## 4.1 Extensive Margin Adjustment Rate is “Sticky” ## Source details **Canonical URL:** [4.1 Extensive Margin Adjustment Rate is “Sticky”](https://www.imf.org/-/media/files/publications/wp/2017/wp17146.pdf) ## Other formats - [Markdown version](/-/media/files/publications/wp/2017/wp17146.pdf.md) - [Structured JSON version](/-/media/files/publications/wp/2017/wp17146.pdf.json) --- ### Stickiness of firm-level product turnover - Objective: test whether a firm’s extensive margin adjustment is “sticky” and assess cross-sectional variation in flexibility across firms. - Methods: - Firms grouped within-industry by creation rates, destruction rates and turnover rates. - Transition matrix (Table 1) reports probabilities of staying in the same group versus switching. - Key findings: - “Stayers” dominate “switchers”, indicating persistence in firms’ product turnover rates. - About 70% of firms with low product turnover rates stay in the same low bin. - Over 90% of firms with high product turnover rates exhibit higher product turnover rates over time than the rest. - Conclusion: extensive margin adjustment can be treated as a firm-specific attribute and used as a propagation/amplification channel for heterogeneous responses to aggregate shocks. ### Implication for propagation mechanisms - Stable product turnover supports using firm-specific turnover rates to model heterogeneous responses to aggregate shocks. --- ### 4.2 Comparison between High vs Low Product Turnover Groups ### Objective and data - Examine firm and consumer characteristics correlated with product turnover to address endogeneity concerns. - Data: Bloomberg balance sheet variables; Nielsen Homescan household demographics and inferred retailer sales. ### Tests and variables - Two-group comparisons (high vs low turnover) on: - firm size, book-to-market ratio, beta, leverage, cash flow, stock turnover, spread; - purchaser characteristics: household income (dummy “income lower than median”), household size, presence of child, education dummies; - retailer sales per firm inferred from household shopping data. ### Findings (selected numeric highlights) - No systematic differences across groups: - size (in m$): Bottom 0.02 Top 0.02 pvalue 0.61 - book-to-market ratio: Bottom 3.78 Top 5.43 pvalue 0.09 - turnover (in m$): Bottom 37.01 Top 123.61 pvalue 0.41 - purchaser income 0. --- ### 5.4 Firm’s decision on optimal product scope: Calvo-type frictions ### Time-dependent reset - A firm in group k can adjust product scope with probability 1−λ_k each period. - Reset problem (optimal ˜N^{f}_{k t}) maximizes expected discounted profits with geometric probability λ_k of being frozen. ### Optimal scope with frictions (Proposition 3) - ˜N^{ (σ_u−1)·σ_u/(σ_u−σ_f) }_{f k t} = [ ∑_{j=0}^∞ λ_k^j E_t [ (Λ_{t+j}/MC_{t+j}) C_{t+j} P_{t+j}^{σ_k} P_{k t+j}^{σ_f−σ_k} MC_{t+j}^{−σ_f} ] ] / [ ∑_{j=0}^∞ λ_k^j E_t [ (Λ_{t+j}/MC_{t+j}) F ] ]. - Interpretation: optimal scope increases with the discounted present value of revenue (PVR) and decreases with the discounted present value of cost (PVC) under locking-in. --- ### 5.5 Model summary (variables and equilibrium conditions) ### Key variables and equilibrium conditions - Competitive equilibrium variables (15): { P_{k t}/P_t, ˜P_{k t}/P_t, ˜N_{k t}, S_{k t}, PVR_{k t}, PVC_{k t}, π_t, C_t, S_t, L_t, i_t, W_t/P_t, Λ_{t+1}/Λ_t } plus exogenous { Z_t, ν_t }. - Important group-specific conditions (examples): - Optimal reset variety: - ˜N_{k t}^{(σ_u−1)·σ_u/(σ_u−σ_f)} = σ^{−σ_f}_f f (σ_f−1)^{−σ_f} (σ_u−1)^{−σ_f} PVR_{k t}/PVC_{k t}. (Equation (4)) - Recursive definitions for PVR_{k t} and PVC_{k t} (Equations (5) and (6)). - Optimal reset price: - ˜P_{k t}/P_t = ˜N_{k t}^{1/(1−σ_u)} · σ_f/(σ_f−1) · W_t/(Z_t P_t). (Equation (7)) - Aggregate conditions include: - aggregate price index (Equation (10)); - price dispersion S_t (Equation (11)); - labor demand C_t = Z_t L_t S_t (Equation (12)); - labor supply χ L_t^{1/φ} = W_t C_t / P_t (Equation (13)); - Euler equation 1 = β E_t [ (1+i_t) Λ_{t+1}/Λ_t ] (Equation (14)); - subjective discount Λ_{t+1}/Λ_t = C_t / (C_{t+1} π_{t+1}) (Equation (15)); - Taylor rule for i_t (Equation (16)). - Exogenous processes: - ln Z_t = ρ_z ln Z_{t−1} + e^z_t (Equation (17)); - ln ν_t = ρ_ν ln ν_{t−1} + e^ν_t (Equation (18)). --- ### 5.6 Return rate and asset pricing of firms ### Valuation - Value of a firm with permanent product scope N^{f}_{k t} (real, deflated): - VS_t(N^{f}_{k t}) = N^{1−σ_f}_{f k t} / (1−σ_u) · RS_{k t} − N^{f}_{k t} · CS_{k t}. (Equation (19)) - RS_{k t} and CS_{k t} defined recursively with λ_k and stochastic discount factors. ### Return rate construction - Firms differ by realizations of their path {N}; return rate R_t(N^{f}_{k t}, N^{f}_{k t−1}) derived from dividends and changes in V and VS, accounting for π_t. - Recursive valuation for V_t(˜N^{f}_{k t}) incorporates Div_t, continuation values, and VS differences reflecting future