2 Utility and Its Representations Free Online FlashCards

What does a utility function represent?

A utility function assigns numbers to consumption bundles so that the numbers preserve the consumer’s preference ranking.

What is the marginal utility of a good?

Marginal utility is the change in utility from a small increase in one good’s consumption, holding other goods constant.

How is the MRS of $x$ for $y$ calculated?

The marginal rate of substitution is the ratio of marginal utilities: $MRS=MU_x/MU_y$. Along an indifference curve, it describes the local trade-off between the goods.

What is the standard Cobb–Douglas utility form?

A Cobb–Douglas utility function has the form $U(x,y)=x^\alpha y^\beta$, where $\alpha$ and $\beta$ are positive constants.

What utility function commonly represents perfect substitutes?

Perfect substitutes can be represented by $U(x,y)=ax+by$, where $a$ and $b$ are positive constants.

How does a consumer compare perfect substitutes under a budget constraint?

Compare marginal utility per dollar: $a/p_x$ for $x$ and $b/p_y$ for $y$. The good with the higher ratio gives more utility per dollar.

What does it mean that utility is ordinal?

Utility is ordinal: it records the order of preferences, not a measurable amount of happiness. Any strictly increasing transformation represents the same preferences.

For $U(x,y)=xy$, what are $MU_x$ and $MU_y$?

For $U(x,y)=xy$, marginal utility is $MU_x=y$ and $MU_y=x$.

What happens to the MRS under an increasing utility transformation?

A differentiable, strictly increasing transformation changes marginal-utility values but leaves their ratio, and thus the MRS, unchanged.

What is the MRS for Cobb–Douglas utility?

For $U(x,y)=x^\alpha y^\beta$, the MRS is $\frac{\alpha}{\beta}\frac{y}{x}$.

How does the Cobb–Douglas trade-off change as $x$ increases relative to $y$?

As the consumer has more $x$ relative to $y$, the amount of $y$ they are willing to give up for another unit of $x$ falls.

Why is the logarithmic form equivalent to $U(x,y)=x^{1/2}y^{1/2}$?

For positive quantities, $x^{1/2}y^{1/2}$ and $\frac{1}{2}\ln x+\frac{1}{2}\ln y$ represent the same preferences because the logarithm is strictly increasing.