A structural model of purchases, returns, and return-based targeting strategies

Abstract

Despite the importance of product returns, several less well understood issues still remain. In this paper, we propose a dynamic structural model that characterizes consumers’ sequential shopping trips for purchases and/or returns. We calibrate our proposed model to individual panel data in which we observe each consumer’s sequential shopping trips, including the number of products purchased and/or returned at each trip. Through our simulations, we find that product returns can also lead to desirable outcomes for retailers and that the ability to return purchased products incentivizes consumers to make more shopping trips and thus purchase more. Motivated by these observations, we also examine two possible return-based targeting policies: (1) targeted reduction of return cost and (2) targeted price promotion on return trips.

Appendix: Implementation

Computational details

Here, we specify our implementation of the IJC framework. We denote the parameter set as θ_i, which includes (1) shopping trip fixed cost (v_i), (2) parameters related to purchase utility (i.e., mean (d_i) and variance of match value signals in-store (ϑ_i) and from usage experience (ϑ_{i, r})), (3) decay parameters related to purchase utility (δ_i) and return cost (R_i), and (4) coefficients related to the RFM measures. Individual-specific parameters come from $\left\{{\theta}_i\right\}\sim N\left(\overset{\sim }{\theta },{\sigma}_{\theta}^2\right)$.

Imai et al. (2009) propose a less computationally intensive approach to approximate value function using realizations of the past 100 iterations (N = 100). Suppose that we are at iteration r and have accumulated ${H}^r={\left\{{\left\{{x}_i^k,\kern0.5em {\overset{\sim }{V}}^k\left({x}_i^k;{\theta}_i^k\right)\right\}}_{i=1}^I\right\}}_{k=r-N}^{r-1}$ through past iterations, where k denotes a given iteration before the current r^th iteration, ${x}_i^k$ and ${\theta}_i^k$are the set of state variables and parameters stored at each past iteration (to be discussed subsequently), and ${\overset{\sim }{V}}^k\left({x}_i^k;{\theta}_i^k\right)$ is the value function associated with the stored state variables and parameters at each past iteration. Given that our model accommodates individual-specific parameters, in theory it requires us to store all state variables, parameters, and realized value functions for all consumers across all past iterations, which is computational costly. Thus, we adopt Imai et al.’s (2009) procedure to store one realization from one randomly selected consumer during each iteration. This simplification reduces the requirement for computer memory considerably. Within the rth iteration, the estimation procedure is extended as follows:

Step 1: Update * {θ _i}

• 1. We first draw a new set of parameter values of {θ_i} for each consumer using random walk. We denote the candidate draw as $\left\{{\theta}_i^c\right\}$.

• 2. To assess the value function, we must compute the EV(x_{i, t + 1}). Following Imai et al. (2009), we can compute EV(x_{i, t + 1}) using all past state variables, parameters, and realizations of value functions, as follows:

$$ \overset{\sim }{EV}\left({x}_{i,t+1}\right)=\sum \limits_{k=r-N}^{r-1}{\overset{\sim }{V}}^k\ast \frac{K_{h\theta}\left({\theta}_i^c-{\theta}_i^k\right){K}_{h INV}\left({INV}_i-{INV}^k\right){K}_{h RFM}\left({RFM}_i-{RFM}^k\right)f\left({p}_i-{p}^k\right)f\left({d}_{ig,r}-{d_{ig,r}}^k\right)}{\sum \limits_{k=r-N}^{r-1}{K}_{h\theta}\left({\theta}_i^c-{\theta}_i^k\right){K}_{h INV}\left({INV}_i-{INV}^k\right){K}_{h RFM}\left({RFM}_i-{RFM}^k\right)f\left({d}_{ig,r}-{d_{ig,r}}^k\right)}, $$

where INV_i is the total number of products available for return and RFM_i is the set of RFM measures. Given consumers’ current-period purchase and return decisions, we can compute INV_i and RFM_i deterministically; K_hINV and K_hRFM are their kernel density functions accordingly. The other two state variables—average price of purchased products, p_i, and match value from usage experience d_{ig, r}—evolve stochastically.¹²

• 3. To compute the likelihood function, we must also integrate out the random match values. However, the complexity of our proposed model prevents us from analytically integrating out these two random match values. Following prior research (e.g., Erdem and Keane 1996), we draw 100 sets of random match values from their distributions (as specified in the main body of the paper) to integrate out the likelihood function.

• 4. With the likelihood obtained with ${\theta}_i^c$and ${\theta}_i^{r-1}$, we determine whether or not to accept the candidate parameter draw for a given individual.

Next, we repeat Step 1 to update individual-level parameters for every consumer.

Step 2: Update ${\overset{\sim }{\boldsymbol{V}}}^{\boldsymbol{r}}\left({\boldsymbol{x}}_{\boldsymbol{i}}^{\boldsymbol{r}};{\boldsymbol{\theta}}^{\boldsymbol{r}}\right)$

• 5. The last step for iteration r involves computing the pseudo-value function ${\overset{\sim }{V}}^r\left({x}_i^r;{\theta}_i^r\right)$, which can be done as follows:

  • i Make one draw from the state space;

  • ii Compute the expected future value;

  • iii Apply the Bellman operator once.

• 6. Proceed to the next iteration

State variables

Here, we now discuss the specification and law of motion associated with our state variables.

Number of products available for return

We specify the number of products available for return as INV_it = INV_{i, t − 1} + E_it − G_it. For simplicity of notation, we denote INV_{i, t − 1} as the number of products purchased before the current trips that are still within the window of return, E_it as the number of newly purchased items, and G_it as the number of items returned during the current visit. As we do not observe consumers’ purchases from the very beginning, our proposed model is subject to a left-truncation issue (i.e., we do not observe INV_i0). To address this issue, we first estimate a preliminary model, which has the same structure of our proposed model but with the assumption that INV_i0 equals 0 (i.e., consumers have no items that are available to return at the beginning). With the model estimates, we then forward-simulate consumers’ purchases and returns 100 times randomly draw a set of initial INV_i0 to estimate our proposed model.

RFM measures

Again, in our model, we include the following RFM measures: (1) the number of weeks from the most recent purchase (recency), (2) the number of shopping trips with purchases made in the recent quarter (frequency), and (3) the number of net spending in the recent quarter (monetary). Similar to inventory, all RFM measures are subject to left-truncation. We therefore use the same approach, as described previously, to handle the left-truncation issue with RFM measure.