Lecture 3: Trading from an Endowment

Click here for the quiz on this reading.

Last week we analyzed preferences and indifference curves, first for individual agents and then for an Edgeworth Box.

This week we’ll develop a model of competitive equilibrium in an exchange model. We’re going to assume that each agent in the exchange economy starts with an endowment. They can then trade goods with other agents, either by direct barter or by selling off some of their endowment for money and using that money to buy other goods, to end up with a bundle they prefer. The diagram below shows this kind of contract; drag the two bundles around to get an intuition for the kind of transaction we’ll be discussing:

See interactive graph online here.

Review: Utility Maximization with Exogenous Income

Let’s think about Bob from the last lecture. In Econ 50, we might have assumed that Bob would start out with a certain amount of money – say, $m = 120$ dollars – and then buy two goods with that money. For example, suppose $p_1 = 10$ dollars per unit of good 1 and $p_2 = 5$ dollars per unit of good 2; then if he spent all his money on good 1, he could afford $x_1^\text{max} = {m \over p_1} = {120 \text{ dollars} \over 10 \text{ dollars per unit of good 1}} = 12 \text{ units of good 1}$ Likewise, if he spent all his money on good 2, he could afford $x_2^\text{max} = {m \over p_2} = {120 \text{ dollars} \over 5 \text{ dollars per unit of good 2}} = 24 \text{ units of good 2}$ His budget line therefore extends from $(12,0)$ to $(0,24)$, with a slope of $-2$ units of good 2 per unit of good 1:

See interactive graph online here.

This slope represents the opportunity cost of buying another unit of good 1, in terms of good 2: that is, if he spends $€10$ on another unit of good 1, he has to give up 2 units of good 2 (which would cost $€5$ each). Mathematically, the magnitude of this slope is the price ratio: $|\text{slope of budget line}| = {p_1 \text{ €/unit of good 1} \over p_2 \text{ €/unit of good 2}} = {p_1 \over p_2} {\text{units of good 2} \over \text{units of good 1}}$ which in this case is $p_1/p_2 = 10/5 = 2$.

Solving the consumer’s problem

The central constrained optimization problem for a consumer is to find the utility-maximizing bundle in their budget set. For this course we’ll restrict ourselves to strictly monotonic preferences, which means that more of every good is always preferred. It follows logically from this that the optimal bundle for such preferences must always lie along the constraint, since for any bundle in the interior of the budget set, there must always be a strictly preferred bundle which is also affordable. Therefore, we can reduce our constrained optimization problem to finding the utility-maximizing bundle along their budget line.

To visualize this problem, we can think about plotting the budget line and the utility function in the same 3D graph. In the left-hand graph below, we visualize the constraint as a kind of “fence” over the utility function “hill.” In the right-hand graph, we show the utility you reach at different points along the budget constraint. You can drag the point left and right; the relevant bundle is shown below. Note that the left-hand side of this graph represents the vertical intercept of your budget constraint (where you’re spending all your money on good 2), and the right-hand side represents the horizontal intercept of your budget constraint (where you’re spending all your money on good 1).

See interactive graph online here.

The usual way we would find the peak of a curve like this would be to analyze it using (univariate) calculus. In particular, if we let $m_1$ be the amount of money spent on good 1, and ($m - m_1$) be the amount spent on good 2, then the utility as a function of the amount spent on good 1 is $\hat u(m_1) = u(x_1(m_1),x_2(m_1))$ What happens if we spend a little more money on good 1? Mathematically, by the chain rule, we have $\frac{d \hat u(m_1)}{dm_1} = \frac{\partial u}{\partial x_1} \times \frac{dx_1}{dm_1} + \frac{\partial u}{\partial x_2} \times \frac{dx_2}{dm_1}$ Since each dollar spent on good 1 increases the consumption of good 1 by $1/p_1$ units of good 1, and decreases consumption of good 2 by $1/p_2$ units of good 2, we can write this perhaps more simply as $\frac{d \hat u(m_1)}{dm_1} = {MU_1 \over p_1} - {MU_2 \over p_2}$ We sometimes call the term $MU_i/p_i$ the “bang for your buck for good $i$:” that is, the additional utility you get from another dollar spent on good $i$. Thus the first term is the utility gain per each dollar spend on good 1, and the second term is the utility loss per dollar not spent on good 2.

