## 1. Introduction

## 2. Model

#### Monopolistic Profits

## 3. Price Leadership

**Definition**

**1**

- I.
- At any period $t\ge 0$, the informed firm sets a price ${p}_{I}^{t}$ equal to ${p}_{l}$ if the state is ${s}_{l}$ and equal to ${p}_{h}$ if the state is ${s}_{h}$;
- U.
- the uninformed firm starts by setting the price ${p}_{h}$ at $t=0$ and after that always sets a price ${p}_{U}^{t}$ that matches the informed firm’s previous price, that is, ${p}_{U}^{t}={p}_{I}^{t-1}$.

#### 3.1. Payoffs from Price Leadership

- If the informed firm’s previous price was ${p}_{l}$, the uninformed is setting the price ${p}_{l}$ today. Furthermore, it must be the case that the market size was ${s}_{l}$ in the previous period so the state today is ${s}_{l}$ with probability $(1-\varphi )$ and ${s}_{h}$ with probability $\varphi $. If the current state is ${s}_{l}$ again, both firms set a price ${p}_{l}$ and split the market today and the uninformed firm derives a continuation value of ${V}_{\mathbf{p}}^{U}\left({p}_{l}\right)$. On the other hand, if the current state is ${s}_{h}$, the informed firm sets a price ${p}_{h}$ and the uninformed gets the whole market today at a price ${p}_{l}$ and a continuation value of ${V}_{\mathbf{p}}^{U}\left({p}_{h}\right)$. That is,$$\begin{array}{c}{V}_{\mathbf{p}}^{U}\left({p}_{l}\right)=(1-\delta )\left[(1-\varphi )\frac{\pi ({p}_{l};{s}_{l})}{2}+\varphi \pi ({p}_{l};{s}_{h})\right]+\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}\hfill \\ \hfill \mathrm{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}\delta \left[(1-\varphi )\phantom{\rule{0.166667em}{0ex}}{V}_{\mathbf{p}}^{U}\left({p}_{l}\right)+\varphi \phantom{\rule{0.166667em}{0ex}}{V}_{\mathbf{p}}^{U}\left({p}_{h}\right)\right].\end{array}$$
- If the informed firm’s previous price was ${p}_{h}$, the expected discounted payoff for the uninformed firm is given by$${V}_{\mathbf{p}}^{U}\left({p}_{h}\right)=(1-\delta )\left[(1-\varphi )\frac{\pi ({p}_{h};{s}_{h})}{2}\right]+\delta \left[\varphi {V}_{\mathbf{p}}^{U}\left({p}_{l}\right)+(1-\varphi ){V}_{\mathbf{p}}^{U}\left({p}_{h}\right)\right].$$

- When both the previous and the current states are low, the informed firm knows that the uninformed is going to set a price equal to ${p}_{l}$ because the previous state (and the informed firm previous price) was low. Hence, because the state today is also ${s}_{l}$, the informed firm also sets a price equal to ${p}_{l}$ and both firms equally split the market today. The next state is ${s}_{l}$ with probability $(1-\varphi )$ in which case the continuation value of firm I is ${V}_{\mathbf{p}}^{I}({s}_{l},{s}_{l})$. With probability $\varphi $, the next state is ${s}_{h}$ in which case the informed gets a continuation payoff of ${V}_{\mathbf{p}}^{I}({s}_{l},{s}_{h})$. Then,$${V}_{\mathbf{p}}^{I}({s}_{l},{s}_{l})=(1-\delta )\frac{\pi ({p}_{l};{s}_{l})}{2}+\delta \left[(1-\varphi ){V}_{\mathbf{p}}^{I}({s}_{l},{s}_{l})+\varphi {V}_{\mathbf{p}}^{I}({s}_{l},{s}_{h})\right].$$
- When the previous state was low and the current state is high, the expected discounted payoff for the informed firm is$${V}_{\mathbf{p}}^{I}({s}_{l},{s}_{h})=\delta \left[\varphi {V}_{\mathbf{p}}^{I}({s}_{h},{s}_{l})+(1-\varphi ){V}_{\mathbf{p}}^{I}({s}_{h},{s}_{h})\right].$$
- When the previous state was high and the current state is low, the expected discounted payoff for the informed firm is$${V}_{\mathbf{p}}^{I}({s}_{h},{s}_{l})=(1-\delta )\pi ({p}_{l};{s}_{l})+\delta \left[(1-\varphi ){V}_{\mathbf{p}}^{I}({s}_{l},{s}_{l})+\varphi {V}_{\mathbf{p}}^{I}({s}_{l},{s}_{h})\right].$$
- When both the previous and the current states are high, the expected discounted payoff for the informed firm is$${V}_{\mathbf{p}}^{I}({s}_{h},{s}_{h})=(1-\delta )\frac{\pi ({p}_{h};{s}_{h})}{2}+\delta \left[\varphi {V}_{\mathbf{p}}^{I}({s}_{h},{s}_{l})+(1-\varphi ){V}_{\mathbf{p}}^{I}({s}_{h},{s}_{h})\right].$$

