Mathematical Proof of Diminishing MRS

A consumer's preferences for consumption (c) and free time (t) are described by the utility function U(c, t) = c * t^(1/2). To determine if the marginal rate of substitution (MRS) is diminishing, one can analyze the second derivative of the indifference curve equation, d²c/dt², where consumption is expressed as a function of free time for a constant utility level. Based on this mathematical test for the given utility function, what can be concluded about the shape of the indifference curves and the behavior of the MRS?

Analyzing Consumer Preferences for Convexity

For an indifference curve where consumption (c) is expressed as a function of free time (t), a finding that the second derivative d²c/dt² is negative confirms that the curve is convex and the marginal rate of substitution is diminishing.

An indifference curve expresses consumption (c) as a function of free time (t). Match each mathematical condition for the second derivative of this function (d²c/dt²) with its corresponding economic interpretation regarding the shape of the curve and the behavior of the Marginal Rate of Substitution (MRS).

A consumer's preferences for consumption (c) and free time (t) are represented by the utility function U(c, t) = ln(c) + t. To verify if the indifference curves for these preferences are convex to the origin, we analyze the second derivative of consumption with respect to free time, d²c/dt², holding utility constant. The sign of this second derivative is always ____.

A microeconomist wants to mathematically confirm that a consumer's preferences, represented by a utility function, exhibit a diminishing Marginal Rate of Substitution (MRS). Arrange the following steps in the correct logical order to conduct this proof using the second derivative test.

Connecting Calculus to Economic Intuition

An economics student is analyzing a consumer's preferences represented by the utility function U(c, t) = c + t², where c is consumption and t is free time. The student performs the following steps to determine if the Marginal Rate of Substitution (MRS) is diminishing:

1. Express c as a function of t for a constant utility level U_bar: c = U_bar - t².
2. Calculate the first derivative: dc/dt = -2t.
3. Calculate the second derivative: d²c/dt² = -2.
4. Conclude: "Since the second derivative is negative, this confirms a diminishing MRS."

Which of the following statements best identifies the fundamental error in the student's analysis?

Mathematical Verification of Preference Convexity