If this is positive, then you get more “bang for your buck” from good 1 than good 2, so you increase your utility by moving to the right along the budget line; if it’s negative, the opposite holds. That is,

${MU_1 \over p_1} > {MU_2 \over p_2} \iff \text{ buy more good 1, less good 2}$ ${MU_1 \over p_1} < {MU_2 \over p_2} \iff \text{ buy less good 1, more good 2}$

Example

The graph above is built from the following example: suppose your preferences over apples (good 1) and bananas (good 2) may be represented by the utility function $u(x_1,x_2) = x_1^{3 \over 4}x_2^{1 \over 4}$ For this utility function, the marginal utilities are given by $\begin{aligned} MU_1(x_1,x_2) &= \tfrac{3}{4}x_1^{-{1 \over 4}}x_2^{1 \over 4}\\ MU_2(x_1,x_2) &= \tfrac{1}{4}x_1^{3 \over 4}x_2^{-{3 \over 4}} \end{aligned}$ For the budget constraint, let’s assume:

you have 48 dollars ($m = 48$) to spend on apples and bananas
apples cost 4 dollars per pound ($p_1 = 4$)
bananas cost 2 dollars per pound ($p_2 = 2$)

Now let’s examine the point at which you spend half your money on each good. This would mean buying the bundle $(6,12)$. At this point, your marginal utility from apples is $MU_1(6,12) = \tfrac{3}{4}6^{-{1 \over 4}}12^{1 \over 4} \approx 0.9 \frac{\text{utils}}{\text{apple}}$and your marginal utility from bananas is $MU_2(6,12) = \tfrac{1}{4}6^{3 \over 4}12^{-{3 \over 4}} \approx 0.15 \frac{\text{utils}}{\text{banana}}$ If you spent one more dollar on apples, you could buy 1/4 more pounds of apples (since apples cost $p_1 = 4$ dollars/pound). This would increase your utility by approximately $\text{Utility gain} = {1 \over 4} \text{ apples} \times 0.9 \frac{\text{utils}}{\text{apple}} = +0.225 \text{ utils}$ At the same time, if you spent one less dollar on bananas, you would have to buy 1/2 fewer pounds of bananas (since bananas cost $p_2 = 2$ dollars per pound). This would decrease your utility by approximately $\text{Utility loss} = {1 \over 2} \text{ bananas} \times 0.15 \frac{\text{utils}}{\text{banana}} = -0.075 \text{ utils}$ Since the utility gain from buying a dollar more apples is greater than the utility loss from buying a dollar fewer bananas your utility at this point is increasing as you move to the right along the budget constraint.

The MRS and the Price Ratio

Recall that “Good 1 - Good 2 space,” any slope represents a tradeoff between good 1 and good 2, and is measured in units of good 2 per unit of good 1. We have two slopes which are of interest to us:

The marginal rate of substitution (MRS) measures the amount of good 2 a consumer is willing to give up to get another unit of good 1. Visually, it is the slope of the indifference curve.
The price ratio measures the amount of good 2 the market requires consumers to give up to get another unit of good 1. Visually, it is the slope of the budget line.

We found above that you should spend more money on good 1 if you got more “bang for your buck” from good 1 than good 2: ${MU_1 \over p_1} > {MU_2 \over p_2}$ Note that we can rearrange this to read ${MU_1 \over MU_2} > {p_1 \over p_2}$ But note that the left-hand side of this equation is just the MRS, while the right-hand side of this equation is the price ratio. Therefore, another way of thinking about the consumer problem is to say:

If the MRS is greater than the price ratio, utility is increasing as you move to the right along the budget constraint, so the consumer should buy more good 1 and less good 2.
If the MRS is less than the price ratio, utility is decreasing as you move to the right along the budget constraint, so the consumer should buy less good 1 and more good 2.

