Temptation, horizontal differentiation and monopoly pricing

Abstract

We study the implications for monopoly pricing strategies and product diversity of consumers’ temptation when the differentiation of the product is horizontal. Consumers have an ex-ante ideal product (“commitment preferences”), but they and the monopolist are aware that consumers may fall prey to “temptation preferences” ex-post with some probability. Our results indicate that when consumers are aware of their dynamic change in preferences, the firm cannot take advantage of consumers’ temptation but instead, in order to attract them into the store, the firm must compensate ex-ante consumers for the possibility of yielding to temptation once inside the store. As a result, this paper shows that the firm narrows the variety of products, not offering those products close to temptation preferences. Moreover, it is shown that product prices and firm’s profits decrease with the probability of temptation and with the consumers’ awareness of their dynamic inconsistency.

Appendix

Proof of Lemma 2

We show that a gap with no products in $\left( {\underline{\theta }, \bar{\theta }} \right) \subseteq \left[ {{\theta }_{k},1} \right]$ can be improved upon by designing a product with $\hat{q} \in \left( {\underline{\theta },\bar{\theta }} \right)$. Let $\underline{p}$ be the price of the product $\underline{\theta }$ and $\bar{p}$ be the price of product $\bar{\theta }$. Assume with no loss of generality that consumer $\hat{\theta } \in \left( {\underline{\theta } ,\bar{\theta }} \right)$ buys product $\bar{\theta }$. Then, product $\hat{x} =\left( {\hat{\theta },\hat{p} } \right)$ where $\hat{p} ={\max }\left\{ {\underline{p} +{t}\left( {\hat{\theta } ,\underline{\theta } } \right) ;\bar{p}+t\left( \hat{\theta },\bar{\theta }\right) } \right\}$ is feasible and yields more profits. Thus, repeating this argument we get that, in equilibrium, $q(\theta )=\theta$ for all ${\theta }\in (\underline{\theta },\bar{\theta })$.

Since $q(\theta )=\theta$ for all $\theta \ge \theta _{k}$ by U-IC constraints, we have that $p(\theta )=p^{*}$ for all $\theta \ge \theta _{k}$.

Finally, by V-IC constraint, $p^{*}- p^{v }\ge t(q^{v}, \theta ^{v}) - t(\theta _{k}, \theta ^{v}) \ge 0$, where the last inequality follows from the fact that $q^{v }\le \theta _{k}$. $\square$

Proof of Lemma 3

We show that, if $q^{v }< \theta _{k}$, the monopolist finds it optimal to decrease $\theta _{k}$. By definition of $\theta _{k}, U(x(\theta _{k}); \theta _{k})=U(x^{v}; \theta _{k})$. Thus, using Lemma 2, $p^{v}=p^{*}- t(q^{v}, \theta _{k})$. Therefore, the monopolist’s profits are

$$\begin{aligned}&\Pi =(1-\pi )(\theta _{k}(p^{*}-t(q^{v},\theta _{k}))+ (1-\theta _{k})p^{*})+\pi (p^{*}-t(q^{v}, \theta _{k}))\\&\quad =p^{*}-t(q^{v},\theta _{k})(\pi +(1-\pi )\theta _{k}), \end{aligned}$$

where $\left( {\frac{{\mathrm{d}\Pi }}{{\mathrm{d}\theta }_{k} }} \right) <0$ for all $q^{v}<\theta _{k}$. Thus, in equilibrium, $q^{v}=\theta _{k}$, which implies $p^{v}=p ^{*}$ by incentive compatibility. $\square$

Proof of Corollary 1

Note that social welfare is given by

$$\begin{aligned} \mathcal {W}&= \Pi +\mathop {\int }\limits _0^1 {W}\left( {\bar{M} ;{\theta }} \right) {\mathrm{d}\theta }=s-{t}\left( {{q}^{{v}},0} \right) +\mathop {\int } \limits _0^{{\theta }_{k} } \left[ {{t}\left( {{q}^{{v}},0} \right) -{t}\left( {{q}^{{v}},{\theta }} \right) } \right] {\mathrm{d}\theta }\\&+{\pi }\mathop \int \limits _{{\theta }_{k} }^1 \left[ {{t}\left( {{q}^{{v}},0} \right) -{t}\left( {{q}^{{v}},{\theta }} \right) } \right] {\mathrm{d}\theta }. \end{aligned}$$