reset opportunities. --- ### 5.7 Parametrization ### Parameter choices (selected) - σ_k group elasticity of substitution 2 - σ_f firm elasticity of substitution 3.9 Hottman et al. (2016) - σ_u product elasticity of substitution 6.9 Hottman et al. (2016) - β time preference 0.99 - φ Frisch elasticity 0.5 - χ1 F fixed cost of product scope (per variety) 1 - ρ_z productivity AR(1) coef. 0.9 - σ_z productivity std 0.01 - ρ_ν monetary policy shock AR(1) coef. 0.9 - σ_ν monetary policy std 0.02 - φ_π Taylor rule 1.5 - φ_y Taylor rule 0.5 - λ_1 Probability of not able to change product scope (1st tertile) 0.275 Nielsen - λ_2 Probability ... (2nd tertile) 0.785 Nielsen - λ_3 Probability ... (3rd tertile) 0.921 Nielsen --- ### 5.8 Simulation ### Design and measurement - Simulate 100 firms of 3 groups for 1000 periods with first 800 periods burn-in. - Measure return rate: average return rates of the 200 periods. - Measure asset volatility: standard deviations of the return rates across the periods. ### Simulation procedure (summary) - 1) store RS, CS, VS(˜N), V(˜N), ˜N for each group k from Dynare solution (second-order perturbation); - 2) generate path of {N} from {˜N} following binomial distribution of λ_k; each sequence {N} represents a firm; - 3) calculate RS, CS, Div, V for each firm at each period and then calculate return rate. ### Key simulation findings - Positive correlation between rigidity in product-scope adjustment and return rates: “as firms become more rigid, the return rate increases.” - Covariance between inverse consumption growth and return rate aligns with empirical patterns: covariance decreases with rigidity in product-scope adjustment. - Simulation results qualitatively match empirical relationships: higher λ_k (less frequent resetting) → higher expected returns and higher volatility. --- ### Conclusion and policy-relevant implications ### Main empirical and theoretical findings - Flexibility in firms’ extensive margin adjustment (product creation/destruction) improves firms’ financial risk diversification. - Empirical regularities: - large cross-firm variation in product turnover rates; - product turnover rates are “sticky” over time; - firms with higher flexibility (higher creation/turnover rates) exhibit lower excess stock returns and lower asset volatility. - The multi-product firm model with time-dependent reset frictions and cannibalization rationalizes empirical findings by linking product-scope flexibility to asset pricing and risk diversification. ### Quantitative highlights - One standard deviation increase (about 20%) in firm’s product turnover rate leads to a decline of a 2 percentage point in excess asset returns (empirical summary). - Selected summary statistics (Table 4): - destruction rate (%) N 1413 mean 17.1 sd 15.3 - creation rate (%) N 1413 mean 13.9 sd 16.3 - turnover rate (%) N 1413 mean 26.5 sd 18.2 - No. Firms 203 ### Suggested avenues for future research and policy considerations - Link firms’ financial risk diversification to labor adjustment behaviors or investment decisions (particularly R&D investment). - Investigate how policies could enhance firms’ production flexibility and encourage R&D investment for product creation. *wp17146 - 4.1 Extensive Margin Adjustment Rate is “Sticky”* ### 4.1 Extensive Margin Adjustment Rate is “Sticky” ............................................................13 ### wp17146 - 4.1 Extensive Margin Adjustment Rate is “Sticky” ............................................................13 ### Section inventory and headings - 4.1 Extensive Margin Adjustment Rate is “Sticky” ............................................................13 - 4.2 Comparison between High vs Low Product Turnover Groups .....................................13 - 4.3 Portfolio Analysis ..........................................................................................................15 - 4.4 Extensive margin Adjustment and Risk ........................................................................16 - 4.5 Robustness Check .........................................................................................................18 - 5 A Dynamic Model of Extensive Margin Adjustment.......................................................20 - 5.1 Household ......................................................................................................................21 - 5.2 Price of the Product .......................................................................................................22 - 5.3 Firm’s Decision on Optimal Product Scope: Without Friction .....................................22 - 5.4 Firm’s Decision on Optimal Product Scope: A Calvo-type Model................................23 - 5.5 Model Summary ............................................................................................................24 - 5.6 Return Rate ....................................................................................................................25 - 5.7 Parametrization ..............................................................................................................27 - 5.8 Simulation .....................................................................................................................28 - 6 Conclusion ..........................................................................................................................29 ### Purpose of this content unit (as implied by headings) - Documents empirical findings and checks related to the extensive margin adjustment rate (Section 4.1–4.5). - Develops and characterizes a dynamic model of extensive margin adjustment (Section 5 and subsections). - Provides model calibration details and simulation results (Sections 5.7–5.8). - Concludes the chapter with summarizing remarks (Section 6). *Source: wp17146 - 4.1 Extensive Margin Adjustment Rate is “Sticky” ............................................................13* ### References ............................................................................................................ ### wp17146 - References ### Introduction and research question - Firms in U.S. manufacturing are predominantly multi-product firms; multi-product firms regularly update product portfolios through introduction of new products and discontinuation of existing products ("product turnover", "product scope adjustment", "extensive margin adjustment"). - Central question: How does flexibility in firms’ product scope adjustment affect firms’ performance and productivity, particularly via financial risk diversification? - Main finding summary: - Firms with higher flexibility in adjusting product scopes have lower financial risks, reflected by lower excess asset returns and lower asset volatility. - One standard deviation increase (about 20 %) in firm’s product turnover rate leads to a decline of a 2 percentage point in excess asset returns. ### Risk-absorption channel: intuition from CCAPM - Framework: consumption-based CAPM (CCAPM) with representative household and log-utility; asset excess return determined by covariance between asset return and marginal rate of substitution. - Mechanism: - Product scope N is a parameter in firm profit maximization. A negative shock shifts the profit curve, reducing the optimal product scope from N* to ̃N*. - Firms differ by rigidity parameter λ (probability of not being able to change product scope). - λ=0: firm can freely re-optimize product scope and maintain profit; covariance term ≈ 0 ⇒ excess asset return ≈ 0. - λ=1: firm cannot adjust product scope, suffers profit loss; covariance with inverse consumption growth increases ⇒ higher excess asset return. - Interpretation: flexibility in product turnover attenuates firm-level exposure to economy-wide shocks by allowing profit re-optimization, thereby reducing the price of risk. ### Theoretical modeling components - Model incorporates a time-dependent (Calvo-type) firm-specific product turnover rate: parameter governs timing of firms’ product scope changes rather than timing of price changes. - Cannibalization effect modeled with a CES aggregator at three layers (within-firm, across-firms, across-sectors elasticities of substitution) as in Hottman, Redding and Weinstein (forthcoming). - Cannibalization effect imposes an upper bound on optimal product scope; beyond a certain number of goods, additional products lower firm profit by eroding market share. - The model generates a negative correlation between firm flexibility in product scope adjustment and both excess returns and volatility of asset returns, consistent with empirical findings. ### Data sources and construction of firm-level turnover measures - Data pooled from AC Nielsen Homescan Data, Bloomberg, and CRSP. - AC Nielsen Homescan: - Tracks purchases from retail grocery stores at daily frequency; UPC identifies each product in households’ consumption baskets. - Samples about 40,000 to 60,000 households over 20,000 zipcodes in the U.S. from 2004 to 2014. - Product coverage: about 700,000 UPCs every quarter and 3.2 million UPCs in total. - Expenditure in Homescan constitutes about one third of the total consumer expenditure. - Manufacturer identification via GS1 Data Hub. - Firm-level aggregation: - Products grouped by manufacturer to construct firm-level creation rate, destruction rate, and turnover rate following Broda and Weinstein (2010). - Definitions: - creation rate_t = (Value of New UPCs_t / Total Value_t) - destruction rate_t = (Value of Disappearing UPCs_t / Total Value_{t−1}) - turnover rate_t = creation rate_t + destruction rate_t - Annual frequency used to avoid seasonal patterns. - Financial data: - Stock return information from CRSP; firm-specific financial variables from Bloomberg; other controls from CRSP (leverage ratio, market value, book-to-market ratio, market beta, cash flow, R&D expenditure). - Annual return rate for asset i computed from monthly excess returns r_1 ... r_12 as: return rate_it = (1+r_1)(1+r_2)...