Derivation of the MRS for a Quasi-Linear Utility Function
The Marginal Rate of Substitution (MRS) for a quasi-linear utility function, , is given by the formula . This result can be derived in two primary ways. The first method uses the ratio of marginal utilities: since the marginal utility of free time is and the marginal utility of consumption is , the MRS is . Alternatively, one can start from the indifference curve equation, . By rearranging to express consumption as a function of time, , and then differentiating with respect to , the slope of the indifference curve is found to be . The MRS, which is the absolute value of this slope, is therefore also . Both methods confirm that for quasi-linear preferences, the MRS depends only on the amount of free time, .
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