The effects of loss aversion on deceptive advertising policies

Abstract

We extend the deceptive advertising model of Piccolo et al. (RAND J Econ 46(3):611–624, 2015) to a framework in which consumers may be loss averse. There are two sellers, competing on prices and offering experience goods with some differences in quality. Prospective customers may be harmed by deceptive advertising: a marketing practice that can induce them to make bad purchases. We show that although deceptive advertising occurs depending on the degree of consumers’ loss aversion, this behavioral bias does not reflect on firms’ prices. Nevertheless, the presence of loss-averse consumers crucially changes the optimal deterrence rule that a Public Authority should adopt against false claims and misleading advertising. Unlike Piccolo et al. (2015), in this more general model, strong enforcements may improve the buyer welfare according to the degree of loss aversion.

Appendix

Proof of Proposition 1

In a separating equilibrium, firms make different offers on the base of their products’ quality. Accordingly, consumers can infer products’ quality and prices reflect the information transmitted. This implies that $p_{\text {h}}\le \nu _{\text {h}}$, while $p_{\text {l}}\le \nu _{\text {l}}$ with $p_{\text {h}}\ne p_{\text {l}}$. The proof proceeds by establishing (i) the separating equilibrium prices; (ii) the region in which a separating equilibrium exists.

• (i) When products are vertically differentiated, price competition is relaxed and the difference in prices is given by the difference in products’ quality—i.e., $p_{\text {h}}-p_{\text {l}}=\varDelta$. Following the standard undercutting logic, $p_{\text {h}}=\varDelta$ and $p_{\text {l}}=0$. In fact, if the high-quality seller posted a price $p_{\text {h}}=\varDelta +\varepsilon$, with $\varepsilon \in \left( 0,\nu _{\text {l}}\right)$ (the right-hand restriction comes from the consumers’ participation constraint, according to which $p_{\text {h}}\le \nu _{\text {h}}$), the low-quality seller would profitably set a price $p_{\text {l}}=\omega$, with $\omega \in \left( 0,\varepsilon \right)$, so that consumers would be more willing to buy the low-quality product because $\nu _{\text {l}}-\omega >\nu _{\text {h}}-\left( \varDelta +\varepsilon \right)$. In the same vein, if the low-quality seller posted a price $p_{\text {l}}=\varepsilon$, the high-quality seller would profitably set a price $p_{\text {h}}=\varDelta +\omega$, so that consumers would be more willing to buy the high-quality product because $\nu _{\text {h}}-\left( \varDelta +\omega \right) >\nu _{\text {l}}-\varepsilon$.

• (ii) An equilibrium exists if both firms have no incentive to deviate. Under assumption A3, the high-quality seller has no incentive to deviate because it would signal a low-quality product to consumers, therefore, lowering its profit. Instead, the low-quality seller may send a message $m=m_{\text {h}}$, by claiming misleading ads, and it may set a deviation price $p_{\text {d}}=\varDelta$. In this case, consumers are not able to distinguish the high-quality from the low-quality product. Indeed, their posterior beliefs about products’ quality are $50-50$. This implies a consumers’ utility function equal to Eq. (2) in the text. As a consequence, the low-quality seller has no incentive to deviate because of two alternative reasons: (1) given the price $p=\varDelta$, consumers are not willing to buy any product without knowing its quality; (2) deviating from the equilibrium path is not profitable. Case (1) occurs if consumers’ utility function (2) is lower than zero, which happens if and only if

  $$\begin{aligned} \varDelta >\varDelta ^{*} ( \nu _{\text {l}},\eta ,\lambda ) \equiv 2\nu _{\text {l}}\left( 1+\frac{\eta ( \lambda -1 ) }{2}\right) ^{-1}. \end{aligned}$$

  Differentiating $\varDelta ^{*} ( \nu _{\text {l}},\eta ,\lambda )$ with respect to $\lambda$ yields

  $$\begin{aligned} \frac{\partial \varDelta ^{*} ( \nu _{\text {l}},\eta ,\lambda ) }{\partial \lambda }=-4\nu _{\text {l}}\eta \left( 2+\eta ( \lambda -1 ) \right) ^{-2}<0. \end{aligned}$$