(1+r_12)−1 - Matching and sample: - Matching performed via manufacturer identifiers across datasets. - Final matched sample: 203 firms. ### Empirical facts and stylized findings - Cross-sectional variation: product turnover rates vary across firms. - Persistence: a firm’s product turnover rate is sticky over time — firms with high (low) turnover tend to continue to have high (low) turnover. - Robustness: - No statistical difference in firms’ product turnover rates across household income groups. - Results robust to controlling for household-specific turnover rates, standard financial indicators, industry characteristics, and R&D expenditure. - Quantitative result: - One standard deviation increase (about 20 %) in firm’s product turnover rate → decline of a 2 percentage point in excess asset returns. - Time-series patterns: - Creation rates exhibit more cyclical behavior compared to destruction rates. ### Relation to literature - Connects to finance literature on risk diversification and portfolio choice (e.g., Markowitz (1952), Sharpe (1964)) by showing that firms’ extensive margin adjustment in product portfolios provides a channel for risk diversification. - Contributes to multi-product firms literature by focusing on dynamics of number of goods per firm over time, in contrast to work emphasizing price-setting given product counts. - Links to endogenous growth and technological diversification literature (Romer (1990); Grossman and Helpman (1991); Koren and Tenreyro (2013)) where input/product variety reduces firm- and aggregate-level volatility. - Relates to micro-data asset-pricing studies on nominal rigidity and costs of rigidity (Gorodnichenko and Weber (2016); Weber (2014); Herskovic (2015)). *Source: wp17146 - References (extracted content).* ### 4.1 Extensive Margin Adjustment Rate is ”Sticky” ### 4.1 Extensive Margin Adjustment Rate is ”Sticky” ### Stickiness of firm-level product turnover - Aim: test whether a firm’s extensive margin adjustment is ’sticky’—i.e., a firm’s ability to adjust its product scope is stable over time—and whether there is cross-sectional variation in flexibility across firms. - Method: firms are divided into groups based on creation rates, destruction rates and turnover rates; grouping is performed within-industry to control for sectoral factors. Table 1 reports the transition matrix showing probabilities of staying in the same group (diagonal terms) versus switching (off-diagonal terms). - Key findings: - The ’stayers’ are more dominant than ’switchers’, indicating persistence in firms’ product turnover rates. - About 70% of firms with low product turnover rates stay in the same low bin. - Over 90% of firms with high product turnover rates exhibit higher product turnover rates over time than the rest. - Conclusion: there is a stable distribution of firms when ranked by flexibility of extensive margin adjustments; extensive margin adjustment can be considered a firm-specific attribute and a potential amplification channel for heterogeneous responses to aggregate shocks. ### Implication for propagation mechanisms - Stability in product turnover at the firm-level supports using firm-specific product turnover rates as a source of propagation and amplification of aggregate shocks through heterogeneous firm responses. ### Source *wp17146 - 4.1 Extensive Margin Adjustment Rate is ”Sticky”* --- ### 4.2 Comparison between High vs Low Product Turnover Groups ### Objective - Examine firm and consumer characteristics potentially correlated with product turnover rates to address endogeneity concerns (e.g., healthier balance sheets, clientele wealth, retailer-manufacturer matching). ### Data and controls - Firm-level balance sheet information from Bloomberg. - Nielsen Homescan: household demographics (income, composition, race, ages, education, occupation, facilities), and retailer information inferred from panelist shopping trips to compute effective sales per retailer. ### Tests and construction - Two-group comparison: firms in high vs low product turnover bins; comparisons based on firm size, book-to-market ratio, beta, leverage, cash flow, stock turnover, spread (Table 2, first panel). - Purchaser characteristics: household income, size, presence of child, household head education; original multiple-choice variables summarized into dummies (e.g., “income lower than median”, “higher than high school”) (Table 2, second panel). - Retailer sales compared using household shopping data. ### Findings - Firm financials: high product turnover group is not significantly and systematically different from low turnover group across key financial indicators. - Purchaser characteristics: p-values indicate the two groups are not distinguishable by demand-side features; representative