time-varying markup channel missing from standard NKPCs. We further show that this specification accounts for two prominent episodes--the 2009-2011 missing disinflation and the COVID-era surge--underscoring the role of inventories as an indicator of inflationary pressures. Note that this section provides a reduced-form empirical assessment of aggregate inflation dynamics; causal effects of the inventory channel are established earlier in the industry-level regressions.
We estimate three reduced-form (semi-structural) Phillips curve specifications with theory-motivated exclusion and equality restrictions:
| $$\displaystyle \hat{\pi}_t$$ | $$\displaystyle = \alpha_0 + \alpha_1 \hat{\pi}_{t-1} + \alpha_2 \mathbb{E}_t\hat{\pi}_{t+1} + \alpha_3 \widehat{mc}_t + \alpha_4(\mathbb{E}_{t}\hat{r}_{t,t+1} + \mathbb{E}_{t}\widehat{mc}_{t+1}) + \alpha_5 \widehat{is}_{t} + \varepsilon_{t},$$ | (17) |
where $$\varepsilon_{t}$$ denotes the regression error. The first sets $$\alpha_{4}=\alpha_{5}=0$$, yielding the canonical NKPC without inventories, as in Galí and Gertler (1999), with both backward- and forward-looking inflation expectations. The second sets $$\alpha_{3}=\alpha_{5}=0$$, corresponding to the sales-based NKPC under CES demand (Proposition 1). The third imposes the equality restriction $$\alpha_{3} = - \alpha_{4}$$, delivering the stock-based NKPC (Proposition 2) with expected marginal cost growth expressed under zero inventory depreciation and no average timediscounting.11
To estimate these regressions, we compile macroeconomic variables at quarterly frequency. Since retail inventory management is primarily relevant within the goods sector, our analysis focuses on accounting for goods inflation, though we also report results using headline inflation for comparison. Accordingly, our baseline regression defines $$\hat{\pi}_{t}$$ as the annualized quarterly inflation of the goods component of Personal Consumption Expenditures (PCE). For expected inflation $$\mathbb{E}_{t}\hat{\pi}_{t+1}$$, we use both the mean professional inflation forecast from the Survey of Professional Forecasters (SPF) and the mean household inflation forecast from the Michigan Survey, each reflecting year-ahead expectations. To our knowledge, no survey provides separate forecasts specifically for goods inflation. By estimating an unrestricted coefficient on inflation expectations, we allow for proportionality between expectations of headline and goods inflation.
To proxy for the real marginal cost, $$\widehat{mc}_t$$, we use the unemployment gap (scaled by -1), a commonly used measure of economic slack (e.g., Stock and Watson, 2020). Specifically, we define the unemployment gap as the difference between the quarterly unemployment rate and the quarterly estimate of the Non-Accelerating Inflation Rate of Unemployment (NAIRU) from the Congressional Budget Office (CBO). Because our Phillips curve specification includes expected future real marginal cost, $$\mathbb{E}_{t}\widehat{mc}_{t+1}$$, using the unemployment gap is particularly useful, as we can construct a comparable expectation-based proxy. We define the expected future unemployment gap as the difference between the SPF's year-ahead unemployment forecast and the year-ahead NAIRU estimate from the CBO.
Alternative proxies for the real marginal cost used in the literature include the output gap, the labor share, the vacancy-to-unemployment ratio, and unit labor cost. Because the NKPC model with inventories requires a measure of expected future real marginal cost, however, constructing such a measure using these alternatives is more challenging.12
We define the expected real interest rate, $$\mathbb{E}_{t}\hat{r}_{t,t+1}$$, as the effective Federal Funds rate minus the SPF's year-ahead inflation forecast.
Finally, for the inventory-sales ratio, $$\widehat{is}_{t}$$, we aggregate inventory and sales data from the retail and wholesale sectors. We detrend this variable using the Hamilton (2018) filter.
The sample spans 1983q3-2025q1. The inventory-sales ratio is constructed from 1980q4 onward--when wholesale inventories become available--and this earlier start is used to implement the Hamilton filter, which regresses the ratio on its 8th-11th lags.
We now present regression estimates for the three Phillips curve specifications.
Table 3: NKPC Estimates for Goods and Headline Inflation (SPF Expectations)
| Goods PCE Inflation (1) | Goods PCE Inflation (2) | Goods PCE Inflation (3) | Headline PCE Inflation (4) | Headline PCE Inflation (5) | Headline PCE Inflation (6) | |
|---|---|---|---|---|---|---|
| Lagged inflation | 0.342**, (0.131) | 0.343***, (0.130) | 0.047, (0.100) | 0.455***, (0.135) | 0.463***, (0.140) | 0.197, (0.120) |
| Exp. inflation (SPF) | 0.369, (0.284) | 0.373, (0.250) | 1.173**, (0.459) | 0.321**, (0.160) | 0.263**, (0.128) | 0.665***, (0.225) |
| Marginal cost | 0.030, (0.145) | 0.074, (0.063) | ||||
| Exp. marginal cost | 0.001, (0.092) | 0.038, (0.047) | ||||
| Exp. marginal cost growth | 0.048, (0.166) | 0.089, (0.094) | ||||
| IS ratio | -0.430***, (0.092) | -0.189***, (0.048) | ||||
| Constant | -0.282, (0.868) | -0.319, (0.761) | -2.299**, (1.158) | 0.461, (0.313) | 0.548*, (0.299) | -0.028, (0.530) |
| Observations | 167 | 167 | 167 | 167 | 167 | 167 |
| Adjusted R-squared | 0.130 | 0.130 | 0.375 | 0.315 | 0.313 | 0.500 |
Notes: Variable definitions are provided in the main text. Expected inflation is from the SPF; results using the Michigan Survey are reported in the Appendix. Newey–West standard errors are reported in parentheses. * p<0.10, ** p<0.05, *** p<0.01.
Our derivation of multiple NKPC representations is useful because, under CES demand, the sales-based and stock-based specifications should be empirically equivalent. If their empirical fit differs in favor of the stock-based specification, this provides evidence against CES and, through the lens of our model, implies that aggregate inflation dynamics require channels of time-varying markups beyond those captured by nominal rigidity.
Table 3 reports OLS estimates of the three specifications in equation (17), using goods and headline PCE inflation as dependent variables and SPF inflation expectations. For goods inflation, the stock-based NKPC in column (3) attains a noticeably higher adjusted R-squared than both the canonical specification (column 1) and the sales-based NKPC under CES demand (column 2). The same pattern holds for headline inflation: column (6) dominates columns (4) and (5). The better fit of the stock-based NKPC relative to the CES-demand specification indicates that the data favor our non-CES specification with time-varying markups. Taken together with the industry-level evidence, these results provide plausible evidence that time-varying markups matter for inflation dynamics.13
Why does the stock-based NKPC fit better? Inspecting columns (3) and (6), the inventory-sales ratio enters with a negative and statistically significant coefficient, as predicted by equation (16), while variables related to marginal cost dynamics--including both current and expected marginal cost--are not significant. The magnitudes are economically meaningful: a one-standard-deviation decline in the inventory-sales ratio is associated with roughly a 2 percentage-point increase in goods inflation and a 0.9 percentage-point increase in headline inflation. By comparison, a one-standard-deviation rise in SPF expected inflation is associated with increases of about 1.1 and 0.6 percentage points, respectively. These results indicate that the inventory-sales ratio plays an important role in accounting for inflation dynamics, likely operating through goods prices.
In column (3), once the inventory-sales ratio is included, the coefficient on SPF expectations strengthens substantially, while the lagged goods inflation term becomes negligible. This suggests that lagged goods inflation had been proxying for omitted inventory dynamics, attenuating the role of expectations. With inventories included, the SPF term becomes more informative, consistent with goods-price dynamics being shaped jointly by professional forecasts and retailers' inventory management decisions.
Using the Michigan Survey for inflation expectations yields a complementary message, as shown in the Appendix. The coefficients of the goods NKPC are stable, and lagged goods inflation is not significant even in specifications without inventories, consistent with household expectations already playing an important role in the Phillips curve (Coibion et al., 2018). Adding the inventory-sales ratio in this setting leaves coefficients largely unchanged but increases explanatory power, indicating that inventories provide incremental information largely orthogonal to the channel captured by Michigan expectations. In this sense, inventories complement household expectations by improving fit without undermining the structural stability of the NKPC.
Our findings are robust across a wide range of specifications and data choices.14 Re-estimating the model with alternative marginal cost proxies commonly used in the literature confirms that the inventory-sales ratio remains a strong and significant predictor of inflation.15 We further estimate the NKPC using core goods inflation--adjusting the inventory-sales ratio accordingly--and again obtain results consistent with our main conclusion. Finally, testing a nonlinear specification of the inventory term shows that the linear mechanism continues to dominate.
Table 4: Semipartial R-squared (Unique Contribution of Regressors): Panel A: SPF Expectations
| Goods PCE Inflation | Headline PCE Inflation | |
|---|---|---|
| Lagged inflation | 0.16 | 2.39 |
| Exp. inflation | 4.24 | 5.03 |
| Exp. marginal cost growth | 0.04 | 0.56 |
| IS ratio | 24.43 | 18.86 |
| Total R-squared | 39.03 | 51.18 |
Notes: Semipartial R-squared values report each regressor's marginal contribution conditional on all others. All values are percentages.
