https://file.aiquickdraw.com/mathbot/file/1727952949971jwjegp2s.jpg

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:

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

###

A firm maximizes the following profit function by deciding the optimal quantity q: π = p∫[−∞,q] ξf(ξ)dξ − h∫[−∞,q] (q − ξ)f(ξ)dξ + p∫[q,∞] qf(ξ)dξ − b∫[q,∞] (ξ − q)f(ξ)dξ where p, h, b are coefficients. Please show the optimal q analytically.

Explore how to analytically solve a firm's profit maximization problem by finding the optimal quantity q. This solution involves applying the first-order condition and the Leibniz rule for differentiation under the integral. Ideal for students in advanced economics or mathematics courses.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation