Habits in frequency of purchase models: the case of fish in France

ABSTRACT

The quantity purchased in a period is the result of two decisions: the frequency of purchase and the average purchase on each occasion. We introduce habit formation into demand systems modelling each of these decisions. An econometric model is estimated by Bayesian methods. The data generating processes of the frequency and quantity decisions are assumed to follow, respectively, a multivariate Poisson log-normal distribution and a multivariate gamma log-normal distribution. We estimate the systems using French scanner data for purchases of fish. The results suggest that habits in purchase frequencies are important, while habits in average purchased quantities are less important. Furthermore, we find that price changes are important for explaining average quantities purchased but have only minor effects on purchase frequencies.

KEYWORDS:

• Demand system
• fish
• France
• frequency of purchase
• habit formation

JEL CLASSIFICATION:

• C34
• D12

Acknowledgments

The authors thank Pierre Combris and INRA, Paris for help with obtaining and preparing the data. The Research Council of Norway (Grant 199564) provided financial support for this research.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

¹ Even though count data models have not been used frequently in demand analysis of consumer goods they have frequently been used in health economics, environmental economics, and marketing. The dependent variable in these studies are typically the number of visits to doctors, the number of trips to a recreational site or frequency of shopping (Deb and Trivedi 2002; Egan and Herriges 2006; Uncles and Lee 2006).

² This model was introduced by Aitchison & Ho (1981).

³ This simplifying assumption could lead to an incorrect specification of the variance if the two systems are stochastically correlated, but it does not lead to inconsistent or biased estimates of the parameters.

⁴ Discussion on wild versus farmed fish can be found in, for example, Herrmann, Mittelhammer, and Lin (1993), Asche et al. (2005), Asche and Guttormsen (2014), Rickertsen et al. (2017).

⁵ The durability of good $x_{it}$ could also be modelled by a decay function. However, for simplicity we use one duration parameter.

⁶ We describe how we estimate the duration parameter from Equation (1)(1) $Z_{it}=\sum _{\tau =0}^{\mathrm{\infty }}{d}_i^{\tau }ln{x}_{it-\tau }=ln{x}_{it}+d_i{Z}_{it-1}$(1) using the initial conditions of the service stock in Section 4.1.

⁷ Under the myopic assumption, the consumer does not account for the user cost of stocks.

⁸ The conditional mean of $n_i$ and $q_i$ are given as follows: $\mathrm{E}\left(n_i|{\beta }_i,C,b_{Ni}\right)$, $\mathrm{E}\left(q_i|{\alpha }_i,C,b_{Qi}\right)=\mathrm{P}\mathrm{r}\left(N>0|{\beta }_i,C\right)\mathrm{E}\left(q_i|{\alpha }_i,C,n_i>0,b_{Qi}\right)$. The marginal effects of $\mathrm{E}\left(q_i|{\alpha }_i,C,b_{Qi}\right)$ are given by:$\frac{\mathrm{\partial }\mathrm{E}\left(q_i|{\alpha }_i,C,b_{Qi}\right)}{\mathrm{\partial }{C}_i}=\frac{\mathrm{\partial }\mathrm{E}\left(q_i|{\alpha }_i,C,n_i>0,b_{Qi}\right)}{\mathrm{\partial }{C}_i}\mathrm{P}\mathrm{r}\left(N>0|{\beta }_i,C\right).$.

⁹ For less complex problems, the Gaussian-quadrature could be used.

¹⁰ This assumption could lead to incorrect variance specification but will not result in inconsistent parameter estimates.

¹¹ The mean and variance of the marginal distribution of $q_i$ have, as far as we know, not been derived in the literature.

¹² For a discussion of the random walk Metropolis algorithm, see for example Roberts, Gelman, and Gilks (1997).

¹³ The problem of zero observations is also reduced when the data generating process is discrete as in the Poisson distribution with a positive probability of observing a zero.

¹⁴ The semi-logarithmic demand equations in Equations (32) and (33) are integrable when the restrictions ${\beta }_{is}=0\mathrm{\forall }i\ne s$ and ${\theta }_i=\theta \mathrm{\forall }i$ are imposed (LaFrance and Hanemann 1989).

¹⁵ This method is not without problems since prices will be influenced by choices of quality of fish or store, which potentially introduce endogeneity problems.

¹⁶ The Geweke convergence test is a test of stationarity of the Markov chains. For a discussion of this test, see Nylander et al. (2008).

¹⁷ To find the habit parameter for wild fish, we add the estimated stock parameters from both the frequency and average quantity part given in Table 2, which is 0.00045. Next, we add this parameter (0.00045) to the predicted duration parameter presented in the previous subsection, which is 0.599 and the resulting parameter is 0.59945 or about 0.60.

Additional information

Funding

This work was supported by the The Research Council of Norway [199564].