Low price signals high capacity

Abstract

We study pricing in a model where buyers are homogeneous and sellers have either capacity one or two. We show that if buyers observe prices but not capacities then an equilibrium exists where sellers of capacity two post lower prices than sellers of capacity one. The equilibrium satisfies the intuitive criterion.

Appendix

Proof of Proposition 1

The proof is divided into steps. First we establish two technical results which guarantee that certain variables are well-defined. Lemma 5 establishes the equilibrium price of the low-capacity sellers in any separating equilibrium. Lemmata 6 and 7 demonstrate that $\overline{p}_{2}$ is an equilibrium price of the high-capacity sellers. We need different techniques to show that deviations upward and downward are not profitable; the former is shown in Lemma 6 and the latter in Lemma 7. Finally, in Lemma 8 we show that there exists a multitude of separating equilibria. In all proofs we take it as granted that the buyers’ equilibrium behaviour is given by their indifference between the sellers.

Lemma 3

Assume that a separating equilibrium exists with $p_{2}<p_{1}$. Then the queue length for high-capacity sellers is larger than for low-capacity sellers, i.e., $\alpha >\beta$.

Proof

In the buyers’ indifference condition (6) the LHS is decreasing in $\beta$ and the RHS is decreasing in $\alpha$. Even when the prices are the same the RHS is greater than the LHS when $\alpha =\beta$. Consequently, equality requires that $\alpha >\beta$. $\square$

Lemma 4

Assume that a separating equilibrium exists and (16) holds. Then for any $p_{1}$ and $p_{2}$ such that $p_{2}<p_{1}$, $\alpha$ and $\beta$ are uniquely determined$.$.

Proof

We prove that a unique solution exists to (6) such that $\alpha >\beta >0$. Differentiating the LHS with respect to $x$ it is easily seen to be positive. Differentiating the RHS with respect to $x$ it is easily seen to be negative. When $x=y$ we have ${\alpha }={\beta }={\theta }$. Then the LHS is less than the RHS. Increasing $x$ either produces equality for a unique value $x\in (0,1)$ or the LHS remains less than the RHS even when $x=1$, as the price of low-capacity sellers is so high that no buyer visits them. $\square$

Lemma 5

If there exists a separating equilibrium where the deviators are expected to be low-capacity sellers, then the low-capacity sellers’ equilibrium price is given by

$$\begin{aligned} p_{1}=\frac{1-e^{-\beta }-\beta e^{-\beta }}{1-e^{-\beta }} \end{aligned}$$

Proof

In equilibrium the market utility that a consumer expects is $\bar{U}$. The profit maximisation problem of a low-capacity seller is

$$\begin{aligned} \max _{p_{1}}\pi _{1}=p_{1}(1-e^{-\beta }) \text{ s.t. } (1-p_{1})\frac{ 1-e^{-\beta }}{\beta }=\bar{U}, \end{aligned}$$

Notice that the constraint gives a one-to-one correspondence between $p_{1}$ and $\beta$. Consequently, we can solve $p_{1}$ from the constraint and the problem may be written as

$$\begin{aligned} \max _{\beta }\left( 1-\frac{\beta \bar{U}}{1-e^{-\beta }}\right) (1-e^{-\beta }), \end{aligned}$$

The first-order condition with respect to $\beta$ yields

$$\begin{aligned} e^{-\beta }=\bar{U}, \end{aligned}$$

which we can substitute into the constraint to get

$$\begin{aligned} p_{1}=\frac{1-e^{-\beta }-\beta e^{-\beta }}{1-e^{-\beta }} \end{aligned}$$

Note that the second-order condition is $-\beta e^{-\beta }<0$ which guarantees that $p_{1}$ is a good candidate for and equilibrium price, given the buyers’ indifference condition. That the price constitutes a unique symmetric equilibrium is shown in many works, e.g. Kultti (2011). $\square$

Lemma 6

Assume that low-capacity sellers post price $p_{1}=\frac{1-e^{-\beta }-\beta e^{-\beta }}{1-e^{-\beta }}$ and high-capacity sellers price $\overline{p} _{2}=\frac{1-e^{-\beta }-\beta e^{-\beta }}{1-e^{-\alpha }}$. If the buyers believe that any other price is by a low-capacity seller, then it is not profitable to a high-capacity seller to post a price $p>\overline{p}_{2}$.

