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  • Marginal Rate of Substitution as the Ratio of Marginal Utilities

  • Marginal Utility of Free Time for a Quasi-Linear Function

  • Indifference Curve Equation for a Quasi-Linear Function

  • Marginal Utility of Income in a Quasi-Linear Function

Derivation of the MRS for a Quasi-Linear Utility Function

The Marginal Rate of Substitution (MRS) for a quasi-linear utility function, u(t,c)=v(t)+cu(t, c) = v(t) + c, is given by the formula v′(t)v'(t). 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 ∂u∂t=v′(t)\frac{\partial u}{\partial t} = v'(t) and the marginal utility of consumption is ∂u∂c=1\frac{\partial u}{\partial c} = 1, the MRS is v′(t)1=v′(t)\frac{v'(t)}{1} = v'(t). Alternatively, one can start from the indifference curve equation, v(t)+c=u0v(t) + c = u_0. By rearranging to express consumption as a function of time, c=u0−v(t)c = u_0 - v(t), and then differentiating with respect to tt, the slope of the indifference curve is found to be dcdt=−v′(t)\frac{dc}{dt} = -v'(t). The MRS, which is the absolute value of this slope, is therefore also v′(t)v'(t). Both methods confirm that for quasi-linear preferences, the MRS depends only on the amount of free time, tt.

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