#### 3.2. Price Leadership as an Equilibrium

#### 3.2.1. Price Leadership Equilibrium with Monopolistic Prices

**Proposition**

**1.**

- When $\frac{{s}_{l}}{{s}_{h}}$ is low, the monopolistic price for the low state, ${p}_{l}^{M}$, is low relative to ${p}_{h}^{M}$ and that makes the option of deviating to set ${p}_{l}^{M}$ less desirable.
- When $\varphi $ is low, the demand is persistent, the market size is likely to stay high in the next period and deviating would lead to a low price by the uninformed firm.

**Lemma**

**1.**

**Proposition**

**2.**

- Exists a $\overline{\varphi}>0$, such that for any $\varphi <\overline{\varphi}$, price leadership with monopolistic profits is a PBE.
- The ex-ante expected joint profits go to the monopolistic profits as ϕ goes to 0.

#### 3.2.2. Price Leadership When Monopolistic Prices Are Not Sustainable

## 4. Discussion

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Potentially Profitable Deviations from Price Leadership

- If the informed firm previous price was ${p}_{l}^{M}$, the uninformed firm believes that the previous market size was ${s}_{l}$ and that the current state is ${s}_{l}$ with probability $(1-\varphi )$ and ${s}_{h}$ with probability $\varphi $. Then, the uninformed firm can deviate by:
- -
- Charging a slightly lower price than ${p}_{l}^{M}$. In that case, the uninformed firm gets the whole market and believes that with probability $(1-\varphi )$ its stage payoff will be arbitrarily close to $\pi ({p}_{l}^{M};{s}_{l})$ and with probability $\varphi $ its stage payoff will be arbitrarily close to $\pi ({p}_{l}^{M};{s}_{h})$. This deviation triggers a Nash reversal. This type of deviation is not profitable if:$${V}_{{\mathbf{p}}^{M}}^{U}\left({p}_{l}^{M}\right)\ge (1-\delta )[(1-\varphi )\pi ({p}_{l}^{M};{s}_{l})+\varphi \pi ({p}_{l}^{M};{s}_{h})].$$
- -
- Charging a slightly lower price than ${p}_{h}^{M}$ (and above ${p}_{l}^{M}$). In that case, the uninformed firm sells only if the current state is ${s}_{h}$, a scenario the uninformed firm believes to occur with probability $\varphi $ and will lead to a stage payoff of $\pi ({p}_{h}^{M};{s}_{h})$. Again, this deviation triggers a Nash reversal and is not profitable as long as,$${V}_{{\mathbf{p}}^{M}}^{U}\left({p}_{l}^{M}\right)\ge (1-\delta )\varphi \pi ({p}_{h}^{M};{s}_{h}).$$

- Similarly, if the informed firm previous price was ${p}_{h}$, the uninformed firm believes that the current state is ${s}_{l}$ with probability $\varphi $ and ${s}_{h}$ with probability $(1-\varphi )$ and can deviate by:
- -
- Charging a slightly lower price than ${p}_{l}^{M}$. This type of deviation is not profitable as long as,$${V}_{{\mathbf{p}}^{M}}^{U}\left({p}_{h}^{M}\right)\ge (1-\delta )[\varphi \pi ({p}_{l}^{M};{s}_{l})+(1-\varphi )\pi ({p}_{l}^{M};{s}_{h})].$$
- -
- Charging a slightly lower price than ${}^{M}{p}_{h}$. This type of deviation is not profitable if:$${V}_{{\mathbf{p}}^{M}}^{U}\left({p}_{h}^{M}\right)\ge (1-\delta )(1-\varphi )\pi ({p}_{h}^{M};{s}_{h}).$$