Table 4: Semipartial R-squared (Unique Contribution of Regressors): Panel B: Michigan Expectations
| Goods PCE Inflation | Headline PCE Inflation | |
|---|---|---|
| Lagged inflation | 0.02 | 0.08 |
| Exp. inflation | 9.08 | 12.23 |
| Exp. marginal cost growth | 1.54 | 5.04 |
| IS ratio | 15.07 | 10.62 |
| Total R-squared | 43.87 | 58.38 |
Notes: Semipartial R-squared values report each regressor's marginal contribution conditional on all others. All values are percentages.
To isolate the contribution of the inventory–sales ratio in the stock-based NKPC and compare it with other regressors, we report the semipartial R-squared. Following the Frisch-Waugh-Lovell logic, we orthogonalize each regressor with respect to the other controls and then computing its explanatory power. This statistic captures the regressor's unique (orthogonal) contribution and, in general, does not sum to total R-squared across regressors. Recall that, in our framework, the component of the inventory-sales ratio orthogonal to expected marginal cost growth primarily reflects movements in time-varying markups; accordingly, its semipartial R-squared provides a conservative measure of the markup channel.
Table 4 reports results for both goods and headline inflation using expectations from either the SPF or the Michigan Survey. For SPF expectations (Panel A), the inventory-sales ratio accounts for a substantial 24.4 percent, far exceeding the marginal cost and inflation expectations channels in terms of orthogonal contributions. This indicates that variation in the inventory-sales ratio orthogonal to marginal cost and inflation expectations is quantitatively important for goods inflation. A similar, though smaller, role appears for headline inflation, where the inventory-sales ratio's semipartial R-squared is about 18.9 percent.
With Michigan expectations (Panel B), expected inflation absorbs a larger share of variation in goods inflation (about 9.1 percent). Even so, the inventory-sales ratio remains the single largest semipartial R-squared component (roughly 15 percent). For headline inflation, expected inflation contributes more (about 12.2 percent) than the inventory-sales ratio (about 10.6 percent), but the latter remains economically meaningful. In both cases, the inventory-sales ratio adds distinct, orthogonal information not contained in Michigan expectations and therefore complements them in accounting for inflation dynamics.
Having established the superior fit of the stock-based NKPC in the full sample, we next examine whether the framework can also account for two specific inflation episodes that have attracted considerable attention in the literature.
The 2009-2011 missing disinflation puzzle has been widely discussed in the literature, with a prominent resolution proposed by Coibion and Gorodnichenko (2015). Using the SPF, they show that a version of the canonical NKPC substantially underpredicts inflation over this period. However, when substituting inflation expectations from the Michigan Survey--where household expectations rose more sharply--the puzzle largely disappears. This implies that the puzzle is conditional on the choice of the inflation expectation measure, particularly the SPF.
We offer an alternative explanation using the stock-based NKPC while still conditioning on SPF expectations. Panel C of Figure 2 plots actual headline PCE inflation alongside inflation predicted by both the canonical and stock-based NKPCs. The canonical specification predicts four-quarter inflation to remain below 2 percent between 2009 and 2011 (the first shaded region), while observed inflation reached 3 percent. In contrast, the stock-based NKPC successfully tracks the actual inflation path. This improvement coincides with a period of below-trend inventory-sales ratios, consistent with widespread inventory liquidation by financially constrained firms following the 2008 credit shock, as analyzed by Kim (2021).
For comparison, Panel D uses a specification similar to Coibion and Gorodnichenko (2015) with Michigan inflation expectations. In this case, the canonical NKPC also aligns more closely with observed inflation, and the stock-based NKPC remains consistent with the data.
Figure 2: Model Fit
The stock-based NKPC also captures the surge in inflation--particularly for goods--during the COVID period. As shown in Panels A and B of Figure 2, goods inflation rose to 10 percent between 2021 and 2022 (the second shaded region), a surge that the stock-based NKPC better tracks in both panels. In contrast, the canonical NKPC explains only a smaller portion of the rise, predicting a peak inflation of roughly 4 percent when using SPF expectations (Panel A) and 6 percent with Michigan expectations (Panel B). Moreover, the stock-based NKPC better captures the sharp reversal in goods inflation following the peak, underscoring the complementary role of the inventory-sales ratio in accounting for goods inflation dynamics, especially during periods of swift adjustment.
Turning to headline inflation, the tight inventory-sales ratio, which played a central role in the surge in goods inflation, also contributed significantly to the rise in headline inflation. The stock-based NKPC not only better matches the magnitude of the headline surge than the canonical specification, but also more accurately predicts its earlier timing--showing a peak that arrives sooner and aligns more closely with the data. This improvement is pronounced when using SPF expectations (Panel C), and while the magnitude remains similar with Michigan expectations (Panel D), the earlier predicted peak remains consistent with the observed timing.16
This paper investigates whether the retail price margin channel--whereby inventory scarcity amplifies retail price margins--can help explain observed inflation dynamics. By modeling the retailer's joint decision over pricing and inventory management, and constructing industry-level instruments from supply chain disruptions that plausibly affect only the retailer's stocking behavior, we identify the elasticity of price margins with respect to inventory stocks. We estimate this elasticity to be negative, implying that tighter retail inventories are systematically associated with higher price margins. This evidence supports the view that retail prices are shaped not only by cost-push factors but also by the strategic pricing behavior of firms facing inventory constraints.
Integrating this mechanism into a NKPC framework, we show that the inventory-sales ratio serves as a powerful and observable proxy for retail markups. The resulting stock-based NKPC delivers a substantially improved empirical fit, successfully capturing key inflationary episodes such as the 2009-2011 missing disinflation and the COVID-era surge that originated from the goods sector. These findings suggest that incorporating retail inventory dynamics enhances our understanding of inflation--particularly during periods when supply chains are constrained--by capturing a critical source of time-varying markups often overlooked in standard NKPC models.
Looking ahead, several avenues for further research remain. One is to examine how inventory dynamics interact with nonlinear NKPC specifications, for example by allowing the slope of the Phillips curve to vary across regimes of inventory tightness or demand pressure. Another is to investigate whether the inventory-based mechanism can account for the time variation in the slope of the NKPC documented in the empirical literature. More broadly, extending the framework to incorporate sectoral heterogeneity or to model the interaction between retail inventories and upstream supply chain frictions could yield new insights into how microeconomic bottlenecks shape aggregate inflation dynamics.
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Online Appendix to Retail inventories and inflation dynamics: The price margin channel
Neil Mehrotra17 Hyunseung Oh18 Julio Ortiz19
To calculate the share of retail and wholesale trade margins in the purchasers' value of PCE goods, we use the PCE bridge summary table published by the BEA. Figure A.1 shows the annual composition between 2007 and 2023. Producers' value accounts for roughly 50 percent of purchasers' value, a share that has gradually declined over time. Retail margins represent between 30 and 40 percent, with their share rising in recent years. Wholesale margins remain around 10 percent, while transportation costs account for only a small share.
Figure A.1: Breakdown of Purchasers' Value for PCE Goods
In this section, we detail the data used in our empirical analysis, the sources through which we collect this information, and steps taken to clean or construct certain variables.
The two aggregate time series used in our analysis are the ACR and demand-driven inflation.
Bai et al. (2024) ACR measure. The ACR is a weighted average of the congestion rates for the top 50 container ports worldwide. The congestion rate for a given port is defined as the number of delayed ship visits relative to the total number of ship visits.
Shapiro (Forthcoming) demand-driven inflation. We use the monthly series that reflects the demand-driven component of year-over-year core PCE inflation. Shapiro (Forthcoming) estimates this component by assuming sign restrictions of various categories of inflation. Accordingly, this series tracks the share of categories that are experiencing at least a demand shock.
BLS. We collect monthly PPI data from the BLS.
BEA. We construct our measure of import exposure for manufacturing industries by using the BEA's import matrix, a supplement to the I-O tables. The import matrix provides us with the dollar value imports of intermediate inputs from U.S. manufacturing industries. Using this information, we construct import shares in the year 2007.
For the construction of our industry-specific demand proxy, detailed below, we also collect industry-level quarterly gross output quantities and prices from the BEA.
Census Bureau. Plants responding to the QSPC report the reasons why they produce below capacity. We focus on responses citing "insufficient material supply" as the constraint. These data are publicly available at a quarterly frequency beginning in 2013q1, and available upon request back to 2008q1. Among the 21 three-digit NAICS manufacturing industries, 14 have a complete time series for the share of plants operating below capacity and reporting insufficient material supply, while the remaining 7 industries have breaks in coverage. To ensure that we can accurately estimate and residualize the demand-driven and serially correlated components of the data, we restrict our analysis to the 14 industries with complete time series. These industries are:
Constructing industry-specific demand. Our QSPC shift variable is constructed by residualizing the raw QSPC data on an industry-specific demand proxy. This demand proxy is constructed using principal component analysis. For each manufacturing industry, we collect quarterly gross output quantities and prices from the industry accounts available through the BEA. We calculate the four-quarter percent change in gross output quantities and prices, and extract a component that loads positively onto these two series. Intuitively, a demand shock would lead prices and quantities to move in the same direction. A four-quarter percent change allows us to extract a relatively smoother demand proxy for each industry and it also aligns with the aggregate demand proxy that we use (detailed above). Our QSPC IV results are largely unchanged though slightly noisier when we specify a one-quarter percent change instead.