Visually, if the MRS is not equal to the price ratio at some bundle along the budget line, then the indifference curve passing through that point cuts through the budget line, meaning that there’s a region of affordable bundles which are strictly preferred to that point. If the MRS is greater than the price ratio, then that region must lie to the right of the point; if the MRS is less than the price ratio, then that region must lie to the left of that point:

See interactive graph online here.

Example, continued

In our example, the MRS at the point $(6, 12)$ would be $MRS(6, 12) = {MU_1(6,12) \over MU_2(6,12)} = {0.9 \text{ utils/apple} \over 0.15 \text{ utils/banana}} = 6\ {\text{bananas} \over \text{apple}}$ Likewise, the price ratio was ${p_1 \over p_2} = {4 \text{ dollars/apple} \over 2 \text{ dollars/banana}} = 2\ {\text{bananas} \over \text{apple}}$

Since you’re willing to give up 6 bananas per apple, and their market prices mean you only need to give up 2 bananas per apple, you’re better off buying more apples and fewer bananas.

Of course, the MRS changes as you move along the budget line: in particular, because preferences are “well behaved” (strictly monotonic and strictly convex), the MRS is decreasing as you increase $x_1$ and decrease $x_2$.

We can, in fact, plot the indifference curve/budget line diagram, the total utility along the budget line, and a new graph showing the MRS vs the price ratio along the budget line all together. Note that the ideal point here is at (9, 6). Everywhere to the left of that point, the MRS is greater than the price ratio drawing the consumer to the right; everywhere to the right of that point, the MRS is less than the price ratio, drawing the consumer to the left.

See interactive graph online here.

Key Takeaway

The “gravitational pull” argument outlined above applies to any point along any budget constraint. It tells you, relative to that point, which direction along the constraint corresponds to increasing utility.

In the example above, we can see that at the optimal point, the MRS equals the price ratio, meaning the indifference curve passing through the optimal bundle is tangent to the budget line at that bundle. In some cases, that will characterize the optimal bundle. However, in others, the MRS and the price ratio may be different (as in the case of a corner solution), or may even not exist at all (as in the case of a solution at a kink in a constraint).

In a way, it might be helpful to go back to our initial exercise of plotting utility along the budget line. All we’ve found is how to characterize where utility is increasing or decreasing along a constraint. Just as finding the highest point of a curve isn’t quite as simple as setting $f’(x) = 0$, finding the optimal bundle is not always as simple as setting the MRS equal to the price ratio.

Buying and selling from an endowment

Now suppose, as we did in the last lecture, that instead of starting with an income of 120 dollars, Bob has endowment of 8 units of good 1 and 8 units of good 2: that is, he’s starting a consumer optimization problem from a bundle within good 1 - good 2 space. We’ll call this point $E$ for “endowment,” and denote the quantities of goods 1 and 2 in that bundle as $e_1$ and $e_2$.

Let’s assume that Bob can buy or sell goods 1 and 2 at market prices. For example, suppose again that the price of good 1 is $p_1 = 10$ and the price of good 2 is $p_2 = 5$. If Bob wanted to consume more than $e_2 = 8$ units of good 2, he could sell some of his good 1 to buy more good 2, as shown below:

See interactive graph online here.

Geometry of the endowment budget line

As in the case with exogenous income, we can draw Bob’s budget line; you can add it to the diagram above by checking the box. How do we get the equation of this budget line?

Using the logic from the example above, we can say that if Bob sells some amount $\Delta x_1$ of good 1, he will earn $p_1 \times \Delta x_1$ from the sale. If he uses the proceeds to buy $\Delta x_2$ units of good 2, that will cost him $p_2 \times \Delta x_2$. Assuming the amount he earns is exactly the same as the amount he spends (he doesn’t pocket any money), therefore, we must have the amount he spends be exactly equal to the amount he earns: $p_2 \times \Delta x_2 = p_1 \times \Delta x_1$ Since $\Delta x_1 = e_1 - x_1$ and $\Delta x_2 = x_2 - e_2$, we can write this equation as $p_2(x_2 - e_2) = p_1(e_1 - x_1)$ Collecting the $x$ terms on the left-hand side and the $e$ terms on the right hand side, we can write this as $p_1x_1 + p_2x_2 = p_1e_1 + p_2e_2$ If we compare this to the budget line with income from above, we can see that the left-hand side is the same. We can interpret the right-hand side as the “monetary value” (or “liquidation value”) of the endowment: $\hat m = p_1e_1 + p_2e_2$ In other words, one way of thinking about the endowment budget constraint is that Bob could sell all his endowment for $\hat m$ dollars, and then go back and spend the money on goods 1 and 2 as usual.

