The Dynamics of Pass-through with Expected and Unexpected Costs

Transcription

1 The Dynamics of Pass-through with Expected and Unexpected Costs Alexander MacKay and Marc Remer March 20, 2017 PRELIMINARY AND INCOMPLETE: NOT FOR GENERAL DISTRIBUTION Abstract Prices in many industries adjust slowly to changes in marginal costs. We consider the implications of price dynamics in the context of pass-through and demand estimation. Empirically, we find that a firm s response depends on whether cost changes are expected or unexpected. To account for these price dynamics, we develop an empirical model of dynamic demand that builds on the standard discrete-choice model and requires no supply-side assumptions. We estimate the model and compare its predictions to an alternative Bajari et al. (2007) estimator. We demonstrate that welfare analysis depends critically on properly accounting for demand dynamics. 1 Introduction Prices in a wide array of industries have been documented to react slowly to cost changes. Grocery stores (Peltzman, 2000), retail gasoline (Borenstein et al., 1997), and banking (Neumark and Sharpe, 1992) are just a few examples of markets where firms gradually pass through marginal cost shocks to consumers. While dynamically adjusting prices are an important feature of these markets, the demand estimation literature has largely ignored this property. The consequences of ignoring dynamics are three-fold. First, demand estimation may rely upon misspecified static optimization conditions, which can bias parameter estimates. Second, static models will be unable to reproduce relevant short-run price dynamics, which can have important consumer welfare consequences in markets with frequently changing costs. Finally, pass-through estimates may fail to properly capture the completeness of the price adjustment when anticipation is not accounted for. Harvard Kennedy School, 79 JFK Street, Cambridge, MA amackay@hbs.edu. Swarthmore College, 500 College Ave., Swarthmore, PA, mremer1@swarthmore.edu. 1

2 In this article, we develop and estimate a micro-founded model of dynamic demand, which results in profit-maximizing firms adjusting prices over many time periods in response to a cost change. The model also produces other important pricing patterns, such as price responses prior to anticipated future cost changes. The demand-side model includes a subset of consumers that become attached to the firm from which they purchased in the previous period, creating an incentive for firms to sacrifice current profits for future gains from the attached consumers. Our model of attachment can be used to describe brand loyalty, habit formation, lock-in, or search costs. This feature, which we call lock-in for the remainder of the paper, leads to firms in oligopolistic markets to slowly adjust prices to changes in cost. The demand model is a straightforward extension of the standard logit model common in industrial organization. We advance the empirical literature by developing an estimation method that accommodates dynamic demand, but does not require the assumptions used in standard dynamic estimation techniques, such as those developed in Bajari et al. (2007) (BBL), to identify the dynamic parameters. Importantly, we do not require supply-side assumptions about the competitive game or the expectations of the firms. As supply-side behavior depends crucially on which future shocks are expected or unexpected, we view this as a significant advantage. The data needed to estimate the model are commonly used in the demand estimation literature, namely, prices, shares and cost shifters (or, alternatively, product characteristics). Nonetheless, we show that with firm-level shares (across all customer types), it is possible to separately identify, each firms current shares of locked-in and non-locked-in customers, as well as the probability that a consumer becomes locked-in after purchasing. Identification of locked-in customers allows for the presence of a serially correlated state variable for each firm. In other words, we estimate a model with endogenous unobserved heterogeneity. This flexibility has traditionally been a challenge for the estimation of dynamic models. Note: All results in this draft are preliminary. We estimate the model using a rich panel data set of prices, shares, and costs for retail gasoline stations. In this context, the model is best interpreted as one of habit-formation or search costs, wherein some consumers return to the gas station from which they previously purchased without considering alternative sellers. We estimate that about 80 percent of consumers only consider the gas station they previously purchased on a week-to-week basis, without searching competitors. On average, 98 percent of locked-in customers purchase gasoline, compared to only 34 percent of customers that are not locked-in ("free agents"). Locked-in customers display a much lower price sensitivity. Free-agent customers have an average elasticity of , whereas locked-in customers have an average elasticity of To highlight the importance of accounting for dynamic factors when estimating perperiod firm payoffs, we compare our results to an alternative BBL estimator [to be completed]. We use our estimated demand model and measures of unexpected and expected 2

