To solve this problem analytically for the optimal value of $q$, we need to find the first-order condition for the profit function $\pi$ with respect to $q$ and solve for the value of $q$ that maximizes profit. Let's break down the problem and proceed step by step.

Profit Function

The profit function is given by:

$\pi = p \int_{-\infty}^q \xi f(\xi) d\xi - h \int_{-\infty}^q (q - \xi) f(\xi) d\xi + p \int_q^\infty q f(\xi) d\xi - b \int_q^\infty (\xi - q) f(\xi) d\xi$

Where:

• $p, h, b$ are coefficients,
• $f(\xi)$ is a probability density function,
• $q$ is the decision variable for which we want to find the optimal value.

First-Order Condition

To maximize $\pi$, we need to take the derivative of $\pi$ with respect to $q$ and set it equal to zero:

$\frac{d\pi}{dq} = 0$

Let's denote this derivative as $\pi'(q)$.

Derivatives of Integrals with Respect to $q$

Since $q$ appears as the upper or lower limit in multiple integrals, we use the Leibniz rule for differentiation under the integral sign. Consider the different components of $\pi$:

1. First Term: $p \int_{-\infty}^q \xi f(\xi) d\xi$

   The derivative with respect to $q$ is:

   $\frac{d}{dq} \left( p \int_{-\infty}^q \xi f(\xi) d\xi \right) = p \cdot q f(q)$

2. Second Term: $-h \int_{-\infty}^q (q - \xi) f(\xi) d\xi$

   Expanding $(q - \xi)$ inside the integral and differentiating:

   $\frac{d}{dq} \left( -h \int_{-\infty}^q (q - \xi) f(\xi) d\xi \right) = -h \int_{-\infty}^q f(\xi) d\xi + h \cdot q f(q)$

3. Third Term: $p \int_q^\infty q f(\xi) d\xi$

   Since $q$ is a constant inside the integrand:

   $\frac{d}{dq} \left( p \int_q^\infty q f(\xi) d\xi \right) = p f(q) (-q)$