Lecture Notes: Brand Loyalty and Demand for Experience Goods

Transcription

1 Lecture Notes: Brand Loyalty and Demand for Experience Goods Jean-François Houde Cornell University November 14,

2 Demand for experience goods and brand loyalty Introduction: Measuring state dependence in consumer choice behavior. In marketing and economics it is frequently observed that demand for some products exhibit time dependence: Examples: Switching cost Brand loyalty Pr(d ijt = 1 d ijt = 1) > Pr(d ijt = 1 d ijt = 0). Persistent unobserved heterogeneity Many papers have be written to empirically distinguish between true statedependence and unobserved heterogeneity (e.g. Keane (1997)): Empirical challenges: U ijt = X ijt β + λh ijt + ɛ ijt (1) where ɛ ijt = ρɛ ijt 1 + ν ijt t 1 H ijt = g(τ)d ijτ τ=t 0 Initial condition problem (i.e. d t0 is endogenous and/or unobserved). The presence of persistent unobserved heterogeneity can generates spurious state-dependence. Multi-dimension integration when ɛ ijt are correlated accross options and time: Pr(d i d t0 ) = Pr(U ijt > U ikt, k j, d ijt = 1, t = 1...T ) = Pr(ɛ ijt ɛ ikt > (X ijt X ikt )β (H ijt H ikt )λ, k j, d ijt = 1, t = 1...T ), i.e. dimension is T (J 1). Cannot be evaluated with standard methods. We must use simulation. 2

3 GHK Simulator Simpler cross-sectional example with 4 choices: U ij = X ij β + ɛ ij, ɛ ij N(0, Ω) (2) Then the probability of choosing option 4 is: Pr(d i4 = 1) = Pr(ɛ ij ɛ i4 < (X ij X i4 )β, k 4) = df (ɛ i1 ɛ i4, ɛ i1 ɛ i3, ɛ i3 ɛ i4 ) A 1 A 2 A 3 where A j = {ɛ ij ɛ i4 < (X ij X i4 )β} Lets redefine the variables to compute the probability of choosing option 4: ν ij = ɛ ij ɛ i4 Xij = X ij X i4 ν N(0, Σ), Σ 3 3 = C C ν = C η, η N(0, I). Standard Monte-Carlo integration (i.e. Accept-reject): 1. Draw M vectors η m i (0, I), 2. Keep draw m if C η m i A 1 A 2 A 3. Otherwise reject. 3. Compute simulated choice probability: Problems and limitations: Pr(d i4 ) = Number accepted draws M (3) Non-smooth simulator (i.e. cannot use gradient methods) Require a high number of draws to avoid P r(d i4 ) = 0. 3

4 If the dimension of integration is large: infeasible (e.g. Panel data). We can alleviate the non-smooth problem by smoothing the Accept/Reject probability: Pr(d i4 ) = 1 M...still very bias if M is too small. m k exp(( X ik β νm ij )/ρ) (4) The GHK simulator avoids the main problems of the standard MC-AR method by drawing only from the accepted region. Recall that: C = c 11 c 12 c 13 0 c 22 c c 33 In order to compute Pr(d i4 = 1) we proceed sequentially: 1. Draw νi1 m: Compute Φ i1 = Pr(ν i1 < Xi1 β). Draw ηm 1 η Φ( X i1 c 11 ) How? (a) Draw λ i U[0, 1] (b) Set λ i1 = λ i Φ i1 (c) Set η m i1 = Φ 1 (λ i1 ) (d) Finally ν i1 = c 11 η m i1 2. Draw ν m i2 from a truncated normal (conditional on νm i1 ): η m 2 ν2 m = c 12 η1 m + c 22 η 2 < Xi2β ( X Φ i2 β c 12 ηi1 m ) Φ i2 c 22 ( Thus we first draw ηi2 m from Φ X i2 β c 12 ηi1 m νi2 m = c 12ηi1 m + c 22ηi2 m. 4 c 22 ) from a truncated normal: as before, and compute