  By contrast, if $\varDelta \le \varDelta ^{*} ( \nu _{\text {l}},\eta ,\lambda )$, consumers are still willing to buy one of the products on sale. Hence, a separating equilibrium exists if case (2) occurs, i.e., deviating from the equilibrium path entails non-positive expected profits, i.e.,

  $$\begin{aligned} \pi ^{d} ( p=\varDelta ,m=m_{\text {h}} ) =\frac{\varDelta -f}{2}\le 0, \end{aligned}$$

  ²³ which happens if and only if $f\ge \varDelta$. $\square$

Proof of Proposition 2

In a pooling equilibrium, firms make the same offer independently of their products’ quality. Let us consider a candidate equilibrium in which both sellers send a message $m=m_{\text {h}}$ and set a price p. To prove the existence of a pooling equilibrium, we need to prove that consumers are willing to purchase at price p (consumers’ participation constraint), sellers make non-negative expected profits (sellers’ participation constraints) and they are willing to charge the equilibrium price p rather than a different price (sellers’ incentive compatibility constraint).

The consumers’ utility function is

$$\begin{aligned} u ( p,m=m_{\text {h}} ) ={\mathbb {E}}\left[ \nu \right] -\frac{\varDelta }{4}\eta ( \lambda -1 ) -p, \end{aligned}$$

which is larger than zero—i.e., it satisfies the consumers’ participation constraint—if and only if

$$\begin{aligned} p\le \nu _{\text {l}}+\frac{\varDelta }{2}\left( 1-\frac{\eta ( \lambda -1 ) }{2}\right) . \end{aligned}$$

(9)

Under assumption A2, consumers equally randomize between products when they cannot distinguish the high-quality and the low-quality product. Hence, the participation constraint for high- and low-quality sellers is

$$\begin{aligned} \pi _{\text {h}} ( p,m=m_{\text {h}} )&=\frac{p}{2}\ge 0,\\ \pi _{\text {l}} ( p,m=m_{\text {h}} )&=\frac{p-f}{2}\ge 0. \end{aligned}$$

While the high-quality seller has an expected profit which is non-negative for any price $p\ge 0$, the equilibrium price p must be larger than monetary sanction f to satisfy the low-quality seller’s participation constraint, i.e.,

$$\begin{aligned} p\ge f. \end{aligned}$$

(10)

Under assumption A3, deviating from the equilibrium path gives a “bad” signal to consumers about product’s quality. The expected profit for any deviating firm is, therefore, independent of the actual value of $\nu$. Since deviation profits are the same across sellers, the high-quality seller has the higher expected profit in equilibrium, the incentives that really matter are those of the low-quality seller. Assumption A3 implies also that the best deviation for the low-quality seller consists in no advertising and setting a price $p_{\text {d}}<p$. In fact, to attract consumers, the low-quality seller must charge $p_{\text {d}}$ such that $\nu _{\text {l}}-p_{\text {d}}>\nu _{\text {h}}-p,$ which implies $p_{\text {d}}<p-\varDelta$. The low-quality seller must offer a discount with respect to the equilibrium price by setting $p_{\text {d}}\in \left[ 0,p-\varDelta \right)$. Clearly, it maximizes its profit if it slightly undercuts $p-\varDelta$. With a slight abuse of notation, we approximate $p_{\text {d}}=p-\varDelta$. The deviation profit is, therefore, equal to $\pi ^{d}=p-\varDelta$, because it does not entail any monetary sanction for deceptive advertising. Hence, the low-quality seller has no incentive to deviate from the equilibrium path if such a deviation is not profitable, i.e.,

$$\begin{aligned} \frac{p-f}{2}\ge p-\varDelta , \end{aligned}$$

which implies the following restriction (the incentive compatibility constraint) on the equilibrium price

$$\begin{aligned} p\le 2\varDelta -f. \end{aligned}$$

(11)

Taking together all the restrictions on the equilibrium price—i.e., (9)–(11)—yields the following set of equilibrium prices:

$$\begin{aligned} p\in \left[ f;\min \left( \nu _{\text {l}}+\frac{\varDelta }{2}\left( 1-\frac{\eta \left( \lambda -1\right) }{2}\right) ;2\varDelta -f\right) \right] , \end{aligned}$$

(12)

where the most competitive equilibrium, in which sellers post the lowest possible price, is $p=f$.

