A non-linear approach to alcohol consumption decisions: monopoly versus competition

Abstract

Aim

One of the public health objectives of the Swedish and Danish alcohol policies is to reduce harm to individuals from the risky consumption of alcohol. In furtherance of this objective, it is interesting to research the purchasing and consumption patterns of drinkers, with a particular focus on clarifying the purchasing behavior of heavy drinkers relative to moderate and light drinkers. Thus, this article examines demand for alcoholic beverages in Denmark and Sweden.

Subjects and methods

Since there are significant differences in alcohol policy in Denmark and Sweden, it is interesting to study a comparative analysis of consumer behavior. Our study included a randomly drawn sample of the alcohol-buying population in both countries. A proportional odds model was applied to capture the natural ordering of dependent variables and any inherent nonlinearities.

Results

The findings show that individual demand for alcoholic beverages depends on economic, regional, and socio-demographic variables but that there is also a heterogeneity in consumer response to alcohol consumption under competition and monopoly.

Conclusion

This study provides some evidence and support to the notion that people can generally be characterized by certain factors associated with alcohol demand. This information can help policymakers when they discuss concepts related to public health issues.

Appendices

Appendix 1

Table 3. Percentage distribution of study variables in Sweden and Denmark

Table 4. Correlation matrix for explanatory variables in Sweden and Denmark

Appendix 2

Proportional odds model (POM)

Since our response, i.e., the observations on our dependent variable, was a three-level ordinal variable, it is wise to consider the natural ordering to the response levels when modeling the effects of the explanatory variables on consumer behavior (Agresti 2007). Let:

$$ {\uptheta}_1\ \left({x}_i\right)={\uppi}_1\left({x}_i\right), $$

(1)

and

$$ {\uptheta}_2\ \left({x}_i\right)={\uppi}_1\ \left({x}_i\right)+{\uppi}_2\left({x}_i\right), $$

(2)

where π₁(x_i) is the probability of being a heavy alcohol consumer (HAC) at the ith setting of values of k explanatory variables x_i = (x_1i,..., x_ki), while π₂(x_i) and π₃(x_i) are the probabilities of being a moderate alcohol consumer (MAC) and being a low alcohol consumer (LAC), respectively. Thus, θ₁(x_i) and θ₂(x_i) represent cumulative probabilities: θ₁(x_i) is the probability of being an HAC, and θ₂(x_i) is the probability of being an MAC or even being an HAC. Let us define the two cumulative logits:

$$ \mathrm{logit}\left[{\theta}_1\left({x}_i\right)\right]=\log \left[\frac{\pi_1\left({x}_i\right)}{\pi_2\left({x}_i\right)+{\pi}_3\left({x}_i\right)}\right], $$

(3)

and

$$ \mathrm{logit}\left[{\theta}_2\left({x}_i\right)\right]=\log \left[\frac{\pi_1\left({x}_i\right)+{\pi}_2\left({x}_i\right)}{\pi_3\left({x}_i\right)}\right] $$

(4)

The first cumulative logit should be interpreted as the log odds of being an HAC compared with being an MAC or an LAC, and the second logit is the log odds of being an MAC or an HAC compared with an LAC. By assuming that the log odds are linear functions of the explanatory variables, we can write:

$$ \mathrm{logit}\left[{\theta}_j\left({x}_i\right)\right]={\alpha}_j+{x}_i^{\hbox{'}}{\beta}_j,\kern0.5em j=1,2 $$

(5)

We maximized the log of the likelihood function to obtain maximum likelihood estimates:

$$ L={\varPi}_{i=1}^n{\left[{\pi}_1\left({x}_i\right)\right]}^{d1i}{\left[{\pi}_2\left({x}_i\right)\right]}^{d2i}{\left[{\pi}_3\left({x}_i\right)\right]}^{d_{3i}} $$

(6)

subject to Eq, (5), where d_hi = 1 if the ith individual gets the hth purchasing option i = 1, 2 …, n, h = 1, 2, 3 d_hi = 0 otherwise.

However, this approach does not take into account the ordinal scale of the response variable. Thus, we suggest using the following, more parsimonious, model:

$$ \mathrm{logit}\left[{\theta}_j\left({x}_i\right)\right]={\alpha}_j+{x}_i^{\hbox{'}}\beta, \kern3.5em j=1,2. $$

(7)

Hence, the effect of an explanatory variable on the log odds of being an HAC compared with being an MAC or LAC is the same as the log odds of being an MAC or HAC compared with being an LAC. Furthermore, to better grasp the consequences implied by the restriction, let x1 and x2 be two different settings of the explanatory variables. We then have the following result:

$$ \mathrm{logit}\left[{\uptheta}_j\left({x}_1\right)\right]\hbox{--} \mathrm{logit}\left[{\theta}_j\left({x}_2\right)\right]=\left({x}_1^{\hbox{'}}-{x}_2^{\hbox{'}}\right)\beta, \kern1em j=1,2 $$

(8)

The log cumulative odds ratios are proportional to the distance between the values of the explanatory variables. This feature has also given the model its name as the POM model. However, maximizing the log of the likelihood function given by Eq. (6) subject to the constraints in Eq. (7) yields the parameter estimates of α₁, α₂, and β.