Critique of a Mathematical Derivation
An economics student is working through a proof to show that two different models of the reservation wage curve are equivalent. After several algebraic steps, they arrive at the equation:
P_α(α^N) = P_α(w - (m/q) * [1 - P(w)])
(Where P_α is the cumulative distribution function of unemployment utility, α^N is the utility of not working, w is the wage, m is the job meeting rate, q is the quit rate, and P(w) is the cumulative distribution of wages.)
The student concludes: "Since the function P_α appears on both sides of the equation, we can simply cancel it out, which leaves us with α^N = w - (m/q) * [1 - P(w)]. This completes the proof."
Critically evaluate the student's final step and conclusion. Is their reasoning for simplifying the equation sound? Explain why or why not, specifying the essential property of the P_α function that is required to justify the simplification.
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An economist is demonstrating that two different mathematical expressions for the reservation wage curve are equivalent. After several substitutions, they arrive at the following equation:
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