A Simple Model of Demand Accumulation

A Simple Model of Demand Accumulation Igal Hendel Aviv Nevo Northwestern University December 14, 2009 Hendel & Nevo (Northwestern University) A Simple Model of Demand Accumulation December 14, 2009 1 / 33

Motivation Demand estimation is a key part of: empirical work in IO and trade many anti-trust cases (e.g., mergers and price xing) For the most part the demand models used are static There is ample evidence of dynamic demand Neglecting demand dynamics may lead to inconsistent estimates In this paper we present a simple dynamic model to estimate demand/test for dynamics analyze supply We also explore common alternative "solutions" Hendel & Nevo (Northwestern University) A Simple Model of Demand Accumulation December 14, 2009 2 / 33

1 price.8 1 1.2 1.4 1.6 1.8 0 10 20 30 40 50 week Hendel & Nevo (Northwestern University) A Simple Model of Demand Accumulation December 14, 2009 3 / 33

Evidence of Demand Accumulation Pesendorfer (2002): aggregate demand depends on duration from previous sale Hendel and Nevo (2006a): demand accumulation and demand anticipation e ects HH frequency of purchases on sales correlated with proxies of storage costs when purchasing on sale, longer duration to next purchase (within and across HH) proxies for inventory is (i) negatively correlated with quantity purchased and (ii) the probability of purchasing Sun, Neslin and Srinivsan (2001), Erdem, Imai and Keane (2004), Hendel and Nevo (2006b) estimate structural models of consumer inventory behavior Extensive Marketing Literature: Post-Promotion Dip Puzzle (Blattberg and Neslin, 1990) Hendel & Nevo (Northwestern University) A Simple Model of Demand Accumulation December 14, 2009 4 / 33

Just in case you doubt... Table 1: Quantity of 2-Liter Bottles of Coke Sold S t 1 = 0 S t 1 = 1 S t = 0 247.8 199.4 227.0 S t = 1 763.4 531.9 622.6 465.0 398.9 Hendel & Nevo (Northwestern University) A Simple Model of Demand Accumulation December 14, 2009 5 / 33

Why are there "sales"? 1 A change in (static) cost, demand or retailer inventory cost 2 Search and mixed strategies (Varian, 1980; Salop and Stiglitz, 1982) 3 Retailer behavior: multi-category pricing 4 Intertemporal price discrimination (Sobel, 1984; Conlisk Gerstner and Sobel, 1984; Pesendorfer, 2002; Narasimhan and Jeuland, 1985; Hong, McAfee and Nayyar, 2002) 5 Intertemporal NL pricing: Hendel, Lizzeri and Nevo (2006); Hendel & Nevo (Northwestern University) A Simple Model of Demand Accumulation December 14, 2009 6 / 33

Implications for demand estimation Assume we use weekly data to estimate (static) demand Price variation is needed for demand estimation, but neglecting dynamics may be problematic econometric bias: 1 lagged quantities/prices omitted, can be correlated with the current price; 2 as we discuss below, "wrong" prices; SR vs LR elasticities regardless of the econometric bias static demand can recover SR elasticities, for most applications we want LR elasticities. Hendel & Nevo (Northwestern University) A Simple Model of Demand Accumulation December 14, 2009 7 / 33

Outline of a Model with Demand Accumulation Quadratic preferences U(q, m) = Aq q 0 Bq + m q = [q 1, q 2,..., q N ] and m is the outside good. Absent storage: linear demand system q t i (p) = α βp t i + γp t j Multi-period set up with storage: consumers can anticipate purchases for future consumption Denote purchases by x and consumption by q. Assume: A1: two prices: sale and non-sale A2: free storage A3: inventory lasts for T periods (for now assume T = 1) A4: proportion ω of consumers does not store Hendel & Nevo (Northwestern University) A Simple Model of Demand Accumulation December 14, 2009 8 / 33

Purchasing Patterns Consumers (who store) purchase for following period s consumption when the product is on sale For now assume product A1-A4 plus perfect foresight imply i is not storable four states de ned by sale last and current period SS, NN, NS and NS where S=sale and N=non-sale Purchases by storers 8 (1 ω)q i (pi >< t, pt i ) x i (p t 0 ) = (1 ω)(q >: i (pi t, pt i ) + q i (pi t, pt+1 i )) (1 ω)q i (pi t, pt+1 i ) Non-storers always contribute ωq i (p t i, pt i ) if NN SN NS SS Hendel & Nevo (Northwestern University) A Simple Model of Demand Accumulation December 14, 2009 9 / 33

