# Gross substitutes vs. net substitutes

Wikipedia explains the difference between products that are "gross substitutes" and products that are "net substitutes". However, the mathematical explanation doesn't give much intuition about these concepts. So my questions are:

1. What is the intuitive difference between gross and net substitutes?
2. What are good real-life examples of these?
3. Why is the requirement for competitive equilibrium, "gross substitute" and not "net substitute"? I.e., why does a competitive equilibrium not exist if the products are net-substitute?

Intuitively, a higher price for pears means that I have to give up more apples to be able to afford an extra pear (or, conversely, if I give up one pear then the number of extra apples that I can afford increases). This is going to make me want to reduce my pear consumption and increase my apple consumption (in orther words, to substitute away from pears towards apples). Graphically, this corresponds to a change in the slope of the budget constraint and is known as the (Hicksian) substitution effect.

This substitution effect is, however, moderated by a second consideration. If I increase one or more prices then the total amount of "stuff" that I can afford to buy decreases, so it is as if my income has decreased (which would correspond to a shift of the budget constraint towards the origin). This is known as the income effect and will usually mean that my consumption of apples decreases if the price of pears rises.

When the income and substitutes effects are put together, you get the total effect of an increase in the pear price upon the demand for apples. This total effect gives rise the the notion of gross substitutes: apples and pears are gross substitutes if the following is true

increasing the price of pears causes the consumer to demand more apples.

Formally, this is written as $$\frac{\partial X_a}{\partial p_P}>0,$$ where $X_A$ and $p_P$ are the demand for apples and the price of pears respectively.

To arrive at the notion of net substitutes we simply take a price change and shut-down the income effect. The hypothetical exercise works like this:

1. we increase the price of pears, which induces both an income effect and a substitution effect.
2. to compensate the consumer for the income effect, we give him exactly enough extra money to ensure that (even though one of the prices is higher) he can still afford to get onto the same indifference curve as before. Hence his utility will remain unchanged.
3. We look at how his demand changed, which now depends only on the substitution effect (since we have compensated the consumer for any change in real income).

Two goods are net substitutes if, after making this adjustment, we find that the demand for apples has increased. Thus, we say that two produces are net substitutes if

Increasing the price of pears while compensating the consumer for the resulting decrease in his real income causes the consumer to demand more apples.

Formally, $$\left.\frac{\partial X_a}{\partial p_P}\right|_{\text{constant }U}>0,$$ where $X_A$ and $p_P$ are the demand for apples and the price of pears respectively, and $U$ is the consumer's utility.

It is not true to say that "a competitive equilibrium [does] not exist if the products are net-substitute". Indeed, most of the time, products that are gross substitutes will also be net substitutes as well. Thus, most examples of gross substitute preferences supporting a competitive equilibrium will also be examples of net substitutes doing the same.

The reason why it doesn't make sense to state the existence condition for competitive equilibrium in terms of net substitutes is that net substitutes is a purely hypothetical construction in which a fictitious agent intervenes to shut down the income effect and keep the consumer's utility constant. The whole point of a competitive equilibrium is that there is no such intervention: the equilibrium is entirely decentralised and is sustained purely by finding prices such that the market clears when consumers pick their optimal demand.