3 costs to address how accurately Bertrand-Nash pricing rationalizes the observed prices, given our demand model. We then evaluate the relative importance of accurately capturing firm expectations compared to imposing assumptions commonly employed in the estimation of dynamic models. Prior to developing our model, we present reduced-form pass-through results on how prices respond to cost shocks in retail gasoline markets. We use the data to separate out expected and unexpected costs, and we show that firms respond differentially to these two measures. Importantly, firms begin raising prices in anticipation of higher costs approximately 28 days prior to a cost shock. Firms exhibit full pass-through for expected cost changes, raising prices by 1.02 dollars for each dollar increase in cost. We find that the the pass-through of unexpected costs is limited, with firms raising prices by only 0.66 cents for an unexpected dollar increase in costs. We explore heterogeneity in firm pass-through and how it relates to the competitive conditions in the market. 2 The Data The analysis relies upon daily, regular fuel retail prices for nearly every gas station in the states of Kentucky and Virginia, which totals almost six thousand stations. As a measure of marginal cost, the data include the brand-specific, daily wholesale rack price charged to each retailer. We therefore almost perfectly observe each gas station s marginal cost changes, except for privately negotiated discounts per-gallon, which are likely fixed over the course of a year. 1 The data ranges from September 25th, 2013 through September 30th, The data was obtained directly from the Oil Price Information Service (OPIS), which routinely supplies data used in academic studies (e.g. Lewis and Noel 2011; Chandra and Tappata 2011; Remer 2015). OPIS also supplied the market share data, which it obtains directly from actual purchases that fleet drivers charge to their Wright Express Universal card. The data is specified at the weekly, county/gasoline-brand level. 2 Due to contractual limitations, OPIS only provided each brand s share of sales, not the actual volume. Thus, to account for temporal changes in market-level demand, we supplement the share data with monthly, state-level consumption data from the Energy Information Administration (EIA). 1 The data also include all federal, state, and local taxes. 2 In some instances, the brand of gasoline may differ from the brand of the station. For example, some 7-Eleven stations in the data are identified as selling Exxon branded gasoline. 3

4 3 Evidence of Dynamic Pricing To motivate the structural model, we first provide evidence on dynamically adjusting retail gasoline prices. A host of previous studies have found that retail gasoline prices may take multiple weeks to fully incorporate a change in marginal cost. 3 One innovation of our study is that we use separate measures of unexpected and expected costs to see if, consistent with forward-looking behavior, firms respond differentially to these two types of costs. 3.1 Identifying Expected and Unexpected Costs To disentangle the reaction to anticipated and unanticipated cost changes, we leverage data on wholesale gasoline futures traded on the New York Mercantile Stock Exchange (NYMEX). The presence of a futures market allows us to project expectations of future wholesale costs for the firms in our market. To make these projections, we assume that firms are engaging in regression-like predictions of future wholesale costs, and we choose the 30-day ahead cost as our benchmark. 4 Using station-specific wholesale costs, we regress the 30-day lead wholesale cost on the current wholesale cost and the 30-day ahead future. In particular, we estimate the following equation. c it+30 = α 1 c it + α 2 F 30 t + γ i + ɛ it (1) Here, c it+30 is the 30-day-ahead wholesale cost for firm i, Ft 30 is the 30-day ahead forward contract price at date t, and γ i is a station fixed effect. We use the estimated parameters to construct expected 30-day ahead costs for all firms: ĉ it+30 = ˆα 1 c it + ˆα 2 Ft 30 unexpected cost, or cost shock, is the residual: c it+30 = c it+30 ĉ it ˆγ i. The For robustness, we construct a number of alternative estimates of expected costs, including a specification that makes use of all four available futures. However, we found that these alternative specifications were subject to overfit; the estimates performed substantially worse out-of-sample when we ran the regression on a subset of the data. Our chosen specification is remarkably stable, with a mean absolute difference of one percent when we use only the first half of the panel to estimate the model. Expected costs constitute 74.6 percent of the variation in costs (R 2 ) in our two-year sample, which includes a large decline in wholesale costs due to several supply shocks in Note on the 30-Day Ahead Expectation One of the challenges in discussing expectations is that they change each day with new information. News about a cost shock 30 days from now may arrive anytime within the 3 See Eckert (2013) for a comprehensive review of the literature. 4 Futures are specified in terms of first-of-the-month delivery dates. To convert these to 30-day ahead prices, we use the average between the two futures, weighted by the relative number of days to the delivery date. 4

5 next 30 days, if it has not arrived already. Therefore, any discussion of an unexpected cost shock must always be qualified with an as of when. Given previous findings in the gasoline literature indicating that prices take approximately four weeks to adjust, a 30- day ahead window seems an appropriate one to capture most of any anticipatory pricing behavior. Additionally, our findings support this window as being reasonable in this context. We see no relationship between unexpected costs or expected costs and the price 30 days prior. We do find a small anticipatory effect of our measure of unexpected costs on price 14 days in advance, but is less than 5 percent of the total price adjustment. We interpret this correlation to arise from the underlying correlation in unobserved cost shocks. 3.2 Pass-through Regressions Further highlighting the temporal component of cost pass-though, we separately estimate how gas stations react to expected versus unexpected cost changes. Beyond motivating the structural model, these results also demonstrate the importance of capturing firms anticipated price responses when estimating cost pass-through rates. For example, to analyze how much of a tax increase firms will pass-on to consumers, it is imperative to recognize that firms may begin to adjust their prices prior to the tax increase being enacted; failure to account for this response may lead to underestimating pass-through rates. We employ our measures of expected and unexpected costs to study how firms differentially respond to these costs. We incorporate the main components of marginal costs for retail gasoline, which include the wholesale cost of gasoline and the per-unit sales tax. We estimate the following model: p it = 50 s= 50 β s ĉ it s + 50 s= 50 γ s c it s + 50 s= 50 φ s τ it s + ψ i + ε it. (2) Here, p it, is the price observed at gas station i at time t. ĉ it s and c it s are the expected and unexpected wholesale costs observed with lag s, and τ it s is the state-level sales tax. 5 Using the estimated coefficients on the cost measures, we construct cumulative response functions to track the path of price adjustment to a one time, one unit cost change at time t = 0. We incorporate 50 leads and lags to capture the full range of the dynamic response. We focus our results on unexpected and expected costs, as we do not have enough tax changes in our data to estimate a consistent pattern of response. 6 Figure 1 plots the cumulative response functions for unexpected and expected costs. Panel (a) displays the results for unexpected costs. Prices react suddenly and quickly at 5 To more easily incorporate future anticipated costs into the regression, we do not estimate an error-correction model (Engle and Granger, 1987), which is commonly used to estimate pass-through in the retail gasoline literature. 6 As a robustness check, we also estimated the price response to expected and unexpected costs using the errorcorrection model, and we found nearly identical results. 5

