Small Increments and the Condition for Constant Utility
For a utility function of two variables, , the small increments formula approximates the total change in utility () that results from small changes in both free time () and consumption (). The formula sums the individual effects of these changes. Critically, if these small changes are made in such a way that one moves along a single indifference curve, the level of utility remains constant by definition. Therefore, the total change in utility must be zero (). This sets up the mathematical condition needed for further analysis of the indifference curve's properties, such as its slope. For a more detailed mathematical explanation of the small increments formula, refer to Section 14.2 of 'Mathematics for Economists: An Introductory Textbook' by Malcolm Pemberton and Nicholas Rau (4th ed., 2015 or 5th ed., 2023).
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