BLS. We collect monthly average hourly earnings, hours worked, and PPI data from the BLS. Note that the PPI data for retail and wholesale industries is the gross margin (i.e., the difference between the current selling price and the current aquisiton price for a good).
The PPI series for NAICS retailer 449 (Furniture, home furnishings, electronics, and appliance retailers) is only available starting in December 2022 (based on NAICS 2022 definitions). To obtain a full history of observations for this 3-digit industry, we combine the PPI of NAICS 442 (furniture stores, from the 2017 definitions) and NAICS 443 (electronic stores, from the 2017 definitions). The combined series is a weighted average of the PPI of NAICS 442 and 443 where we use a constant weight to construct the average. Specifically, we take the following steps:
BEA. We collect real sales as well as real inventory-sales ratios from the National Income and Product Accounts Underlying Tables (Table 2BU and 3BU), available through the BEA.
In addition, we collect our measure of retail exposure from the 2007 I-O use table. From the I-O use table, we obtain the value of inputs purchased by a given retail industry from a given manufacturing industry in 2007. We construct the retail exposure share by dividing by the total value of inputs (manufacturing and non-manufacturing) used by a given retailer.
Table B.1 presents the summary statistics for each variable and lists the wholesale and retail industries included in our analysis.
Table B.1: Summary Statistics and Industry Names -- Panel A: Summary Statistics
| Summary Statistics | Mean | Median | Std. |
|---|---|---|---|
| One-quarter Difference: Inventory-sales ratio | -0.217 | -0.134 | 8.451 |
| One-quarter Difference: PPI (gross margin) | 0.723 | 0.569 | 2.888 |
| One-quarter Difference: Average hourly earnings | 0.724 | 0.632 | 1.2575 |
| One-quarter Difference: Real sales | 0.664 | 0.589 | 5.949 |
| One-quarter Difference: Hours worked | -0.061 | 0.140 | 3.298 |
| One-quarter Difference: Exposure-weighted mfg. PPI | 0.108 | 0.092 | 0.362 |
| Two-quarter Difference: Inventory-sales ratio | -0.412 | -0.276 | 9.975 |
| Two-quarter Difference: PPI (gross margin) | 1.456 | 1.161 | 3.994 |
| Two-quarter Difference: Average hourly earnings | 1.437 | 1.254 | 1.785 |
| Two-quarter Difference: Real sales | 1.207 | 1.165 | 6.948 |
| Two-quarter Difference: Hours worked | -0.169 | 0.227 | 4.002 |
| Two-quarter Difference: Manufacturing PPI | 0.223 | 0.139 | 0.616 |
Notes: Summary statistics are based on data for the sample period spanning January 2008 to March 2025. Industry classifications follow NAICS 2022 definitions.
Table B.1: Summary Statistics and Industry Names -- Panel B: Industry Names
| Industry | NAICS code |
|---|---|
| Merchant wholesalers, durable goods | 423 |
| Merchant wholesalers, nondurable goods | 424 |
| Motor vehicle & parts dealers | 441 |
| Building materials & garden equipment & supply dealers | 444 |
| Food and beverage stores | 445 |
| Furniture, home furnishings, electronics, & appliance retailers | 449 |
| General merchandise stores | 455 |
| Clothing & clothing accessories retailers | 458 |
Notes: Summary statistics are based on data for the sample period spanning January 2008 to March 2025. Industry classifications follow NAICS 2022 definitions.
Table B.2: Shift-share IV summary statistics -- Panel A: One-quarter Difference $$(k=1)$$
| Mean | Median | Std. | |
|---|---|---|---|
| QSPC shift variable | -4.7e-11 | -0.005 | 0.064 |
| ACR shift variable | 1.5e-8 | -0.020 | 1.794 |
| QSPC shift-share variable | -0.0003 | -0.0005 | 0.005 |
| ACR shift-share variable | 3.7e-10 | -0.0005 | 0.046 |
Notes: Summary statistics are based on data for the sample period spanning January 2008 to March 2025. Industry classifications follow NAICS 2022 definitions.
Table B.2: Shift-share IV summary statistics -- Panel B: Two-quarter Difference $$(k=2)$$
| Mean | Median | Std. | |
|---|---|---|---|
| QSPC shift variable | 0.0007 | -0.009 | 0.090 |
| ACR shift variable | -0.147 | -0.664 | 2.723 |
| QSPC shift-share variable | 0.00002 | -0.002 | 0.009 |
| ACR shift-share variable | -0.004 | -0.015 | 0.070 |
Notes: Summary statistics are based on data for the sample period spanning January 2008 to March 2025. Industry classifications follow NAICS 2022 definitions.
Table B.2: Shift-share IV summary statistics -- Panel C: Share variables (percent)
| Mean | Median | Std. | |
|---|---|---|---|
| Retail exposure | 0.612 | 0.256 | 1.317 |
| Composite retail-mfg-import exposure | 0.115 | 0.040 | 0.270 |
Notes: Summary statistics are based on data for the sample period spanning January 2008 to March 2025. Industry classifications follow NAICS 2022 definitions.
In the first-stage regressions corresponding to Table 1 (reported in Table B.3), both QSPC material shortages and shipping congestion (ACR)--instruments designed to isolate supply-side disruptions--significantly predict declines in the retail inventory-sales ratio.
Table B.3: First-stage Regression Results -- Panel A: One-quarter Difference ($$k=1$$)
| Instrument | Dependent variable: Inventory-sales ratio (QSPC) | Dependent variable: Inventory-sales ratio (ACR) |
|---|---|---|
| Coefficient | -2.123**, (0.895) | -41.738***, (13.539) |
| Observations | 536 | 232 |
| Adjusted R-squared | 0.91 | 0.94 |
| Sample | 2008q1–2025q1 | 2017q2–2024q2 |
Notes: Driscoll–Kraay standard errors in parentheses. $$^{*}p<0.10$$, $$^{**}p<0.05$$, $$^{***}p<0.01$$.
Table B.3: First-stage Regression Results -- Panel B: Two-quarter Difference ($$k=2$$)
| Instrument | Dependent variable: Inventory-sales ratio (QSPC) | Dependent variable: Inventory-sales ratio (ACR) |
|---|---|---|
| Coefficient | -2.982***, (0.895) | -2.982***, (13.480) |
| Observations | 528 | 224 |
| Adjusted R-squared | 0.81 | 0.89 |
| Sample | 2008q2–2025q1 | 2017q3–2024q2 |
Notes: Driscoll–Kraay standard errors in parentheses. $$^{*}p<0.10$$, $$^{**}p<0.05$$, $$^{***}p<0.01$$.
To further support the validity of our QSPC-based shift-share instrument, Table B.4 reports a set of falsification tests at both the manufacturer and retailer levels.
One concern is that the QSPC-based shift variable may not adequately control for demand conditions at the manufacturer level. To assess this, we regress two standard demand proxies--new orders-to-shipments and shipments--on the QSPC shift variable. Panel A reports the results at one- and two-quarter horizons, and in all cases we find no statistically significant relationship.
Another concern is that the instrument may not fully account for potential confounders at the retailer level. To address this, we regress three retail outcomes--real sales, non-capital and non-labor input costs, and labor productivity--on the QSPC shift-share variable.20 The first three rows of Panel B show that none of these variables are related to the instrument.
Finally, we conduct a pre-trend test by regressing lagged outcomes on the QSPC instrument. The fourth row of Panel B reports these results, which indicate that the QSPC-based shift-share instrument does not predict outcomes prior to the shifts.
Table B.4: Balance Tests -- Panel A: Manufacturer-level
| One-quarter Difference: Coefficient | One-quarter Difference: Std. error | One-quarter Difference: Obs. | Two-quarter Difference: Coefficient | Two-quarter Difference: Std. error | Two-quarter Difference: Obs. | |
|---|---|---|---|---|---|---|
| New orders-to-shipments | 0.004 | (0.032) | 470 | 0.033 | (0.040) | 460 |
| Shipments | -0.015 | (0.026) | 936 | -0.057 | (0.038) | 916 |
Notes: Each row reports regressions at three-month and six-month horizons in which the row variable is the dependent variable. Panel A reports regressions of the row variable against the QSPC shift variable, controlling for manufacturing industry fixed effects and time fixed effects. Panel B reports regressions of the row variable against the QSPC shift-share variable, controlling for the same controls as in regression (3) of the main text. Driscoll-Kraay standard errors are reported in the std. error column. * p<0.10, ** p<0.05, *** p<0.01.
Table B.4: Balance Tests -- Panel B: Retailer-level
| One-quarter Difference: Coefficient | One-quarter Difference: Std. error | One-quarter Difference: Obs. | Two-quarter Difference: Coefficient | Two-quarter Difference: Std. error | Two-quarter Difference: Obs. | |
|---|---|---|---|---|---|---|
| Real sales | -0.035 | (0.498) | 528 | 0.236 | (0.733) | 528 |
| Non-labor, non-capital costs | 0.121 | (0.435) | 96 | -0.254 | (0.375) | 91 |
| Labor productivity | 0.272 | (0.250) | 528 | 0.144 | (0.104) | 528 |
| Lagged gross price margin | 0.818 | (0.509) | 520 | 0.962 | (0.854) | 512 |
Notes: Each row reports regressions at three-month and six-month horizons in which the row variable is the dependent variable. Panel A reports regressions of the row variable against the QSPC shift variable, controlling for manufacturing industry fixed effects and time fixed effects. Panel B reports regressions of the row variable against the QSPC shift-share variable, controlling for the same controls as in regression (3) of the main text. Driscoll-Kraay standard errors are reported in the std. error column. * p<0.10, ** p<0.05, *** p<0.01.
