The optimal choice for an individual is found where their subjective trade-off (MRS) equals the objective trade-off (MRT). This well-known optimality rule, expressed as the equation `MRS = MRT`, represents the first-order condition for a constrained optimization problem. This condition can be derived directly from calculus, where the MRS and MRT are represented as the derivatives of the utility and feasible frontier functions, respectively (e.g., $v'(t)$ and $g'(t)$). Equating them ensures that the highest possible utility is achieved given the constraints.

The Optimality Condition (MRS = MRT)

In the economic model of work-leisure choice with a constant wage, the Marginal Rate of Transformation (MRT) represents the objective market trade-off between consumption and free time. It is defined as the absolute value of the slope of the feasible frontier (the budget constraint). This value is constant and equal to the wage rate ($w$).

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Mathematical solutions to constrained choice problems, such as calculus-based approaches using the substitution method or the MRS = MRT condition, offer more precision than graphical analysis. These general methods are robust and can be applied to solve optimization problems for various types of utility functions, not limited to just quasi-linear ones. For a more detailed mathematical explanation of these techniques, Sections 8.1 to 8.3 of the textbook 'Mathematics for Economists: An Introductory Textbook' by Pemberton and Rau are a recommended resource.

Mathematical Methods for Solving Constrained Choice Problems

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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One mathematical approach to solving a constrained choice problem is the substitution method. This process involves using the constraint equation to express one variable in terms of the other (e.g., expressing consumption $c$ as a function of free time $t$). This expression is then substituted into the utility function, transforming it from a function of two variables, $u(t, c)$, into a function of a single variable, $t$. This simplification makes the optimization problem solvable using single-variable calculus.

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