Analyzing Changes in Optimal Choice via Differentiation
To determine how the optimal choice of consumption and free time responds to a change in an external factor, one can employ differentiation. This method of comparative statics involves differentiating the functions for the optimal choice (e.g., and ) with respect to the parameter of interest (e.g., wage, ), while holding other parameters (e.g., unearned income, ) constant.
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Analyzing Changes in Optimal Choice via Differentiation
An individual's preferences for consumption (c) and free time (t) are represented by the utility function U(c, t) = c * t. The individual has a total time endowment of T hours, earns a wage of 'w' per hour worked, and receives unearned income of 'I'. Which of the following expressions correctly represents the individual's optimal choices for consumption and free time as functions of their wage and unearned income?
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An individual's optimal choices for free time (t*) and consumption (c*) are described by the functions t*(w, I) = (wT + I) / (2w) and c*(w, I) = (wT + I) / 2, where 'w' is the wage rate, 'I' is unearned income, and 'T' is total time available. If this individual's unearned income 'I' increases, while their wage 'w' and total time 'T' remain unchanged, what is the predicted effect on their optimal choices?
Statement: For any individual who considers both consumption and free time to be normal goods, an increase in the wage rate will unambiguously lead them to choose less free time.
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An individual's optimal choice of free time is represented by the function
t*(w, I), wherewis the wage rate andIis unearned income. To determine the rate at which the optimal amount of free time changes as the wage rate changes, holding unearned income constant, one must calculate the ______ derivative of the function with respect tow.An economist wants to determine how an individual's optimal choice of consumption responds to a change in their unearned income, assuming their wage rate stays the same. Arrange the following steps in the correct logical order to conduct this analysis using differentiation.
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t*, is described by the functiont*(w, I) = 12 + I / (2w), wherewis the hourly wage andIis daily unearned income. Based on this function, which of the following statements correctly analyzes how the choice of free time responds to a change in the wage rate?An individual's optimal consumption,
c*, is a function of their wage (w) and unearned income (I). If consumption is considered a normal good for this individual, then the partial derivative∂c*/∂Iwill be negative.