income levels for the two categories are very close; household size differs by about 0.01 standard deviation. - Retailer sales: no significant difference between top and bottom groups based on firms’ product turnover rates. - Overall conclusion: high product turnover firms are not intrinsically or systematically different from low turnover firms in financial health, consumer types, or retailer relationships. ### Source *wp17146 - 4.1 Extensive Margin Adjustment Rate is ”Sticky”* --- ### 4.3 Portfolio Analysis ### Method - Rank firms separately by creation rates, destruction rates, and product turnover rates; form three portfolios from least flexible to most flexible. - Construct portfolio market price as weighted average of underlying stocks’ prices (weights = individual stock market value). - Table 5 reports average return rates from CAPM, Fama-French 3-factor and 5-factor models for bottom (bottom third turnover rates) and top (top third turnover rates) bins; last two columns show differences and p-values. ### Findings - Creation rates: - Difference in excess return rates between high-creation and low-creation firms is statistically significant. - Firms with higher creation rates exhibit low asset returns and lower excess returns compared to firms with lower creation rates. - Finding robust to CAPM, Fama-French 3-factor and 5-factor models. - Destruction rates: - Differences across firm groups disappear when categorizing by destruction rates—no statistically significant differences in return rates between destruction-rate bins. - Robust to alternative portfolio analyses. - Consistent with limited dynamics of destruction rates over time and prior literature (product turnover mainly driven by creation rates, not destruction rates, per Broda and Weinstein (2010)). ### Source *wp17146 - 4.1 Extensive Margin Adjustment Rate is ”Sticky”* --- ### 4.4 Extensive Margin Adjustment and Risk ### Panel regressions: specification and variables - Baseline specification (1): return rate_it = β1 creation rate_it + β2 destruction rate_it + X_it + e_it - Main dependent variables: market-based measures of firm financial riskiness — excess stock returns and asset volatility. - Main independent variables: creation rates, destruction rates, turnover rates. - Controls: standard Bloomberg financial indicators — leverage, book-to-market ratio, price-to-cost margin, cash flow, market beta, spread, gross margin and industry portfolio rate of return and book-to-market ratio. Year fixed effects included in benchmark. ### Excess returns (Table 6) — main results - Higher creation rate is significantly negatively correlated with excess asset returns. - Destruction rate: relationship not statistically significant. - Robustness: - Column (1): benchmark with year fixed effects. - Additional controls (industry-specific characteristics Column (2), R&D expenditure Column (3), total number of UPCs Column (4)) maintain negative and significant coefficient on creation rate. - Column (4): total number of UPCs also negative and significant, implying product-scope magnitude matters for risk diversification. - Column (5): using turnover rate as main independent variable yields that firms with high turnover rates exhibit lower excess asset returns. - Quantitative statement: one standard deviation decrease in product creation rate (about 16%) leads to about a 1.6 percentage point increase in excess return. ### Asset return volatility (Table 7) — main results - Firms with higher creation and turnover rates show significantly lower volatility of asset returns, controlling for firm financial variables. - Quantitative statement: a decline of creation rate by one standard deviation leads to 0.16 percentage point decline in volatility of firm’s asset returns. - Destruction rate: not meaningfully related to asset volatility. - Robustness checks analogous to excess return regressions (industry portfolio, R&D, total number of products). Column (5) shows higher product turnover rate is related to lower asset volatility with magnitude similar to introduction rate. ### CCAPM-style covariance test (specification (2)) - Specification (2): cov(R_i, ΔC/C) = β1 creation rate_i + β2 destruction rate_i + X_i + e_i - Covariance computed using monthly firm returns and monthly private consumption in the U.S. from FRED. - Table 8 results: - One standard deviation increase in product introduction rate leads to an increase of asset volatility by about one standard deviation increase in covariance. - Coefficient on creation rate is consistently positive and significant controlling for financial variables and R&D (Column (2)). - Column (3): total number of products matters in addition to creation rate. - Columns (4) and (5): higher product turnover rate (total number of UPCs over time) is related to higher covariance; magnitude similar to introduction rate. ### Interpretation - Firms with greater flexibility in extensive