8. Empirical Models of Learning Dynamics: A Survey of Recent Developments

Given habit persistence or learning, past purchase creates exposure to a product, which directly affects a consumer’s perceived utility of a product. In an inventory model, past purchase matters because it affects current inventory, but also, more subtly, because the prices at which past purchases are made affect the reference price of the product.

In order to implement their model, Erdem and Keane (1996) used the approximate solution methods for dynamic programming models developed in Keane and Wolpin (1994). See Ching et al. (2013) for a complete explanation of the procedure. The simplifying assumptions in Eckstein et al. (1988) allowed them to use the Gittin’s index to find the solution of their model (see Appendix A of Ching et al. 2013).

For instance, a diaper may hold all of a baby’s urine on some occasions but not on others (depending on how much milk the baby drank), so one use may not fully reveal its quality.

Note that this is equivalent to say \(I_{it}\) consists of all past signals consumer i has received before he/she makes the purchase at time t, given the Bayesian learning framework.

Still, heterogeneity in S _it may persist over time, because: (i) both brands and consumers are finitely lived, (ii) as people gather more information the value of trial purchases diminishes, and so eventually learning about unfamiliar products will become slow; (iii) there is a flow of new brands and new consumers entering a market.

Ching et al. (2013) explain how to extend the above basic framework to allow consumers to learn from multiple information sources (such as advertising, word-of-mouth and so on).

In almost all cases no purchase is also an option. Erdem and Keane (1996) denote the no purchase option as j = 0, and simply set the expected utility of no purchase to \(E\left[ {U_{i0} |S_{it} } \right] = e_{i0t}\).

Hendrick et al. (2012) propose a similar framework to study how consumers choose a product among J > 2 alternatives. Newberry (2016) extends this framework to study the role of pricing in observational learning using data from an online market for music.

The models in these papers largely adopt the framework discussed so far, and hence we will not devote space to explicitly discussing their structure.

The likelihood of their model involves multiple integrals because the explanatory variables include lagged latent dependent variables and serially correlated errors, but they show that the GHK simulator remains tractable for this generalized framework (see Keane 1994).

Ching (2010b) allows for multiple generic firms. In addition, generic firms’ entry decisions are also endogenous. But since the focus of this chapter is learning, we abstract away the entry decisions when describing the model.

Although this paper does not explicitly model a dynamic game, the dynamic demand model is very useful in evaluating the consequences of Marlboro’s strategic response to the competition of generic brands.

The basic idea of this model is that players in a game vary in their depth of strategic thinking. A completely naïve player will choose actions by completely ignoring the presence of other players (level zero). A level one player believes that other players will not react to his choice, and his action is the best response with respect to this belief. A level two player believes that all other players are level one, and so on and so forth. This model captures bounded rationality, and can explain players’ behavior in games that cannot be rationalized by standard game theory.

The efficiency ratio measures how well a drug converts reduced cholesterol levels to reduced heart disease risks.

Note that the basic idea of search models is that consumers need to compare the expected gain from searching vs. the cost of search. In contrast, standard choice models with learning assume that consumers learn about an attribute by buying the product multiple times because information signals are noisy. These models also assume there are a fixed number of alternatives to choose from. In search models with an unknown price (or attribute) distribution, consumers learn about the parameters that characterize the distribution. For a normal distribution, that would be simply learning its mean and standard deviation. But, for a Dirichlet distribution, a consumer needs to use the whole history of price realizations and the initial parameters that characterize the prior to construct his posterior.

This is in contrast to static models, where the current expected utilities, \(E[U(Q_{jt}^{E} ,P_{jt} )|I_{t} ]\), alone determines choice probabilities.

In labor economics, researchers may argue that wages capture much of the current payoff. Or, researchers can control current payoffs in a lab experiment (e.g., Houser et al. 2004).

Each sub-problem can be characterized as follows. A consumer either chooses a fixed reward in each period forever, or chooses brand j this period. If he/she chooses brand j this period, a noisy quality signal about brand j will be revealed, and then the consumer faces these two choices again next period. The reasons the index method provides significant computational gains are: (a) it reduces the size of the state space from N ^J to J × N, where N is the number of state points associated with each alternative, and (b) solving for the index strategy for J optimal stopping problems is much less costly compared with solving one J dimensional dynamic programming problem.

To see this, compare Eq. (8.28) with Eqs. (8.29)–(8.30). Clearly, the GK approach amounts to choosing a parameterization for the \(G(j,0,I_{t} )\) function.

Another example is that the advent of internet retailing has made it possible to do comparison shopping from home, thus arguably reducing the costs of gathering information. An interesting hypothesis is that this change in the environment may have caused consumers to engage in more comparison shopping.

One way to interpret our argument is that only structural models attempt to predict what decision rules consumers will adopt in a new environment (indeed, this is precisely what structural models are designed to do). But that doesn’t mean their predictions will necessarily be correct. It is important to keep in mind the point that a structural model is only invariant to all conceivable environmental changes if it is perfectly correctly specified—that is, if it is in fact the “true model.” As all models are ultimately false (as they are simplifications), a completely policy invariant model is an aspirational goal, not a reality. The best we can do in practice is to incrementally validate a structural model by showing that it predicts well across a range of policy environments. This may give us confidence in using the model to predict in a new environment. But we can never be certain that the new environment won’t be the one that reveals the flaw in the model! As a practical matter, the best we can hope for is to build structural models we are confident in using for certain types of policy predictions, but perhaps not for others (i.e., it is perfectly possible that a structural model can reliably predict responses to some types of policy changes but not others—just as we see with commonly used models in the physical sciences and engineering). See Keane (2010) for further discussion of these issues.

Interestingly, higher-income borrowers learn twice as fast, and forget twice as slowly, as lower-income borrowers.

An exception is Yang and Ching (2014) who develop and estimate a consumer life-cycle model to explain the adoption decision of a new technology.

In the optimal solution consumers should consider a category in every period regardless of their inventory. Even if inventory is high, a low enough price would make it optimal to stock up even more.