Ideally, we would observe the extent to which capacity utilization declines among plants operating below capacity. Instead, our QSPC variable of interest is constructed as the share of plants reporting material shortages relative to all plants operating below capacity. As a result, it is possible, at least in principle, for our QSPC measure to rise even as overall capacity utilization increases. To address this potential measurement concern, we normalize the QSPC variable by capacity utilization in each manufacturing industry, yielding an "adjusted" QSPC variable. We then residualize this adjusted variable as described above. This adjustment mildly strengthens the instrument, as reported in Table B.5, suggesting that our baseline QSPC instrument is relatively robust to potential mismeasurement.
Table B.5: Industry Regression Results: Adjusted QSPC -- Panel A: One-quarter Difference ($$k=1$$)
| Estimation method | Dependent variable: Gross price margin (OLS) | Dependent variable: Gross price margin (IV) | Dependent variable: Gross price margin (IV) |
|---|---|---|---|
| Instrument | Adj. QSPC | Adj. QSPC, ACR | |
| IS ratio | -0.143***, (0.038) | -0.518***, (0.205) | -0.616***, (0.186) |
| First-stage F-stat | 6.77 | 13.24 | |
| AR Wald F (p-value) | 0.070 | <0.001 | |
| Hansen J (p-value) | 0.910 | ||
| Observations | 536 | 536 | 232 |
| Sample | 2008q1–2025q1 | 2008q1–2025q1 | 2017q2–2024q2 |
Notes: Driscoll–Kraay standard errors are reported in parentheses. Instruments for the IV columns are: reported material supply shortages from the Quarterly Survey of Plant Capacity (QSPC) and Average Congestion Rate (ACR). * p<0.10, ** p<0.05, *** p<0.01.
Table B.5: Industry Regression Results: Adjusted QSPC -- Panel B: Two-quarter Difference ($$k=2$$)
| Estimation method | Dependent variable: Gross price margin (OLS) | Dependent variable: Gross price margin (IV) | Dependent variable: Gross price margin (IV) |
|---|---|---|---|
| Instrument | Adj. QSPC | Adj. QSPC, ACR | |
| IS ratio | -0.121**, (0.048) | -0.725***, (0.198) | -0.595***, (0.068) |
| First-stage F-stat | 16.46 | 14.35 | |
| AR Wald F (p-value) | 0.005 | <0.001 | |
| Hansen J (p-value) | 0.200 | ||
| Observations | 536 | 528 | 224 |
| Sample | 2008q1–2025q1 | 2008q2-2025q1 | 2017q3–2024q2 |
Notes: Driscoll–Kraay standard errors are reported in parentheses. Instruments for the IV columns are: reported material supply shortages from the Quarterly Survey of Plant Capacity (QSPC) and Average Congestion Rate (ACR). * p<0.10, ** p<0.05, *** p<0.01.
In this section, we provide details on how we derived the non-CES demand function and several necessary equations based on that. We also provide the retailers' full optimality conditions.
Households minimize the cost in choosing each variety to satisfy the habit-adjusted consumption variety:
| $$\displaystyle c_{j,t} \equiv s_{j,t} - \theta s_{j,t-1},$$ |
taking as given the variety price as well as habits formed in the previous period:
| $$\displaystyle \min_{c_{j,t}} \int^{1}_{0}p_{j,t}c_{j,t}dj$$ subject to$$\displaystyle \quad \int^{1}_{0}v_{j,t}\Upsilon\left(\frac{c_{j,t}}{v_{j,t}C_{t}}\right) dj = 1.$$ |
The first order condition for each variety can be written as
| $$\displaystyle p_{j,t} = \frac{\lambda^{p}_{t}}{C_{t}}\Upsilon'\left(\frac{c_{j,t}}{v_{j,t}C_{t}}\right),$$ |
where $$\lambda^{p}_{t}$$ is the lagrange multiplier of the implicit function. If we define $$P_{t}$$ as a price index that satisfies
| $$\displaystyle P_{t}C_{t} = \int^{1}_{0}p_{j,t}c_{j,t}dj$$ |
in the optimum, we can invoke the Envelope theorem to get
| $$\displaystyle P_{t}C_{t} = \lambda^{p}_{t}\int^{1}_{0}\Upsilon'\left(\frac{c_{j,t}}{v_{j,t}C_{t}}\right)\frac{c_{j,t}}{C_{t}}dj.$$ |
Substituting out $$\lambda^{p}_{t}$$, we obtain
| $$\displaystyle \Upsilon'\left(\frac{c_{j,t}}{v_{j,t}C_{t}}\right) = \frac{p_{j,t}}{P_{t}}\int^{1}_{0}\Upsilon'\left(\frac{c_{j,t}}{v_{j,t}C_{t}}\right)\frac{c_{j,t}}{C_{t}}dj.$$ |
Therefore, the demand function could be expressed as
| $$\displaystyle c_{j,t} = v_{j,t}C_{t}\Upsilon'^{-1}\left(\frac{p_{j,t}}{P_{t}}D_{t}\right),$$ |
where
| $$\displaystyle D_{t} = \int^{1}_{0}\Upsilon'\left(\frac{c_{j,t}}{v_{j,t}C_{t}}\right)\frac{c_{j,t}}{C_{t}}dj.$$ |
In terms of real sales, the demand function is
| $$\displaystyle s_{j,t} = v_{j,t}C_{t}\Upsilon'^{-1}\left(\frac{p_{j,t}}{P_{t}}D_{t}\right) + \theta s_{j,t-1}.$$ | (C.1) |
Note that the value of $$P_{t}$$ could only be implicitly derived by plugging the demand function back into the implicit function for the aggregator $$C_{t}$$:
| $$\displaystyle 1 = \int^{1}_{0}v_{j,t}\Upsilon\left(\frac{c_{j,t}}{v_{j,t}C_{t}}\right)dj = \int^{1}_{0}v_{j,t}\Upsilon\left(\Upsilon'^{-1}\left(\frac{p_{j,t}}{P_{t}}D_{t}\right)\right)dj.$$ |
Here, we derive several necessary equations of the Klenow and Willis (2016) specification of the Kimball (1995) aggregator. Note that the Kimball aggregator satisfies
| $$\displaystyle \int^{1}_{0}v_{j}\Upsilon(x_{j})dj = 1,$$ |
where $$x_{j} = c_{j}/(v_{j}C)$$. In the case of CES,
| $$\displaystyle \Upsilon_{CES}(x) = x^{\frac{\eta-1}{\eta}},$$ |
which implies that
| $$\displaystyle C = \left(\int^{1}_{0}v_{j}^{\frac{1}{\eta}}c_{j}^{\frac{\eta-1}{\eta}}dj\right)^{\frac{\eta}{\eta-1}}.$$ |
The following equations are useful benchmarks for the CES:
| $$\displaystyle \Upsilon_{CES}'(x)$$ | $$\displaystyle = \frac{\eta-1}{\eta}x^{-\frac{1}{\eta}},$$ | |
| $$\displaystyle \Upsilon_{CES}'^{-1}(x)$$ | $$\displaystyle = \left(\frac{\eta-1}{\eta x}\right)^{\eta},$$ | |
| $$\displaystyle (\Upsilon_{CES}'^{-1})'(x)$$ | $$\displaystyle = -\frac{\eta}{x}\left(\frac{\eta-1}{\eta x}\right)^{\eta},$$ |
In the case of Klenow and Willis (2016), we have
| $$\displaystyle \Upsilon_{KW}'(x)$$ | $$\displaystyle = \frac{\eta-1}{\eta} \exp{\left(\frac{1-x^{\frac{\psi}{\eta}}}{\psi}\right)},$$ | |
| $$\displaystyle \Upsilon_{KW}'^{-1}(x)$$ | $$\displaystyle = \left(1+\psi \ln{\left(\frac{\eta-1}{\eta x}\right)}\right)^{\frac{\eta}{\psi}},$$ | |
| $$\displaystyle (\Upsilon_{KW}'^{-1})'(x)$$ | $$\displaystyle = -\frac{\eta}{x}\left(1+\psi \ln{\left(\frac{\eta-1}{\eta x}\right)}\right)^{-\frac{\psi-\eta}{\psi}}.$$ |
We can verify that the above expressions are consistent with CES when $$\psi\rightarrow 0$$.
