Define and characterize equilibria of the following game

Question

Consider the following game between a monopolist firm and a consumer. Consumer's income is $1$, and he needs to allocate it between period 1 and period 2 consumption to maximize his utility $u(c_1,c_2)=c_1+c_2$, where $c_t$ is the consumer's period $t$ consumption. Period 2 prices are given exogenously by $P$. Assume that $P$ is random and takes value in $\{\bar{p},\underline{p}\}$ with equal probability where $\bar{p}>\underline{p}>0$. It is common knowledge that the firm knows $P$, and the consumer knows $P$ with probability $\alpha$. The timing of the game is as follow. The firm chooses the first period price $p$ to maximize its profits in period 1. After observing $p$, the consumer decides how to allocate his income between $c_1$ and $c_2$.

1) Define an equilibrium for this game

2) Characterize the equilibria.

I am having difficulties in formally setting up dynamic games and I would therefore appreciate if someone could check out the solution I provided below, and most importantly whether the definition of equilibrium I provide (strategy sets, strategies, optimality conditions, etc) is correct, and if not why? Thank You

Answer 1

SOLUTION:

1) DEFINITION OF EQUILIBRIUM

\begin{equation} \begin{split} &p \in \mathbb{R}_+\text{ is the price chosen by the firm in period 1}\\ &a_U \in [0,1] \text{ is the share of income the Uninformed consumer allocates in period 1}\\ &a_I \in [0,1] \text{ is the share of income the Informed consumer allocates in period 1}\\ &\Theta = \{I,U\} \text{ is the type of consumer: Informed, Uninformed}\\ & S_I = \{ a : \{\underline{p},\bar{p}\} \times \mathbb{R}_+ \to [0,1] \} \text{ is the strategy set of the I. consumer}\\ & S_U = \{ a : \mathbb{R}_+ \to [0,1] \} \text{ is the strategy set of the U. consumer}\\ & S_f = \{ p : \{\bar{p},\underline{p}\} \to \mathbb{R}_+ \} \text{ is the strategy set of the firm}\\ &\mu_U(p) = Pr(P =\bar{p}|p) \text{ : consumer of type U belief that the second period price is } \bar{p}, \text{ given } p\\ &\mu_I(P) = Pr(P =\bar{p}|p) \text{ : consumer of type I belief that the second period price is } \bar{p}, \text{ given } p\\ \end{split} \end{equation}

An equilibrium of the game is a triple of strategies, $s^*=(a^*_I, a^*_U,p^*)$ for each type of consumer and the monopolist, together with a system of belief $\mu$ such that: for each player $i$, $s_i^*$ is sequentially rational, and the beliefs are determined by the strategy profile and Bayes rule whenever it applies. \

In particular for each price $p$, the consumer of type $i \in \{U.I\}$ solves:
\begin{equation} \begin{split} &a_i^*(p) = \text{ argmax } \sum_p \big[ ( \frac{a}{p} + \frac{1-a}{\bar{p}} )\mu_I(P) + ( \frac{a}{p} + \frac{1-a}{\underline{p}} )(1-\mu_I(P)) \big] \end{split} \end{equation}

Also, in the case of the Informed consumer, we know that \begin{equation} \begin{split} &\mu_I(P) = Pr(P =\bar{p}|p) = \begin{cases} 1 &\text{ if } P =\bar{p}\\ 0 &\text{ if } P = \underline{p}\\ \end{cases} \\ \end{split} \end{equation}

given the consumers optimal strategies and $P$, $p^*(P)$ maximizes the monopolist first period profits:

\begin{equation} \begin{split} &\pi(P)^* = \text{ argmax } \alpha [a^*_I(p) p] + (1-\alpha) [a^*_U(p) p]\\ \end{split} \end{equation}

