MRS in Quasi-Linear Preferences Depends Only on the Non-Linear Good

Explaining the MRS for Quasi-Linear Preferences

An individual's preferences for hours of free time (t) and a composite consumption good (c) are described by the utility function $u(t, c) = 4\sqrt{t} + c$. Calculate this individual's marginal rate of substitution (MRS), defined as the rate at which they are willing to give up units of consumption for an additional hour of free time.

An analyst is attempting to derive the Marginal Rate of Substitution (MRS) for a consumer with the quasi-linear utility function u(t, c) = ln(t) + c, where 't' is free time and 'c' is consumption. Their derivation is shown below:

Step 1: Set up the indifference curve equation: ln(t) + c = u₀ Step 2: Isolate 't' as a function of 'c': t = e^(u₀ - c) Step 3: Differentiate with respect to 'c' to find the slope: dt/dc = -e^(u₀ - c) Step 4: Conclude that the MRS is equal to the result from Step 3.

Identify the fundamental error in the analyst's reasoning.

An economist wants to derive the Marginal Rate of Substitution (MRS) for a consumer with quasi-linear preferences, represented by the utility function u(t, c) = v(t) + c, by using the slope of the indifference curve. Arrange the following steps of the derivation into the correct logical order.

Consider a person whose preferences for free time (t) and consumption (c) can be represented by a utility function of the form u(t, c) = v(t) + c. If this person's willingness to trade consumption for an extra hour of free time increases as their amount of free time increases, this implies that the function v(t) must be concave (i.e., its second derivative is negative).

Comparing Derivation Methods for Quasi-Linear MRS

For a consumer with preferences described by the general quasi-linear utility function u(t, c) = v(t) + c, where 't' is free time and 'c' is consumption, match each conceptual component with its correct mathematical expression.

Inferring Utility Function Structure from Behavior

For a consumer whose preferences for hours of free time (t) and a consumption good (c) are represented by the utility function u(t, c) = 20t - t^2 + c, the marginal rate of substitution (the amount of consumption they are willing to give up for one more hour of free time) is given by the expression ______.

Modeling Consumer Preferences with Quasi-Linear Functions