Equivalent variation

So far, these lecture notes have considered (potentially) **observable** outcomes — income and wealth, earnings and wages. These outcomes all measure different things. Depending on what we are ultimately interested in, consideration of one or the other of these outcomes might be the most relevant object to study.

Suppose now that we are ultimately interested in some notion of **welfare**. Great discussions on various notions of welfare and how it might be measured can be
found in Sen (1995) and Roemer (1998). How could we possibly measure welfare? Do any of the observable outcomes considered so far correspond to welfare? Economists like to think of welfare in terms of realized **utility**. Utility, however, is **not observable**. It is therefore not obvious whether "utility" provides a meaningful concept of welfare at all.

Rather than attempting to directly measure the level of utility, we might **change the question**. Instead of the level we might try to measure changes in utility, induced by a given change in prices, wages, taxes, or some other policy.
And instead of measuring changes in utility itself, we might try to measure how these changes in utility compare to changes that would be induced by a simple transfer of money. We have thus shifted the question in two ways:

**Changes**in utility, rather than**levels**of utility.- Transfers of
**money**that would induce similar changes of utility, rather than changes in**utility**itself.

It turns out that, when we modify the question in this way, then it has a well-defined and surprisingly simple answer — at least if we assume that individuals are utility maximizing. In this chapter, we will derive this answer for the case of changing prices of consumer goods. We will show that a change \(d p_j\) of the price \(p_j\) of good \(j\) has the same effect on utility of individual \(i\) as a reduction of income by \(d p_j \cdot x_{j,i}\), where \(x_{j,i}\) is the individual's current consumption of good \(j\). In the next chapter, we will use the same idea to talk about redistributive taxation.

Standard economic theory assumes that individuals choose their **consumption** to **maximize** their **utility**. They do so subject to the constraint that their expenses do not exceed their income. Denote individuals by \(i\). Assume there are two consumption goods, good 1 and good 2, with prices \(p_1\) and \(p_2\). Individual \(i\) has an income \(y_i\) and chooses her consumption \(x_i =(x_{1,i}, x_{2,i})\) to maximize her utility \(u_i\). Formally,

subject to the **budget constraint**

The utility \(v_i\) that a household can achieve for given prices and income is equal to the utility of the chosen consumption bundle,

We can rewrite the individual's budget constraint (assuming that it holds with equality) to express consumption of good \(1\) in terms of the other variables,

$$ x_{1,i} = \tfrac{1}{p_1} (y_i - x_{2,i} \cdot p_2 ) $$

We can next substitute the budget constraint, rewritten in this form, into the optimization problem, to get an unconstrained problem:

$$ x_{2,i} = {\arg\!\max}_{x_2} u_i\left (\tfrac{1}{p_1} (y_i - x_2 \cdot p_2 ) , x_2\right ). $$

The solution to this unconstrained problem has to satisfy the **first order condition**

$$ \frac{\partial}{\partial x_{2}} \left [ u_i\left (\tfrac{1}{p_1} (y_i - x_2 \cdot p_2 ) , x_2\right )\right ] =0, $$

which we can rewrite as

This first order condition is sometimes interpreted as saying that the ratio of marginal benefits to marginal costs has to be the same for both goods.

When the price of good 2 changes, how does that affect the welfare of different individuals? Formally, what is \(\partial_{p_2} v_i(p, y_i) \)?

To make the notation less cluttered, we drop the subscript \(i\) from the following derivation. You should not forget that everything is different for different individuals, though. We can calculate the **welfare effect of changing** \(p_2\) using (i) the chain rule, (ii) substituting for \(\partial_{p_2} x_{1} \) using the rewritten budget constraint, (iii) rearranging, and (iv) using the first order condition of utility maximization:

$$ \begin{align*} \partial_{p_2} v(p, y) &= \partial_{x_1} u(x) \cdot \partial_{p_2} x_{1} + \partial_{x_2} u(x) \cdot \partial_{p_2} x_{2} \\ &= \partial_{x_1} u(x) \cdot\left ( -\frac{x_{2}}{p_1} - \partial_{p_2} x_{2} \frac{p_2}{p_1} \right ) + \partial_{x_2} u(x) \cdot \partial_{p_2} x_{2} \\ &=-\partial_{x_1} u(x) \cdot \frac{x_{2}}{p_1} +\left (-\frac{\partial_{x_1} u(x)}{ p_1} + \frac{\partial_{x_2} u(x)}{ p_2}\right ) \cdot p_2\cdot \partial_{p_2} x_{2}\\ &=-x_{2} \cdot \frac{\partial_{x_1} u(x)}{p_1} . \end{align*} $$