2) CHARACTERIZATION OF EQUILIBRIA

for any realization of $P$ in period 2, we have that \begin{equation} \begin{split} a_I(p)^* = \begin{cases} 1 &\text{ if } p < P\\ 0 &\text{ if } p >P\\ \in [0,1] &\text{ if } p = P \\ \end{cases} \\ \end{split} \end{equation}

\begin{equation} \begin{split} a_U(p)^* = \begin{cases} 1 &\text{ if } p < \mu_U(p) \bar{p} + (1-\mu_U(p))\underline{p}\\ 0 &\text{ if } p > \mu_U(p) \bar{p} + (1-\mu_U(p))\underline{p}\\ \in [0,1] &\text{ if } p = \mu_U(p) \bar{p} + (1-\mu_U(p))\underline{p} \end{cases} \\ \end{split} \end{equation}

To define the optimal strategy of the monopolist consider the following cases, where $\pi$ denotes the monopolist first period profits:

a)Suppose $p=\bar{p}, \forall P$, then By Bayes rule $\mu_U(p) = 1/2$; also suppose that off the equilibrium path $\mu_U(p) = 1/2, \forall p \neq\bar{p}$ ; then the monopolist profits are given by:

\begin{equation} \begin{split} \pi(P) = \begin{cases} \alpha \bar{p} a_I^*(\bar{p}) &\text{ if } P=\bar{p} \\ 0 &\text{ if } P = \underline{p} \\ \end{cases} \\ \end{split} \end{equation} This is not an equilibrium strategy because the monopolist can deviate and increase its profits when $P=\underline{p}$ by setting, for example, $\hat{p} = 1/2\bar{p} + 1/2\underline{p}$; this will give him a profit of $\alpha a_U^*(\hat{p}) >0$. Similarly it is possible to prove that no-matter what the off the equilibrium path are, there is always a profitable deviation for the monopolist. \

b)Suppose $p=\underline{p}, \forall P$, then $\mu_U(p) = 1/2$ and also suppose that off the equilibrium path $\mu(p) = 1/2, \forall p \neq\underline{p}$ ; then the monopolist profits are given by:

\begin{equation} \begin{split} \pi(P) = \begin{cases} \underline{p} &\text{ if } P=\bar{p} \\ \underline{p} (\alpha a_I^*(\underline{p}) +(1-\alpha) ) &\text{ if } P = \underline{p}\\ \end{cases} \\ \end{split} \end{equation} this is not an equilibrium strategy because $\exists \epsilon > 0 $ such that monopolist will deviate and increase its profits when $P=\bar{p}$; for example by setting $\hat{p} = 1/2\bar{p} + 1/2\underline{p} -\epsilon>\underline{p}$ which will give him a profit of $\hat{p}> \underline{p}$ .

c)Suppose $p=\underline{p}, \text{ if } P= \underline{p}$, $p=\bar{p}, \text{ if } P= \bar{p}$ and $\mu_U(p) = 1, \forall p > \underline{p}$ then

\begin{equation} \begin{split} \pi_1^*(P) = \begin{cases} \bar{p}(\alpha a_I^*(\bar{p}) +(1-\alpha) a_U^*(\bar{p})) &\text{ if } P=\bar{p} \\ \underline{p} (\alpha a_I^*(\underline{p}) +(1-\alpha) a_U^*(\underline{p})) &\text{ if } P = \underline{p}\\ \end{cases} \\ \end{split} \end{equation} this represents an equilibrium as long as $a_I^*(\bar{p}) = a_U^*(\bar{p}) = 1$; otherwise the monopolist can set a price $\hat{p}< \bar{p}$ and can achieve a profit of $\bar{p} - \epsilon >\alpha a_I^*(\bar{p}) +(1-\alpha) a_U^*(\bar{p})$