Using the optimal menu, $t(q^{v},0)=\pi t(q^{v},1)$ and $q^{v}=\theta _{k}$, we get

$$\begin{aligned} \mathcal {W}={s}-\mathop {\int } \limits _0^{{q}^{{v}}} {t}\left( {{q}^{{v}},{\theta }} \right) {\mathrm{d}\theta } +{\pi }\mathop \int \limits _{{q}^{{v}}}^1 {t}\left( {{q}^{{v}},{\theta }} \right) {\mathrm{d}\theta }. \end{aligned}$$

Let T be the antiderivative of t. Then, applying the fundamental theorem of calculus, we can write,

$$\begin{aligned} \mathcal {W}=s-\left[ {\left. {-{T}\left( {{q}^{{v}},{\theta }} \right) } \right] _0^{{q}^{{v}}} +{\pi }\left. {{T}\left( {{q}^{{v}},{\theta }} \right) } \right] _{{q}^{{v}}}^1} \right] ={s}-\left[ {{T}\left( {{q}^{{v}},0} \right) +{\pi T}\left( {{q}^{{v}},1} \right) } \right] , \end{aligned}$$

where

$$\begin{aligned} \frac{{\mathrm{d}}\mathcal {W}}{{\mathrm{d}\pi }}=\frac{{\mathrm{d}q}^{{v}}}{{\mathrm{d}\pi }}\left[ {{t}\left( {{q}^{{v}},0} \right) -{\pi t}\left( {{q}^{{v}},1} \right) } \right] +{T}\left( {{q}^{{v}},1} \right) ={T}\left( {{q}^{{v}},1} \right) >0, \end{aligned}$$

where the last equality holds because $t(q^{v},0)=\pi t(q^{v},1)$ at the optimal menu. $\square$

Proof of Lemma 4

It is immediate, from the analysis in Sect. 4, that in equilibrium $V(x^{v})={W}(\bar{M} ;{\theta }^{{NFC}})=0$, so $t(q^{v}(\pi , \theta ^{NFC}),0)={\pi }t(q^{v}(\pi , \theta ^{NFC}), \theta ^{NFC})$. Hence, the tempting choice when the market is non-fully covered depends on the temptation probability $\pi$ and the number of consumers served $\theta ^{NFC}$. Where, $\frac{{\mathrm{d}q}^{{v}}({\pi },{\theta }^{{NFC}})}{{\mathrm{d}\pi }}>0$ and $\frac{{\mathrm{d}q}^{{v}}({\pi },{\theta }^{{NFC}})}{{\mathrm{d}\theta }^{{NFC}}}>0$.

To compute the optimal menu, we just need to calculate ${\theta }^{{NFC}}$ solving the following maximization problem:

$$\begin{aligned} \mathop {\max }\limits _{{\theta }^{{NFC}}} {p\theta }^{{NFC}} \end{aligned}$$

st.,

$$\begin{aligned} p=s-t(q^{v}(\pi , \theta ^{NFC}), 0). \end{aligned}$$

Thus, the implicit equation for ${\theta }^{{NFC}}$ is given by

$$\begin{aligned} s-t(q^{v}(\pi , \theta ^{NFC}), 0)= \theta ^{NFC} t\hbox {'}(q^{v}(\pi , \theta ^{NFC}), 0)\frac{{\mathrm{d}q}^{{v}}({\pi },{\theta }^{{NFC}})}{{\mathrm{d}\theta }^{{NFC}}}. \end{aligned}$$

