Consumer education: why the market doesn’t work

Abstract

A growing literature studies the interactions between fully rational profit maximizing firms, on one side, and biased consumers, on the other side. Along these lines, this paper focuses on the consequences of quality misperception on the market equilibrium, by raising the following question: when quality bias affects consumer choice, do firms have incentives to educate their competitor’s customers in order to attract them? To tackle this issue, I incorporate consumer misperception in a Cournot-type duopoly model and consider the consequences on the market outcome. I focus on the two polar cases, when both firms either exploit consumer misperception, or educate completely their rival’s customers. I show that the market exerts conflicting forces on the firms’ incentives, such as a curse of debiasing might occur even in the presence of substitute goods. Consequently, the opportunity of a legal intervention to trigger consumer education is a key issue.

Appendix

1.1 The market outcome absent consumer education

The choice of focusing on price competition is consistent with the fact that I constrain the analysis to substitute goods. Indeed, Singh and Vives (1984) show that in the framework with substitute commodities, it is a dominant strategy for each firm to choose the quantity rather than the price. Nonetheless, the price competition model deserves to be mentioned. In the next paragraphs, I study the market outcome under quantity and then under price competition.

The market outcome under quantity competition: I study a duopoly where two competing firms $1$ and $2$ offer one commodity each. Let $q_1$ and $q_2$ represent the quantities of good offered repsectively by firms $1$ and $2$. In this framework, consumers maximize $U(q_1, q_2)-\sum \limits _{i=1}^{2} p_i q_i$, where $q_i$ is the amount of good $i$ and $p_i$ its price.

Each consumer has the following utility function: $V=y+U(q_1, q_2)$, where $y$ designates the composite good, whose price $p_y$ satisfies $p_y=1$. The consumer’s budgetary constraint can then be written as follows:

$$\begin{aligned} R=y+p_1q_1+p_2q_2 \end{aligned}$$

(15)

I specify the utility function:

$$\begin{aligned} U(q_1,q_2)=\hat{\alpha }_1q_1+\hat{\alpha }_2q_2-1/2(\beta q_1^2+\beta q_2^2+2\gamma q_1q_2) \end{aligned}$$

This function yields the system of linear inverse demands:

$$\begin{aligned} \left\{ \begin{array}{l} p_1=\hat{\alpha }_1-\beta q_1-\gamma q_2 \\ p_2=\hat{\alpha }_2-\beta q_2-\gamma q_1 \\ \end{array} \right. \end{aligned}$$

(16)

To study the case of quantity competition, I solve the following problem for $i\,\mathrm{and} \,j \in (1,2) \,\mathrm{and} \, i \ne j$: $\max _{q_i}\Pi _i^c=\max _{q_i}(\alpha _i-\beta q_i-\gamma q_j)q_i$. In equilibrium, one obtains the following prices and quantities:

$$\begin{aligned} q_i^c= \frac{2\beta \hat{\alpha }_i-\gamma \hat{\alpha }_j}{4\beta ^2-\gamma ^2} \,\mathrm{and} \,p_i^c=\frac{2\beta ^2\hat{\alpha }_i-\beta \gamma \hat{\alpha }_j}{4\beta ^2-\gamma ^2} \end{aligned}$$

The firms’ profits are then equal to:

$$\begin{aligned} \hat{\Pi }_i^c=\frac{\beta (2\hat{\alpha }_i\beta -\gamma \hat{\alpha }_j)^2}{(4\beta ^2-\gamma ^2)^2} \end{aligned}$$

The market outcome under price competition: Note that this model can easily be extended to a Bertrand-type price competition duopoly. The utility function $U(q_1,q_2)=\hat{\alpha }_1q_1+\hat{\alpha }_2q_2-1/2(\beta q_1^2+\beta q_2^2+2\gamma q_1q_2)$ entails the following system of direct demand functions:

$$\begin{aligned} \left\{ \begin{array}{rcr} q_1 = \hat{a}_1-b p_1+gp_2 \\ q_2= \hat{a}_2-b p_2+gp_1\\ \end{array} \right. \end{aligned}$$

With:

$$\begin{aligned} \hat{a}_i=\frac{\beta \hat{\alpha }_i-\gamma \hat{\alpha }_j}{\beta ^2-\gamma ^2} \,; \,b=\frac{\beta }{\beta ^2-\gamma ^2} \,; \,g=\frac{\gamma }{\beta ^2-\gamma ^2} \end{aligned}$$

One can verify that at the symmetric equilibrium, the quantities and prices are equal to:

$$\begin{aligned} q_i^b=\frac{2b^2 \hat{a}_i + b g \hat{a}_j}{4b^2- g^2} \,\mathrm{and} \,p_i^b=\frac{2b \hat{a}_i+g \hat{a}_j}{4b^2-g^2}. \end{aligned}$$