d) Suppose $p = 1/2\bar{p} + 1/2\underline{p}, \forall P$ then $\mu_U(p)=1/2$ and and also suppose that off the equilibrium path $\mu(p) = 1/2, \forall p \neq ( 1/2\bar{p} + 1/2\underline{p}) $ \begin{equation} \begin{split} \pi_2^*(P) = \begin{cases} (1/2\bar{p} + 1/2\underline{p}) (\alpha +(1-\alpha) a_U^*(p)) &\text{ if } P=\bar{p} \\ ( 1/2\bar{p} + 1/2\underline{p} ) ( 0 + (1-\alpha)a_U^*(p)) &\text{ if } P = \underline{p}\\ \end{cases} \\ \end{split} \end{equation} This is an equilibrium strategy as long as there are no profitable deviation; notice that the monopolist can set $p=\bar{p}$ when $P=\bar{p}$ and given $\mu_U = 1/2$ its profits would equal $\alpha \bar{P}a_I^*(\bar{p})$. Also, when $P=\underline{p}$ the monopolist could set $p=\underline{p}$ and get a profit of $\underline{p}$. So we need to insure that neither deviation is possible.

e) ?

We can therefore characterize the following (Perfect Bayesian) Equlibria:

1ST EQUILIBRIUM) \begin{equation} \begin{split} &a_I(p)^* = 1, a_U(p)^* = 1 \\ &\pi_1^*(P) \text{ defined as above } \\ &\mu_U(p) = 1 \text{ both "on and off" the equilibrium path } \\ &\text{(that is, if the monopolist deviates } \\ &\text{from the equilibrium strategy } \mu_U(p)=1 \text{ still holds} \\ \end{split} \end{equation}

SET OF EQUILIBRIA) \begin{equation} \begin{split} &a_I(p)^* =1, a_U(p)^* \in[0,1] \\ &\pi_2^*(P) \text{ defined as above } \\ &\mu_U(p) = 1/2 \text{ both "on and off" the equilibrium path } \\ &\text{(that is, if the monopolist deviates } \\ &\text{from the equilibrium strategy } \mu_U(p)=1/2 \text{ still holds} \\ & \text{Also, The following two conditions below hold:}\\ &\alpha \bar{P}a_I^*(\bar{p}) < (1/2\bar{p} + 1/2\underline{p}) (\alpha a_U^*(\bar{p}) +(1-\alpha) ) \\ &\underline{p} < ( 1/2\bar{p} + 1/2\underline{p} ) (\alpha a_U^*(\bar{p}) +0)\\ \end{split} \end{equation}

Answer 2

Hint: Notice that there is no discount factor between the two periods. Consumption between the two periods are perfect substitutes; if the consumer, they will spend all their income in the lower price period, if possible.

So there are two cases the firm considers. Either the consumer does not know the space that price is draw from, $\{\bar{p},\underline{p}\}$, with probability $1 -\alpha$, or the consumer does know this with probability $\alpha$.

If the consumer "doesn't know" $P$, then the rest of this question depends on whether the consumer knows the distribution of $P$, but not the value of $P$ itself in the second period, which is unclear to me given the question. If the consumer does know the distribution of $P$ (which is what I'm assuming), but not the value, then they won't consume anything in period 1 if $\frac{w}{p_1} < \frac{1}{2}(\frac{w}{\bar{p}}+\frac{w}{\underline{p}})$, since the expected amount of goods you could buy with wealth $w$ next period will be lower, so spending all income in that period would maximize expected utility. Note it wouldn't simply be $p_1 > \frac{1}{2}(\bar{p}+\underline{p})$ being the condition.

Then think about what happens if the consumer knows the value of $P$ next period and how they would act. Finally, the firm, knowing this and the weight of $\alpha$, will try to maximize expected profit (we don't have a cost function, so expected revenue) with respect to $p_1 \in \mathbb{R}$.

$$\max_{p_1} \ \Pi(p_1, P) = p_1E(c_1) + E(P)E(c_2)$$

Edit: Oh, if you were instructed specifically to use a Bellman, good luck. It looks like you're on the right track in general.