Utility Function u(t,c) = t(c+600) Where a Wage Increase Always Reduces Free Time
For an individual whose preferences are represented by the utility function , any increase in wages will consistently lead to a reduction in free time. This is mathematically demonstrated by the derivative of the optimal free time function with respect to wages, which is always negative: . This result shows that for this specific type of utility, the negative substitution effect always dominates the positive income effect.
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Introduction to Microeconomics Course
The Economy 2.0 Microeconomics @ CORE Econ
Ch.3 Doing the best you can: Scarcity, wellbeing, and working hours - The Economy 2.0 Microeconomics @ CORE Econ
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Utility Function u(t,c) = t(c+600) Where a Wage Increase Always Reduces Free Time
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An individual's preferences for consumption (c) and free time (t) are represented by the utility function u(t,c) = t(c+600). The mathematical relationship describing how their optimal choice of free time (t*) changes in response to a change in their wage rate (w) is given by the derivative: ∂t*/∂w = - (I + 600) / (2w^2), where I is non-labor income, which is non-negative (I ≥ 0). Given this relationship, for any increase in the wage rate, the change in the individual's optimal amount of free time will always be __________. (Enter 'positive', 'negative', or 'zero').
An individual's preferences for consumption (c) and free time (t) are represented by the utility function u(t,c) = t(c+600). When this individual's wage rate increases, there are two opposing pressures on their choice of free time:
- The pressure to take less free time, because each hour of leisure now has a higher opportunity cost in terms of foregone earnings.
- The pressure to take more free time, because their higher overall income allows them to afford more of all goods, including leisure.
For an individual with these specific preferences, how do these two pressures interact as their wage changes?