Money Metric Utility Function: Budget constraints become Utility Functions?

Question

Im currently reading up on the "money metric utility function" (also known as the minimum income function or direct compensation function).

By definition it is defined as:

$$m(\text{p},\text{x})\equiv e(\text{p},u(\text{x}))$$

Hal Varian writes in Microeconomic analysis (page 109):

it is easy to see that for a fixed $\text{x}$, $u(\text{x})$ is fixed, so $m(\text{p},\text{x})$ behaves exactly like an expenditure function: its monotonic, homogenous, concave in $\text{p}$, and so on. What is not so obvious is that when when $\text{p}$ is fixed, $m(\text{p},\text{x})$ is in fact a utility function.

The proof is simple: for fixed prices the expenditure function is increasing in the level of utility: if you want a higher utility level, you have to spend more money. In fact, the expenditure function is strictly increasing in $u$ for continuous, local non-satiated preferences.

Hence for fixed $\text{p}$, $m(\text{p},\text{x})$ is a monotonic transform of the utility function therefore itself a utility.

Does this (the bolded statement) mean that we essentially convert our consumer's budget constraint/budget line into his indifference curves when prices are fixed?

Answer 1

$m(\mathbf p,\mathbf x)$ specifies the minimum amount of money required for a consumer to attain the same utility as consuming bundle $\mathbf x$, taking prices $\mathbf p$ as given. In other words, since all bundles on the same indifference curve as $\mathbf x$ have the same utility value, and to achieve this utility, one needs at least an income of $m(\mathbf p,\mathbf x)$, we can thus establish a one-to-one mapping between utility levels and income levels. As utility is ordinal, we could as well use income as our utility measure.

This does not, however, suggest that the consumer's budget line is converted to her indifference curve; the two remain distinct objects. To see this, consider a two-goods case. With fixed prices, the slope of the budget line is $p_2/p_1$, which is constant for all $\mathbf x$'s. In contrast, the slope of the indifference curves is generally a non-constant function over $\mathbf x$: \begin{equation} \frac{\partial x_2}{\partial x_1}=\frac{\partial e(\mathbf p,u(\mathbf x))/\partial x_1}{\partial e(\mathbf p,u(\mathbf x))/\partial x_2}=\frac{MU_1(\mathbf x)}{MU_2(\mathbf x)}\,. \end{equation}

The money metric utility function simply changes the unit of utility measurement from utils to dollars, which is legitimate thanks to the ordinality of utility. It does not change the underlying mathematical objects in any substantive way, though.