How can we interpret the endpoints of the budget line? Mathematically they’re given by $\begin{aligned}x_1^\text{max} &= {\hat m \over p_1} = e_1 + {p_2e_2 \over p_1}\\ \\ x_2^\text{max} &= {\hat m \over p_2} = e_2 + {p_1e_1 \over p_2} \end{aligned}$ In the context of trading from an endowment, each of these represents the total amount of a good Bob could afford if he sold all his other goods and used the proceeds to buy that good. For example, the first expression for the maximum amount of good 1 says that if Bob sold all his good 2, he could get $p_2e_2$ for it, and use the proceeds to buy $p_2e_2/p_1$ units of good 1; so his final consumption would be his initial endowment $e_1$ plus $(p_2/p_1)e_2$.

Effect of changes in prices

In the budget line with exogenous income, an increase in the price of either good would lead to a reduction in the size of the budget set, and a decrease would lead to an increase in the size of the budget set. This is because income was fixed, and didn’t change due to a change in prices. With an endowment, the situation is a little more complicated.

We just saw that the monetary value of Bob’s endowment was $p_1e_1 + p_2e_2$. An increase in the price of a good therefore leads to an increase in the monetary value of Bob’s endowment; to the extent that Bob has a lot of good 1 which he might want to sell, this is good for him. However, it also means that buying additional good 1 will be more expensive than it was before. It’s also possible that Bob doesn’t own any good 1, in the which case this is unquestionably bad for him.

Geometrically, one way of thinking about the endowment budget line is that it’s a line that passes through the endowment point and has a slope equal to the price ratio. Therefore:

an increase in the price ratio (an increase in $p_1$ and/or a decrease in $p_2$) results in a clockwise pivot of the budget line around the endowment point, a lower horizontal intercept, and a higher vertical intercept
a decrease in the price ratio (a decrease in $p_1$ and/or an increase in $p_2$) results in a counter-clockwise pivot of the budget line around the endowment point, a higher horizontal intercept, and a lower vertical intercept.

The graph below summarizes all of this. The diagram in its initial state shows the situation with $p_1 = 10$ and $p_1 = 5$. If you change the prices, a green dotted line shows this original budget line, so you can see the effect of the change:

See interactive graph online here.

A few things to try:

Double the price of good 1, so $p_1 = 20$ and $p_2 = 5$. In the left-hand graph the budget set shrinks, while in the right-hand graph there’s an area of newly affordable bundles above and to the left of the endowment and an area of newly unaffordable bundles below and to the right. That’s because good 1 is worth more if you sell it, but costs more if you buy it.
Now double the price of good 2 as well, so $p_1 = 20$ and $p_2 =10$. Relative to the original budget line, the left-hand budget line has a parallel shift in, but the right-hand budget line is unchanged. That’s because the value of the endowment doubles when prices go up, leaving the budget set unchanged.

Optimization from an Endowment

Once we have an endowment budget line, the optimization problem is no different than with an exogenous income: we’re trying to get to the highest possible indifference curve along the budget constraint. However, while our optimal bundle was previously a function of prices and income, now the optimal bundle will be a function of prices and the endowment.