Finally, we show the region of parameters in which the set of equilibrium prices (12) exists. To exist, the monetary sanction f must be weakly lower than the upper bound of (12), i.e.,

$$\begin{aligned} f\le \min \left\{ \varDelta ;\nu _{\text {l}}+\frac{\varDelta }{2}\left( 1-\frac{\eta \left( \lambda -1\right) }{2}\right) \right\} . \end{aligned}$$

Notice that there exists a threshold $\varDelta ^{*}\left( \nu _{\text {l}} ,\eta ,\lambda \right)$ in which

$$\begin{aligned} \varDelta =\nu _{\text {l}}+\frac{\varDelta }{2}\left( 1-\frac{\eta \left( \lambda -1\right) }{2}\right) \Rightarrow \varDelta ^{*} ( \nu _{\text {l}},\eta ,\lambda ) =2\nu _{\text {l}}\left( 1+\frac{\eta ( \lambda -1 ) }{2}\right) ^{-1}, \end{aligned}$$

such that $\varDelta \le \nu _{\text {l}}+\frac{\varDelta }{2}\left( 1-\frac{\eta ( \lambda -1 ) }{2}\right)$ if $\varDelta \le \varDelta ^{*} ( \nu _{\text {l}},\eta ,\lambda )$ and $\varDelta >\nu _{\text {l}}+\frac{\varDelta }{2}\left( 1-\frac{\eta ( \lambda -1 ) }{2}\right)$ otherwise. Hence, for $\varDelta \le \varDelta ^{*} ( \nu _{\text {l}},\eta ,\lambda )$, a pooling equilibrium exists if $f\le \varDelta$. Instead, for $\varDelta >\varDelta ^{*} ( \nu _{\text {l}},\eta ,\lambda )$, a pooling equilibrium exists if $f\le \nu _{\text {l}}+\frac{\varDelta }{2}\left( 1-\frac{\eta ( \lambda -1 ) }{2}\right)$. $\square$

Proof of Proposition 3

Given the equilibrium prices, the expected consumer surplus (ECS) is

$$\begin{aligned} \mathrm{ECS}( m_{i}=m_{\text {l}},m_{j}=m_{\text {h}} ) =\nu _{\text {h}}-\varDelta =\nu _{\text {l}} \end{aligned}$$

if deceptive advertising does not occur (separating equilibrium) and

$$\begin{aligned} \mathrm{ECS} ( m_{i,j}=m_{\text {h}} ) =\nu _{\text {l}}+\frac{\varDelta }{2}\left( 1-\frac{\eta ( \lambda -1 ) }{2}\right) -f \end{aligned}$$

otherwise (pooling equilibrium).

Notice that, in pooling equilibrium, ECS depends, among other things, on the monetary sanction for deceptive advertising f. ECS is larger in pooling rather than in separating equilibrium if and only if

$$\begin{aligned} \nu _{\text {l}}+\frac{\varDelta }{2}\left( 1-\frac{\eta ( \lambda -1 ) }{2}\right) -f>\nu _{\text {l}}, \end{aligned}$$

(13)

which implies

$$\begin{aligned} f<\frac{\varDelta }{2}\left( 1-\frac{\eta ( \lambda -1 ) }{2}\right) . \end{aligned}$$

In this region of parameters, a pooling equilibrium always exists. In fact, a pooling equilibrium exists if

However, for $\varDelta >\varDelta ^{*} ( \nu _{\text {l}},\eta ,\lambda )$ both pooling and separating equilibria may exist. Under assumption A4, it prevails the separating equilibrium, because it yields the largest ex ante profit ($\frac{\varDelta }{2}$ in separating equilibrium versus $\frac{f}{2}$ in pooling equilibrium). This implies that deceptive advertising does not occur if $\varDelta >\varDelta ^{*} ( \nu _{\text {l}},\eta ,\lambda )$, regardless of the monetary sanction f.