Purchasing Patterns - Cross Price First, we need a measure for p t+1 i when consumers buy for next period We will use the realized future prices (perfect foresight) Could use expected future prices (rational expectations) Second, assume other products are storable In principle, interaction in storage costs however given our storage technology this is simple (see discussion below) In principle, state space more complex: depends on past sales of other products In practice, all we need is to "correct" the relevant cross price. For example: suppose NN after a sale of other product then q i (p t i, pt 1 i ) As a side, using current prices is the wrong control Hendel & Nevo (Northwestern University) A Simple Model of Demand Accumulation December 14, 2009 10 / 33

Comments on the Model s Assumptions By assuming 2 prices we can simply look at sale vs non-sale Two-price support assumption is restrictive not essential, really need: periods of accumulation to be well de ned fraction of non-storers, ω, does not depend on price More complicated pricing can be accommodated by tweaking the model Hendel & Nevo (Northwestern University) A Simple Model of Demand Accumulation December 14, 2009 11 / 33

Comments on the Model s Assumptions (cont) Storage technology allows us to simplify the state space (relative to inventory model): there are no left overs to carry as a state variable no need to keep long history of prices (alternatively, we could get this from a rst order Markov assumption on the price process) No need to compute how much consumers store (when they store) It also detaches the storage decision of di erent products The link between products is captured by e ective prices Storage technology: taken literally, ts perishable products (or cost of transporting the products) Easy to allow for T > 1 can also allow for heterogeneity across consumers in T slightly more complex Hendel & Nevo (Northwestern University) A Simple Model of Demand Accumulation December 14, 2009 12 / 33

Static Demand Estimation If price changes in the data are permanent static estimation will yield consistent estimates of LR demand If price changes are temporary then: The own price e ects are over-estimated: Response to a sale attributed to consumption, not storage (and a reduction in future purchases) The price increase after a sale coincides with a decline in purchases, which is mis-attributed as a decline in consumption The cross product e ects are understated instead In some periods the current and e ective prices di er Consider the period after a sale of a competing product The "e ective" (cross) price is the sale period purchase price The observed price is higher, which is not accompanied by a decline in purchases We can view problem as using the wrong price or missing control Hendel & Nevo (Northwestern University) A Simple Model of Demand Accumulation December 14, 2009 13 / 33

How Well Does the Model Approximate an Inventory Model? We simulated data from a full-blown dynamic model: linear demand β = 4 2 prices: prob of sale is 20% ω = 0.5 never store (behave statically), others solve DP storage cost linear in inventory rational expectations about future prices and demand shocks Varied the cost parameter Estimated the various models using the simulated data Hendel & Nevo (Northwestern University) A Simple Model of Demand Accumulation December 14, 2009 14 / 33

Figure 2: Optimal Dynamic Behavior as a Function of Storage Costs Hendel & Nevo (Northwestern University) A Simple Model of Demand Accumulation December 14, 2009 15 / 33 Simulated Consumption and Storage 14 12 10 8 6 Storage Consumption 4 2 0 0.08 0.11 0.14 0.17 0.2 0.23 0.26 0.29 0.32 0.35 0.38 0.41 0.44 0.47 0.5 Storage Cost

Figure 3: Percent Bias in Estimated Slope Parameter Hendel & Nevo (Northwestern University) A Simple Model of Demand Accumulation December 14, 2009 16 / 33 Bias in Price Coefficient Percent Bias 180 160 140 120 100 80 60 40 20 Fix T=1 OLS Fix T=2 0 20 0.08 0.11 0.14 0.17 0.2 0.23 0.26 0.29 0.32 0.35 0.38 0.41 0.44 0.47 0.5 Storage Cost

Table 2: Monte Carlo Simulations Mean β MSE β Simulated Data OLS T = 1 T = 2 OLS T = 1 T = 2 c Consumption Storage N=100 0.08 4.80 16.20 10.57 5.82 4.73 43.98 3.59 0.66 0.17 4.19 9.31 8.36 4.89 4.07 19.26 0.92 0.13 0.29 3.64 5.80 6.50 4.08 4.30 6.39 0.14 0.35 0.38 3.54 5.05 6.16 3.91 4.37 4.75 0.15 0.38 0.50 2.97 0 3.99 4.00 3.99 0.05 0.20 0.19 Hendel & Nevo (Northwestern University) A Simple Model of Demand Accumulation December 14, 2009 17 / 33