5 3. Compute Φ i3 similarly. 4. Compute Pr(d i4 = 1): Pr(d i4 = 1) = Advantages of the GHK: 1 ( X Φ i1 β ) ( X Φ i2 β c 12 η m ) i1 M c m 11 c 22 ( X Φ i3 β c 12 ηi1 m c 23ηi2 m ) c 33 Highly accurate even with high dimension integrals Differentiable Require fewer draws In order to applied to panel data with AR(1) correlation in the ɛ ij and arbitrary correlation across options we need to simulate the probability of observing a sequence of choices {j it } t=1...t. The Algorithm is the same... just longer (i.e. there are (J 1)T sequential draws to make for each m and i). To see this, let redefine the variables in the following way: Ũ kt = U kt U jt t ɛ kt = ɛ kt ɛ jt t Where, ɛ N(0, Σ), Where Σ = C C is a (J 1)T (J 1)T covariance matrix appropriately transformed to reflect the vector of choices j t, t = 1...T. With this transformation, i j it if Ũkt 0, k and ɛ i = C η i. 5

6 To compute Pr(j i ): Period 1: Draw ɛ m i1 : 1: Draw η m i11 from a truncated normal s.th: Ũ i11 (η m i11) < 0. 2: Draw η m i21 from a truncated normal s.th: Ũ i21 (η m i11, η m i21) < j i1 : Skip ν m ij i Period t: Draw ɛ m it : 1: Draw η m i1t from a truncated normal s.th: Ũ i11 (η m i11,.., η m ijt 1, η m i1t) < j i1 1: Draw η m ij it 1t from a truncated normal s.th: Ũ iji1 11(η m i11,.., η m ijt 1, η m i1t,..., η m ij it 1t < 0). j i1 : Skip η m ij it t... Finally compute Pr(j i ) by taking the product of each component and averaging over m. 6

7 Back to Keane (1997): Modeling heterogeneity and state dependence in consumer choice behavior Keane (1997) estimates many different specifications of equation 1 using SMS with GHK and finds strong evidences of true state-dependence for Ketchup. The goal is to estimate a statistical model of product choice that flexibly account for state-dependence and rich unobserved preference heterogeneity Discrete-choice model: U ijt = x it β j + p jt (φ 0 + x it φ 1 + ν i ) + GL(H ijt, α)λ + A u jtω ijt { αgl(h ij,t 1, α) + (1 α)d ijt If t > 1, State dependence: GL(H ijt, α) = 0 If t = 1. Unobserved heterogeneity: A u Ω ijt = L j W i + κγ ij + δ ijt Decomposition of unobserved heterogeneity: Consumer heterogeneity: κγ ij where Γ ij N(0, I). Product taste heterogeneity: L j W i = Product FE Unobserved taste for products, where W i N(0, I). Time-varying preferences: δ ijt = ρp j ξ ij,t 1 + ρɛ ij,t 1 + η ijt, where P j is a vector of product FE, ξ ijt N(0, 1), η ijt N(0, 1) and ɛ ijt N(0, 1). Estimation: Method of Simulated Moments (MSM) m(θ) = N i=1 T t=1 J j=1 W ijt [d ijt ˆP ] GHK (d ijt d ij1,..., d ij,t 1, X i, A, P, θ) = 0 Where, ˆPGHK (d ijt d ij1,..., d ij,t 1, X i, A, P, θ) = ˆP GHK (d ij1,..., d ij,t ) ˆP GHK (d ij1,..., d ij,t 1 ) 7

8 Estimation Results: Basic Specifications Model 2: Nearly 50% of the error is due to time-invariant consumer heterogeneity (κ) Model 3-4: Adding state-dependence reduces the importance of unobserved heterogeneity to 30%, and heterogeneity in taste across brand (L j ) further reduces it to less than 20% (between 11 18%). 8

9 Estimation Results: Richest Specifications Model 12: Adding product-specific time-varying heterogeneity suggest rich dynamic switching/loyalty patterns. The contribution time-invariant consumer heterogeneity is roughly 10% of the error. Model 13-16: The AR(1) coefficient is significantly different from zero (reject constant correlation in taste over time). Model 13-16: The state-dependence parameters (λ = 1.3 and α = 0.9) are fairly stable across specifications, and suggest that brand loyalty is an important phenomenon. A one-time switch is equivalent to a short-run price increase of roughly 5 cents. Model 16: Accounting for rich heterogeneity is crucial to accurately measure brand loyalty. one-time switch leads to the following drop in utility: A t t + 1 t + 2 t + 10 Model Model Model 16: Allowing for state-dependence and serial-correlation substantially reduce the importance consumer heterogeneity (κ): between 4% and 22% of the residual variance. 9