Instead, for $\varDelta \le \varDelta ^{*}\left( \nu _{\text {l}},\eta ,\lambda \right)$, deceptive advertising occurs if $f<\varDelta$ ($f=\varDelta$ is excluded under assumption A4) and it is welfare increasing in the subset $f<\frac{\varDelta }{2}\left( 1-\frac{\eta \left( \lambda -1\right) }{2}\right)$. Let us normalize $\eta =1$, then the latter condition can be written also as a restriction on the degree of consumers’ loss aversion:

$$\begin{aligned} \lambda <\lambda ^{*} ( \varDelta ,f ) \equiv 3-\frac{4f}{\varDelta }. \end{aligned}$$

Now we show the optimal enforcement in this region of parameters. In pooling equilibrium, ECS is clearly decreasing in f. This implies that the optimal enforcement if $\lambda <\lambda ^{*} ( \varDelta ,f )$ coincides with $f=0$.

By contrast, for $\varDelta \le \varDelta ^{*} ( \nu _{\text {l}},\eta ,\lambda )$ and $\lambda \ge \lambda ^{*} ( \varDelta ,f )$, consumers are better off without deceptive advertising. This implies that the monetary sanction must be set sufficiently high to discourage deceptive advertising, i.e., $f\ge \varDelta$. Since ECS does not depend on f in separating equilibrium, it is maximized for any $f\ge \varDelta$. $\square$

Proof of Proposition 4

In this proof, we characterize the region of parameters in which deceptive advertising occurs and minimizes ECS. It follows immediately from the proof of Proposition 3 that (i) deceptive advertising does not occur if $\varDelta >\varDelta ^{*} ( \nu _{\text {l}},\eta ,\lambda )$; (ii) for $\varDelta \le \varDelta ^{*} ( \nu _{\text {l}},\eta ,\lambda )$, ECS is decreasing in the monetary sanction f if $f<\varDelta$, while it is constant and equal to $\nu _{\text {l}}$ if $f\ge \varDelta$. To show the region in which ECS is minimized, we need to compare ECS in pooling and separating equilibrium when $\varDelta \le \varDelta ^{*}\left( \nu _{\text {l}},\eta ,\lambda \right)$. Deceptive advertising is welfare decreasing if and only if

$$\begin{aligned} \nu _{\text {l}}+\frac{\varDelta }{2}\left( 1-\frac{\eta ( \lambda -1) }{2}\right) -f<\nu _{\text {l}} \end{aligned}$$

which implies

$$\begin{aligned} f>\frac{\varDelta }{2}\left( 1-\frac{\eta ( \lambda -1 ) }{2}\right) . \end{aligned}$$

By normalizing $\eta =1$, this condition can be written as follows:

$$\begin{aligned} f>f^{*} ( \varDelta ,\lambda ) \equiv \frac{\varDelta }{2}\left( 1-\frac{\lambda -1}{2}\right) , \end{aligned}$$

where $f^{*} ( \varDelta ,\lambda )$ is always lower than $\varDelta$, which means that a pooling equilibrium exists when $\varDelta \le \varDelta ^{*} ( \nu _{\text {l}},\eta ,\lambda )$. Rearranging the latter inequality $\varDelta \le \varDelta ^{*} ( \nu _{\text {l}},\eta ,\lambda )$, it follows that a pooling equilibrium may exist if

$$\begin{aligned} \varDelta \le 2\nu _{\text {l}}\left( 1+\frac{\eta ( \lambda -1 ) }{2}\right) ^{-1} \Rightarrow \lambda \le 1+\frac{1}{\eta }\left( \frac{4\nu _{\text {l}} }{\varDelta }-2\right) . \end{aligned}$$

By normalizing $\eta =1$, this condition can be written as follows:

$$\begin{aligned} \lambda \le \lambda ^{*}\left( \nu _{\text {l}},\varDelta \right) \equiv \frac{4\nu _{\text {l}} }{\varDelta }-1. \end{aligned}$$

Hence, if $\lambda \le \lambda \left( \nu _{\text {l}},\varDelta \right)$ and $f\in ( f^{*} ( \varDelta ,\lambda ) ,\varDelta )$, deceptive advertising minimizes ECS. $\square$