Moreover, $q_i^b$ and $p_i^b$ yield the following profit :

$$\begin{aligned} \Pi _i^b=\frac{b(2b \hat{a}_i+g \hat{a}_j)^2}{(4b^2-g^2)^2}. \end{aligned}$$

However, the market outcome under price competition does not need to be calculated. Indeed, the duality of the quantity and price problems allows us to solve the latter by simply replacing $\hat{\alpha }_i$ by $\hat{a}_i, \beta$ by $b$ and $\gamma$ by $-g$. This duality which was first pointed out by Sonnenschein (1968), is due to the fact that the firms’ problems under quantity and quality competition are the dual of each other: in the first case, the problem for firm $I$ is $\max _{q_i}\Pi _i^c=\max _{q_i}(\hat{\alpha }_i-\beta q_i-\gamma q_j)q_i$, whereas under price competition the problem is $\max _{p_i}\Pi _i^b=\max _{p_i}(\hat{a}_i-b q_i+gq_j)p_i$.

While the duality between the price and the quantity models allows for a generalization of the results as far as the market outcome is concerned, the symmetry no longer holds when one turns to the issue of consumer education. When the goods are substitutes, firms have less capacity to raise their prices above marginal cost in Bertrand competition. This entails lower prices in Bertand than in Cournot, which could modify substantially the firms’ incentives to educate consumers. Indeed, one would expect consumer debiasing to be less likely in Bertrand competition, as firms would not reap any profit from such a strategy. However, this intuition could be challenged by the fact that the index $\frac{\gamma }{\beta }$ has an ambiguous effect on the firms incentives. Although this paper focuses on the Cournot duopoly, the case of price competition is be worthy of further exploration.

1.2 Proof of propositions 1 and 2

In the presence of debiasing, and according to (8), $\Pi ^c_i=\frac{\beta (2\beta \hat{\alpha }_i-\gamma \hat{\alpha }_j)^2}{(4\beta ^2-\gamma ^2)^2}-c_i$, for $i \in (1,2)$. Recall that according to (17).

$$\begin{aligned} \left\{ \begin{array}{rcr} \,\hat{\alpha }_1= \mathrm{max} (\bar{\alpha }_1-ac_2 ; \underline{\alpha }_1) \\ \,\hat{\alpha }_2= \mathrm{max} ( \bar{\alpha }_2-dc_1 ; \underline{\alpha }_2) \\ \end{array} \right. \end{aligned}$$

(17)

Wherefrom, for $i=1$:

$$\begin{aligned} \frac{\partial \Pi _1}{\partial c_1}&= \frac{2d\beta \gamma }{(4\beta ^2-\gamma ^2)^2}(2\beta (\bar{\alpha }_1-ac_2) -\gamma (\bar{\alpha }_2-dc_1))-1 \end{aligned}$$

(18)

$$\begin{aligned} \frac{\partial ^2\Pi _1}{\partial c_1^2}&= 2\beta \left[ \frac{d\gamma }{(4\beta ^2-\gamma ^2)}\right] ^2 \end{aligned}$$

(19)

And for $i=2$:

$$\begin{aligned} \frac{\partial \Pi _2}{\partial c_2}&= \frac{2a\beta \gamma }{(4\beta ^2-\gamma ^2)^2}(2\beta (\bar{\alpha }_2-dc_1) -\gamma (\bar{\alpha }_1-ac_2))-1 \end{aligned}$$

(20)

$$\begin{aligned} \frac{\partial ^2\Pi _2}{\partial c_2^2}&= 2\beta \left[ \frac{a\gamma }{(4\beta ^2-\gamma ^2)}\right] ^2 \end{aligned}$$

(21)

As $\frac{\partial ^2\Pi _i}{\partial c_i^2}$ is always positive, the profit function is convex with respect to costs. Therefore, if the profit function is strictly increasing (respectively decreasing) the firm’s best response will be to choose $c_i=\bar{c}_i$ (respectively $c_i=0$). To finish the proof of propositions 1 and 2, I can now study the sign of the first derivative of the profit with respect to $c_i$.

Proof of proposition 1

I begin with the proof of proposition 1, that is to say the symmetric Nash equilibrium with consumer education. I focus on the case when $\frac{\partial \Pi _i}{\partial c_i}>0$. Let us focus first on $i=1$.

$$\begin{aligned} \frac{\partial \Pi _1}{\partial c_1}&> 0 \Leftrightarrow \frac{2d\beta \gamma }{(4\beta ^2-\gamma ^2)^2}(2\beta (\bar{\alpha }_1-ac_2) -\gamma (\bar{\alpha }_2-dc_1))>1 \end{aligned}$$