To illustrate this point, let’s think of the simple Cobb-Douglas utility function $u(x_1,x_2) = x_1x_2$. The optimal bundle occurs at the point along the budget line where $MRS = x_2/x_1 = p_1/p_2$; solving for $x_2$, this gives us the tangency condition $x_2 = {p_1 \over p_2}x_1$ If we plug this into the income budget constraint with exogenous income, we get $\begin{aligned} p_1x_1 + p_2\left[{p_1 \over p_2}x_1 \right] &= m\\ 2p_1x_1 &= m\\ x_1^\star &= {1 \over 2}{m \over p_1} \end{aligned}$ and therefore $x_2^\star = {p_1 \over p_2}x_1^\star = {1 \over 2}{m \over p_2}$ If we plug the tangency condition into the endowment budget constraint for endowment $(e_1,e_2)$, we get $\begin{aligned} p_1x_1 + p_2\left[{p_1 \over p_2}x_1 \right] &= p_1e_1 + p_2e_2\\ 2p_1x_1 &= p_1e_1 + p_2e_2\\ x_1^\star &= {1 \over 2}{p_1e_1 + p_2e_2 \over p_1} = {1 \over 2}\left(e_1 + {p_2 \over p_1}e_2\right) \end{aligned}$ and therefore $x_2^\star = {p_1 \over p_2}x_1^\star = {1 \over 2}\left(e_2 + {p_1 \over p_2}e_1\right)$

Note that these are really equivalent expressions, if we use monetary value of the endowment $\hat m = p_1e_1 + p_2e_2$ as the “income.” That’s a perfectly fine shortcut to use in solving these problems; just be sure to substitute the endowments back in at the end!

Finally, note that the expression for the optimal bundle given an endowment budget constraint depends only on the ratio of the prices, not on the individual prices. In other words, doubling all prices doesn’t change the location of the budget constraint (it still goes through the endowment with a slope of the price ratio), so it can’t affect the optimal bundle.

The graph below illustrates the optimal bundles for this utility function, given the two kinds of budget constraints:

See interactive graph online here.

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:

See interactive graph online here.

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:

See interactive graph online here.

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:

See interactive graph online here.

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

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 $p_1 = p_2$.

See interactive graph online here.

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 $p_1 = 18/6 = 3$.

Application: Intertemporal Choice

Last time we talked about preferences over time. Let’s now think about optimization in that same context.

The Intertemporal Budget Constraint

As before, to model the tradeoff between present and future consumption, we think of “good 1” as “present consumption,” denoted $c_1$; and “good 2” as “future consumption,” denoted $c_2$. For the purposes of interest rates, we might think of this as “consumption this year” and “consumption next year,” or “consumption in period 1” vs “consumption in period 2,” or just more generally “consumption today” and “consumption tomorrow,” in the metaphorical sense of “tomorrow.”

We’ll assume that an agent — let’s call her Rita — has an income stream of a certain amount of money now, and a certain amount she expects to receive in the future. We can think of this as, for example, a weekly paycheck; or that she has a certain amount of money in the bank now, and expects a future payment. Either way, we can represent her income stream as her endowment point of $m_1$ dollars today and $m_2$ dollars in the future, as shown as the point $M$ in the graph below.

If Rita has no access to banking or financial markets, she has a simple choice: she can either choose to spend all $m_1$ of her current income today, or save some of it as cash to increase her consumption tomorrow. That is, if she saves $s$ dollars today, she can consume $c_1 = m_1 - s$ dollars today and $c_2 = m_2 + s$ dollars tomorrow; that is, $c_2 = m_2 + (m_1 - c_1)$ or more simply $c_1 + c_2 = m_1 + m_2$ This is just an endowment budget line $p_1x_1 + p_2x_2 = p_1e_1 + p_2e_2$ with the variables

if we interpret the variables as follows: $\begin{aligned} x_1 &= c_1 & \text{ (present consumption)}\\ x_2 &= c_2 & \text{ (future consumption)}\\ e_1 &= m_1 & \text{ (present income)}\\ e_2 &= m_2 & \text{ (future income)} \end{aligned}$ where the price ratio is 1, since a dollar saved today is a dollar consumed tomorrow.

See interactive graph online here.

Saving with interest

Now suppose that Rita has a bank account that will pay her an interest rate of $r$ on her money: that is, if she saves $s$ at interest rate $r$, in the future she will receive $(1 + r)s$. Now her future consumption will be $c_2 = m_2 + (1 + r)s = m_2 + (1 + r)(m_1 - c_1)$ Collecting the $c$ terms on the left, this gives us $(1+r)c_1 + c_2 = (1+r)m_1 + m_2$ Here we have our endowment budget constraint again, though now the “price” of present consumption is $1 + r$ because spending one dollar today means giving up $1 + r$ dollars in the future.

