Price Discrimination The following equation arose in a...

Price Discrimination The following equation arose in a problem of price discrimination in economics found in an article by Sailors and colleagues. ${ }^{25}$ $$ 1-\left(\frac{m}{b}\right)^{x-1}=\left(1-\frac{k}{m}\right)(x-1) $$ where $x>1$ and $b>k$. Show that there is a unique solution for $m$ on the interval $(k, b)$.

Price Discrimination The following equation arose in a problem of price discrimination in economics found in an article by Sailors and colleagues. ${ }^{25}$
$$
1-\left(\frac{m}{b}\right)^{x-1}=\left(1-\frac{k}{m}\right)(x-1)
$$
where $x>1$ and $b>k$. Show that there is a unique solution for $m$ on the interval $(k, b)$.

Key Concepts

Fixed Point Analysis

Fixed point analysis involves rewriting an equation so that the solution is a point that remains unchanged under a particular function. By analyzing the conditions under which a function behaves as a contraction or is monotonically increasing or decreasing, one can prove both the existence and the uniqueness of such fixed points, which correspond to solutions of the original equation.

Existence and Uniqueness of Solutions

To prove that an equation has a solution, one often shows that a related function is continuous and applies the Intermediate Value Theorem to establish existence. Additionally, by demonstrating that the function is monotonic or satisfies conditions of a contraction, one can show that the solution is unique, eliminating the possibility of multiple roots.

Monotonicity

Monotonicity refers to a function being either entirely non-increasing or non-decreasing over an interval. When a function is strictly monotonic, any zero or fixed point of the function is unique, since the function cannot oscillate and produce multiple crossings of a given value in the interval.

Intermediate Value Theorem

The Intermediate Value Theorem is a cornerstone result in calculus which states that if a continuous function takes on two values of opposite sign over an interval, then it must cross zero (or any intermediate value) somewhere in that interval. This theorem is used to establish the existence of at least one solution of an equation within a particular interval.

Continuity

Continuity is a fundamental property of functions whereby small changes in the input lead to small changes in the output. This property is crucial when applying mathematical theorems that guarantee the existence of solutions, ensuring that the behavior of the function does not have sudden jumps or breaks over an interval.

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