Search and Ripoff Externalities

Abstract

This paper surveys models of markets in which only some consumers are “savvy”. I discuss when the presence of savvy consumers improves the deals available to all consumers (the case of search externalities), and when the non-savvy consumers fund generous deals for all consumers (ripoff externalities). I also discuss when the two groups of consumers have aligned or divergent views about market interventions. The analysis focusses on two kinds of models: (1) an indivisible product in a market with price dispersion; and (2) products that involve add-on pricing.

Appendix

Details for the model presented in Sect. 2.3: There are two kinds of equilibria to consider: those where sellers only ever serve the savvy consumers, and those where sellers sometimes serve non-savvy consumers.

First, look for an equilibrium where the minimum price satisfies $p_{\min }>\alpha v$, so that non-savvy consumers never participate in the market. As discussed in the main text, the equilibrium profit for a seller is $\sigma \alpha v(1-\alpha )$. If a seller deviated and instead set the lower price $p=\alpha v$ it would sell to $1-\sigma +\sigma \alpha$ consumers, and so obtain profit $\alpha v[1-\sigma +\alpha \sigma ]$. For this deviation to be unprofitable we require $\sigma \alpha v(1-\alpha )\ge \alpha v[1-\sigma +\alpha \sigma ]$, i.e., that condition (5) in the main text is violated. The CDF for a seller’s mixed strategy for price, F, satisfies

$$\begin{aligned} \sigma \alpha \left[ 1-\alpha +\alpha (1-F(p))\right] p\equiv \sigma \alpha v(1-\alpha ), \end{aligned}$$

(12)

which does not depend on $\sigma$. The minimum price in the support, which makes $F=0$, is therefore

$$\begin{aligned} p_{\min }=(1-\alpha )v, \end{aligned}$$

which given that (5) is violated is indeed greater than $\alpha v$. Here, the support of prices is $[p_{\min },v]$, which has no gap. Subject to $\sigma$ remaining outside the range (5), prices do not depend on $\sigma$ and there is neither a search nor a ripoff externality present.

Next, look for an equilibrium where the minimum price satisfies $p_{\min }<\alpha v$, so that non-savvy consumers are sometimes served. In this case, the price support will consist of two disjoint segments, with a gap in between, so that the support takes the form $[p_{\min },\alpha v]\cup [p^{*},v]$ where $p^{*}>\alpha v$. Again, a seller obtains equilibrium profit $\sigma \alpha v(1-\alpha )$, and so $p_{\min }$ satisfies

$$\begin{aligned} \left[ 1-\sigma +\sigma \alpha \right] p_{\min }=\sigma \alpha v(1-\alpha )\ \end{aligned}$$

which indeed implies that $p_{\min }\le \alpha v$ when (5) holds. The CDF for a seller’s mixed strategy for price, F, again satisfies (12) in the upper segment $[p^{*},v]$, while in the lower segment $[p_{\min },\alpha v]$ it satisfies

$$\begin{aligned}{}[\sigma \alpha (1-\alpha )+(\sigma \alpha ^{2}+1-\sigma )(1-F(p))]p=\sigma \alpha v(1-\alpha ). \end{aligned}$$

(13)

Since $F(\alpha v)=F(p^{*})$, expression (13) implies that

$$\begin{aligned}{}[\sigma \alpha (1-\alpha )+(\sigma \alpha ^{2}+1-\sigma )(1-F(p^{*}))]\alpha v=\sigma \alpha v(1-\alpha ), \end{aligned}$$

so that the probability a seller caters only to the savvy, $1-F(p^{*})$, is

$$\begin{aligned} 1-F(p^{*})=\frac{\sigma (1-\alpha )^{2}}{1-\sigma (1-\alpha ^{2})}. \end{aligned}$$

(Note that $1-F(p^{*})=1$ when (5) just binds, while $1-F(p^{*})=0$ when $\sigma =0$.) It follows from (12) that $p^{*}$ satisfies

$$\begin{aligned} p^{*}=\frac{1-\sigma +\alpha ^{2}\sigma }{1-\sigma +\alpha \sigma }v, \end{aligned}$$

(14)

which lies between $\alpha v$ and v. This completes the derivation of the equilibrium.

Note that $p^{*}$ in (14) decreases with $\sigma$. Moreover, one can check that F in (13) decreases with $\sigma$. This implies that a seller’s price weakly increases with $\sigma$ in the sense of first-order stochastic dominance. (If (5) does not hold, F does not depend on $\sigma$, while if (5) holds prices strictly increase with $\sigma$.) Because of this, a consumer of either type is weakly better off when $\sigma$ falls, and a ripoff externality is present.

