When simplifying the equation $96t_C^2 = 150t_D^2$ by taking the square root, two algebraic solutions arise. The solution $4t_C = -5t_D$ is rejected because it has no logical economic interpretation. Since free time ($t$) must be a positive quantity, a relationship where one positive value equals a negative one is impossible in this context.

Rejection of the Negative Root in the Utility Equivalence Simplification

The analysis concludes that a hypothetical increase in unearned income to \$1,560, while the wage rate remains at its original level of \$96, allows an individual to achieve the exact same level of utility that results from an actual wage increase from \$96 to \$150.

Equivalence of a $1,560 Unearned Income and a Wage Rise to $150 in Utility Terms (Figure E3.4)

The primary equation for utility equivalence, $96t_C^2 = 150t_D^2$, is simplified to make it easier to solve. By taking the square root of both sides, the expression reduces to the economically meaningful relationship $4t_C = 5t_D$. This simplified form is then used to substitute the expressions for $t_C$ and $t_D$ to solve for the hypothetical income, J.

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To calculate the amount of hypothetical unearned income, J, that would allow an individual to reach the utility level of point D while facing the original wage (w=\$96), an equation must be solved. This equation sets the utility at the hypothetical point C equal to the utility at the final point D. For the specific model in Figure E3.4, this relationship is expressed as $96t_C^2 = 150t_D^2$, which is solved using the expressions for the optimal choice of free time at point C ($t_C$) and point D ($t_D$).

Equation to Determine the Hypothetical Income (J) in Figure E3.4

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).

CORE Econ - The Economy 2.0: Microeconomics

In the analysis of Figure E3.4, Point C represents a hypothetical optimal choice. This point illustrates the bundle of goods and leisure an individual would select if their wage rate remained at the original level ($w=96$), but their income was increased just enough to achieve the same level of utility as the final point (D) reached after the actual wage rise.

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