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Chapter 18 / Trading from an Endowment

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:

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 $p_1 = 18/6 = 3$.

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Copyright (c) Christopher Makler /