margin adjustments (higher creation/turnover rates) exhibit lower excess returns and lower asset volatility, and higher covariance with inverse consumption growth; destruction rates generally do not predict these risk measures. - Results align with a role for product-scope flexibility in risk diversification. ### Source *wp17146 - 4.1 Extensive Margin Adjustment Rate is ”Sticky”* --- ### 4.5 Robustness Check ### Motivation - Address potential confounding roles of retailers and households in firm-level product scope adjustments (retailer pricing/pass-through, household non-homothetic preferences and income-dependent switching). ### Additional controls constructed - Household-specific product creation and destruction rates: constructed using household identifiers and item-level purchase histories linked to manufacturers; households’ time-varying creation/destruction rates derived from manufacturers’ rates of purchased UPCs. - Retailer-specific product creation and destruction rates: constructed by linking items carried by a retailer to manufacturers and their manufacturer-specific creation/destruction rates. ### Tests and results - Table 10: - Column (1): controls for retailer-specific extensive margin measures. - Column (2): adds household-specific extensive margin measures. - Column (3): contains both sets. - Negative coefficients on product turnover measures persist, confirming robustness of the risk-diversification channel after controlling for retailer- and household-specific factors. - Table 11: - Similar robustness checks for CCAPM-related covariance dependent variable; results robust. ### Co-integration–type test for selective treatment by retailers/households - Construct annual turnover rates for each retailer-firm pair and zip code-firm pair t: Turnover Rate_itj = α + Turnover Rate_it,-j + e_itj (3) - Leave-one-out mean of retailer-(household-) firm specific turnover rates computed for each firm-year and correlated with the given retailer (household). - Results (Table 9): - Coefficients for the leave-one-out mean are almost one, strongly supporting no significant differences across retailers (or households) in treating products from the same firm. - Conclusion: main driver of variation in product turnover rates is across-firm differences, not across retailers or households. ### Source *wp17146 - 4.1 Extensive Margin Adjustment Rate is ”Sticky”* --- ### 5 A Dynamic Model of Extensive Margin Adjustment ### Model ingredients and intuition - Two main ingredients: 1. Time-dependent (Calvo-type) product turnover rate: firms can reset optimal product scopes only with a fixed probability, analogous to time-dependent price-setting. 2. Cannibalization effects (Feenstra and Ma (2008); Eckel and Neary (2010); Hottman, Redding and Weinstein (forthcoming)): introduction of new products can reduce firm profits by eroding market shares of existing products; measured as the partial elasticity of existing product sales with respect to introduction of new products, which helps make optimal product scope finite. - Model implication: - Flexibility in product scope adjustment allows firms to diversify and absorb shocks. - Firms with higher rigidity to re-optimize product scopes (high time-dependent parameter) will have higher excess asset returns and higher asset volatility, compensating for inability to reoptimize frequently. - Next steps in the text: model specification and simulation results to evaluate the risk-diversification channel. ### Source *wp17146 - 4.1 Extensive Margin Adjustment Rate is ”Sticky”* ### 5.1 Household ### 5.1 Household ### Household preferences and consumption aggregator - Households have GHH preferences over consumption and labor; lifetime utility: - max ∑_{t=0}^∞ β^t [ ln C_t − χ L_t^{1+1/φ}/(1+1/φ) ]. - Budget constraint and income sources: - P_t C_t + B_{t+1} = W_t L_t + (1+i_t) B_t + ∫_0^1 ∫_0^1 D^{f}_{k t} df dk. - Total consumption C_t is a three-layer CES aggregator: - first layer: groups k ∈ [0,1]; - second layer: firms f ∈ [0,1]; - third layer: intra-firm products u ∈ [0, N^{f}_{k t}]. - Elasticities: - σ_k, σ_f, σ_u represent elasticities across groups, firms, and products respectively. - Assumption: σ_f ≤ σ_u (inter-firm elasticity smaller than intra-firm elasticity). ### Demand structure (nested CES) - Layered consumption indices: - C_t = [ ∫_0^1 C_{k t}^{σ_k−1/σ_k} dk ]^{σ_k/(σ_k−1)}. - C_{k t} = [ ∫_0^1 C_{f k t}^{σ_f−1/σ_f} df ]^{σ_f/(σ_f−1)}. - C_{f k t} = [ ∫_0^{N^{f}_{k t}} C_{u f k t}^{σ_u−1/σ_u} du ]^{σ_u/(σ_u−1)}. ### Role of product differentiation - Products within the same firm are more substitutable than products across different firms, linking product differentiation to cannibalization effects and enabling connection between extensive margin dynamics and asset pricing. --- ### 5.2 Price of the product (benchmark without extensive-margin friction) - Production technology: Y_{u f k t} = Z_t L_{u f k t}. - Demand