6 time zero, but it takes about four weeks for the prices to reach the new long-run equilibrium, reaching a peak of 0.71 after 34 days. Panel (b) displays the cumulative response function for expected costs. Notably, firms begin to react to expected costs approximately 28 days in advance, with a relatively constant adjustment rate until the new long-run equilibrium is reached 21 days after the shock. Though the total duration of adjustment is longer compared to the unexpected cost shock, the firm incorporates the cost more quickly after it is realized. This coincides with substantial anticipation by the firm; the price already captures about a third of the effect of the expected cost shock the day before it arrives. A striking result from these estimates is the difference in the long-run pass-through rates. Expected costs experience approximately full pass-through - a cost increase leads to a corresponding price increase of equal magnitude. On the other hand, unexpected costs demonstrate incomplete pass-through, moving about only 66 cents for each dollar increase in cost. The different response to unexpected and expected costs emphasizes the need for empirical researchers to think carefully about designing proper estimators for pass-through. If costs are anticipated, then a pass-through measure that omits leads will only capture a portion of the overall response. Additionally, if a pass-through estimate is to be used for evaluation, is it important to ensure that the estimator relies on a mix of unexpected and expected shocks that translate to the policy under analysis. In our setting, for example, an analysis that used an unexpected cost shock to predict the impact of a tax rate change would be inappropriate, as the tax change is anticipated and leads to a much greater price response. An important consideration for pass-through when analyzing imperfect competition is the distinction between idiosyncratic costs and common costs. In the current section, we consider only the simple cut between unexpected and expected costs to focus attention on this previously unexplored dimension of pass-through. In Appendix A, we present results for common costs and idiosyncratic costs. In our setting, costs are highly correlated, with common costs tending to dominate idiosyncratic costs at moderate frequencies. Therefore, the results for common costs are very similar to those in this section. One distinction, however, is that pass-through of unexpected common costs is higher than the pass-through for total unexpected costs. For idiosyncratic costs, which we consider by controlling for the prices of rivals, pass-through is on the order of 0.04 to This is not surprising for a highly competitive market such as retail gasoline. 3.3 Heterogeneity We now show that there is significant heterogeneity across firms in the dynamics of cost pass-through, and that it can, in part, be explained by the firm s competitive environment. 6

7 Figure 1: Cumulative Pass-through (a) Unexpected Costs Cumulative Price Change Days After Cost Change (b) Expected Costs Cumulative Price Change Days After Cost Change 7

8 Table 1: Pass-through Heterogeneity Parameter p10 p25 p50 p75 p90 Anticipation Rate Duration Finish LR_PTR Anticipation_Exp Rate_Exp Duration_Exp Finish_Exp LR_PTR_Exp To do so, we estimate station-specific regressions of equation (2), and then construct a cumulative response function for expected and unexpected costs separately for each gas station. To summarize these response functions, we use non-linear least squares to fit a three-segment spline to each firm s cumulative response function. We restrict the slopes of the first and last segments to equal zero. The middle segment captures the duration and magnitude of the price adjustment. This methodology allows to to estimate for each gas station (i) anticipation: how far in advance the firm begins responding to a cost shock, (ii) duration: how many days it takes to reach the new long-run equilibrium, and (iii) rate: what proportion of the cost change is passed through to price each day. We also construct the long-run pass-through rate ( LR_PTR ), which reflects the degree to which prices reflect costs at the end of the adjustment period. Table 1 summarizes the estimated heterogeneity parameters. The five rows correspond to the response to unexpected costs, and the last five rows correspond to expected costs. These summary statistics align with the mean pass-through parameters estimated in the previous section. The median firm anticipate expected costs much earlier (25.4 days in advance) than unexpected costs (4.1 days in advance), and full pass-through is greater for expected costs (1.02) than unexpected costs (0.66). The median firm takes longer to incorporate expected costs, but tends to reach full pass-through around the same time for both types of shocks (21.5 versus 23.0 days after the shock). Comparing the percentile range of the estimates, we find that there is greater variation in the pricing response to unexpected costs. The median rate of adjustment is faster for unexpected costs (0.026/day compared to 0.012), but sees a spread of (0.009, 0.103) for the percentile range, compared to (0.015, 0.051). Likewise, there is greater variation in the timing of when firms fully incorporate cost shocks in price and the full passthrough rates (0.278 to for unexpected, to 1.209). The long-run pass-through estimates for expected costs display a tendency toward homogeneity, as they are clustered 8

