Worker Perspectives on Wages and Hours

You are asked to algebraically verify that the indifference curves derived from a utility function are convex. Arrange the following mathematical steps in the correct logical order to complete this verification.

An individual's preferences are represented by the utility function $u(t, c) = (t-6)^2(c-45)$, where $t$ is free time and $c$ is consumption. To check if the indifference curves are convex, we can express $c$ as a function of $t$ for a fixed utility level (ar{u}) and find the second derivative, rac{d^2c}{dt^2}. The result of this calculation is rac{d^2c}{dt^2} = rac{2ar{u}}{(t-6)^4}. Assuming $t > 6$ and $c > 45$ (which implies ar{u} > 0), what does this result indicate about the shape of the indifference curves and the individual's preferences?

An individual's preferences are represented by the utility function $u(t, c) = (t-6)^2(c-45)$, where $t$ is free time and $c$ is consumption. To check if the indifference curves are convex, we can express $c$ as a function of $t$ for a fixed utility level (ar{u}) and find the second derivative, rac{d^2c}{dt^2}. The result of this calculation is rac{d^2c}{dt^2} = rac{2ar{u}}{(t-6)^4}. Assuming $t > 6$ and $c > 45$ (which implies ar{u} > 0), what does this result indicate about the shape of the indifference curves and the individual's preferences?

Calculating the Slope of an Indifference Curve

Verifying Preference Convexity via Calculus

For a utility function where consumption ($c$) can be expressed as a function of free time ($t$) along an indifference curve, if the first derivative $\frac{dc}{dt}$ is negative and increasing for all valid values of $t$, then the underlying preferences are convex.

Calculating the Second Derivative of an Indifference Curve

Analysis of a Mathematical Proof for Preference Convexity

Error Analysis in Convexity Verification