Note that if $c_1 = 0$, we have $c_2 = (1+r)m_1 + m_2$. This is the vertical intercept of the budget line, and is called future value of the income stream.

Borrowing with interest

Finally, let’s suppose that instead of saving, Rita borrows some amount $b$ in the present. Like most loans, it comes with an interest rate $r$: that is, she needs to repay $(1 + r)b$ in the future. With this, her present consumption can be $c_1 = m_1 + b$ and in the future she’ll have to pay back the loan, so $c_2 = m_2 - (1+r)b$ Substituting $b = c_1 - m_1$ into the second equation gives us $c_2 = m_2 - (1+r)(c_1 - m_1)$ or $(1 + r)c_1 + c_2 = (1 + r)m_1 + m_2$ which is exactly what we had before. Essentially, as long as Rita can borrow or save at the same interest rate $r$, her budget constraint will be a straight line passing through her endowment point with a slope of $1 + r$:

See interactive graph online here.

Notice that if we divide the entire budget constraint through by $1 + r$, we get $c_1 + {c_2 \over 1 + r} = m_1 + {m_2 \over 1 + r}$ The right-hand side of this equation is called the present value of the income stream, and is the horizontal intercept of the budget line. It represents the value of the income stream if it were all spent today: that is, the largest amount Rita could afford to borrow would be $b = m_2/(1+r)$, since that would grow to $(1 + r)b = m_2$ in the future, which is the maximum amount she could pay back with her future income.

Effect of a change in the interest rate

How does a change in the interest rate affect the budget line?

Let’s first think about an increase in the interest rate. If Rita wants to save money, it helps her: for any given amount of savings, $s$, she gets more in the future. On the other hand, if she wants to borrow money, it hurts her: for any amount of loan $b$, she has to repay more interest in the future. Thus when the interest rate rises, the vertical intercept shifts up, and the horizontal intercept shifts in, as the budget line pivots clockwise around her endowment point.

On the other hand, a decrease in the interest rate would have the opposite effect: it would make borrowers better off, and savers worse off.

Optimal Saving and Borrowing

Having established the budget line $c_1 + {c_2 \over 1 + r} = m_1 + {m_2 \over 1 + r}$ we can combine it with the utility function from last time $u(c_1,c_2) = v(c_1) + \beta v(c_2)$ and follow the same procedure for finding the optimal bundle as we developed above: if the conditions for tangency are met, we find the point along the budget line where the MRS is equal to the slope, $1 + r$. If the conditions for tangency are not met, we use the gravitational pull argument to find the solution.

Let’s solve using the Cobb-Douglas utility function $u(c_1,c_2) = \ln c_1 + \beta \ln c_2$. In this case the tangency condition is ${c_2 \over \beta c_1} = 1 + r \Rightarrow c_2 = \beta(1+r)c_1$ Plugging this into the budget constraint gives us, $\begin{aligned} c_1 + {\beta(1+r)c_1 \over 1 + r} &= m_1 + {m_2 \over 1 + r}\\ (1 + \beta)c_1 &= m_1 + {m_2 \over 1 + r}\\ c_1^\star &= {1 \over 1 + \beta}\left(m_1 + {m_2 \over 1 + r}\right) \end{aligned}$ and therefore $c_2^\star = \beta(1+r)c_1^\star = {\beta \over 1 + \beta}[(1+r)m_1 + m_2]$ In other words, as the Cobb-Douglas “trick” would suggest, the consumer will want to consume fraction $1/(1 + \beta)$ of the present value of their income in the first period, and fraction $\beta/(1 + \beta)$ of the future value of their income in the second period:

See interactive graph online here.

Note that the optimal bundle moves to the right along the budget line as $\beta$ decreases: that is, the less patient you are, the more you want to consume now (giving up future consumption).

