When a feasible frontier is a straight line, its slope is constant. Consequently, the Marginal Rate of Transformation (MRT)—the absolute value of the slope—is the same at every point along the frontier. This signifies that the trade-off an individual faces remains unchanged regardless of their position on the frontier. Karim's model in Unit 3 provides a clear example, as his straight-line feasible frontier results in a constant MRT.

MRT for a Straight-Line Feasible Frontier (Budget Constraint)

Figure 3.6 illustrates Karim's budget constraint at an hourly wage of €30. When his affordable combinations of consumption and free time are plotted, they form a downward-sloping straight line, defined by the equation $c = 30(24 - t)$. This line represents the boundary of his feasible set. The figure also includes a table that details the calculations for his maximum consumption based on work hours ranging from 0 to 16 per day.

Figure 3.6: Karim's Budget Constraint, Feasible Set, and Equation at a €30 Wage

When the various combinations of two goods that a person can afford are plotted on a graph, they form a downward-sloping straight line. This line is the graphical representation of the budget constraint and visually demarcates the boundary of an individual's feasible set.

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A budget constraint is an equation that represents all the combinations of goods and services an individual could purchase that would precisely use up all of their available budgetary resources. It effectively defines the boundary of what a person can afford.

Budget Constraint

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

Graphical Representation of the Budget Constraint

This example describes a budgeting problem faced by Zoë, a university student in London. Guided by survey data on student spending, she allocates a fixed budget of \£240 for her term's social life. Zoë must decide how to spend this money on two activities: nights out with friends, costing \£16 each, and cinema trips, with tickets priced at \£10 each. This scenario presents a classic consumer choice problem where a fixed income must be allocated between two different goods.

Zoë's Budget Constraint for Social Activities

This example illustrates a standard constrained choice problem. An individual named Zoë has a fixed budget of £240 to spend on two goods: cinema tickets, which cost £10 each, and nights out, which cost an average of £16 each. Her objective is to maximize her satisfaction by acquiring as many of both goods as she can, but her purchasing power is limited by her budget constraint.

Zoë's Consumer Choice Problem with a Fixed Budget

An individual's budget constraint relates consumption ($c$), wage ($w$), and free time ($t$). Although it can be broadly defined by the inequality $c \leq w(24 - t)$, which represents the entire feasible set, a key simplification is often made for optimization problems. Because utility is assumed to depend positively on both $t$ and $c$ (the 'more is better' principle), a rational individual will always choose a point on the feasible frontier rather than inside it. This justifies writing the constraint as a strict equation, $c = w(24 - t)$, which simplifies the mathematical solution process. This equation plots as a downward-sloping straight line, representing the feasible frontier.

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