- When the demand goes from ${s}_{l}$ to ${s}_{l}$: Because firms are following the price leadership profile, the informed firm previous price was ${p}_{l}^{M}$ implying that the uninformed firm current price is also ${p}_{l}^{M}$. In this case, the only potentially profitable deviation is for the informed firm is to charge a price slightly below ${p}_{l}^{M}$, a scenario in which the informed firm gets a stage payoff arbitrarily close to $\pi ({p}_{l}^{M};{s}_{l})$.This type of deviation is not profitable if:$${V}_{{\mathbf{p}}^{M}}^{I}({s}_{l},{s}_{l})\ge (1-\delta )\pi ({p}_{l}^{M};{s}_{l})$$
- When the demand goes from ${s}_{l}$ to ${s}_{h}$: The informed firm knows that the the uninformed is setting a price equal to ${p}_{l}^{M}$ so the only potentially profitable deviation is for the informed firm is to charge a price slightly below ${p}_{l}^{M}$ and get a stage payoff close to $\pi ({p}_{l}^{M};{s}_{h})$. This type of deviation is not profitable if:$${V}_{{\mathbf{p}}^{M}}^{I}({s}_{l},{s}_{h})\ge (1-\delta )\pi ({p}_{l}^{M};{s}_{h})$$
- When the demand goes from ${s}_{h}$ to ${s}_{l}$: In that case, the informed firm is obtaining the whole monopolistic profits in that period so there is no potential profitable deviation.
- When the demand goes from ${s}_{h}$ to ${s}_{h}$: In that case, the uninformed sets a current price of ${p}_{h}^{M}$ because the informed previous price was ${p}_{h}^{M}$. Then, there are two potential profitable deviations for the informed firm:
- -
- It can charge a price slightly below ${p}_{h}^{M}$ (and above ${p}_{l}^{M}$) and get a stage payoff close to $\pi ({p}_{h}^{M};{s}_{h})$. Such a deviation is not profitable as long as:$${V}_{{\mathbf{p}}^{M}}^{I}({s}_{h},{s}_{h})\ge (1-\delta )\pi ({p}_{h}^{M};{s}_{h})$$
- -
- It can deviate by pretending the state is ${s}_{h}$ by charging a price ${p}_{l}^{M}$ as in Figure 1. The informed firm gets the whole market deriving a stage payoff of $\pi ({p}_{l}^{M};{s}_{h})$. The uninformed firm does not make a sale and therefore is not able to distinguish whether the market size was low or there was a deviation. Therefore, the next period the uninformed firm will set a price equal to ${p}_{l}^{M}$. Furthermore, next period market size is ${s}_{l}$ with probability $\varphi $ and the informed firm will face an identical problem as the case in which market size went from ${s}_{l}$ to ${s}_{l}$. Similarly, next period market size is ${s}_{h}$ with probability $(1-\varphi )$ and the informed firm will face an identical problem to the case in which the market size went from ${s}_{l}$ to ${s}_{h}$. Consequently, if price leadership with monopolistic prices is an equilibrium, it must be the case that:$${V}_{{\mathbf{p}}^{M}}^{I}({s}_{h},{s}_{h})\ge (1-\delta )\pi ({p}_{l}^{M};{s}_{h})+\delta [\varphi {V}_{{\mathbf{p}}^{M}}^{I}({s}_{l},{s}_{l})+(1-\varphi ){V}_{{\mathbf{p}}^{M}}^{I}({s}_{l},{s}_{h})]$$

## Appendix B. Proofs

**Lemma**

**A1.**

**Proof of Lemma A1.**

**Corollary**

**A1.**

**Corollary**

**A2.**

**Lemma**

**A2.**

**Proof of Lemma A2.**

**Lemma**

**A3.**

**Proof of Lemma A3.**

**Proof of Proposition 1.**

**Proof of Lemma 1.**

**Proof of Proposition 2.**

## Notes

1 | The model introduces information asymmetries into a two-firm and two-state version of Kandori [8]. |

2 | If the uninformed firm was unable to infer the state with certainty in any prior period, then the beliefs are derived in a similar manner from the initial distribution and the transition matrix. |

3 | See [1], on p. 446. |

4 | The described setting is a special case of Obara and Zincenko [13]. They consider a Bertrand oligopoly with complete information in which firms can have different discount factors. |

5 | See Section 6.7.1 in [15] for an example. |

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**Figure 1.**Undetected Deviation. In the cooperative phase of the price leadership profile, the informed firm can cut price without being detected.

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