9 around 1. For the specification in this section, our fitted splines consist of three parameters, where we restrict the level of the first segment to be zero. In Appendix B, we consider another version that adds the level of the first segment as a fourth parameter. The results are consistent. 3.4 Competition and Heterogeneity Regressions In this section, we project our estimated parameters onto a simple measure of market competition. In doing so, we demonstrate that competition has an important effect on the dynamics of cost pass-through, which further motivates the dynamic oligopoly model and demonstrates that firm expectation play an important role in determining cost pass-through rates. We perform a series of county-level regressions that relate the estimated parameters in Table 1 to the Herfindahl-Hirschman index ( HHI ) in the county. To do so, we first calculate the median firm-level parameter in each county, and regress it on the HHI and the number of gas stations in the the county. 7 To calculate the HHI, we sum the square of each gasoline brand s weekly county market share. We then take the average across all weeks for each county to use as an independent variable in the regression. 8 We also include the number of gas stations in each county as a regressor. This accounts for variation in population and demand across counties in a simple way; without this control, counties with a large consumer base would, all else equal, have lower HHI. Table 2: Pass-Through and County Competition (1) (2) (3) (4) (5) (6) (7) (8) Dependent Variable LR_PTR LR_PTR_exp Rate Rate_exp Anticipation Anticipation_exp Duration Duration_exp HHI ** ** * (0.138) (0.055) (0.023) (0.007) (4.051) (5.662) (7.212) (8.312) Number of Stations (0.001) (0.000) (0.000) (0.000) (0.017) (0.023) (0.046) (0.034) Constant 0.629*** 1.014*** 0.034*** 0.021*** 5.493*** *** *** *** (0.044) (0.019) (0.008) (0.003) (1.415) (1.848) (2.744) (2.758) Observations Notes: LR_PTR is the long run full pass through rate. "_exp" represents the reaction to an expected cost change, and other variables are reactions to unexpected cost changes. HHI is Herfindahl Hirschman index. Obsevations are at the county level. The dependent variable is the median estimated firm level parameter in each county. Robust standard errors in parentheses. *** p<0.01, ** p<0.05, * p<0.1 The results of the series of regression are presented in Table 2, and demonstrate that competition affects pass-through dynamics. We find a significant relationship between the HHI and reaction to expected changes in cost. Specifically, the anticipated response ( An- 7 Results are qualitatively the same when using the mean parameter value. We choose the median, as the distribution of parameter estimates are slightly skewed and there are a few outliers. 8 We take the weekly average HHI, as we observe market share rather than quantity. 9

10 ticipation_exp ), the rate of adjustment ( Rate_exp ), and duration of adjustment ( Duration_exp ) are all significantly related to the county HHI such that less competition leads to later and quicker responses to anticipated cost changes. Interestingly, we find in this simple regression that the long-run pass through rate ( LR_PTR ) is unaffected by competition. Adding additional controls, however, such as population or mean income leads to positive and significant results, in some specifications. Thus, there is some evidence that less competition also leads to a greater proportion of expected cost changes to be passed-through to consumers On the other hand, we do not estimate a significant relationship between pass-through of unexpected cost shocks and the HHI. This underscores the importance of distinguishing between expected and unexpected cost changes when analyzing the interaction of competition and pass-through dynamics. In the following sections, we develop a structural model to capture these reduced-form pricing dynamics 3.5 Correlation in Shares Over Time Though ultimately the importance of dynamics in our data will be estimated by the model, it is informative to examine the reduced-form relationships between key elements. The dynamic model we develop in the next section is one in which today s quantity depends on the quantity sold last period. As motivation for this model, we present the results from reduced-form regressions of shares on lagged shares in Table 3. The regression indicate that lagged shares are a significant predictor of current shares. In specification (2), lagged shares explain nearly 94 percent of the variance in current shares, and the coefficient is close to one. In specifications (3) and (4), we include measures of competition in the regressions. These measures, which include the mean and standard deviations of competitor prices, are correlated with shares, but lagged shares still are the most important predictor of current shares. In specification (6), we include time and brandcounty fixed effects, as well as a third-order polynomial in own price. The coefficient of on lagged shares in this specification indicates that deviations in shares from the brand-county mean are highly correlated over time. 4 A Oligopoly Model of Dynamic Demand 4.1 Demand Side We now present a dynamic model of oligopolistic competition with product differentiation where consumers may become locked in to the firm from which they previously purchased. Lock-in may be interpreted as customer loyalty, habit formation, or a cost of search. Customers in the model are myopic in that they maximize current period utility rather than a discounted flow of future utility. This assumption is likely a good fit for retail gasoline 10

11 Table 3: Regressions with Share as the Dependent Variable (1) (2) (3) (4) (5) (6) Price (0.001) (0.000) (0.000) (0.002) (0.001) (0.024) Lagged Share (0.001) (0.001) (0.002) (0.002) Comp. Price (Mean) (0.000) (0.001) (0.001) (0.001) Comp. Price (SD) (0.003) (0.005) (0.004) (0.004) Num. Competitors (0.000) (0.000) (0.000) (0.000) Price Squared (0.009) Price Cubed (0.001) County-Brand FEs X X X Time FEs X X X Observations R Standard errors in parentheses p < 0.10, p < 0.05, p <