(22)

$$\begin{aligned} \frac{\partial \Pi _1}{\partial c_1}&> 0 \Leftrightarrow 2\beta (\bar{\alpha }_1-ac_2)-\gamma (\bar{\alpha }_2-dc_1) > \frac{(4\beta ^2-\gamma ^2)^2}{2d\beta \gamma } \end{aligned}$$

(23)

I want to define the conditions which guarantee a symmetric Nash equilibrium such as $c_1= \bar{c}_1$. I suppose that firm 2 sets $c_2$ at its maximal value and determine when firm 1’s best response is to adopt the same strategy. If the profit function it is strictly increasing with regards to the debiasing costs, then $c_1$ necessarily equals $\bar{c}_1$. For $c_2=\bar{c}_2$, one can then write:

$$\begin{aligned} \frac{\partial \Pi _1}{\partial c_1}&> 0 \Leftrightarrow 4\beta (\bar{\alpha }_1-a\bar{c}_2)-2\gamma (\bar{\alpha }_2-d\bar{c}_1) > \frac{(4\beta ^2-\gamma ^2)^2}{d\beta \gamma } \end{aligned}$$

(24)

$$\begin{aligned} \frac{\partial \Pi _1}{\partial c_1}&> 0 \Leftrightarrow 4\beta \underline{\alpha }_1-2\gamma \underline{\alpha }_2 > \frac{(4\beta ^2-\gamma ^2)^2}{d\beta \gamma } \end{aligned}$$

(25)

$$\begin{aligned} \frac{\partial \Pi _1}{\partial c_1}&> 0 \Leftrightarrow 4\underline{\alpha }_1-2\frac{\gamma }{\beta }\underline{\alpha }_2 > \frac{\beta }{d}\frac{[4-(\frac{\gamma }{\beta })^2]^2}{\frac{\gamma }{\beta }} \end{aligned}$$

(26)

By symmetry, one can prove that for $i=2$:

$$\begin{aligned} \frac{\partial \Pi _2}{\partial c_2}>0 \Leftrightarrow 4\underline{\alpha }_2-2\frac{\gamma }{\beta }\underline{\alpha }_1 > \frac{\beta }{a}\frac{[4-(\frac{\gamma }{\beta })^2]^2}{\frac{\gamma }{\beta }} \end{aligned}$$

(27)

Whence, if the following conditions hold, a symmetric Nash equilibrium whereby both firms completely debias their competitor’s customers emerges:

$$\begin{aligned} \left\{ \begin{array}{rcr} \,4\underline{\alpha }_1-2\frac{\gamma }{\beta }\underline{\alpha }_2>\frac{\beta }{d}\frac{[4-(\frac{\gamma }{\beta })^2]^2}{\frac{\gamma }{\beta }} \\ \,4\underline{\alpha }_2-2\frac{\gamma }{\beta }\underline{\alpha }_1>\frac{\beta }{a}\frac{[4-(\frac{\gamma }{\beta })^2]^2}{\frac{\gamma }{\beta }}\\ \end{array} \right. \end{aligned}$$

(28)

Proof of proposition 2

Let us now turn to second symmetric Nash equilibrium, in which neither of the two firms educates customers. This equilibrium prevails if and only if the profit functions are strictly decreasing with regards to the debiasing costs. For $i=1$, this is equivalent to:

$$\begin{aligned} \frac{\partial \Pi _1}{\partial c_1}&< 0 \Leftrightarrow \frac{2d\beta \gamma }{(4\beta ^2-\gamma ^2)^2}(2\beta (\bar{\alpha }_1-ac_2) -\gamma (\bar{\alpha }_2-dc_1))<1 \end{aligned}$$

(29)

$$\begin{aligned} \frac{\partial \Pi _1}{\partial c_1}&< 0 \Leftrightarrow 2\beta (\bar{\alpha }_1-ac_2)-\gamma (\bar{\alpha }_2-dc_1) < \frac{(4\beta ^2-\gamma ^2)^2}{2d\beta \gamma } \end{aligned}$$

(30)

So as to study firm $1$’s best response to its rival’s strategy, I suppose that $c_2=0$ and search for the conditions which entail $c_1=0$. With a similar argument as above, if the profit function is strictly decreasing with regards to the debiasing costs, then $c_1=0$. Therefore, for $c_2=0$ I have:

$$\begin{aligned} \frac{\partial \Pi _1}{\partial c_1}<0 \Leftrightarrow 2\beta \bar{\alpha }_1-\gamma \bar{\alpha }_2 < \frac{(4\beta ^2-\gamma ^2)^2}{2d\beta \gamma } \end{aligned}$$

(31)