How Do We Recover Preferences? For simplicity, assume a single product The model suggests the following purchasing patterns: 8 q(p >< t ) NN x(p t ωq(p ) = t ) SN (2 ω)q(p >: t if ) NS q(p t ) SS two di erent ways to recover parameters from the data use all periods and impose the "accounting" of the model "timing" restrictions: use periods without demand anticipation (NN and SS) Note, that the model is over-identi ed Hendel & Nevo (Northwestern University) A Simple Model of Demand Accumulation December 14, 2009 18 / 33

Back to Table 1 Table 1: Quantity of 2-Liter Bottles of Coke Sold S t 1 = 0 S t 1 = 1 S t = 0 247.8 199.4 227.0 S t = 1 763.4 531.9 622.6 465.0 398.9 Static estimate: β Static = x S x N p = Timing Restriction: β ti min g = x SS Accounting Restrictions: 623 227 0.4 = 988 x NN = p SS p NN 532 248 0.4 = 710 x NN /x SN = ω = 0.8 ; ωx SS x SN = ωβ p ) β = 708 x NS /x SS = 2 ω ) ω = 0.57 ; x NS (2 ω)x NN = (2 ω)β p ) β = 713 Hendel & Nevo (Northwestern University) A Simple Model of Demand Accumulation December 14, 2009 19 / 33

Estimation As the previous example shows the model is over-identi ed (even more so if there is within state price variation) Estimate the model by minimizing the distance b/ observed and predicted purchases The estimation controls for prices (own and competition), can control for other factors, and account for endogenous prices. Can also use GMM/IV Need to account for store xed e ects Can enrich model to allow for T > 1, more types, heterogeneity and more exible demand systems; Hendel & Nevo (Northwestern University) A Simple Model of Demand Accumulation December 14, 2009 20 / 33

An Empirical Example: Demand for Colas Data: Store-level scanner data weekly observations 8 chains in 729 stores in North East Due to data problems will only use 5 chains focus on 2 liter bottles of Coke, Pepsi and Store brands We estimate linear demand, allowing for store xed e ects A sale is de ned as a price $1 Hendel & Nevo (Northwestern University) A Simple Model of Demand Accumulation December 14, 2009 21 / 33

Distribution of the Price of Coke Percent 0 10 20 30.5 1 1.5 2 Price of Coke Hendel & Nevo (Northwestern University) A Simple Model of Demand Accumulation December 14, 2009 22 / 33

Descriptive Statistics Table 3: Descriptive Statistics % of variance explained by: Variable Mean Std chain week chain-week Q Coke 446.2 553.2 5.6 20.4 52.5 Q Pepsi 446.0 597.8 2.8 24.4 46.7 P Coke 1.25 0.25 7.1 29.7 79.9 P Pepsi 1.19 0.23 7.5 30.7 79.8 Coke Sale 0.30 0.46 6.4 30.0 86.6 Pepsi Sale 0.36 0.48 9.3 29.2 89.0 24,674 observations from ve chains. Hendel & Nevo (Northwestern University) A Simple Model of Demand Accumulation December 14, 2009 23 / 33

Demand for Coke Table 4: Demand for Coke FE Timing Only All Restrictions Di erent slopes Rational Exp (1) (2) (3) (4) (5) (6) P Coke -1428.2-743.4-967.4-938.9-522.4-1273.1-767.8 (11.1) (11.8) (11.2) (11.1) (35.1) (16.7) (8.8) P Pepsi 66.5 191.6 82.8 150.1-73.1 145.01 195.4 (11.9) (10.9) (11.0) (11.4) (36.4) (21.3) (12.3) ω (fraction non- storers) 0.53 0.53 0.35 0.57 (0.01) (0.01) (0.01) (0.01) Cross price Corrections No Yes Yes Yes Hendel & Nevo (Northwestern University) A Simple Model of Demand Accumulation December 14, 2009 24 / 33

Demand for Pepsi Table 5: Demand for Pepsi FE Timing Only All Restrictions Di erent Slopes Rational Exp (1) (2) (3) (4) (5) (6) P Coke -20.9 71.8 62.8 106.6-246.2 216.7 140.0 (12.2) (11.0) (10.5) (10.5) (26.9) (13.3) (12.1) P Pepsi -1671.3-994.0-1016.5-996.9-341.3-1255.0-762.3 (13.1) (15.8) (11.6) (11.6) (29.3) (14.8) (8.9) ω (fraction non-storers) 0.44 0.44 0.28 0.47 (0.01) (0.01) (0.01) (0.01) Cross Price Corrections No Yes Yes Yes Hendel & Nevo (Northwestern University) A Simple Model of Demand Accumulation December 14, 2009 25 / 33