Using the demand function (C.1), we have
| $$\displaystyle p_{j} = \Upsilon'\left(\frac{s_{j}-\theta s_{j,-1}}{v_{j}C}\right)PD,$$ |
which implies that
| $$$\displaystyle \frac{d p_{j}}{d s_{j}} = \Upsilon"\left(\frac{s_{j}-\theta s_{j,-1}}{v_{j}C}\right)\frac{PD}{v_{j}C}.$$$ |
Moreover, note that
| $$$\displaystyle -\frac{\Upsilon'(x)}{x\Upsilon"(x)} = \eta x^{-\frac{\psi}{\eta}}.$$$ |
Therefore, the price elasticity of demand could then be written as
| $$\displaystyle \eta(s_{j}) \equiv -\frac{d \ln{s_{j}}}{d \ln{p_{j}}} = -\frac{p_{j}}{s_{j}}\frac{d s_{j}}{d p_{j}} = \eta x_{j}^{-\frac{\psi}{\eta}}\left(\frac{s_{j}-\theta s_{j,-1}}{s_{j}}\right).$$ |
The lagrangian of the retailer is written as follows:
| $$\displaystyle \mathcal{L} = \sum^{\infty}_{t=0}\mathbb{E}_{0}\Lambda_{0,t}\Bigg[p_{j,t}s_{j,t} - Q_{t}y_{j,t} + P_{t}mc_{j,t}\left\{y_{j,t} + (1-\delta)(a_{j,t-1}-s_{j,t-1})-a_{j,t}\right\}$$ | |
| $$\displaystyle +P_{t}\kappa_{j,t}\left\{\left(\frac{a_{j,t}}{A_{t}}\right)^{\zeta}\Upsilon'^{-1}\left(\frac{p_{j,t}}{P_{t}}D_{t}\right)C_{t} + \theta s_{j,t-1} - s_{j,t} \right\} - \frac{\nu_{p}}{2}\left(\frac{p_{j,t}}{p_{j,t-1}}-1\right)^{2}P_{t}S_{t}\Bigg].$$ |
Taking the first order conditions, we get
| $$\displaystyle [p_{j,t}]:$$ | $$\displaystyle \ s_{j,t} = - \kappa_{j,t}D_{t}C_{t}\left(\frac{a_{j,t}}{A_{t}}\right)^{\zeta}(\Upsilon'^{-1})'\left(\frac{p_{j,t}}{P_{t}}D_{t}\right) + \nu_{p}\frac{P_{t}S_{t}}{p_{j,t-1}}\left(\frac{p_{j,t}}{p_{j,t-1}}-1\right)$$ | |
| $$\displaystyle \ - \nu_{p}\mathbb{E}_{t}\frac{\Lambda_{t,t+1}p_{j,t+1}P_{t+1}S_{t+1}}{p_{j,t}^{2}}\left(\frac{p_{j,t+1}}{p_{j,t}}-1\right),$$ | ||
| $$\displaystyle [a_{j,t}]:$$ | $$\displaystyle \ mc_{j,t} = \zeta \kappa_{j,t}\left(\frac{a_{j,t}}{A_{t}}\right)^{\zeta-1}\Upsilon'^{-1}\left(\frac{p_{j,t}}{P_{t}}D_{t}\right)\frac{C_{t}}{A_{t}} + (1-\delta)\mathbb{E}_{t}\frac{\Lambda_{t,t+1}P_{t+1}}{P_{t}}mc_{j,t+1},$$ | |
| $$\displaystyle [s_{j,t}]:$$ | $$\displaystyle \ \frac{p_{j,t}}{P_{t}} = \kappa_{j,t} + \mathbb{E}_{t}\frac{\Lambda_{t,t+1}P_{t+1}}{P_{t}}((1-\delta)mc_{j,t+1} - \theta \kappa_{j,t+1}),$$ | |
| $$\displaystyle [y_{j,t}]:$$ | $$\displaystyle \ mc_{j,t} = \frac{Q_{t}}{P_{t}},$$ | |
| $$\displaystyle [mc_{j,t}]:$$ | $$\displaystyle \ a_{j,t} = y_{j,t} + (1-\delta)(a_{j,t-1}-s_{j,t-1}),$$ | |
| $$\displaystyle [\kappa_{j,t}]:$$ | $$\displaystyle \ s_{j,t} = \left(\frac{a_{j,t}}{A_{t}}\right)^{\zeta}\Upsilon'^{-1}\left(\frac{p_{j,t}}{P_{t}}D_{t}\right)C_{t} + \theta s_{j,t-1}.$$ |
The nominal stochastic discount factor between period $$t$$ and $$t+1$$ could be written as the household's subjective discount factor $$\beta$$, the effective real stochastic discount factor $$r_{t,t+1}$$, and the price indices:
| $$\displaystyle \Lambda_{t,t+1} = \beta r_{t,t+1}\frac{P_{t}}{P_{t+1}}.$$ |
In a symmetric equilibrium, and replacing the stochastic discount factor with the complete market equation, the first-order conditions could be rewritten as
| $$\displaystyle [p_{j,t}]:$$ | $$\displaystyle \ S_{t} = - \kappa_{t}(\Upsilon'^{-1})'(1)C_{t} + \nu_{p}S_{t}\pi_{t}(\pi_{t}-1) - \nu_{p}\beta\mathbb{E}_{t}r_{t,t+1}S_{t+1}\pi_{t+1}(\pi_{t+1}-1),$$ | |
| $$\displaystyle [a_{j,t}]:$$ | $$\displaystyle \ mc_{t} = \zeta \kappa_{t}\Upsilon'^{-1}(1)\frac{C_{t}}{A_{t}} + \beta(1-\delta)\mathbb{E}_{t}r_{t,t+1}mc_{t+1},$$ | |
| $$\displaystyle [s_{j,t}]:$$ | $$\displaystyle \ 1 = \kappa_{t} + \beta\mathbb{E}_{t}r_{t,t+1}((1-\delta)mc_{t+1} - \theta \kappa_{t+1}),$$ | |
| $$\displaystyle [y_{j,t}]:$$ | $$\displaystyle \ mc_{t} = \frac{Q_{t}}{P_{t}},$$ | |
| $$\displaystyle [mc_{j,t}]:$$ | $$\displaystyle \ A_{t} = Y_{t} + (1-\delta)(A_{t-1}-S_{t-1}),$$ | |
| $$\displaystyle [\kappa_{j,t}]:$$ | $$\displaystyle \ S_{t} = \Upsilon'^{-1}(1)C_{t} + \theta S_{t-1}.$$ |
Note that $$D_{t}=1$$ in a symmetric equilibrium and that the following conditions hold:
| $$\displaystyle \Upsilon'^{-1}(1) = \left(1+\psi\ln{\left(\frac{\eta-1}{\eta}\right)}\right)^{\frac{\eta}{\psi}}, \quad (\Upsilon'^{-1})'(1) = -\eta \left(1+\psi\ln{\left(\frac{\eta-1}{\eta}\right)}\right)^{\frac{\eta}{\psi}-1}.$$ |
The key steady-state conditions are
| $$\displaystyle S = -\kappa (\Upsilon'^{-1})'(1)C,$$ | |
| $$\displaystyle (1-\beta(1-\delta))mc = \zeta \kappa \Upsilon'^{-1}(1)\frac{C}{A},$$ | |
| $$\displaystyle 1 = (1-\beta\theta)\kappa + \beta(1-\delta)mc,$$ | |
| $$\displaystyle C = \frac{1-\theta}{\Upsilon'^{-1}(1)}S.$$ |
This implies that
| $$\displaystyle \kappa = - \frac{1}{1-\theta}\frac{\Upsilon'^{-1}(1)}{(\Upsilon'^{-1})'(1)} = \frac{1}{\eta(1-\theta)}\left(1+\psi\ln{\left(\frac{\eta-1}{\eta}\right)}\right),$$ | |
| $$\displaystyle mc = \frac{1-(1-\beta\theta)\kappa}{\beta(1-\delta)},$$ | |
| $$\displaystyle \zeta = -(1-\beta(1-\delta))mc\frac{(\Upsilon'^{-1})'(1)}{\Upsilon'^{-1}(1)}\frac{A}{S} = \frac{(1-\beta(1-\delta))mc}{\eta}\left(1+\psi\ln{\left(\frac{\eta-1}{\eta}\right)}\right)^{-1}\frac{A}{S}.$$ |
Therefore, the optimal pricing condition could be log-linearized as
| $$\displaystyle \hat{\pi}_{t} = \beta \mathbb{E}_{t}\hat{\pi}_{t+1} - \frac{1}{\nu_{p}}(\hat{\kappa}_{t} + \hat{c}_{t}-\hat{s}_{t}).$$ |
The optimal stocking condition could be log-linearized as
| $$\displaystyle \widehat{mc}_{t} = (1-\beta(1-\delta))(\hat{\kappa}_{t}+ \hat{c}_{t}-\hat{a}_{t}) + \beta(1-\delta)\mathbb{E}_{t}(\hat{r}_{t,t+1}+\widehat{mc}_{t+1}).$$ |
Substituting out $$\hat{\kappa}_{t}$$, we get
| $$\displaystyle \hat{\pi}_{t} = \beta \mathbb{E}_{t}\hat{\pi}_{t+1} + \frac{1}{\nu_{p}(1-\beta(1-\delta))}[\beta(1-\delta)\mathbb{E}_{t}(\hat{r}_{t,t+1} + \widehat{mc}_{t+1})-\widehat{mc}_{t}] -\frac{1}{\nu_{p}}(\hat{a}_{t} - \hat{s}_{t}).$$ |
Note that the optimal stocking condition could also be written as
| $$\displaystyle \hat{a}_{t} - \hat{c}_{t} = \hat{\kappa}_{t} +\frac{1}{1-\beta(1-\delta))}[\beta(1-\delta)\mathbb{E}_{t}(\hat{r}_{t,t+1}+\widehat{mc}_{t+1})-\widehat{mc}_{t}],$$ |
meaning that the stock to habit-adjusted sales ratio depends on both the expected marginal cost changes and the shadow value of the sales equation, which is linked to the price markup. In turn, the optimal sales condition could be expressed as