Make sure you understand each step of this proof! The most important step in this derivation is the last one: because we assume that **individuals** maximize utility, the **first order condition** holds. And because the first order condition holds, we can **drop the term involving** \( \partial_{p_2} x_{2}\). As far as their welfare is concerned, it does not really matter how individuals react to price changes!In fact, there is a deep sense in which this is the *only* implication of "welfarism," where welfarism is the idea of evaluating household welfare based on their realized utility. This is sometimes also called utilitarianism.

We can do a completely similar calculation to get the effect of increasing income \(y\):

$$ \begin{align*} \partial_{y} v(p, y) &= \partial_{x_1} u(x) \cdot \partial_{y} x_{1} + \partial_{x_2} u(x) \cdot \partial_{y} x_{2} \\ &= \partial_{x_1} u(x) \cdot\left ( -\frac{1}{p_1} - \partial_{y} x_{2} \frac{p_2}{p_1} \right ) + \partial_{x_2} u(x) \cdot \partial_{y} x_{2} \\ &=- \frac{\partial_{x_1} u(x)}{p_1} +\left (-\frac{\partial_{x_1} u(x)}{ p_1} + \frac{\partial_{x_2} u(x)}{ p_2}\right )\cdot p_2\cdot \partial_{p_2} x_{2}\\ &=- \frac{\partial_{x_1} u(x)}{p_1} . \end{align*} $$

As before: as far as their welfare is concerned, it does not really matter how individuals react to income changes. Now we are almost done. We can calculate how the welfare effect of a price change \(dp_2\) compares to the welfare effect of a change in income. This is called **equivalent variation**, we abbreviate it by \(EV\):

Increasing the price of good 2 by one dollar has the same effect on individual \(i\) as decreasing her income by \(-x_{2,i}\) dollars, where \(x_{2,i}\) is the amount she consumes of good 2.

So far, we have considered a fairly special case. Only the price of good 2 changes, there are only two goods, and individuals are only consumers. All of these are easily generalized:

- There is nothing special about good 2, so we get the same result for good 1:

$$ EV=\frac{\partial_{p_1} v_i(p, y_i) \cdot dp_1}{\partial_{y} v_i(p, y_i) } = - x_{1,i} \cdot dp_1. $$ - There is nothing special about the case of 2 goods. We might just as well assume that there are \(J\) goods, and do the exact same proof. (This would be a good exercise!). When the price of good \(j\) changes by \(dp_j\) for \(j=1\ldots J\), this implies a welfare change of

$$EV=\frac{d v_i(p, y_i) }{\partial_{y} v_i(p, y_i) } = - \sum_j x_{j,i}\cdot dp_j. $$ - We also don't need to assume that individuals are only consumers. Suppose they start out with an endowment
$$
\omega_i = (\omega_{1,i},\ldots,\omega_{J,i}),
$$

which they can either consume or sell on the market. Their

*net*consumption of good \(j\) is equal to \(x_{j,1}-\omega_{j,i}\). In such a setting, we get

$$EV=\frac{d v_i(p, y_i) }{\partial_{y} v_i(p, y_i) } = \sum_j (\omega_{j,i} - x_{j,i})\cdot dp_j. $$

These formulas are the basis of so-called Paasche price indices. They say that we should evaluate price changes by weighting them with an appropriate consumption basket, corresponding to the amounts consumed of each good. This formula gives a different price index for every individual. To evaluate the distributional impact of price changes, all we have to do is to collect information on every individual's net consumption of various goods. Once we have estimates of the welfare changes (in dollar terms) for each individual, we can plot how these welfare changes relate to income or various demographic factors, for instance location of residence.

In fact, the same logic carries us much further. We won't prove it here, but we get similar evaluations of the welfare effect of price changes if:

- Individuals also face discrete choices, rather than just continuous ones, as we have assumed so far.
- Individuals make intertemporal choices, that is choices over time.
- Individuals also face constraints other than their budget constraint, such as informational constraints, credit constraints, etc.

In all of these cases we can still compare the welfare effect of changing prices (or wages, or interest rates, for that matter), to the welfare effect that a lump-sum transfer of money would have. And in all of these cases we derive this welfare effect by essentially ignoring any behavioral responses to a change in prices.