12 markets, where consumers do not choose a gas station anticipating that it will limit their future choice set; rather, some consumers are likely to return to the same gas station due to habit formation or costly search behavior. As we detail below, we introduce differentiated product demand using the standard logit model, and then place the demand model into a dynamic oligopoly setting. The dynamics stem from firms current sales affecting future profits through the accumulation of locked-in consumers. We will then demonstrate how common properties of cost pass-through (such as sticky prices, adjustment to expected cost shocks, and estimates of long-run pass-through) differ in this dynamic setting as compared to static environments. 9 The demand-side is characterized by the standard logit model, 10 but a fraction of consumers, δ, become locked-in to the firm from which they purchased in the previous period. We assume that consumers that purchase the outside good in period t 1 are never locked-in at time t. Consumers that are not locked-in, which we will refer to as free agents, have utility u (n) jt = ξ j αp jt + η jt + ɛ (n) jt from purchasing from any inside firm j {1,.., J} or the outside good, j = 0. The superscript (n) indicates customer-specific values. For lockedin consumers, the utility from purchasing good j conditional on purchasing good i in the previous period is: u (n) jt = ξ j + 1[j = i] ξ (α + 1[j = i]ᾱ)p jt + η jt + ɛ (n) jt. (3) Locked-in customers therefore experience a relative level utility shock ξ and a relative price sensitivity shock ᾱ. The relative utility shock, ξ, may be interpreted as a switching cost, a brand-loyalty effect, or a cost of search where non-searches are more likely to return to their previously visited gas station without checking other prices. For both free agents and locked-in consumers, we normalize the utility of the outside good to be zero. In our model, we include a product-time shock η jt, which allows for potential correlation in unobservable utility shocks over time. We denote the choice probability of a free-agent as s jt (0) and locked-in consumer as s jt (i), where i denotes the firm to which a customer is locked-in and 0 denotes that a consumers is a free-agent. For ease of notation, we define ξ jt = ξ j αp jt +η jt and ξ jt + ξ jt = ξ j + ξ (α + ᾱ)p jt + η jt. Then, given the standard logit assumption of a type 1 extreme value distribution on the utility shock, ɛ (n) jt, the market shares of free agents and locked-in consumers are: 9 Slade (1998) estimates a model of habit-forming consumers and sticky prices. That model, however, explicitly imposes a cost of price-adjustment. Our model does not rely upon a menu cost to explain dynamic price adjustments. 10 In future versions, we plan to generalize the model to either the nested or random coefficients logit model. 12

13 exp(ξ jt ) s jt (0) = 1 + k exp(ξ kt) s jt (i) = (4) exp(ξ jt + 1[j = i] ξ jt ) 1 + k/i exp(ξ kt) + exp(ξ it + ξ jt ). (5) The observable share for firm j is then given by the average of these two components, where each component is weighted by the relative proportion of customers in the free-agent or the locked-in state. 4.2 Supply Side Given the demand model above, a firm j s profit-maximization problem can be written as follows: max p jt {E[ β t Π(p t, r t, c t )]}. t=0 In this equation, p t, r t, and c t are vectors of each firm s prices, total number of lockedin consumers, and marginal costs, respectively, at time t. Thus, firms compete by choosing prices that maximize expected discounted profits, where firms anticipate both future marginal costs and how current choices affect the future state of locked-in consumers. The profit-maximization problem can be rewritten in Bellman equation form. To do so, we first define firm j s total share as: S jt = (1 k r kt )s jt (0) + i=1 r it s jt (i). (6) Thus, a firm s total share of sales can be written as a weighted sum of its share of free agent, s jt (0), and locked-in consumers, s jt (i). Note that firm j will make sales to consumers locked-in to other firms j i, but the probability that such consumers will choose firm j is strictly lower than the choice probability of a free-agent consumer. We then define the number of consumers locked-in the firm j at t + 1 to simply be r jt+1 = δs jt. A fraction of a firms previous period consumers, δ, become locked-in next period, and therefore today s sales affect tomorrow s profits. This is captured in the following Bellman equation: V j (r, c) = { max (pjt c jt )S jt + βe(v j (r t+1, c t+1 ) p t+1, r t+1, c t+1 ) }. (7) p jt p kt jt In this Bellman formulation, firms optimal prices depend upon the number of consumers currently locked-in to each firm, r jt and each firm s marginal cost c jt, which enter as state variables. We therefore must solve for J optimal pricing policies conditional on 2J state 13