| $$\displaystyle \hat{\kappa}_{t} = \theta\beta\mathbb{E}_{t}(\hat{r}_{t,t+1}+\hat{\kappa}_{t+1}) - \left[\eta(1-\theta)\left(1+\psi\ln{\left(\frac{\eta-1}{\eta}\right)}\right)^{-1} - (1-\theta\beta ) \right]\mathbb{E}_{t}(\hat{r}_{t,t+1} +\widehat{mc}_{t+1}).$$ |
The lagrangian of the retailer without inventories is written as follows:
| $$\displaystyle \mathcal{L} = \sum^{\infty}_{t=0}\mathbb{E}_{0}\Lambda_{0,t}\Bigg[p_{j,t}s_{j,t} - Q_{t}y_{j,t} + P_{t}mc_{j,t}\left\{y_{j,t} - s_{j,t}\right\}$$ | |
| $$\displaystyle +P_{t}\kappa_{j,t}\left\{\Upsilon'^{-1}\left(\frac{p_{j,t}}{P_{t}}D_{t}\right)C_{t} + \theta s_{j,t-1} - s_{j,t} \right\} - \frac{\nu_{p}}{2}\left(\frac{p_{j,t}}{p_{j,t-1}}-1\right)^{2}P_{t}S_{t}\Bigg].$$ |
Taking the first order conditions, we get
| $$\displaystyle [p_{j,t}]:$$ | $$\displaystyle \ s_{j,t} = - \kappa_{j,t}D_{t}C_{t}(\Upsilon'^{-1})'\left(\frac{p_{j,t}}{P_{t}}D_{t}\right) + \nu_{p}\frac{P_{t}S_{t}}{p_{j,t-1}}\left(\frac{p_{j,t}}{p_{j,t-1}}-1\right)$$ | |
| $$\displaystyle \ - \nu_{p}\mathbb{E}_{t}\frac{\Lambda_{t,t+1}p_{j,t+1}P_{t+1}S_{t+1}}{p_{j,t}^{2}}\left(\frac{p_{j,t+1}}{p_{j,t}}-1\right),$$ | ||
| $$\displaystyle [s_{j,t}]:$$ | $$\displaystyle \ \kappa_{j,t} = \frac{p_{j,t}}{P_{t}} -mc_{j,t} + \mathbb{E}_{t}\frac{\Lambda_{t,t+1}P_{t+1}}{P_{t}}\theta \kappa_{j,t+1},$$ | |
| $$\displaystyle [y_{j,t}]:$$ | $$\displaystyle \ mc_{j,t} = \frac{Q_{t}}{P_{t}},$$ | |
| $$\displaystyle [mc_{j,t}]:$$ | $$\displaystyle \ y_{j,t} = s_{j,t},$$ | |
| $$\displaystyle [\kappa_{j,t}]:$$ | $$\displaystyle \ s_{j,t} = \Upsilon'^{-1}\left(\frac{p_{j,t}}{P_{t}}D_{t}\right)C_{t} + \theta s_{j,t-1}.$$ |
In a symmetric equilibrium, the first-order conditions could be rewritten as
| $$\displaystyle [p_{j,t}]:$$ | $$\displaystyle \ S_{t} = - \kappa_{t}(\Upsilon'^{-1})'(1)C_{t} + \nu_{p}S_{t}\pi_{t}(\pi_{t}-1) - \nu_{p}\beta\mathbb{E}_{t}r_{t,t+1}S_{t+1}\pi_{t+1}(\pi_{t+1}-1),$$ | |
| $$\displaystyle [s_{j,t}]:$$ | $$\displaystyle \ \kappa_{t} = 1 - mc_{t} +\theta\beta\mathbb{E}_{t}r_{t,t+1}\kappa_{t+1},$$ | |
| $$\displaystyle [y_{j,t}]:$$ | $$\displaystyle \ mc_{t} = \frac{Q_{t}}{P_{t}},$$ | |
| $$\displaystyle [mc_{j,t}]:$$ | $$\displaystyle \ Y_{t} = S_{t},$$ | |
| $$\displaystyle [\kappa_{j,t}]:$$ | $$\displaystyle \ S_{t} = \Upsilon'^{-1}(1)C_{t} + \theta S_{t-1}.$$ |
The key steady-state conditions are
| $$\displaystyle S = -\kappa (\Upsilon'^{-1})'(1)C,$$ | |
| $$\displaystyle 1 = (1-\beta\theta)\kappa + mc,$$ | |
| $$\displaystyle C = \frac{1-\theta}{\Upsilon'^{-1}(1)}S.$$ |
This implies that
| $$\displaystyle \kappa = - \frac{1}{1-\theta}\frac{\Upsilon'^{-1}(1)}{(\Upsilon'^{-1})'(1)} = \frac{1}{\eta(1-\theta)}\left(1+\psi\ln{\left(\frac{\eta-1}{\eta}\right)}\right),$$ | |
| $$\displaystyle mc = 1-(1-\beta\theta)\kappa.$$ |
Therefore, the optimal pricing condition could be log-linearized as
| $$\displaystyle \hat{\pi}_{t} = \beta \mathbb{E}_{t}\hat{\pi}_{t+1} - \frac{1}{\nu_{p}}(\hat{\kappa}_{t} + \hat{c}_{t}-\hat{s}_{t}).$$ |
The optimal sales condition could be expressed as
| $$\displaystyle \hat{\kappa}_{t} = \theta\beta\mathbb{E}_{t}(\hat{r}_{t,t+1}+\hat{\kappa}_{t+1}) - \left[\eta(1-\theta)\left(1+\psi\ln{\left(\frac{\eta-1}{\eta}\right)}\right)^{-1} - (1-\theta\beta ) \right]\widehat{mc}_{t}.$$ |
In this section, we assess the robustness of our results along four dimensions: (i) replacing SPF-based inflation expectations with those from the Michigan Survey; (ii) employing alternative proxies for real marginal cost; (iii) estimating the NKPC with core goods PCE inflation as the dependent variable; and (iv) introducing a nonlinear inventory specification. Finally, we also examine pseudo-out-of-sample performance.
Our main results are robust to using inflation expectations from the Michigan Survey instead of the SPF. Table D.1 is analogous to Table 3 in the main text but replaces SPF expectations with those from the Michigan Survey. For headline PCE inflation, the Michigan Survey improves the fit of all three NKPC specifications in columns (4)-(6) relative to their SPF-based counterparts, as measured by the adjusted R-squared. Nevertheless, our main results hold: the stock-based NKPC in column (6) delivers the best fit and retains a significantly negative coefficient on the inventory-sales ratio. For goods PCE inflation, the improvement in fit from adding the inventory-sales ratio--comparing column (3) to column (1)--is even larger than in the headline case, as reflected in the adjusted R-squared.
Table D.1: NKPC Estimates for Goods and Headline Inflation (Michigan Expectations)
| Goods PCE Inflation (1) | Goods PCE Inflation (2) | Goods PCE Inflation (3) | Headline PCE Inflation (4) | Headline PCE Inflation (5) | Headline PCE Inflation (6) | |
|---|---|---|---|---|---|---|
| Lagged inflation | 0.137, (0.108) | 0.136, (0.107) | -0.017, (0.104) | 0.160, (0.098) | 0.147, (0.104) | 0.040, (0.102) |
| Exp. inflation (Michigan) | 1.549***, (0.463) | 1.564***, (0.463) | 1.250***, (0.360) | 0.945***, (0.210) | 0.933***, (0.207) | 0.810***, (0.149) |
| Marginal cost | -0.033, (0.160) | 0.090, (0.065) | ||||
| Exp. marginal cost | -0.029, (0.091) | 0.066, (0.046) | ||||
| Exp. marginal cost growth | 0.226, (0.149) | 0.212***, (0.080) | ||||
| IS ratio | -0.332***, (0.080) | -0.137***, (0.039) | ||||
| Constant | -5.256***, (1.687) | -5.285***, (1.703) | -4.048***, (1.341) | -1.687***, (0.644) | 1.680***, (0.659) | -1.105**, (0.494) |
| Observations | 167 | 167 | 167 | 167 | 167 | 167 |
| Adjusted R-squared | 0.274 | 0.275 | 0.425 | 0.465 | 0.471 | 0.574 |
Notes: Variable definitions are provided in the main text. Expected inflation is from the Michigan Survey. Newey–West standard errors are reported in parentheses. * p<0.10, ** p<0.05, *** p<0.01.
Table D.2 reports three model comparison measures: the Akaike (AIC) and Bayesian (BIC) information criteria, which gauge in-sample fit with a penalty for model complexity, and the one-quarter-ahead root mean squared error (RMSE), a pseudo-out-of-sample accuracy measure computed from a recursive regression starting with 1981q3-1995q1 and expanding through 2025q1. For all three metrics, lower values indicate better performance. Across goods and headline inflation, and using either SPF or Michigan inflation expectations, all measures favor the stock-based NKPC.