$\square$

Proof of Lemma 5

Suppose by contradiction that the monopolist designs commitment products in the interval $[\underline{\theta },1]$, where $\underline{\theta } \in [0,1)$. Then, for any commitment product, the price is $\underline{p} ={s}-{t}\left( {\underline{\theta },0} \right)$, which is unique by the incentive compatibility constraints. Therefore, the ex-ante surplus of consumers with $\theta \ge \theta _{k}$, is given by

$$\begin{aligned} {W}\left( {\bar{M} ;{\theta }} \right) =\left\{ {{\begin{array}{l} {\left( {1-{\pi }} \right) (s-\underline{p} -t\left( {{\theta },\underline{\theta } } \right) +\pi \left( {{t}\left( {{q}^{{v}},{\theta }^{{v}}} \right) -{t}\left( {{q}^{{v}},{\theta }} \right) } \right) \hbox {for all }\theta <\underline{\theta },} \\ {\left( {1-{\pi }} \right) ( {{s}-\underline{p} } )+\pi \left( {{t}\left( {{q}^{{v}},{\theta }^{{v}}} \right) -{t}\left( {{q}^{{v}},{\theta }} \right) } \right) \hbox {for all }\theta >\underline{\theta }.} \\ \end{array}}} \right. \end{aligned}$$

Since $\left( \frac{\mathrm{d}W(\bar{W};\theta )}{\mathrm{d}\theta }\right) <0$, for all $\theta >q^{v}$, the “worst consumer type” from the point of view of the monopolist is $\theta =1$. Note that by setting $\underline{\theta } =1$, the monopolist maximizes ${W}(\bar{M},1)$ and thus allows for the lowest $q^{v}$ which yields the highest price for the tempting product and hence the highest profits.

Finally, note that since $\underline{\theta } =1, p = s - t(1,0)$. Moreover, since $V(x^{v})=0, p^{v}= s - t(q^{v},0)$. Therefore, from the definition of $\theta _{k}, U(x(\theta _{k}); \theta _{k})=U(x^{v}; \theta _{k})$, we get that $\theta _{k}=0$. $\square$

Proof of Lemma 6

Since the monopolist wants to charge the highest possible p$^{v}$, we need to check that $p^{v}=s$ (i.e., $q^{v}=\theta ^{v}=0$) is feasible. In our case, it is sufficient to check that ${W}\left( {\bar{M} ;{\theta }} \right) \ge 0$ for all $\theta \in [0,1]$, where

$$\begin{aligned} {W}\left( {\bar{M}} ;\theta \right) =(1-\pi )(s-p-t(q,\theta ))+\pi (s-p^{v}-t(q^v,\theta )). \end{aligned}$$

Thus, applying Lemma 5 and letting $p^{v}=s$, we get

$$\begin{aligned} {W}\left( {\bar{M}};\theta \right)&= (1-\pi )(s-(s-t(1,0)) -t(1,\theta ))+\pi (t(0,0)-t(0,0))\\&= \left( {1-{\pi }} \right) \left( {{t}\left( {1,0} \right) -{t}\left( {1,{\theta }} \right) } \right) -{\pi t}\left( {0,{\theta }} \right) . \end{aligned}$$

Note that

$$\begin{aligned} \frac{{\mathrm{d}W}\left( {\bar{M}};\theta \right) }{{\mathrm{d}\theta }}=0 \hbox { if and only if } \left| \frac{{t^\prime }\left( {1,{\theta }} \right) }{{t}^{{{\prime }}}\left( {0,{\theta }} \right) }\right| =\frac{\pi }{1-\pi }. \end{aligned}$$

Due to the properties of the transportation cost function, this condition implies that for all $\pi \in [0,1], \exists \theta ^{*}\in [0,1]$ such that for all $\theta <\theta ^{*} (\theta >\theta ^{*}), \frac{\mathrm{d}W(\bar{M}; \theta )}{\mathrm{d}\theta }>0 (<0)$. Therefore, since ${W}\left( {\bar{M}}; 0\right) =0$, a necessary and sufficient condition for $W(M;\theta )\ge 0$ for all $\theta \in [0,1]$ is that $W(M;1)\ge 0$. Since

$$\begin{aligned} {W}(\bar{M};1)&= (1-\pi )t(1,0)-\pi {t}(0,1)\\&= \left( {1-{\pi }} \right) {t}\left( {1,0} \right) -{\pi t}\left( {1,0} \right) , \end{aligned}$$

then ${W}\left( {\bar{M}},1 \right) \ge 0$ iff ${\pi }\le \frac{1}{2}$. This implies that, when ${\pi }>\frac{1}{2}$, the monopolist has to locate the tempting choice beyond $\theta ^{v}$, i.e., $q^{v}>0$, which implies charging a $p^{v}<s$ to attract consumer $\theta =1$ into the store. Since ${W}\left( {\bar{M}}; 1\right) =0$, then

$$\begin{aligned} {W}\left( {\bar{M}};1\right) =(1-\pi )t(1,0)-\pi (t(q^v,0)-t(q^v,1))=0. \end{aligned}$$