14 variables. Note also that the optimal choice of prices depends essentially on the expectations of future costs, c t+1. One contribution of the empirical work is to disentangle expected and unexpected costs to shed light on the sensitivity of dynamic welfare analysis to the proper accounting of expectations Steady-State Analysis While solving for the optimal pricing policy in oligopoly markets is analytically intractable, analyzing steady-state prices in a monopoly market (with an outside good) highlights the impact that locked-in consumers have on optimal prices and cost pass-through rates. In the steady state, r jt = r jt+1 = r j and c jt = c jt+1 = c j. To simplify notation in the monopoly case, let the firm s share of locked-in and free-agent consumers be s j and s 0, respectively, and its number of locked-in consumers be r t. It follows that the steady-state number of locked-in consumers, r ss is: r = δ((1 r)s 0 + r s j ) r ss = δs 0 1 δ(s j s 0 ). The steady-state number of locked-in consumers is increasing in the probability of becoming locked in, δ, and the (absolute value) of the gap between the choice probabilities of locked-in customers and free agents, s j s 0. Using the steady-state value of locked-in consumers, we can solve for the steady-state pricing function: The steady state period value is: V ss (r ss, c ss ) = (p ss c ss )((1 r ss )s 0 + r ss s j ) + βv ss = pss c ss 1 β s 0 1 δ(s j s 0 ). This equation represents the monopolists discounted profits, conditional on costs remaining at its current level. Thus, profits are increasing in both δ and the difference in choice probabilities of locked-in and free-agent consumers. These results are straightforward: lock-in is profitable. Also, note that a model with no lock-in is embedded in this formulation, δ = 0 and s j = s 0, and therefore profits are simply the per-unit discounted profits multiplied by the firm s market share. Maximizing the steady-state value with respect to p ss yields the firm s pricing function: 14

15 p ss = c ss s 0 (1 δs j + δs 0 ) + ds 0 dp (1 δs j) + dsj dp δs. (8) 0 }{{} m = markup of price over marginal cost The second term, m, on the right-hand side of equation (8) captures the extent to which the firm prices above marginal cost. As this markup term depends upon choice probabilities, it is implicitly a function of price. Thus, as in the standard logit model, we cannot derive an analytical solution for the steady-state price. Nonetheless, using implicit differentiation we can derive comparative statics to see how markups and cost pass-through are impacted by lock-in. First, we derive conditions under which the steady-state markup of price over cost is higher in the presence of consumer lock-in. Equation (8) reduces to p ss = c ss s0 ds 0/dp when δ = 0 and s j = s 0. Though the steady-state values of s 0 will depend on lock-in parameters, we abstract away from this consideration for now. The below analysis applies for environments in which the markups will be approximately equal, resulting in similar values for s 0. Denoting the markup with lock-in as m l and the markup without lock-in as m n, we have m l = s 0 (1 δs j + δs 0 ) ds 0 dp (1 δs j) + dsj dp δs 0 s 0 ds 0 /dp = m n. Rearranging terms, we obtain a simple condition relating markups in the presence of lock-in to markups in a static model: m l > m n s 0 p > s j p. (9) Thus, if the demand from locked-in consumers is relatively less sensitive to changes in price, then markups are higher. In our model, free-agent consumers are more elastic than locked-in customers, which implies s0 1 p s 0 > sj 1 p s j. However, locked-in customers also have higher shares, so markups may be higher or lower with consumer lock-in. Consider the logit model, where s0 p = αs 0(1 s 0 ) and sj p = (α + ᾱ)s j(1 s j ). Though α(1 s 0 ) > (α + ᾱ)(1 s j ), the locked-in share s j may be high enough relative to s 0 to flip the inequality. 11 The intuition for this result is akin to those summarized in Farrell and Klemperer (2007); with dynamic demand and lock-in, firms face a trade-off between pricing aggressively today and harvesting locked-in consumers in future periods. In the steady state, our model shows that either effect may dominate. In the empirical section, we estimate (α + ᾱ) to be 11 In the logit model, markups increase in the presence of lock-in when s 0 > 0.5. Again, one should account for differences in the steady-state levels of s 0. 15

16 close to zero and the share of locked-in customers to be close to one. Thus, our data support markups being higher in in the presence of lock-in. This finding has important implications for counter-factual exercises, such as merger simulation. Failing to account for customer lock-in will result in inferring incorrect markups from first-order conditions, which is a critical input to merger simulation. Furthermore, antitrust agencies often infer elasticities from markups calculated using accounting data (see, Miller et al. (2013)); our result demonstrates that this will lead to incorrect elasticities and merger price predictions when lock-in (or brand loyalty) is important. 5 Estimation Methodology We now outline our estimation methodology. We divide it in two stages, as demand can be estimated independently of the supply-side assumptions. Our method of demand estimation relies on data that is widely used in static demand estimation: shares, prices, and an instrument. 5.1 First Stage: Demand Estimation Given our dynamic extension of the logit demand system, we obtain the familiar expression for the log ratio of shares from Equation (4): ln s jt (0) ln s 0t (0) = ξ jt (10) Likewise, we obtain the following relation for the shares of locked-in customers: ln s jt (i) ln s 0t (i) = ξ jt + 1[j = i] ξ it (11) We can combine Equations (10) and (11) to obtain the following: ln s jt (i) ln s 0t (i) (ln s jt (0) ln s 0t (0)) = 1[j = i] ξ it (12) When we specify ξ it = ξ ᾱp it, then this empirical relationship depends only on the dynamic parameters ξ and ᾱ along with the observed prices Separating Locked-In Shares from Observed Shares The challenge in dynamic demand estimation is that we do not separately observe the nonlocked-in and free-agent customers. Instead, we observe the aggregate share, S jt, which is a weighted combination of the {s jt (i)} and depends on the number of locked-in customers 16

