The setup is as follows:

A theatre has a monopoly over the rights to screen a new movie over two successive weeks $t = 1, 2$. Customers want to see the movie at most once. There are two types of customers: a fraction $λ ∈ (0, 1)$ are movie buffs, willing to pay $V_h = 10$ to see it in any given week, while the remainder are willing to pay at most $V_l = 5$ to see it. It costs nothing (at the margin) to the theatre to screen the movie, and it has enough capacity to screen it for all customers in any given week. Each customer is indifferent over which day within any given week it sees the movie, but prefers to see it in the earlier week. Their impatience is represented by a common discount factor $δ ∈ (0, 1)$, so seeing it in the second week at a price $p$ generates present value utility of $δ(V_i − p)$ to a type $i$ customer, while seeing it in the first week at price $p$ gives present value utility of $V_i − p$. The theatre also discounts profits across weeks at the same rate. In what follows, restrict attention to pure strategies, and assume that customers see the movie whenever indifferent between seeing it or not.

The question is as follows:

Suppose that the theatre can show it over two successive weeks and can commit to admission prices $p_1$, $p_2$ for the two weeks in advance. Customers then decide whether to see it, and if so, when to see it. Set up the problem of deriving the optimal pricing policy for the theatre assuming it wants to serve both types of customers.

What i have so far:

We want to maximize the revenue. Let $U_{B1} = 10 - p_1, U_{B2} = \delta(10 - p_2), U_{R1} = 5 - p_1$ and $U_{R2} = \delta(10 - p_2)$ be the utility functions of week 1 and 2 for buffs and regulars respectively.
Since both buffs and regular movie-goers have a uniform utility function and share a common discount factor, they will collectively make a decision to attend in the first week or the second week based on which option gives them higher utility. There isn't a scenario where some buffs or regulars choose the first week while others choose the second; they will act as a group because they are modeled to have the same preferences and face the same prices. We thereby form the following maximization problem: \begin{align*} \max\limits_{p_1,p_2} \quad & (\lambda \cdot p_1 \cdot I(U_{B1} \geq U_{B2}) + (1 - \lambda) \cdot p_1 \cdot I(U_{R1} \geq U_{R2})) + \delta(\lambda \cdot p_2 \cdot I(U_{B1} < U_{B2}) + (1 - \lambda) p_2 \cdot I(U_{R1} < U_{R2})) \\ \text{s.t.} \end{align*} Here $I$ is an indicator function that is 1 if the statement is true and 0 otherwise.

Im uncertain if my way of setting up the revenue function is correct and if so what constraints i should give it. Can someone guide me in the correct direction?



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