Finally, I determine by symmetry a condition relative to $c_2=0$. To conclude, there is a symmetric Nash equilibrium in which both firms choose not to educate consumers if:

$$\begin{aligned} \left\{ \begin{array}{l} \,4\bar{\alpha }_1-2\frac{\gamma }{\beta }\bar{\alpha }_2< \frac{\beta }{d}\frac{[4-(\frac{\gamma }{\beta })^2]^2}{\frac{\gamma }{\beta }}\\ \,4\bar{\alpha }_2-2\frac{\gamma }{\beta }\bar{\alpha }_1< \frac{\beta }{a}\frac{[4-(\frac{\gamma }{\beta })^2]^2}{\frac{\gamma }{\beta }} \end{array} \right. \end{aligned}$$

(32)

1.3 Welfare analysis

I want to determine when it is socially efficient to force firms to debias their rival’s customers. In practical terms, the issue is to define under which conditions a mandatory debiasing policy results in an increase in social welfare. To grasp the variation of social welfare ensuing from a debiasing policy, I choose to focus on the monetary savings freshly educated consumers make, ensuing from a debiasing scheme. In this perspective, I first calculate $\bar{q}_i$, the quantity of good $x_i$ a biased consumer buys. I then turn to the quantity $\hat{q}_i$, which corresponds to the amount of good $x_i$ a debiased consumer would buy at the same price. Comparing the two quantities allows me to measure the increase in consumer welfare resulting from debiasing. In order to express this welfare variation in monetary terms, I consider that prices remain unchanged and are equal to $\bar{p}_i^c$. The choice of keeping $\bar{p}_i^c$ constant throughout the welfare analysis is reasonable since it corresponds to the price consumers would actually pay absent any debiasing strategy.

Let us first remind that, for $i \in (1,2), j \in (1,2)$ and $i\ne j$:

$$\begin{aligned} \bar{q}_i= \frac{2\beta \bar{\alpha }_i-\gamma \bar{\alpha }_j}{4\beta ^2-\gamma ^2} \,\mathrm{and} \,\hat{q}_i = \frac{2\beta \hat{\alpha }_i-\gamma \hat{\alpha }_j}{4\beta ^2-\gamma ^2}. \end{aligned}$$

Moreover, according to (17), $\hat{\alpha }_1=\bar{\alpha }_1-ac_2$ and $\hat{\alpha }_2=\bar{\alpha }_2-dc_1$.

Consequently, one can show that a general consumer education scheme results in a demand decrease $\Delta q_i$, such as:

$$\begin{aligned} \Delta q_1= \frac{2\beta ac_2-\gamma dc_1}{4\beta ^2-\gamma ^2} \,\mathrm{and} \,\Delta q_2= \frac{2\beta dc_1 - \gamma a c_2}{4\beta ^2-\gamma ^2} \end{aligned}$$

(33)

Let us then denote $G(q_i)$ the increase in consumer welfare ensuing from a lesser consumption of good $i$. As I want to evaluate in monetary terms the savings educated consumers could enjoy, I naturally obtain $G(q_i)= \Delta q_i \bar{p}_i^c$. According to (33), for $i=1$ this is equivalent to:

$$\begin{aligned} G(q_1)=\frac{2\beta ac_2-\gamma dc_1}{4\beta ^2-\gamma ^2} \bar{p}_1^c \end{aligned}$$

(34)

Similarly, one can see that for $i=2$:

$$\begin{aligned} G(q_2)=\frac{2\beta dc_1-\gamma ac_2}{4\beta ^2-\gamma ^2} \bar{p}_2^c \end{aligned}$$

(35)

Finally, when both firms educate simultaneously their rival’s customers, consumers globally enjoy a welfare increase $\Delta W$ such as:

$$\begin{aligned} \Delta W&= \frac{2\Delta \alpha _i-\frac{\gamma }{\beta } \Delta \alpha _j}{\beta -\frac{\gamma }{\beta }\gamma } \bar{p}_1^c + \frac{2 \Delta \alpha _j-\frac{\gamma }{\beta } \Delta \alpha _i}{\beta -\frac{\gamma }{\beta }\gamma } \bar{p}_2^c \end{aligned}$$

(36)

$$\begin{aligned} \Delta W&= \frac{2ac_2-\frac{\gamma }{\beta }dc_1}{\beta -\frac{\gamma }{\beta }\gamma } \bar{p}_1^c + \frac{2dc_1-\frac{\gamma }{\beta } ac_2}{\beta -\frac{\gamma }{\beta }\gamma } \bar{p}_2^c \end{aligned}$$

(37)

I next want to determine when debiasing is socially efficient, that is to say when the benefits exceeds the costs. The welfare condition can be written :

$$\begin{aligned} c_1 + c_2 < \frac{2ac_2-\frac{\gamma }{\beta }dc_1}{\beta -\frac{\gamma }{\beta }\gamma } \bar{p}_1^c + \frac{2dc_1-\frac{\gamma }{\beta } ac_2}{\beta -\frac{\gamma }{\beta }\gamma } \bar{p}_2^c \end{aligned}$$

(38)