17 for each product {r jt }. Observed shares are determined by the following: S jt = (1 k r kt ) s jt (0) + i r it s jt (i). To separate out s jt (i) from S jt, we leverage the structure of our model. First, we note that r jt is proportional to the observed shares in the previous period: r jt = δs jt 1. Second, we use the relations in the previous section to obtain the following expressions: s jt (0) = ( ) s0t (0) s 0t (j) 1 1 exp( ξ jt ) 1 s jt (i) = s 0t (i) s jt(0) s 0t (0) exp ( 1[j = i] ξ it ) That is, the J + J 2 unknowns {s jt (i)}, can be expressed in terms of the J + 1 unknowns {s 0t (j)} and s 0t (0). The observed share equation gives us J restrictions: S jt = (1 ( ) s0t (0) r kt ) s 0t (j) 1 1 exp( ξ k jt ) 1 + s jt(0) s 0t (0) r it s 0t (i) exp ( ) 1[j = i] ξ it i And the final restriction, 1 k s jt(0) s 0t (0) = 0identifies the shares {s jt (i)}, conditional on the dynamic parameters Computational Simplicity We can further reduce the burden of solving for the J +1 parameters {s 0t (j)}in each market by transforming the restrictions into a quadratic function of two parameters in each market. Using the fact that r kt s 0t (k) exp ( ) 1[j = k] ξ kt = r kt s 0t (k) + ( exp( ξ jt ) 1 ) r jt s 0t (j) k k We can write 17

18 0 = [ exp( ξ jt ) 1 ] r jt s 0t (0) s 0t(j) 2 + s 0t (j) 0,k [ exp( ξ jt ) 1 ] (S jt r jt ) + 1 s 0t (0) r kt s 0t (k) r kt s 0t (k) 0,k and solve for {s 0t (t)} as a quadratic function of observable variables and the two parameters s 0t (0) and 0,k r kts 0t (k) Identification of the Dynamic Parameters As we have shown above how to decompose shares for any values of the dynamic parameters, we need additional restrictions to identify the dynamic parameters. In order to do so, we assume that any idiosyncratic product-time shocks, after allowing for a periodspecific fixed effect, are uncorrelated over time. These shocks may contain real demand shocks or measurement error from our data. That is, we assume η jt = φ t + κ jt, and Cov(κ jt, κ j(t+1) ) = Implementation To implement our estimator, we use a nested regression approach with the following steps: 1. First, pick values for δ, ξ, and ᾱ. 2. Calculate r jt = δs jt 1 for all periods except the first. 3. In each market, solve for s 0t (0) and 0,k r kts 0t (k) using the non-linear system of equations obtained previously. Find s jt (0) for each firm. 4. Run the regression implied by equation (10) using the {s jt (0)} obtained in the previous step. Calculate the correlation of the residuals Cor(ˆκ jt, ˆκ j(t+1) ). 5. Repeat 1-5 to find the dynamic parameters that set the correlation to zero. The regression for Equation (10) may involve instrumental variables and the use of panel data methods such as fixed effects. In our empirical application, we make use of both. The estimation methodology employs two tricks to speed up the computation of the dynamic model. First, the explicit formula for {s jt (0)} means that the non-linear solver only has to find two parameters, s 0t (0) and 0,k r kts 0t (k), for each market-period. The quadratic form for the remaining unknowns results in fast calculation. Second, the linear form for the nested regression allows for a quick calculation of the inner part of the routine. 18

19 5.2 Second-Stage: Supply-Side Estimation [To be completed] Given our demand estimates, we construct the components in each firm s Bellman equation from (7). This allows us to directly estimate the derivative of the continuation value, βe[v (r, c ) r, c, p]. We use the expected and unexpected costs estimated in our passthrough section to estimate a model-restricted function for the continuation value. We then compare our estimates to an approach that implements supply-side assumptions in estimation in the manner of Bajari et al. (2007), which first constructs the continuation values and uses the continuation values to estimate the dynamic parameters. We use the error in the first-order conditions to evaluate the importance of successfully estimating expectations. 6 Estimation Results For the empirical application, we implement the methodology described in Section (5.1). Specifically, we supplement our EIA-adjusted weekly brand-county share measures with the average prices for the brand in a week-county. To reduce the occurrence of zero shares, which do not arise in the logit model, we use a simple linear interpolation for gaps up to four weeks. For any gap greater than four weeks, we assume the station was not in the choice set for that gap. We drop any observations that have missing prices, missing shares, or missing shares in the previous week. This includes dropping the first week of data, for which we do not have previous shares. We retain 177,032 out of 194,275 observations. Our regression equation takes the following form: ln(s jmt /s 0mt ) = ξ jm αp jmt + γn jmt + η jmt Here, we have added the subscript m to denote the market (county) and M to denote the larger region (state). We have shares and prices at the brand-county-week level. We model a within county share as depending on the number of stations for the brand in that market, N jmt, after accounting for a brand-state fixed effect, ξ jm. We allow for endogeneity in pricing behavior by instrumenting for p imt with predicted deviations in wholesale costs, where the predictions are obtained from a regression of deviations of wholesale costs (from the brand-state average) on the interaction of US production of crude oil (obtained from EIA) with the average wholesale cost for the brand in the state. 12 This gives us brand-statespecific time variation in our instrument, and it is plausibly tied to variation in the wholesale cost and not linked to demand. We chose this measure, rather than instrumenting directly with brand-state wholesale costs, to account for the possibility that wholesalers respond to local, brand-specific demand shocks. 12 Our measure of the average brand-state wholesale cost is the fixed effect obtained by a regression of wholesale costs on brand state and weekly fixed effects, thereby accounting for compositional differences in brand-states across time. 19