Details for the model with rational consumers presented in Sect. 3.1: Given $\sigma$, suppose that $(P^{*},p^{*})$ is the equilibrium price pair in this market, and write $U=s(p^{*})-P^{*}$ for consumer surplus gross of transport costs in this equilibrium. Suppose one seller considers choosing the price pair (P, p) , while its rival plays the equilibrium strategy $(P^{*},p^{*})$. Savvy consumers see both of this seller’s prices, and so the fraction of them who buy from it is

$$\begin{aligned} \frac{1}{2}+\frac{s(p)-P-U}{2t}. \end{aligned}$$

Non-savvy consumers do not observe p, and form rational expectations about this price given P. Suppose they anticipate add-on price $\hat{p}$, in which case the fraction of non-savvy who buy from the seller is

$$\begin{aligned} \frac{1}{2}+\frac{s(\hat{p})-P-U}{2t}. \end{aligned}$$

Similarly to expression (6), the seller’s total profit is therefore

$$\begin{aligned} \left[ \sigma \left( \frac{1}{2}+\frac{s(p)-P-U}{2t}\right) +(1-\sigma )\left( \frac{1}{2}+\frac{s(\hat{p})-P-U}{2t}\right) \right] \times [P-C+\pi (p)]. \end{aligned}$$

(15)

Given P and $\hat{p}$, the seller chooses its add-on price p to maximize this profit. For the non-savvy consumers to have rational expectations about the add-on price, we require that choosing $p=\hat{p}$ maximizes profit in (15). This implies that the equilibrium add-on price given P satisfies the first-order condition

$$\begin{aligned} (t+s(p)-P-U)\pi ^{\prime }(p)-\sigma q(p)(P-C+\pi (p))=0. \end{aligned}$$

(16)

Given equilibrium surplus U, expression (16) characterizes the relationship off the equilibrium path between a seller’s core price and its rationally anticipated add-on price.

Note that if $\sigma =0$, the equilibrium add-on price satisfies $\pi ^{\prime }(p)=0$, so that $p=p^{M}$ regardless of the core price P and the equilibrium add-on price is $p^{*}=p^{M}$. At the other extreme, if $\sigma =1$ the outcome as discussed in the text involves $p^{*}=c, P^{*}=C+t$ and $U=s(p^{*})-P^{*}$, which satisfies (16).

Expression (16) can be written as

$$\begin{aligned} (1-\eta (p))(t+s(p)-P-U)=\sigma (P-C+\pi (p)), \end{aligned}$$

(17)

where $\eta (p)\equiv -pq^{\prime }(p)/q(p)$ is the elasticity of add-on demand (which is increasing given that q is logconcave). Since $\eta$ is increasing, the left-hand side of (17) is decreasing with p (at least in the range $c\le p\le p^{M}$), while the right-hand side is increasing. Thus, there is a unique price p which solves (17). Since the left-hand side of (17) decreases with P, while the right-hand side is increasing with P, it follows that the equilibrium add-on price p is a decreasing function of P. Intuitively, when the seller undercuts the equilibrium core price, this reduces its incentive to gain market share from the savvy consumers with a low add-on price. For this reason, the market is less competitive when sellers face a mixed population of consumers.

If $\sigma >0$, from (16) we can write the core price P which implements a given add-on price p as

$$\begin{aligned} P(p)=\frac{\pi ^{\prime }(p)(t+s(p)-U)+\sigma q(p)(C-\pi (p))}{\pi ^{\prime }(p)+\sigma q(p)}. \end{aligned}$$

(18)

The seller then chooses its add-on price p and associated core price P(p) to maximize its profit

$$\begin{aligned} \left[ \frac{1}{2}+\frac{s(p)-P(p)-U}{2t}\right] \times [P(p)-C+\pi (p)]. \end{aligned}$$

(19)

The remaining condition is to ensure that U in (18–19) is the equilibrium level of consumer surplus, i.e., that $U=s(p^{*})-P^{*}$. In sum, $(P^{*},p^{*})$ constitutes a symmetric equilibrium if:

• (1) $p^{*}$ maximizes (19), where P(p) is defined in (18);

• (2) $P^{*}=P(p^{*})$, and

• (3) $U=s(p^{*})-P^{*}$.