Implications Static Pricing Demand estimates are typically used to compue markups from a (static) FOC Static estimates margin of 25% for Coke and 22% percent for Pepsi, implies a marginal cost of 0.94 for Coke and 0.92 for Pepsi. Dynamic estimates (col 4): margins are 38% implied marginal costs are 0.77 for Coke and 0.73 for Pepsi. But why use a FOC from static supply if demand is dynamic Hendel & Nevo (Northwestern University) A Simple Model of Demand Accumulation December 14, 2009 26 / 33

Implications Dynamic Pricing Dynamic pricing by monopolist ps and p NS monopoly prices that max static pro ts from selling to the populations of storers and non-storers pnd the monopoly price of a non-discriminating monopolist. The monopolist will pick a pair of prices, p and p, to maximize (ωq NS (p) + 2(1 ω)q s (p))(p c) + ωq NS (p)(p c) If pns > p S then p = p NS while p is the price charged by a non-discriminating monopolist who faces demand ωq NS (p) + 2(1 ω)q s (p), ps < p < p ND. Optimal pricing price cycle is of length T + 1. Dynamic estimates with hetro (col 5) non-sale margins 61% and mc is 0.54. sale margin 43% and mc is 0.53. Hendel & Nevo (Northwestern University) A Simple Model of Demand Accumulation December 14, 2009 27 / 33

Extensions: Rational Expectations consumer maximizes E t (U t+1 (q, m)) when Coke on sale and Pepsi not, decide how much Coke to purchase (for t + 1) solution of 3 FOC: q C, q P and q P q C given by α βpi t + γe (p t+1 i.) q P and q P α βηpi t + γp t i + ζe (pt+1 i.) linear in own price (but di erent coe ) same cross price e ect depends on actual and expected cross price take q C as given (since it was decided in the pervious period before Pepsi prices where revealed). de nition of the state involves the prices at t and t products. e.g., SN, Pepsi NN, demand for Coke is q(p C S, pp NS ) + qc. In principle 16 states, e ectively 8 states. Estimates (col 6) very similar to PF 1 of both (all) Hendel & Nevo (Northwestern University) A Simple Model of Demand Accumulation December 14, 2009 28 / 33

Extensions: Discrete Choice the utility consumer i gets from product j is given by u ijt = δ ij α i p jt + ε ijt Consume single unit; PF of prices and ε Demand by storers 8 >< x i (p t ) = >: αp jt e δ j k e δ k α min(p kt 1,p kt ) 0 ( e δ j αp jt k e δ k α min(p kt 1,p kt ) + e δj αpjt k e δ k α min(p kt,p kt+1 ) ) e δ j αp jt k e δ k α min(p kt,p kt+1 ) Can be extended to more exible demand (e.g. BLP) RE of prices: the consumer considers the option of waiting if NN SN NS SS Hendel & Nevo (Northwestern University) A Simple Model of Demand Accumulation December 14, 2009 29 / 33

Alternative Corrections: Lagged Prices Include lagged prices as additional regressors The LR e ect is computed by tracing the e ect of a price change The e ect of lagged prices could be complex (might depend on the current state) The model helps guide the correct way to control for lags Hendel & Nevo (Northwestern University) A Simple Model of Demand Accumulation December 14, 2009 30 / 33

Aggregating Purchases Over time A common approach to deal with inventory is to aggregate data over time. For example, from weekly to monthly In our example aggregation can solve the problem only under some special conditions. In general, will not work (see example in the Appendix) unit values are the wrong price to get correct prices need a model Hendel & Nevo (Northwestern University) A Simple Model of Demand Accumulation December 14, 2009 31 / 33

Demand for Coke: Alternative Corrections Table 6: Demand for Coke Alternative Corrections FE Our lag 1 p lag 4 p agg bi-w agg month (1) (2) (3) (4) (5) (6) P Coke -1428.2-938.9-1131.3-1239.3-1130.2-911.2 (11.1) (11.2) (15.1) (22.9) (9.8) (14.5) P Pepsi 66.5 150.1 63.3-252.5-31.5-9.0 (11.9) (11.4) (17.1) (25.6) (11.1) (16.0) Hendel & Nevo (Northwestern University) A Simple Model of Demand Accumulation December 14, 2009 32 / 33

Concluding Comments We o er a simple model to deal with dynamics that is easy to estimate Our simple model can be generalized to allow: more types longer periods of storage more exible demand systems Seems to work well on Simulated data Cola data Estimated demand parameters are consistent with sales being driven by PD Manageable supply side implies higher markups because: elasticities are lower dynamic pricing implies higher markups Hendel & Nevo (Northwestern University) A Simple Model of Demand Accumulation December 14, 2009 33 / 33