Net demand for present consumption

Recall that “net demand” refers to the amount by which your optimal bundle exceeds your endowment. In particular, if we subtract the first-period income $m_1$ from the optimal consumption in period 1, $c_1^\star$, we get the net demand for borrowing — that is, the amount by which the consumer would like to exceed their current income when facing current interest rates: $\begin{aligned} c_1^\star - m_1 &= {1 \over 1 + \beta}\left(m_1 + {m_2 \over 1 + r}\right)-m_1\\ &= {1 \over 1 + \beta}\left({m_2 \over 1 + r}\right)-{\beta \over 1 + \beta}m_1 \end{aligned}$ When this is negative – when you want to consume less money now than your present income – then it becomes your net supply of savings.

We can see that this is:

positive when ${m_2 \over \beta m_1} > 1 + r$, and negative when ${m_2 \over \beta m_1} < 1 + r$: you want to borrow when the MRS at your endowment is greater than the slope of the budget constraint $(1 + r)$, and save when it’s less.
decreasing in $r$, $m_1$ and $\beta$: the higher the interest rate, more income you have now, or the more patient you are, the less you want to borrow/more you want to save
increasing in $m_2$: the more income you have in the future, the more you want to borrow/less you want to save

You can use the graph below to play with any of these comparative statics. See how the net demand for borrowing, and the net supply of savings, is affected by changing each of the parameters:

See interactive graph online here.

Try using the above diagrams to answer the questions:

If you’re completely patient ($\beta = 1$) and $m_1 > m_2$, is there any interest rate for which you would borrow money? Why or why not?
If you’re very impatient ($\beta = 0.25$), for what sort of income stream $(m_1,m_2)$ would you save money?

Summary and next steps

In this lecture we established the equation for the budget line when agents are endowed with goods rather than money; solved their optimization problem for such a budget line; derived expressions for net demand and net supply; and then applied this to the setting of intertemporal choice.

Things you should know

Optimization

Endowment $(e_1,e_2)$: the “starting bundle” an agent is “endowed” with
Endowment budget line : the budget line for an agent with an endowment $(e_1,e_2)$ who can buy and sell the two goods for prices $(p_1,p_2)$; given by the equation $p_1x_1 + p_2x_2 = p_1e_1 + p_2e_2$ with intercepts $x_1^\text{max} = e_1 + \frac{p_2}{p_1}e_2$ $x_2^\text{max} = e_2 + \frac{p_1}{p_2}e_1$
MOST IMPORTANT: the equation of the endowment budget line ONLY depends on the price ratio $p = \frac{p_1}{p_2}$, not the individual prices. If you double all prices, the budget line doesn’t change.

Comparative statics

Offer curve: a parametric plot showing the optimal consumption bundle (gross demands) at various prices. Because an agent is not forced to trade, their offer curve must lie ON or ABOVE the indifference curve passing through their endowment.
Gross demand for good j ($x_j^\star$): the quantity of good $j$ an agent chooses to consume facing an endowment budget line
Net demand for good j ($x_j^\star - e_j$): the amount by which an agent’s gross demand for good $j$ exceeds their endowment of good $j$.
Net supply of good j ($e_j - x_j^\star$): the amount by which an agent’s endowment of good $j$ exceeds their gross demand for good $j$.
Sometimes we use net demand for everything, and allow it to be negative; sometimes we say that net demand shows only the prices where the agent wants to buy more of a good than they’re endowed with, and net supply shows only the prices where the agent wants to sell some of their endowment of the good.
MOST IMPORTANT: An agent will supply good 1 and demand good 2 if their $MRS(e_1,e_2) < \frac{p_1}{p_2}$ and demand good 1 and supply good 2 if $MRS(e_1,e_2) > \frac{p_1}{p_2}$

Things you should be able to do

Given an endowment $(e_1,e_2)$ and constant prices $p_1$ and $p_2$, draw the endowment budget line.
Illustrate how a change in endowment and/or prices affects a budget line.
Given an endowment and any utility function (not just Cobb-Douglas!), find the optimal bundle to which the agent will trade.
Given an endowment and any utility function, derive and plot the offer curve, gross demand, net demand, and net supply for either good as a function of the market prices.
Given different prices for buying and selling, draw the budget constraint and determine whether the agent will buy, sell, or remain at their endowment.

Reading Quiz

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