Proof

Assume that a high-capacity seller deviates and posts price $p\ne \overline{ p}_{2}$. The queue length $\gamma$ that this price generates is given by

$$\begin{aligned} \frac{1-e^{-\beta }}{\beta }(1-p_{1})=e^{-\beta }=\frac{2-2e^{-\alpha }-\alpha e^{-\alpha }}{\alpha }(1-\overline{p}_{2})=\frac{1-e^{-\gamma }}{ \gamma }(1-p) \end{aligned}$$

(17)

As $p>\overline{p}_{2}$ it must be the case that $\gamma <\alpha$. The deviator’s profit is less than the equilibrium profit if

$$\begin{aligned} \left( 2-2e^{-\gamma }-\gamma e^{-\gamma }\right) p\le \left( 2-2e^{-\alpha }-\alpha e^{-\alpha }\right) \overline{p}_{2}. \end{aligned}$$

(18)

Substituting from (17) this is equivalent to

$$\begin{aligned} \left( 2-2e^{-\gamma }-\gamma e^{-\gamma }\right) \left[ 1-\frac{\gamma }{ 1-e^{-\gamma }}\frac{2-2e^{-\alpha }-\alpha e^{-\alpha }}{\alpha }(1- \overline{p}_{2})\right] \le \left( 2-2e^{-\alpha }-\alpha e^{-\alpha }\right) \overline{p}_{2}. \end{aligned}$$

Adopting the notation $\Omega _{z}\equiv 2-2e^{-z}-ze^{-z}$ this, in turn, is equivalent to

$$\begin{aligned} \Omega _{\gamma }\left[ 1-\frac{\gamma }{1-e^{-\gamma }}\frac{\Omega _{\alpha }}{\alpha }(1-\overline{p}_{2})\right] \le \Omega _{\alpha } \overline{p}_{2}. \end{aligned}$$

Next we substitute from (17) for $\frac{\Omega _{\alpha }}{\alpha } (1-\overline{p}_{2})$ to get

$$\begin{aligned} \Omega _{\gamma }\left[ 1-\frac{\gamma }{1-e^{-\gamma }}e^{-\beta }\right] \le \Omega _{\alpha }\overline{p}_{2}, \end{aligned}$$

or

$$\begin{aligned} \overline{p}_{2}\ge \frac{\Omega _{\gamma }}{\Omega _{\alpha }}\left[ 1- \frac{\gamma }{1-e^{-\gamma }}e^{-\beta }\right] . \end{aligned}$$

This is equivalent to

$$\begin{aligned} \frac{1-e^{-\beta }-\beta e^{-\beta }}{1-e^{-\alpha }}\ge \frac{\Omega _{\gamma }}{\Omega _{\alpha }}\left[ 1-\frac{\gamma }{1-e^{-\gamma }} e^{-\beta }\right] , \end{aligned}$$

which is equivalent to

$$\begin{aligned} \frac{\Omega _{\alpha }}{1-e^{-\alpha }}\left( 1-e^{-\beta }-\beta e^{-\beta }\right) \ge \frac{\Omega _{\gamma }}{1-e^{-\gamma }}\left( 1-e^{-\gamma }-\gamma e^{-\beta }\right) . \end{aligned}$$

As $1-e^{-\gamma }-\gamma e^{-\beta }$ reaches its maximum at $\gamma =\beta$ it is enough to show that $\frac{\Omega _{\alpha }}{1-e^{-\alpha }}>\frac{ \Omega _{\gamma }}{1-e^{-\gamma }}$ but this is the case as $\frac{\Omega _{z}}{1-e^{-z}}$ is increasing in $z$. $\square$

Lemma 7

Assume that low-capacity sellers post price $p_{1}=\frac{1-e^{-\beta }-\beta e^{-\beta }}{1-e^{-\beta }}$, and high-capacity sellers price $\overline{p} _{2}=\frac{1-e^{-\beta }-\beta e^{-\beta }}{1-e^{-\alpha }}$. If the buyers believe that any other price is by a low-capacity seller, then it is not profitable to a high-capacity seller to post a price $p<\overline{p}_{2}$.

Proof

Again (17) holds and we want to show that

$$\begin{aligned} \left( 2-2e^{-\gamma }-\gamma e^{-\gamma }\right) p\le \left( 2-2e^{-\alpha }-\alpha e^{-\alpha }\right) \overline{p}_{2}. \end{aligned}$$

(19)

Consider the following process: Find price $q_{2}^{1}<\overline{p}_{2}$ that generates the same queue length as $\overline{p}_{2}$, namely $\gamma ^{0}\equiv \alpha$. Any price $p\in \left[ q_{2}^{1},\overline{p} _{2}\right)$ generates less than equilibrium profits. Then find queue length $\gamma ^{1}$ that generates equilibrium profits at price $q_{2}^{1}$. Notice that $\gamma ^{1}>\alpha$. Next find price $q_{2}^{2}$ that generates queue length $\gamma ^{1}$. Clearly $q_{2}^{2}<q_{2}^{1}$, and any price $p\in \left[ q_{2}^{2},q_{2}^{1}\right)$ generates less than equilibrium profits. Find then queue length $\gamma ^{2}$ that generates the equilibrium profits at price $q_{2}^{2}$. Notice that $\gamma ^{2}>\gamma ^{1}.$ Find price $q_{2}^{3}$ that generates queue length $\gamma ^{2}$. Clearly $q_{2}^{3}<q_{2}^{2}$, and any price $p\in \left[ q_{2}^{3},q_{2}^{2}\right)$ generates less than equilibrium profits. In the sequence $\left\{ \left( q_{2}^{i},\gamma ^{i}\right) \right\}$ the first co-ordinate is decreasing and the second increasing and both co-ordinate sequences clearly converge or are not defined: The latter happens if at some point even an infinite queue length (or trading two units with probability one) is not enough to generate equilibrium profits. In this case there does not exist a profitable deviation. So assume that the sequence converges to $\left( \overline{q},\overline{\gamma }\right)$.

