# 18.5 Gross Demands and Net Demands

The expressions for the agent’s optimal bundle as a function of prices and their endowment are called their **gross demand functions**; for example, for the Cobb-Douglas utility function $u(x_1,x_2) = x_1x_2$, we showed that the gross demand function for good 1 would be
\(x_1^\star(p_1, p_2, e_1, e_2) = {1 \over 2}\left(e_1 + {p_2 \over p_1}e_2\right)\)
We can plot out a gross demand curve for good 1 as a function of $p_1$ as follows: in the upper graph we have the endowment optimization problem in good 1 - good 2 space. In the lower graph we plot the gross demand for good 1 as a function of the price of good 1:

However, since we’re starting from an endowment, we’re really interested in the agent’s *optimal transaction* rather than the total amount they end up consuming. Furthermore, we’re interested in whether they optimally sell good 1 and buy good 2, or buy good 1 and sell good 2?

To answer this, we compute their **net demand function**: that is, the amount they would like to end up with (their gross demand), *minus the amount they start with in their endowment*:
\(\text{Net demand for good 1 }\equiv x_1^\star(p_1, p_2, e_1, e_2) - e_1\)
Continuing with the Cobb-Douglas example, we have
\(\text{Net demand for good 1 }\equiv {1 \over 2}\left(e_1 + {p_2 \over p_1}e_2\right) - e_1 = {1 \over 2}\left({p_2 \over p_1}e_2 - e_1\right)\)
We can visualize this by changing the scale of the demand curve to be the *difference from endowment of good 1.* This is the same graph as above, but we’re looking at how much the agent wants to *buy or sell* of good 1:

Note that with this formulation, net demand may be either positive (when $x_1^\star > e_1$, so the agent wants to buy more good 1) or negative (when $x_1^\star < e_1$ the agent wants to sell some of their good 1). We’re often interested in the *supply* and *demands* for goods, so we’ll plot “net demand” for just the positive part of the net demand curve, and plot a separate “net supply” curve:

Mathematically, we can write these net demand and net supply functions as \(\begin{aligned} d_1(p_1 | p_2) &= \begin{cases}0 & \text{ if } x_1^\star \le e_1\\x_1^\star - e_1 & \text{ if } x_1^\star \ge e_1\end{cases} \\ \\ s_1(p_1 | p_2) &= \begin{cases}e_1 - x_1^\star & \text{ if } x_1^\star \le e_1 \\ 0 & \text{ if } x_1^\star \ge e_1\end{cases} \end{aligned}\)

## The “gravitational pull” and net demand and supply

So when does someone want to demand more of a good, and when do they want to supply some of their endowment to the market? The answer comes down to the *relationship between the price ratio and the agent’s MRS at their endowment*.

Note that the net demand in the above case is positive when \(\begin{aligned} {1 \over 2}\left({p_2 \over p_1}e_2 - e_1\right) & \gt 0\\ {p_2 \over p_1}e_2 & \gt e_1\\ {e_2 \over e_1} & \gt {p_1 \over p_2} \end{aligned}\) The right-hand side of the expression is the price ratio. The left-hand side is the MRS at the endowment $(e_1,e_2)$ for the utility function $u(x_1,x_2) = x_1x_2$. In other words:

- When $MRS > p_1/p_2$ at the endowment, the agent is a
**net demander**of good 1 (and a**net supplier**of good 2). - When $MRS < p_1/p_2$ at the endowment, the agent is a
**net supplier**of good 1 (and a**net demander**of good 2). - When $MRS = p_1/p_2$ at the endowment, the agent is already optimizing, and so will neither demand nore supply either good.

For example, for Bob’s endowment of $E = (8,8)$, the $MRS = 8/8 = 1$. Therefore Bob will be a net supplier of good 1 if $p_1 > p_2$, a net demander if $p_1 < p_2$, and not trade at all if $p_1 = p_2$.

This relationship between the MRS at the endowment and the price ratio is true for *any* utility function, not just Cobb-Douglas! And it makes intuitive sense: the MRS, after all, is measuring your willingness to give up good 2 to get more good 1. In the case of trading from an endowment, we’re literally talking about selling (or trading) some of your good 2 to get more good 1. So if you’re more willing to give up good 2 than the market requires – that is, if your MRS is greater than the price ratio – then you’ll trade away some of your good 2 to get some additional good 1.

## Shifts in net demand and supply

The net demand and supply curves show the relationship between $p_1$ and the amount of good 1 someone wants to buy or sell, holding their endowment and $p_2$ constant. What happens, then, if $p_2$ or their endowment changes? How does this shift the net demand and supply curves?

Since the vertical intercept of both the net demand and net supply curves is $p_2 \times MRS(e_1,e_2)$, an increase in the price of good 2 *increases* this intercept, effectively shifting the net demand curve to the right (and the net supply curve to the left). Intuitively, the more expensive/valuable good 2 is, the more likely someone is to sell good 2 to buy good 1. Of course, if $p_2$ decreases, the opposite happens: the net demand curve shifts to the left, and the net supply curve shifts to the right.

As long as the agent’s preferences are “well behaved” — that is, strictly monotonic and strictly convex — then the MRS will be increasing in $x_2$ and decreasing in $x_1$. Intuitively, the more good 2 you have, the more you’re willing to give it up to get more good 1, and vice versa. So, for example, increasing one’s endowment of good 1 shifts the net supply of good 1 to the right and the net demand for good 1 to the left.

To illustrate this, let’s think of another agent — let’s call her Alison — who has the same utility function as Bob, but an endowment of $(12,2)$. Her MRS at her endowment is therefore \(MRS(12,2) = {2 \over 12} = {1 \over 6}\) so she’ll be a net supplier of good 1 for any price $p_1 > {1 \over 6}p_2$. You can see this in the diagrams above by setting $E$ to $(12,6)$ and $p_2 = 18$, and confirming that Alison would neither buy nor sell if $p_1 = 18/6 = 3$.