10 Ackerberg (2001): Informative versus persuasive advertising Question: How to distinguish between informative and persuasive advertisement? Two views: Information: Ads inform consumers about the quality and/or price of products Persuasion: Ads interact with consumption of brands (i.e. complementarity), and create a prestige effect that increases demand for the good. Why do we care? The two theories have different implications for the welfare effect of advertising (wasteful investment?) From an antitrust point of view, the two theories have different implications for the effect of advertising on market-power. Informative advertising typically has a pro-competitive effect, while persuasive advertising can create barriers to entry. Identification problems: Even with the best natural experiment, the effect of ads on demand does not tell us much about the underlying mechanism. Solution: Exploit fairly long panel of consumers repeated purchases to measure the effect of advertising on experienced and inexperienced consumers. Initial conditions? Usually, the experience level of consumers is highly endogenous. Ackerberg solves this problem by studying the life-cycle of a new product (Yoplait 150) from its introduction to its steady-state. 10

11 Simple test: Effect of ads on aggregate new purchases versus repeated purchased. Discrete-choice model: Repeated logit with random effect { α i + x i β x β p p jt + ɛ ijt If j = 1 U ijt = ɛ i0t If j = 0 Likelihood function: Results: L(d i, x i θ) = T t=1 Pr(d it x it, α i, θ)f(α i )α i 11

12 Erdem and Keane (1996): Decision-Making under uncertainty Estimate a dynamic bayesian learning model of demand for detergent using the A.C. Nielsen public scanner data set (i.e 3000 households over 3 years). The data-sets for detergent, ketchup, margarine and canned soup are available at: Why? Market with frequent brand introduction. Reasonable to assume that consumers learn about the quality of the product only through experience and advertising. Provide and economic interpretation for brand loyalty: If consumers are risk averse experiencing too frequently is costly (i.e. endogenous switching cost). Measure the information content of advertising messages (see also Ackerberg (2001) and Ackerberg (2003)). 12

13 Model Finite Horizon DP problem: V j (I(t), d j = 1) = V (I(t)) = { E[U j (t) I(t)] + e jt + βe[v (I(t + 1)) I(t), d j (t)] E[U j (T ) I(T )] + e jt max V j(i(t), d j ) d j,j=1..j if t < T Where the expectation is taken over the evolution of the information set I(t) and the idiosyncratic shock e jt. Components of the expected utility: 1. Atribute of good j if experienced at t: A E jt = A j + δ jt. 2. Utility after realization of δ jt and e jt : Where 3. Expected utility: Outside options: U jt = ω p P jt + ω A A E jt ω A ra E jt2 + ejt r = risk aversion coefficient e jt = logit utility shock P jt = Price of j (stochastic) else E[U jt I(t)] = ω p P jt + ω A E[A E jt I(t)] ω A re[a E jt2 I(t)] [ ] ω A re (A E jt E[A E jt I(t)]) 2 ] I(t) + e jt E[U 0t I(t)] = Φ 0 + Ψ 0 t + e 0t E[U NP t I(t)] = Φ NP + Ψ NP t + e NP t 13

14 Signals and components of the information set: 1. Experience: A E jt = A j + δ jt, δ jt N(0, σδ) 2 The experience signals are unbiased: δ is mean zero. 2. Priors on A j : A j N(A, σ ν (0) 2 ). 3. Advertising message (with probablity p S j estimated from the data): S jt = A j + η jt, η jt N(0, σ 2 η), The information content of advertising messages is measured by σ 2 η Advertising is exogenous and strictly informative: η jt is mean zero. Bayesian Updating: Update expectation about good j s attribute E[A j I(t)] = E[A j I(t 1)] + d jt β 1jt [ A E jt E[A E jt I(t 1)] ] +ad jt β 2jt [ Sjt E[S jt I(t 1)] ] Where the updating weights are given by: β 1jt = σ2 ν j (t) σ 2 ν j (t) + σ 2 δ & β 2jt = σ2 ν j (t) σ 2 ν j (t) + σ 2 η From the econometrician point of view (since A j is a parameter), we can rewrite the problem in terms of expectation errors ν j (t) = E[A E jt I(t)] A j. This generates a first-order markov process payoff in relevant state variables: With ν j (0) = A A j, j. ν j (t) = ν j (t 1) + d jt β 1jt [ νj (t 1) + δ jt ] +ad jt β 2jt [ νj (t 1) + η jt ] (5) 14

