Revenue Optimization with Poisson Demand

Daily rental demand for a single apartment is modeled as $Q \sim \text{Poisson}(\lambda_0 e^{-\beta p})$, where $p \geq 0$ is the nightly price, and $\lambda_0 > 0$, $\beta > 0$ are parameters. 1. Choose $p$ to maximize expected revenue $E[pQ]$. Find the optimal price $p^{*}$. 2. Now suppose you must also keep the expected vacancy rate below $v_{\max} \in (0, 1)$, meaning $P(Q = 0) \leq v_{\max}$. Add this constraint and solve for the optimal price. Characterize when the vacancy constraint binds.