Thus,

$$\begin{aligned} {t}\left( {{q}^{{v}},1} \right) -{t}\left( {{q}^{{v}},0} \right) =\frac{1-{\pi }}{{\pi }}{t}\left( {1,0} \right) . \end{aligned}$$

$\square$

Proof of Lemma 7

To prove this lemma, we use some of the results and intuition of our basic model. Let us start with the candidate of optimal menu with self-control preferences (See Fig. 7). Thus, consumers in the interval $(2q^{v},1]$ purchase a product which is a compromise between commitment and temptation ideal points $({q}\left( {\theta } \right) =\frac{{\theta }}{2})$, while all consumers in the interval $[0,2q^{v}]$ purchase the tempting choice $x^{v}=(q^{v},p^{v})$. The proof consists of several steps.

Step 1 Note that IC constraints of consumers $\theta \in (q^{v},1]$ imply that $p(\theta )=p$.

Step 2 We need to check that in equilibrium consumers with $\theta \in (2q^{v},1]$ do not want to deviate and choose the offer $x^{v}$. By the IC constraint of consumer $\theta =2q^{v}$, this implies that $p^{v}=p$.

Step 3 In equilibrium ${W}^{{SC}}\left( {\bar{M}}; 1 \right) =0$, because $\theta =1$ is the consumer with the highest self control cost. Since ${W}^{{SC}}\left( {\bar{M}}; 1 \right) =s-p-f({q}^{{v}},1)$, we get that in equilibrium $p=s-f(q^{v},1)$.

Step 4 Let us define $\hat{\theta } \in \left[ {{q}^{{v}},\frac{1}{2}} \right]$. We need to check that consumer $\hat{\theta }$ do not want to deviate and consume ${q}_{d} =\hat{\theta }$. Note that ${W}^{{SC}}\left( {\bar{M}}; \hat{\theta } \right) =s-p-t(q^{v},\hat{\theta })$ while under the deviation ${W}_{d}^{{SC}} \left( {{M}^{{d}};\hat{\theta }} \right) ={s}-{p}-\left[ {{t}\left( {\hat{\theta },0} \right) -{t}\left( {{q}^{{v}},0} \right) } \right]$. Where ${W}^{{SC}}\left( {\bar{M}};\theta \right) >W^{SC}_{d}(M^{d};\theta )$ iff ${t}\left( {{q}^{{v}},0} \right) +{t}\left( {{q}^{{v}},\hat{\theta } } \right) <t\left( {\hat{\theta },0} \right)$ which is true because of strict superadditivity.

Step 5: Moreover, we need to assure participation of the low tail of the distribution $({W}^{{SC}}\left( {\bar{M}};0 \right) \ge 0)$. Note that ${W}^{{SC}}\left( {\bar{M}}, 0 \right) =s-p-t(q^{v},0)$, and hence applying Step 3, we get ${W}^{{SC}}\left( {\bar{M}},0 \right) =f\left( q^{v},1\right) -t(q^{v},0)= t\left( \frac{1}{2},1\right) +t\left( \frac{1}{2},0\right) -2t(q^{v},0)= 2\left[ t\left( \frac{1}{2},0\right) -t(q^{v},0)\right]$, where the last equality comes from the symmetry of the transportation cost function. Therefore, ${W}^{{SC}}\left( {\bar{M}},0\right) \ge 0$ for all ${q}^{{v}}\le \frac{1}{2}$.

Step 6: Finally, since $p=s-f(q^{v},1)$ and ${q}^{{v}}\le \frac{1}{2}$, the firm maximizes profits offering a single product ${x}^{{SC}}=\left( {\frac{1}{2},{s}-{t}\left( {\frac{1}{2},1} \right) } \right)$, which, as we have shown in the previous steps, is a feasible menu. $\square$