Consider a model of inter-temporal choice, with utility function $u\left(c_{1}, c_{2}\right)$. If $c_{1}$ and $c_{2}$ are substitutes then which of the following must be true? A) $\frac{\partial u\left(c_{1}, c_{2}\right)}{\partial c_{1}}0$. B) $\frac{\partial u\left(c_{1}, c_{2}\right)}{\partial c_{2}}0$. D) $\frac{\partial^{2} u\left(c_{1}, c_{2}\right)}{\partial c_{1} \partial c_{2}}0$

Question

Consider a model of inter-temporal choice, with utility function $u\left(c_{1}, c_{2}\right)$. If $c_{1}$ and $c_{2}$ are substitutes then which of the following must be true?
A) $\frac{\partial u\left(c_{1}, c_{2}\right)}{\partial c_{1}}>0$.
B) $\frac{\partial u\left(c_{1}, c_{2}\right)}{\partial c_{2}}<0$
C) $\frac{\partial^{2} u\left(c_{1}, c_{2}\right)}{\partial c_{1} \partial c_{2}}>0$.
D) $\frac{\partial^{2} u\left(c_{1}, c_{2}\right)}{\partial c_{1} \partial c_{2}}>0$

Quizplus · Accepted Answer

If $c_{1}$ and $c_{2}$ are substitutes, the cross-partial derivative of the utility function with respect to $c_{1}$ and $c_{2}$, $\frac{\partial^{2} u\left(c_{1}, c_{2}\right)}{\partial c_{1} \partial c_{2}}$, should be positive. This indicates that an increase in one good leads to an increase in the marginal utility of the other, consistent with the behavior of substitutes.

Consider a Model of Inter-Temporal Choice, with Utility Function u(c1,c2)u\left(c_{1}, c_{2}\right)u(c1​,c2​)

Consider a Model of Inter-Temporal Choice, with Utility Function $u\left(c_{1}, c_{2}\right)$