15 Similarly, the precision of signals is updated using the following markov process: [ 1 σ νj (t) = σ ν (0) + s t d js s t σδ 2 + ad ] 1 js (6) σν 2 Timing: t = 0 Purchasing decision based on ν j (0). New signals: δ j0 (if d j0 = 1) and η j0 (if ad j0 = 1). Update ν j (1) according to equation 5 t = 1 Purchasing decision based on ν j (1). New signals: δ j1 (if d j1 = 1) and η jt (if ad j1 = 1). Update ν j (2) according to equation 5... Therefore at any period t the payoff relevant state variables are: { I(t) = d js, } ad js, ν jt s t 1 s t 1 } {{ } Discrete }{{} Continuous j=1...j Solution/Estimation: Nested fixed-point estimation algorithm where the DP is solved by backward induction. Challenges: Size of the state-space: Impossible to solve exactly. Choice probabilities: ( ) exp EU j (I(t)) + βe[v (I(t + 1), t + 1) Pr j (I(t)) = ( )f(ν)dν k exp EU j (I(t)) + βe[v (I(t + 1), t + 1) Integration is complicated by the fact that ν jt is serially correlated: Must integrate the sequence of past ν js s using simulation method: simulate M sequences of ν jt, just like in Hendel and Nevo (2006). 15

16 Initial condition problem: Do not observe the initial level of purchasing history and attribute expectation (problem if the products are not newly introduced as in Ackerberg (2003)). Solution: Set I(0) = {0, 0, ν j0 } j=1...,j, and simulate the model for the first two years of the data. Use the last two years for the estimation (i.e. T = 100 weeks). 16

17 Solution Method: Keane and Wolpin (1994) General Idea: Solve the value function exactly only at a subset I (t) of the states and interpolate between them using least-squares to compute EV (I(t)) at I(t) / I. Backward induction algorithm for a fix grid I : T : 1. Calculate EV T (I(T )) for all I(T ) I : { EV T (I(T )) = max EU jt (I(T )) + e jt }df (e) j ( ( ) ) = log exp EU jt (I(T )) 2. Run the following regression: j EV T (I(T )) = G(I(T ))θ T + ν = ÊV T(I(T )) + u, where G(I(T )) is vector containing flexible transformations of the state variables, where u is a regression error. Note: For the approximation to work, the R 2 of the regressions must be very high. Alternative methods exist to improve the quality of the interpolation (e.g. kernels, polynomials, etc). T 1: 1. Draw M random variables: {δ m 1,..., δ m J, ηm j,...ηm J, adm 1,..., ad m J }. 2. For each state I(T 1) I compute the expected value of choosing brand j in T 1: E [ V T (I(T )) I(T 1), d jt 1 = 1 ] = 1 M ÊV T (I m (T )) Where I m (T ) is the state corresponding to the mth draw and d jt 1 = 1. If I m (T ) I use the exact solution, otherwise use G(I m (T ))θ T. m 17

18 3. For each I(T 1) I calculate V (I(T 1)): ( EV T 1 (I(T 1)) = log exp ( EU jt 1 (I(T 1)) + 4. Run the following regression: j βe [ V T (I(T )) I(T 1), d jt 1 = 1 ])) EV T 1 (I(T 1)) = G(I(T 1))θ T 1 + ν,... Repeat steps 1-4 for the remaining periods t = T 2,..., 0. Note: In Erdem and Keane (1996) G(I(t)) includes the expected attributed level of each brand and the perception error variances. To estimate the model, the model needs to be solve using the Interpolation/Simulation algorithm for each parameter values. The choice probabilities are computed by simulating a sequence of states for each households. 18

19 Results Risk-aversion coefficient is large and negative: Important switching cost of experimenting. The variance of the advertizing signal is much much larger than the variance of the experience signal: Consumers don t get much from TV ads! Initial priors are very precise: Little uncertainty in this market. Forward looking model fits better... 19