for product u of firm f in group k: - C_{u f k t} = C_t P_t^{−σ_k} P_{k t}^{σ_k−σ_f} P_{f k t}^{σ_f−σ_u} P_{u f k t}^{−σ_u}. - Proposition 1 (optimal price with flexible pricing): - P_{u f k t} = σ_f/(σ_f−1) · W_t / Z_t. - Markup is based on firm elasticity σ_f (not product elasticity σ_u) because of within-firm substitution (cannibalization). --- ### 5.3 Firm’s decision on optimal product scope: without friction - Maintenance cost of a variety: F units of effective labor per period (interpreted as fully depreciating capital associated with the product). - Marginal revenue of changing product scope N has two parts: - product-level consumption change due to substitution between surviving and newly introduced/disappearing products; - consumption associated specifically with the marginal product. - Proposition 2 (cannibalization effect): - product level: - −(d C_{u f k t}/d N^{f}_{k t}) · (N^{f}_{k t}/C_{u f k t}) = (σ_u − σ_f)/(σ_u − 1). - firm level: - (d C_{f k t}/d N^{f}_{k t}) · (N^{f}_{k t}/C_{f k t}) = σ_f/(σ_u − 1). - Special cases noted: - product-level cannibalization is zero when σ_u = σ_f; - product-level cannibalization is 100% when σ_u → ∞ (products perfect substitutes within firm). - Optimal product scope under full flexibility: - N^{f}_{k t} = [ C_{f k t} / F(σ_u − 1) ]^{(σ_u−1)/σ_u}. --- ### 5.4 Firm’s decision on optimal product scope: Calvo-type frictions - Time-dependent (Calvo) friction: a firm in group k can adjust product scope with probability 1−λ_k each period. - Firms that can reset choose ˜N^{f}_{k t} to maximize expected discounted profits: - max_{˜N^{f}_{k t}} ∑_{j=0}^∞ λ_k^j E_t { (Λ_{t+j}/Λ_t) [ ∫_0^{˜N^{f}_{k t}} (P_{u f k t+j} − MC_{t+j}) C_{u f k t+j} du − F ˜N^{f}_{k t} MC_{t+j} ] }. - Proposition 3 (optimal product scope with frictions): - ˜N^{ (σ_u−1)·σ_u/(σ_u−σ_f) }_{f k t} = [ ∑_{j=0}^∞ λ_k^j E_t [ (Λ_{t+j}/MC_{t+j}) C_{t+j} P_{t+j}^{σ_k} P_{k t+j}^{σ_f−σ_k} MC_{t+j}^{−σ_f} ] ] / [ ∑_{j=0}^∞ λ_k^j E_t [ (Λ_{t+j}/MC_{t+j}) F ] ]. - Interpretation: - Optimal product scope is positively correlated with the discounted present value of revenue when the scope cannot be changed forever, and negatively correlated with the discounted present value of marginal cost when the scope is permanently set. --- ### 5.5 Model summary (variables and equilibrium conditions) - Competitive equilibrium variables (15): - { P_{k t}/P_t, ˜P_{k t}/P_t, ˜N_{k t}, S_{k t}, PVR_{k t}, PVC_{k t}, π_t (≡ P_t/P_{t−1}), C_t, S_t, L_t, i_t, W_t/P_t, Λ_{t+1}/Λ_t } plus exogenous processes { Z_t, ν_t }. - Key group-specific conditions: - optimal reset variety: - ˜N_{k t}^{(σ_u−1)·σ_u/(σ_u−σ_f)} = σ^{−σ_f}_f f (σ_f−1)^{−σ_f} (σ_u−1)^{−σ_f} PVR_{k t}/PVC_{k t}. (Equation (4)) - PVR_{k t} and PVC_{k t} defined with recursive expectations (Equations (5) and (6)): - PVR_{k t} = (W_t/Z_t/P_t)^{1−σ_f} (P_{k t}/P_t)^{σ_f−σ_k} C_t + β λ_k E_t [ (Λ_{t+1}/Λ_t) π_{t+1} PVR_{k t+1} ]. - PVC_{k t} = W_t F/(Z_t P_t) + β λ_k E_t [ (Λ_{t+1}/Λ_t) π_{t+1} PVC_{k t+1} ]. - optimal reset price: - ˜P_{k t}/P_t = ˜N_{k t}^{1/(1−σ_u)} · σ_f/(σ_f−1) · W_t/(Z_t P_t). (Equation (7)) - group price aggregation and dispersion (Equations (8) and (9)). - Aggregate conditions: - aggregate price index: 1 = [ ∫_0^1 (P_{k t}/P_t)^{1−σ_k} dk ]^{1/(1−σ_k)}. (Equation (10)) - aggregate price dispersion: S_t ≡ ∫_0^1 (P_{k t}/P_t)^{−σ_k} S_{k t} dk. (Equation (11)) - labor demand: C_t = Z_t L_t S_t. (Equation (12)) - labor supply: χ L_t^{1/φ} = W_t C_t / P_t. (Equation (13)) - Euler equation: 1 = β E_t [ (1+i_t) Λ_{t+1}/Λ_t ]. (Equation (14)) - subjective discount factor: Λ_{t+1}/Λ_t = C_t / (C_{t+1} π_{t+1}). (Equation (15)) - Taylor rule: - ln i_t = φ_i ln i_{t−1} + (1−φ_i) ln [ ̄i + φ_π (P_t/P_{t−1} − 1) + φ_y (C_t − ̄C) ] + ln ν_t. (Equation (16)) - Exogenous processes: - ln Z_t = ρ_z ln Z_{t−1} + e^z_t. (Equation (17)) - ln ν_t = ρ_ν ln ν_{t−1} + e^ν_t. (Equation (18)) --- ### 5.6 Return rate and asset pricing of firms - Value of a firm path with permanent product scope N^{f}_{k t} (real value deflated by P_t): - VS_t(N^{f}_{k t}) = N^{1−σ_f}_{f k t} / (1−σ_u) · RS_{k t} − N^{f}_{k t} · CS_{k t}. (Equation (19)) - Definitions: - RS_{k t} = σ_f^{−σ_f} f (σ_f−1)^{−σ_f} MC_t^{1−σ_f} P_t^{σ_k−1} P_{k t}^{σ_f−σ_k} C_t + β λ_k (Λ_{t+1}/Λ_t) π_{t+1} RS_{k t+1}. - CS_{k t} = F · MC_t / P_t + β λ_k (Λ_{t+1}/Λ_t) π_{t+1} CS_{k t+1}. - Value relation across two firms differing only by never getting the opportunity to change scope: - V_t(N′_k) = V_t(N″_k) − VS_t(N″_k) + VS_t(N′_k). (Equation (20)) - Dividend specification (Div_t(N)) equals firm profit and is used to form return rate: - Return rate R_t(N^{f}_{k t}, N^{f}_{k t−1}) expression provided in text (depends on V_t, VS_t, Div_{t−1}, and π_t). - Recursive valuation for V_t(˜N^{f}_{k t}) given potential future resets and λ_k (equations show decomposition into Div_t, continuation value, and terms involving VS differences). --- ### 5.7 