The text describes a case where the derivative of the optimal free time function with respect to wage, $\frac{dt^*}{dw}$, is negative for all wage levels and any positive unearned income ($I > 0$). It can be concluded that for an individual with this specific type of utility function, any wage increase will invariably cause them to choose less free time. This outcome implies that for these preferences, the substitution effect of a wage change consistently outweighs the income effect.

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The sign of the derivative of the optimal free time function ($t^*$) with respect to the wage ($w$) reveals how an individual's choice of free time responds to a marginal change in their wage. A positive sign for the derivative $\frac{dt^*}{dw}$ indicates that a wage increase leads to more free time, while a negative sign signifies that a wage increase leads to less free time.

Interpreting the Sign of the Derivative of the Optimal Free Time Function

To cite the ebook, use The CORE Econ Team 2023 The Economy 2.0: Microeconomics Open access e-text https://core-econ.org/the-economy/.

To cite a unit, use The CORE Econ Team 2023 The Economy 2.0: Microeconomics Open access e-text https://core-econ.org/the-economy/ Unit [unit number] (url).

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