Table D.2: In-sample Fit and One-quarter-ahead Forecast Performance -- Panel A: Goods PCE Inflation - SPF
| Canonical NKPC | CES-demand NKPC | Stock-based NKPC | |
|---|---|---|---|
| AIC | 865.98 | 866.02 | 811.72 |
| BIC | 878.46 | 878.49 | 827.31 |
| RMSE | 3.73 | 3.73 | 3.45 |
Notes: The CES-demand NKPC corresponds to the sales-based specification under CES demand. AIC denotes the Akaike Information Criterion and BIC the Bayesian Information Criterion. RMSE is the one-quarter-ahead root mean squared error, computed from a recursive regression beginning with the sample period 1981q3-1995q1. Lower AIC and BIC values indicate better in-sample fit, while lower RMSE indicates superior out-of-sample performance.
Table D.2: In-sample Fit and One-quarter-ahead Forecast Performance -- Panel B: Goods PCE Inflation - Michigan
| Canonical NKPC | CES-demand NKPC | Stock-based NKPC | |
|---|---|---|---|
| AIC | 835.72 | 835.63 | 797.92 |
| BIC | 848.20 | 848.10 | 813.51 |
| RMSE | 3.87 | 3.88 | 3.69 |
Notes: The CES-demand NKPC corresponds to the sales-based specification under CES demand. AIC denotes the Akaike Information Criterion and BIC the Bayesian Information Criterion. RMSE is the one-quarter-ahead root mean squared error, computed from a recursive regression beginning with the sample period 1981q3-1995q1. Lower AIC and BIC values indicate better in-sample fit, while lower RMSE indicates superior out-of-sample performance.
Table D.2: In-sample Fit and One-quarter-ahead Forecast Performance -- Panel C: Headline PCE Inflation - SPF
| Canonical NKPC | CES-demand NKPC | Stock-based NKPC | |
|---|---|---|---|
| AIC | 595.83 | 596.17 | 544.21 |
| BIC | 608.30 | 608.64 | 559.80 |
| RMSE | 1.72 | 1.72 | 1.58 |
Notes: The CES-demand NKPC corresponds to the sales-based specification under CES demand. AIC denotes the Akaike Information Criterion and BIC the Bayesian Information Criterion. RMSE is the one-quarter-ahead root mean squared error, computed from a recursive regression beginning with the sample period 1981q3-1995q1. Lower AIC and BIC values indicate better in-sample fit, while lower RMSE indicates superior out-of-sample performance.
Table D.2: In-sample Fit and One-quarter-ahead Forecast Performance -- Panel D: Headline PCE Inflation - Michigan
| Canonical NKPC | CES-demand NKPC | Stock-based NKPC | |
|---|---|---|---|
| AIC | 554.61 | 552.73 | 517.58 |
| BIC | 567.08 | 565.20 | 533.17 |
| RMSE | 1.77 | 1.74 | 1.61 |
Notes: The CES-demand NKPC corresponds to the sales-based specification under CES demand. AIC denotes the Akaike Information Criterion and BIC the Bayesian Information Criterion. RMSE is the one-quarter-ahead root mean squared error, computed from a recursive regression beginning with the sample period 1981q3-1995q1. Lower AIC and BIC values indicate better in-sample fit, while lower RMSE indicates superior out-of-sample performance.
For a formal pseudo-out-of-sample predictive-accuracy comparison between the stock-based and CES-demand NKPCs, we report the Diebold-Mariano (DM) statistics. Again, we re-estimate each specification recursively, starting with 1981q3-1995q and expanding through 2025q1, and generate one-quarter-ahead forecasts. DM statistics are computed from squared forecast errors. Table D.3 reports results for our four specifications, where positive values favor the stock-based NKPC over the CES-demand NKPC.
In the full sample, DM statistics generally favor the stock-based specification, although power is limited given the reduced pseudo-out-of-sample size. By subsample, the stock-based NKPC delivers significantly higher predictive accuracy for goods inflation in the latter period (2010q1-2025q1). For headline inflation, gains are significant in the earlier period (1995q1-2009q4) and less clear thereafter. Overall, the DM tests tend to favor the stock-based NKPC, with weaker power in shorter windows.
Table D.3: Diebold-Mariano Tests: Recursive Pseudo-out-of-sample (One-quarter-ahead)
| 1995q1-2025q1 | 1995q1-2009q4 | 2010q1-2025q1 | |
|---|---|---|---|
| Goods PCE - SPF | 1.234 | -0.263 | 1.373* |
| Goods PCE - Michigan | 1.138 | -1.049 | 1.733** |
| Headline PCE - SPF | 1.172 | 1.734** | 0.683 |
| Headline PCE - Michigan | 1.486* | 1.810** | 1.346* |
Notes: The table reports Diebold-Mariano statistics. Positive values indicate lower squared forecast errors for the stock-based NKPC relative to the CES-demand NKPC. One-quarter-ahead forecasts are generated from recursive estimations starting with 1981q3–1995q1 and expanding through 2025q1. Diebold-Mariano statistics are based on quadratic loss differentials. * p<0.10, ** p<0.05, *** p<0.01.
COVID-era pseudo-out-of-sample forecasts.
Given the limited power of DM tests, the COVID-era inflation surge provides a useful episode for pseudo-out-of-sample comparisons, since the models are estimated without exposure to this period. Figure D.1 plots one-quarter-ahead pseudo-out-of-sample forecasts against realized goods and headline inflation from 2019q4 onward, using both SPF and Michigan inflation expectations. Quarterly forecasts are transformed into four-quarter inflation for the plots. For comparability with the main text, we show the stock-based and canonical NKPCs (the CES-demand NKPC behaves similarly to the canonical NKPC). The patterns mirror our in-sample results: the stock-based NKPC tracks the data more closely, especially during the COVID-era run-up in goods inflation. In particular, for headline inflation with Michigan expectations, the stock-based NKPC also anticipates an earlier pickup and earlier peak--albeit a lower one than the canonical NKPC--followed by a faster reversal.
Figure D.1: Pseudo-out-of-sample Model Fit (One-quarter-ahead)
We estimate the canonical NKPC regression with and without the inventory-sales ratio across several commonly used proxies for real marginal cost. Table D.4 reports results using goods PCE inflation, and Table D.5 reports results using headline PCE inflation. In both tables, columns (1) and (2) use the inverse of the unemployment rate gap, constructed as described in the main text; columns (3) and (4) use the labor share gap; and columns (5) and (6) use the unit labor cost gap, with both gap measures constructed using the Hamilton filter. Finally, columns (7) and (8) use the CBO's output gap. Across all four proxies, including the inventory-sales ratio as an observable improves the fit of the NKPC: the adjusted R-squared value in columns (2), (4), (6), and
(8), are consistently higher than those in columns (1), (3), (5), and (7), respectively.
Table D.4: NKPC Estimates for Goods PCE Inflation (SPF Expectations)
| (1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | |
|---|---|---|---|---|---|---|---|---|
| Lagged inflation | 0.342**, (0.131) | 0.0223, (0.0998) | 0.336**, (0.130) | 0.0178, (0.104) | 0.345***, (0.127) | -0.00727, (0.0933) | 0.329**, (0.132) | 0.0287, (0.102) |
| Exp. inflation (SPF) | 0.369, (0.284) | 1.237***, (0.423) | 0.451, (0.334) | 1.432***, (0.440) | 0.395, (0.333) | 1.399***, (0.440) | 0.323, (0.278) | 1.198***, (0.417) |
| Unemp. rate gap (inverse) | 0.0303, (0.145) | 0.269*, (0.140) | ||||||
| Labor share gap | -0.127, (0.115) | 0.139, (0.106) | ||||||
| Unit labor cost gap | -0.00943, (0.135) | 0.183*, (0.0963) | ||||||
| Output gap (CBO) | 0.223, (0.155) | 0.245*, (0.127) | ||||||
| IS ratio gap | -0.452***, (0.0861) | -0.473***, (0.0856) | -0.469***, (0.0848) | -0.432***, (0.0836) | ||||
| Constant | -0.282, (0.868) | -2.211**, (1.105) | -0.526, (0.978) | -2.913**, (1.141) | -0.379, (0.995) | -2.794**, (1.150) | -0.0325, (0.869) | -2.167**, (1.088) |
| Observations | 167 | 167 | 164 | 164 | 164 | 164 | 167 | 167 |
| Adjusted R-squared | 0.130 | 0.390 | 0.136 | 0.395 | 0.130 | 0.399 | 0.144 | 0.391 |
Notes: Newey-West standard errors are reported in parentheses. * denotes 10% significance, ** denotes 5% significance, *** denotes 1% significance.