Parametrization - Elasticities: - across-group elasticity set to 2. - across-firm and across-product elasticities set to the median estimates from Hottman, Redding and Weinstein (forthcoming). - Time-dependent parameters λ_k for product-scope adjustments set using tertiles of product scope adjustments from Nielsen scanner data. --- ### 5.8 Simulation - Simulation design: - Simulate 100 firms of 3 groups for 1000 periods with the first 800 periods burn-in. - Measurement: - To measure the return rate: take the average return rates of the 200 periods. - To measure asset volatility: take the standard deviations of the return rates throughout the periods. - Key simulation findings (descriptive): - Positive correlation between rigidity in product-scope adjustment and return rates: "as firms become more rigid, the return rate increases." - Covariance between inverse consumption growth and return rate coincides with empirical negative relationship: covariance decreases with rigidity in product-scope adjustment. - Simulation steps (summary as provided): - 1) store RS, CS, VS(˜N), V(˜N), ˜N for each group k from Dynare solution (second-order perturbation); - 2) generate path of {N} from {˜N} following binomial distribution of λ_k; each sequence {N} represents a firm; - 3) calculate RS, CS, Div, V for each firm at each period and then calculate return rate. --- ### Conclusion and policy-relevant implications - Main result: - Flexibility in firms’ extensive margin adjustment (product creation/destruction) improves firms’ financial risk diversification and is associated with lower excess stock returns and lower asset volatility. - Empirical regularities documented from rich retailer-product-level data: - great variation in product turnover rates across firms; - product turnover rates are "sticky" over time (high-turnover firms remain relatively high-turnover; low remain low). - firms with higher flexibility in product-scope adjustments exhibit less financial risk (lower excess returns and asset volatility). - Model interpretation: - The multi-product firm model with product turnover rates rationalizes the empirical findings by linking product-scope flexibility to asset pricing and risk diversification. - Suggested avenues for future research and policy considerations: - connect firms’ financial risk diversification to labor adjustment behaviors or investment decisions (particularly R&D investment); - investigate how policies could enhance firms’ production flexibility and encourage R&D investment for product creation. *Source: wp17146 - 5.1 Household* ### References ### wp17146 - References ### Bibliography - Bems, Rudolfs, and Julian Di Giovanni. 2014. “Income-Induced Expenditure Switch- ing.” - Bergin, Paul R, and Giancarlo Corsetti. 2008. “The extensive margin and monetary policy.” Journal of Monetary Economics, 55(7): 1222–1237. - Bernard, Andrew B., J Bradford Jensen, Stephen J Redding, and Peter K Schott. 2007. “Firms in international trade.” National Bureau of Economic Research. - Bernard, Andrew B., Stephen J. 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The optimal price is - P_u f kt = σ_f / (σ_f − 1) W_t Z_t - Proof sketch includes: - Demand: C_f kt = [∫_0^{N_f kt} C_u^{σ_u−1/σ_u} u f kt du]^{σ_u/(σ_u−1)}, and related expressions for C_u f kt and P_f kt. - First-order condition leads to P_u f kt = σ_f/(σ_f−1) W_t Z_t. - Proposition 2. Cannibalization effect - product level: − (dC_u f kt / dN_f kt) (N_f kt / C_u f kt) = (σ_u − σ_f) / (σ_u − 1) - firm level: (dC_f kt / dN_f kt) (N_f kt / C_f kt) = σ_f / (σ_u − 1) - Proposition 3. Optimal product scope without friction: - N_f kt = [C_f kt / (F(σ_u − 1))]^{(σ_u − 1)/σ_u} - ∂N_f kt / ∂F < 0, ∂N_f kt / ∂C_f kt > 0 - Proposition 4. Optimal product scope with friction: - (σ_u − 1) ˜N_f kt^{(σ_u − σ_f)/(σ_u − 1)} = ∑_{j=0}^∞ λ_j^k E_t[Λ_{t+j} MC_{t+j} C_{t+j} P_{kt+j}^{σ_k} P_{f}^{−σ_k} MC_{t+j}^{−σ_f}] / ∑_{j=0}^∞ λ_j^k E_t[Λ_{t+j} MC_{t+j} F] - Equivalent expression using present value ratios: - ˜N^{(σ_u − σ_f)/(σ_u − 1)}_f kt ≡ σ^{−σ_f}_f f(σ_f − 1)^{−σ_f} (σ_u − 1) PV R_{kt} / PVC_{kt} - Definitions: - PV R_{kt} = 1/P_t ∑_{j=0}^∞ β^j λ_j^k E_t[Λ_{t+j}/Λ_t MC_{t+j} C_{t+j} P_{kt+j}^{σ_f−σ_k} ...] (full expressions retained in source) - PVC_{kt} = 1/P_t ∑_{j=0}^∞ β^j λ_j^k E_t[Λ_{t+j}/Λ_t MC_{t+j} F] - Note: M_{t,t+k} only depends on aggregate C_t which is independent with N since there is no firm entry. ### Tables — Key Tables and Selected Numeric Highlights - Table 1: Transition Matrix (cells preserve concatenated formatting as in source) - Bottom Top | Bottom Top | Bottom Top - Bottom 0.660.34 0.710.29 0.680.32 - Top 0.090.91 0.080.92 0.090.91 - Table 2: High Turnover Firms vs. Low Turnover Firms — selected numeric entries - size (in m$): Bottom 0.02 Top 0.02 pvalue 0.61 - book-to-market ratio: Bottom 3.78 Top 5.43 pvalue 0.09 - turnover (in m$): Bottom 37.01 Top 123.61 pvalue 0.41 - purchaser income