It is hard to make further progress at this level of generality, and so I illustrate the analysis with the example where $q(p)=1-p$ and $C=c=0$, so that $s(p)=\frac{1}{2}(1-p)^{2}, \pi (p)=p(1-p)$ and $p^{M}=\frac{1}{2}$. Expression (18) then implies

$$\begin{aligned} P(p)=\frac{(1-2p)\left( t+\frac{1}{2}(1-p)^{2}-U\right) -\sigma p(1-p)^{2}}{1-2p+\sigma (1-p)}. \end{aligned}$$

Substituting this into profit (19) shows that the seller’s profit with add-on price p is

$$\begin{aligned} \frac{1}{8t}\sigma \left( 1-p\right) (1-2p)\left( \frac{1-p^{2}-2U+2t}{ 1-2p+\sigma (1-p)}\right) ^{2}, \end{aligned}$$

and the (relevant) first-order condition for maximizing this expression is that p satisfies

$$\begin{aligned} \left( (1-p)(1-2p)\right) ^{\prime }\frac{1-p^{2}-2U+2t}{1-2p+\sigma (1-p)} +2(1-p)(1-2p)\left( \frac{1-p^{2}-2U+2t}{1-2p+\sigma (1-p)}\right) ^{\prime }=0 \end{aligned}$$

or

$$\begin{aligned} \left( 4p-3\right) \left( 1-p^{2}-2U+2t\right) +2(1-p)(1-2p)\left( -2p+\left( 2+\sigma \right) \left( \frac{ 1-p^{2}-2U+2t}{1-2p+\sigma (1-p)}\right) \right) =0. \end{aligned}$$

(20)

Given that $U=\frac{1}{2}(1-p)^{2}-P$, expression (18) implies that

$$\begin{aligned} U=\frac{\sigma (1+p)(1-p)^{2}-2t(1-2p)}{2\sigma (1-p)} \end{aligned}$$

(21)

and substituting this value of U into (20) reveals that the equilibrium add-on price $p^{*}$ given $\sigma$ satisfies

$$\begin{aligned} t(1-2p^{*})-\sigma (1-p^{*})\left( t+2p^{*}(1-p^{*})(1-2p^{*})\right) =0. \end{aligned}$$

(22)

When $\sigma =0$, this expression has solution $p^{*}=\frac{1}{2}$, while when $\sigma =1$ the expression has solution $p^{*}=0$. From (22), we can write the $\sigma$ which implements a given add-on price $p^{*}\in [0,\frac{1}{2}]$ as

$$\begin{aligned} \sigma =\frac{t(1-2p^{*})}{\left( 1-p^{*}\right) \left( t+2p^{*}(1-p^{*})(1-2p^{*})\right) }. \end{aligned}$$

(23)

Unless t is very small indeed (below 0.01 or so), the above expression decreases with $p^{*}$ in the range $p^{*}\in [0,\frac{1}{2}]$ , and so there is a one-to-one decreasing relationship between $\sigma$ and the equilibrium add-on price $p^{*}$. When $\sigma$ satisfies (23), equilibrium consumer surplus with add-on price $p^{*}$, which is U in (21), simplifies to

$$\begin{aligned} U=\tfrac{1}{2}\left( 1-8[p^{*}]^{3}+11[p^{*}]^{2}-4p^{*}\right) -t. \end{aligned}$$

This is “U-shaped” in $p^{*}$ in the relevant range $p\in [0,\frac{1}{2}]$, and hence U-shaped as a function of $\sigma$. This market has a ripoff externality when the initial $\sigma$ is small, while for large $\sigma$ the market has a search externality. However, consumer surplus when $\sigma =1$ (i.e., when $p=0$) is above that when $\sigma =0$, i.e., when $p=\frac{1}{2}$. Likewise, industry profit when the equilibrium add-on price is $p^{*}, P(p^{*})+\pi (p^{*})$, in this example is

$$\begin{aligned} 2p^{*}(1-p^{*})(1-2p^{*})+t, \end{aligned}$$

which is hump-shaped in $p^{*}$. Thus, industry profit is maximized with a mixed population of consumers, rather than when all consumers are savvy or all are non-savvy. It is hard analytically to invert expression (23) to obtain the add-on price $p^{*}$ as an explicit function of $\sigma$. It is straightforward to do this numerically, however, and Table 1 in the text reports outcomes for a range of $\sigma$ when $t=1$.