Table D.5: NKPC Estimates for Headline PCE Inflation (SPF Expectations)
| (1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | |
|---|---|---|---|---|---|---|---|---|
| Lagged inflation | 0.455***, (0.135) | 0.123, (0.117) | 0.458***, (0.131) | 0.194, (0.122) | 0.459***, (0.125) | 0.0841, (0.105) | 0.429***, (0.139) | 0.140, (0.122) |
| Exp. inflation (SPF) | 0.321**, (0.160) | 0.842***, (0.203) | 0.369*, (0.188) | 0.863***, (0.239) | 0.308*, (0.171) | 0.891***, (0.208) | 0.316**, (0.159) | 0.804***, (0.199) |
| Unemp. rate gap (inverse) | 0.0744, (0.0630) | 0.226***, (0.0701) | ||||||
| Labor share gap | -0.0830, (0.0550) | 0.0123, (0.0591) | ||||||
| Unit labor cost gap | 0.0431, (0.0537) | 0.178***, (0.0498) | ||||||
| Output gap (CBO) | 0.149**, (0.0718) | 0.187***, (0.0652) | ||||||
| IS ratio gap | -0.210***, (0.0431) | -0.193***, (0.0477) | -0.216***, (0.0433) | -0.192***, (0.0413) | ||||
| Constant | 0.461, (0.313) | -0.106, (0.431) | 0.256, (0.373) | -0.494, (0.495) | 0.424, (0.364) | -0.308, (0.465) | 0.566*, (0.325) | -0.103, (0.424) |
| Observations | 167 | 167 | 164 | 164 | 164 | 164 | 167 | 167 |
| Adjusted R-squared | 0.315 | 0.533 | 0.320 | 0.504 | 0.312 | 0.539 | 0.333 | 0.530 |
Notes: Newey-West standard errors are reported in parentheses. * denotes 10% significance, ** denotes 5% significance, *** denotes 1% significance.
Table D.6 reports NKPC regression results when using core goods inflation as the dependent variable rather than overall goods inflation. To ensure consistency, we remove food and energy components from both wholesale and retail inventories and sales. Specifically, we exclude groceries, alcoholic beverages, and petroleum from wholesale measures, and drop food and beverage stores from retail inventories and sales as well as gasoline stations from retail sales. Constructing these core-consistent measures requires more detailed category-level data, which are only available beginning in 1992, reducing the sample to 122 observations. Within this sample, our results continue to hold for core goods PCE: the coefficient on the inventory-sales ratio remains negative and statistically significant, and the adjusted R-squared improves markedly for the stock-based NKPC relative to the canonical NKPC.
Table D.6: NKPC Estimates for Core Goods PCE Inflation
| (1) | (2) | (3) | (4) | (5) | (6) | |
|---|---|---|---|---|---|---|
| Lagged inflation | 0.606***, (0.143) | 0.592***, (0.148) | 0.330**, (0.127) | 0.398***, (0.129) | 0.373***, (0.128) | 0.256**, (0.121) |
| Exp. inflation (SPF) | -0.366, (0.320) | -0.129, (0.326) | 0.836, (0.522) | |||
| Exp. inflation (Michigan) | 0.558***, (0.175) | 0.596***, (0.185) | 0.442***, (0.123) | |||
| Marginal cost | 0.022, (0.079) | -0.053, (0.078) | ||||
| Exp. marginal cost | -0.031, (0.039) | -0.073*, (0.044) | ||||
| Exp. marginal cost growth | -0.112, (0.092) | -0.077, (0.089) | ||||
| IS ratio | -0.182**, (0.069) | -0.145**, (0.065) | ||||
| Constant | 0.842, (0.882) | 0.250, (0.850) | -2.013, (1.250) | -2.252***, (0.594) | -2.400***, (0.649) | -1.758***, (0.498) |
| Observations | 122 | 122 | 122 | 122 | 122 | 122 |
| Adjusted R-squared | 0.338 | 0.339 | 0.493 | 0.415 | 0.427 | 0.533 |
Notes: All the variables are described in the main text. For example, the marginal cost is proxied by the inverse of the unemployment gap and the expected marginal cost is defined as the sum of the expected real interest rate and the expected inverse of the unemployment gap. The expected marginal cost growth is the difference between the expected marginal cost and the marginal cost. Newey–West standard errors are reported in parentheses. * p<0.10, ** p<0.05, *** p<0.01.
In this section we consider one specific form of nonlinearity in the inventory channel: an interaction between the inventory–sales ratio gap and an indicator for whether the gap is negative or positive. This specification is motivated by the idea that inventory shortages may affect markups differently from inventory surpluses, and thus captures a potentially important asymmetry of interest to the literature. We emphasize, however, that this is only one way of modeling nonlinearities in the NKPC. A fuller exploration of alternative nonlinear specifications is an important question for future research and lies beyond the scope of this paper. Our main contribution is to document robust evidence of a strong linear effect of inventories on inflation dynamics, a relationship that has not been firmly established in the prior literature.
Columns (2) and (5) of Tables D.7 and D.8 report the stock-based NKPC estimates, fitted to goods and headline PCE, respectively, when the regression is augmented with a nonlinear inventory term that interacts the inventory–sales ratio gap with an indicator for whether the gap is negative. The coefficient on the linear inventory-sales ratio gap remains negative in both regressions, consistent with the baseline linear results. The interaction term also enters with a negative sign, suggesting possible nonlinear effects when inventories fall short of trend.
The evidence for such nonlinearity, however, is weak. The nonlinear term is estimated imprecisely, with absolute $$t$$-statistics close to or below one, and its inclusion reduces the precision of the linear inventory coefficient, which loses conventional statistical significance. Nevertheless, the linear inventory-sales ratio gap continues to emerge as the primary inventory-related driver of inflation dynamics in these regressions.
To assess whether the weaker significance reflects collinearity with the nonlinear interaction rather than the disappearance of the linear effect, we re-estimate the specification using an interaction term for positive inventory-sales ratio gaps. In columns (3) and (6) of the tables, the linear inventory coefficient remains negative and statistically significant, while the nonlinear interaction is small and insignificant. Unlike the negative-gap specification, the positive-gap interaction does not appear strongly collinear with the linear inventory term in our data, which is why the linear coefficient remains precisely estimated. These results suggest that the weaker significance in the negative-gap specification reflects multicollinearity, not the absence of the linear inventory channel. Overall, the evidence points to a robust linear effect of inventories on inflation, with only limited and imprecise signs of potential nonlinearities.
Table D.7: Inventory Nonlinearity in Stock-based NKPC (Goods PCE)
| (1) | (2) | (3) | (4) | (5) | (6) | |
|---|---|---|---|---|---|---|
| Lagged inflation | 0.0473, (0.0998) | 0.0412, (0.0909) | 0.0412, (0.0909) | -0.0171, (0.104) | -0.0179, (0.101) | -0.0179, (0.101) |
| Exp. inflation (SPF) | 1.172**, (0.459) | 1.119**, (0.453) | 1.119**, (0.453) | |||
| Exp. inflation (Michigan) | 1.250***, (0.360) | 1.212***, (0.393) | 1.212***, (0.393) | |||
| Exp. marginal cost growth | 0.0475, (0.166) | 0.106, (0.140) | 0.106, (0.140) | 0.226, (0.149) | 0.255*, (0.137) | 0.255*, (0.137) |
| IS ratio gap | -0.430***, (0.0917) | -0.306, (0.185) | -0.511***, (0.0768) | -0.332***, (0.0796) | -0.268, (0.178) | -0.379***, (0.0769) |
| IS ratio gap < 0 | -0.205, (0.193) | -0.111, (0.207) | ||||
| IS ratio gap > 0 | 0.205, (0.193) | 0.111, (0.207) | ||||
| Constant | -2.299**, (1.158) | -2.552**, (1.161) | -2.552**, (1.161) | -4.048***, (1.341) | -4.113***, (1.267) | -4.113***, (1.267) |
| Observations | 167 | 167 | 167 | 167 | 167 | 167 |
| Adjusted R-squared | 0.375 | 0.378 | 0.378 | 0.425 | 0.423 | 0.423 |
Notes: Newey-West standard errors are reported in parentheses. * denotes 10% significance, ** denotes 5% significance, *** denotes 1% significance.
Table D.8: Inventory Nonlinearity in the Stock-based NKPC for Headline PCE Inflation
| (1) | (2) | (3) | (4) | (5) | (6) | |
|---|---|---|---|---|---|---|
| Lagged inflation | 0.197, (0.120) | 0.192*, (0.112) | 0.192*, (0.112) | 0.0404, (0.102) | 0.0404, (0.102) | 0.0404, (0.102) |
| Exp. inflation (SPF) | 0.665***, (0.225) | 0.651***, (0.226) | 0.651***, (0.226) | |||
| Exp. inflation (Michigan) | 0.810***, (0.149) | 0.808***, (0.164) | 0.808***, (0.164) | |||
| Exp. marginal cost growth | 0.0888, (0.0938) | 0.106, (0.0819) | 0.106, (0.0819) | 0.212***, (0.0799) | 0.213***, (0.0672) | 0.213***, (0.0672) |
| IS ratio gap | -0.189***, (0.0475) | -0.153*, (0.0891) | -0.213***, (0.0403) | -0.137***, (0.0393) | -0.134, (0.0821) | -0.139***, (0.0382) |
| IS ratio gap < 0 | -0.0604, (0.0897) | -0.00451, (0.0931) | ||||
| IS ratio gap > 0 | 0.0604, (0.0897) | 0.00451, (0.0931) | ||||
| Constant | -0.0274, (0.530) | -0.0979, (0.518) | -0.0979, (0.518) | -1.105**, (0.494) | -1.107**, (0.485) | -1.107**, (0.485) |
| Observations | 167 | 167 | 167 | 167 | 167 | 167 |
| Adjusted R-squared | 0.500 | 0.499 | 0.499 | 0.574 | 0.571 | 0.571 |
Notes: Newey-West standard errors are reported in parentheses. * denotes 10% significance, ** denotes 5% significance, *** denotes 1% significance.