20 Crawford and Shum (2005): Uncertainty and Learning in Pharmaceutical Demand Quote: Across several treatment lengths and spell transitions, there is a marked decreasing trend in the switching probability at the very beginning of treatment. Two forces explain these switching probabilities in the bayesian learning model: Initial experimentation and risk aversion. Forward looking behavior: Incentive for patients (or doctors) to acquire more information by experimenting. 20

21 Model A treatment is characterized by two match values (contrary to only one in Erdem and Keane (1996)): µ jn Symptomatic or side-effects (enters the utility directly) ν jn Curative properties (enters the recovery probability). Signals about the match values if j use drug n at t: x jnt N(µ jn, σ 2 n) y jnt N(ν jn, τ 2 n) Initial priors about the match values: µ jnt N( µ nk, σ 2 n) ν jnt N( ν nk, τ 2 n) Where k = indexes the severity type of patients (learned perfectly by the initial diagnostic). Expected Utility (CARA): u(x jnt, p n, ɛ jnt ) = exp( rx jnt ) αp n + ɛ ( jnt ) ẼU(µ jn (t), ν jn (t), p n, ɛ jnt ) = exp rµ jn (t) + 1/2r 2 (σn 2 + V jn (t)) αp n + ɛ jnt = EU(µ jn (t), V jn (t), p n ) + ɛ jnt Recovery probability follow a markov process: ( ) hj (t 1) 1 h j (t 1) + d jnt y jnt h j (t) = ) d jnt y jnt ( hj (t 1) 1 h j (t 1) Updating rule for the beliefs regarding the symptomatic and curative match value µ jn (t + 1) and ν jn (t + 1) by equation (7) and (8) (same as in Erdem and Keane). 21

22 State space: s jt = Where l jn (t) = s<t d jnt. { } µ jn (t), ν jn (t), l jn (t), h j (t) n=1...5 Value Function: Infinite horizon problem with absorbing state (i.e. recovery) V (s) = max EU(s) + ɛ n + βe [ (1 h(s ))V (s ) d n = 1, s ] n [ = log exp ( EU(s) + βe [ (1 h(s ))V (s ) d n = 1, s ])] n Solution method: Value function iteration with interpolation and simulation (i.e. Keane and Wolpin (1994)) 1. Define a discrete grid S S. 2. For each state s S make an initial guess at the value function V 0 (s). 3. Run regression: V 0 (s) = G(s) θ 0 + u s 4. Draw M random signals {x m jn, ym jn } 5. Compute the expected value of choosing dug n for each s S : [ ] E V (s d n = 1, s = 1 (1 h(s m ))V 0 (s m ) M Where s m is state corresponding to the random draw m and drug n being chosen, and V 0 (s m ) is evaluated with the interpolation equation if necessary. 6. Update the value function for each s S : [ V 1 (s) = log exp ( [ ] ) ] EU(s) + βe V (s d n = 1, s n 7. Repeat step 3-6 until convergence of the value function at the grid points. 22 m

23 Results Large risk aversion coefficient: Important switching cost Types are horizontally differentiated Policy experiments: Concentration increases in the level of uncertainty (i.e. smaller switching costs) The pooling of types (i.e. poor initial diagnostic) decreases concentration (i.e. products become less differentiated). 23

24 Figure 1: Results 24

25 References Ackerberg, D. (2003). Advertising, learning, and consumer choice in experience good markets: A structural empirical examination. International Economic Review 44, Ackerberg, D. A. (2001, Summer). Empirically distinguishing informative and prestige effects of advertising. RAND Journal of Economics 32 (2), Crawford, G. and M. Shum (2005). Uncertainty and learning in pharmaceutical demand. Econometrica 73, Erdem, T. and M. P. Keane (1996). Decision-making under uncertainty: Capturing dynamic brand choice processes in turbulent consumer goods markets. Marketing Science 15 (1), Hendel, I. and A. Nevo (2006). Measuring the implications of sales and consumer stockpiling behavior. Econometrica 74 (6), Keane, M. P. (1997). Modeling heterogeneity and state dependence in consumer choice behavior. Review of Economics and Statistics 15 (3), Keane, M. P. and K. I. Wolpin (1994). The solution and estimation of discrete choice dynamic programming models by simulation and interpolation: Monte carlo evidence. The Review of Economics and Statistics 76 (4),