20 In addition to the instrument, we employ panel data methods to address other unobservables. We allow for the fact that η jmt may be correlated over time. We let η jmt = φ t + κ jmt, and estimate weekly fixed effects, {φ t }. We also estimate brand-state specific fixed effects, {ξ jm }. Once we incorporate these fixed effects, the exclusion restriction for a valid use of our instrument is that the brand-market-period specific shock κ jmt is uncorrelated with the instrument, after accounting for aggregate period-specific shocks and brand-state level differences. Table 4: Parameter Estimates (a) Dynamic Parameters Variable Parameter Value Lock-In δ Price ᾱ Intercept ξ (b) Static Parameters Variable Parameter Value Price α (0.700) Num. Stations γ (0.002) R (c) Implied Elasticities Mean p25 p50 p75 Free-Agent Locked-In Weighted Note: The dynamic parameters are obtained via an outer loop estimation. The static parameters are estimated from an IV logit regression in the inner loop. The estimates are reported in Table 4. We estimate that 81 percent of customers become locked-in week-over-week after purchasing from a particular station. These customers have low price sensitivity, as the net price sensitivity ᾱ + α is near zero. The average implied elasticity for the customers that are not locked is This is large in magnitude, and it implies that for a 1 percent increase in price (roughly 3 cents), the station will lose 15 percent of the free-agent customers. This high level of price sensitivity for a subset of retail gasoline customers seems plausible, as, anecdotally, some shoppers are known to go well out of the way to save a few cents per gallon. The average weighted elasticity, which weighs free-agent and locked-in customers by their relative proportions, is In contrast with 20

21 a static model, our dynamic model obtains relatively lower average elasticities, due to the presence of locked-in customers. A pure static model would estimate elasticities in line with the free-agent elasticity. For ease of interpretation of the dynamic parameters, we present summary statistics and graphs of the resulting shares. Table 5 displays the means and standard deviations for observed shares and the two components of the shares we identify. We also report the portion of the observed share that a firm realizes from the locked-in customers, which is 78 percent on average. Table 5: Summary Statistics of Shares Variable Name Mean S.D. Obs S jmt Observed Share ,032 s jmt (0) Free-Agent Share ,032 s jmt (j) Locked-In Share ,032 r jmt s jmt(j) S jmt Portion Locked-In ,032 In Figure 2, we display the time series of the average price and the average shares. The shares for the customers that are not locked-in, displayed in Panel (a) are negatively correlated with the price. Intuitively, customers shift to the outside option when prices are higher. On the other hand, the share of locked-in customers, displayed in Panel (b), track the prices very closely. This occurs because prices respond positively to demand shocks, as do the locked-in customers, who we find to be price inelastic. Table 3 displays the elasticities over time. 7 Conclusion We develop a model of dynamic demand that accounts for the slow adjustment of prices to changes in cost. The dynamics result from competing firms optimally setting prices to customers that may become loyal to their current supplier. Using data from retail gasoline markets, we first demonstrate that prices adjust slowly to cost changes, and that path of price adjustment depends upon whether the costs change is expected or unexpected. This finding demonstrates that importance of accounting for firm expectations when estimating pass-through and estimating a demand model that can accommodate these short-run dynamics. We derive a new estimator that can identify dynamic demand parameters using data on price, shares, and an instrument. Preliminary results suggest that about 80 percent of retail gasoline customers become locked in to the firm from which they currently purchase, and that these loyal consumers are extremely price insensitive. Conversely, we find that non-loyal customers are quite price sensitive. In future iterations, we plan to compare the 21

22 dynamic estimates to those obtained using BBL value function estimation, and evaluate the importance of correctly estimating firm expectations. Finally, we will compare our dynamic model to static demand estimation to quantify the importance of capturing short-run price dynamics when quantifying the welfare impacts of cost or tax changes. 22

23 Figure 2: Shares Over Time (a) Non-Locked-In Customers Share oct apr oct apr oct2015 Date Price Share: Non-Locked-In Average Price (b) Locked-In-Customers Share Price 01oct apr oct apr oct2015 Date Share: Locked-In Average Price 23

24 Figure 3: Elasticities Over Time (a) Free Agent and Locked-In Elasticity Locked-In oct apr oct apr oct2015 Date Elasticity: Free Elasticity: Locked-In Elasticity: Free (b) Mean Elasticity Elasticity Share 01oct apr oct apr oct2015 Date Elasticity (Weighted) Share 24

25 Figure 4: Ratio Locked-In Portion Locked-In Price 01oct apr oct apr oct2015 Date Portion Customers Locked-In Average Price 25

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