Consider next price $r_{2}^{1}=0$. This generates queue length $\delta ^{1}$ such that the buyers are indifferent between contacting any of the sellers. Determine then price $r_{2}^{2}$ that at queue length $\delta ^{1}$ generates the equilibrium profits. Then determine the queue length that price $r_{2}^{2}$ generates as well as price $r_{2}^{3}$ that at queue length $\delta ^{2}$ generates the equilibrium profits. This process gives a sequence $\left\{ \left( r_{2}^{i},\delta ^{i}\right) \right\}$ that clearly converges to, say, $\left( \overline{r},\overline{\delta }\right)$. It is again clear that $\overline{r}\le \overline{q}$; otherwise there would be prices larger than $\overline{q}$ that would generate profits $\Omega _{\alpha }\overline{p}_{2}$, but this is impossible by construction. If $\overline{r}=\overline{q}$ there is no profitable deviation to low prices.

We show that $\overline{r}<\overline{q}$ leads to a contradiction. Notice first that these prices, and the corresponding queue lengths, have to satisfy the following conditions

$$\begin{aligned} \frac{1-e^{-\overline{\gamma }}}{\overline{\gamma }}(1-\overline{q})=\frac{ 1-e^{-\overline{\delta }}}{\overline{\delta }}(1-\overline{r})=e^{-\beta }, \end{aligned}$$

and

$$\begin{aligned} \Omega _{\overline{\gamma }}\overline{q}=\Omega _{\overline{\delta }} \overline{r}=\Omega _{\alpha } \overline{p}_{2}. \end{aligned}$$

We can solve

$$\begin{aligned} \overline{q}=\frac{1-e^{-\overline{\gamma }}-\overline{\gamma }e^{-\beta }}{ 1-e^{-\overline{\gamma }}}, \end{aligned}$$

$$\begin{aligned} \overline{r}=\frac{1-e^{-\overline{\delta }}-\overline{\delta }e^{-\beta }}{ 1-e^{-\overline{\delta }}}. \end{aligned}$$

Next we show that $\Omega _{\overline{\gamma }}\overline{q}=\Omega _{ \overline{\delta }}\overline{r}$ cannot hold. Consider expression $\Omega _{z}\frac{1-e^{-z}-ze^{-\beta }}{1-e^{-z}}$. Its derivative is

$$\begin{aligned} e^{-z}(1+z)\frac{1-e^{-z}-ze^{-\beta }}{1-e^{-z}}-\Omega _{z}\frac{e^{-\beta }\left( 1-e^{-z}-ze^{-z}\right) }{\left( 1-e^{-z}\right) ^{2}}, \end{aligned}$$

which is of the same sign as

$$\begin{aligned} \left( 1-e^{-z}\right) e^{-z}(1+z)\left( 1-e^{-z}-ze^{-\beta }\right) -\Omega _{z}e^{-\beta }\left( 1-e^{-z}-ze^{-z}\right) . \end{aligned}$$

At $z=\beta$ this is negative and continues to be negative for all $z>\beta$. This shows our claim. $\square$

The previous results establish that prices $p_{1}=\frac{1-e^{-\beta }-\beta e^{-\beta }}{1-e^{-\beta }}$ and $\overline{p}_{2}=\frac{1-e^{-\beta }-\beta e^{-\beta }}{1-e^{-\alpha }}$ constitute a separating equilibrium. The next result shows that there are infinitely many of them.

Lemma 8

Assume that low-capacity sellers post price $p_{1}=\frac{1-e^{-\beta }-\beta e^{-\beta }}{1-e^{-\beta }}$. There is a continuum of separating equilibria where high-capacity sellers post price $p_{2}\in \left[ \underline{p}_{2}, \overline{p}_{2}\right]$, and where any other price is expected to be by a low-capacity seller.

Proof

The high-capacity sellers are indifferent between posting $p_{1}=\frac{ 1-e^{-\beta }-\beta e^{-\beta }}{1-e^{-\beta }}$ and $\underline{p}_{2}$ if $\underline{p}_{2}\left( 2-2e^{-\alpha }-\alpha e^{-\alpha }\right) =p_{1}\left( 2-2e^{-\beta }-\beta e^{-\beta }\right)$. Based on the proofs of the previous lemmata it is clear that any price $p_{2}\in \left[ \underline{p}_{2},\overline{p}_{2}\right]$ constitutes a separating equilibrium with $p_{1}=\frac{1-e^{-\beta }-\beta e^{-\beta }}{1-e^{-\beta }}$. Note that the value of $\beta$ depends on $p_{2}$. $\square$