Lecture 8: Solutions at Kinks

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Up to now, we’ve been looking at the optimal choice along a budget line, so the price ratio was a constant. However, as we noted in lecture 5, some budget constraints have price ratios that vary along the constraints. With such kinked budget constraints, we need to examine the relationship between the MRS and the price ratio along each of the segments of the budget constraint to solve the consumer’s optimization problem.

Let’s look at the three budget sets we introduced in that lecture, and see how a consumer might optimize their utility subject to each.

Different prices for buying and selling

First, let’s think about the situation of optimizing from an endowment in which you face different prices for buying and selling goods. In the example from lecture 5, we discussed a situation in which you have an “endowment” of (40 concert tickets, €1200). You can buy more tickets for €60 each, but if you go to sell tickets you’ll only get €25 per ticket. This results in a kinked budget constraint:

See interactive graph online here.

What should you do? It depends on your utility function. Let’s suppose your preferences could be represented by a Cobb-Douglas utility function $u(x_1,x_2) = x_1^\alpha x_2^{1-\alpha}$ for some value of $\alpha$. This has an MRS of $MRS(x_1,x_2) = {\alpha \over 1 - \alpha}{x_2 \over x_1}$ So at your endowment of $(40, 1200)$ your MRS is $MRS(40, 1200) = {\alpha \over 1 - \alpha}{1200 \over 40} = {\alpha \over 1 - \alpha} \times 30 \text{ dollars per ticket}$ In other words, if the price you could get from selling a ticket were greater than that, you’d want to sell some of your tickets; and if the price you had to pay to buy more tickets were less than that, you’d want to buy more tickets. In other words, the MRS at your endowment determines your cutoff price for buying and selling tickets.

However, what happens if the price you could sell tickets for is below your cutoff price, while the price you could buy them for is above your cutoff price? Then you might optimally neither sell tickets for a low price, nor buy additional tickets for a high price. This is illustrated in the case below. You can see that when $\alpha = 0.50$, your cutoff price is $p = 30$; so you are neither willing to sell tickets for €25, nor buy additional tickets at €50:

See interactive graph online here.

Try playing around with the graph to determine how your behavior would change:

if $\alpha$ increases or decreases
if your initial endowment of tickets, $e_1$, increases or decreases
if your initial endowment of money, $e_2$, increases or decreases
extra challenge: if you could sell tickets for a high price or buy them for a low price (but not both)

Electricity rates

For another example, recall the case of nonlinear electricity rates: you pay a low rate for the first kilowatt-hours (kWh) of electricity you use, up to a threshold $\hat x$; then you pay a higher rate. The budget constraint is characterized by an initial relatively flat portion reflecting the low initial price, then a steeper portion reflecting the higher price for additional use. Where the slope of the budget constraint changes, there is once again a “kink:”

See interactive graph online here.

It is entirely possible to have an MRS at the kink point that is simultaneously higher than the price ratio to the left of the kink, and lower than the price ratio to the right of the kink. In such a case the gravitational pull will always draw you toward the kink: your utility will always be increasing along the left-hand segment (since the MRS is greater than the price ratio everywhere in that region), while your utility will always be decreasing along the right-hand segment (since the MRS is less than the price ratio everywhere in that region):

See interactive graph online here.

Note, however, that the optimum doesn’t have to be at the kink. You can drag the threshold $\hat x$ in the diagram above to set it higher or lower than 300 kWh. Notice what happens if you do:

If you drag the threshold $\hat x$ all the way to the left, at 120 kWh, the MRS at the kink is 0.45, which is greater than even the higher price of 0.20. This threshold is so low that when the threshold is met, you’re willing to pay the higher price to continue using electricity; you can see that the optimum occurs at a point of tangency to the right of the kink, with $x_1 = 220$.
On the other hand, if you drag the threshold $\hat x$ all the way to the right, at 600 kWh, the MRS at the kink is 0.05, which is less than even the lower price of 0.10. In this case the threshold is set so high that the higher price above that level is irrelevant: the optimum is a point of tangency to the left of the kink, with $x_1 = 400$.

The bottom line is that even when there is a kinked constraint or a kinked indifference curve, it doesn’t follow that the optimum is automatically at the kink itself. You should compare the MRS and the price ratios at each kink (and each corner), to get a sense as to where the optimal choice will be. Then, use the equation for that part of the budget line (which might involve some tricky algebra!) and maximize as if the budget line were a straight line with that equation.

Should you give a gift card or cash?

Let’s think about the gift cards example, in which you can get a certain amount of a good for free, before you have to start paying for it. In particular, let’s assume that you’re considering giving someone a €50 gift card that can only be spent on lattes. They have $m = €100$ in disposable income that they can spend on lattes (good 1) or other things (good 2). The price of a latte is $p_1 = 5$; the price of a dollar spent on other things is, as usual, $p_2 = 1$.

If you give them €50 in cash, their budget set would be $5x_1 + x_2 = 150$ However, if you give them a €50 gift card that can only be spent on lattes, their budget set would be kinked: they couldn’t spend more than $m = €100$ on other things, but they could get up to 10 lattes for free. Therefore, it would have a horizontal portion from (0, 100) to (10, 100); but then have a slope of $-5$, since for every latte beyond the 10th, they would have to spend their own money.

So, which is better? It depends on how much they like lattes! Suppose your intended recipient had preferences which could be represented by the quasilinear utility function $u(x_1,x_2) = a \ln x_1 + x_2$ For this utility function, $MRS = {MU_1 \over MU_2} = {a/x_1 \over 1} = {a \over x_1}$ The tangency condition sets this equal to the price ratio: $\begin{aligned} MRS &= {p_1 \over p_2}\\ {a \over x_1} &= {5 \over 1}\\ x_1^\star &= {a \over 5} \end{aligned}$ Note that this implies that the amount this person would like to spend on lattes is $p_1x_1^\star = 5 \times {a \over 5} = a$ So, for example, if $a = 100$, then their optimal bundle would involve spending €100 on lattes, and therefore buying 20 lattes, as shown in the diagram below.

See interactive graph online here.

So what happens if you give them a €50 gift card that can only be spent on lattes? If $a > 50$, then it doesn’t matter if you to this, or give them €50 in cash; they would spend the money on lattes either way. But if $a < 50$, they’ll spend the entire gift card on 10 lattes, but they would have been better off if you’d just given them cash!

To see this, drag the parameter $a$ in the diagram above to the left until $a < 50$…and notice that the optimal bundle in the larger triangular budget set, representing what the affordable bundles if you just gave them cash, lies on a higher indifference curve that the bundle at the “kink” of the budget set if you just give them a gift card…

Would you ever not get more of something if it’s free?

The above analysis might make you think that if you have a situation like a gift card, where the first units of a good are free, you should always take all the free stuff. However, this clearly isn’t true: there are lots of things that are free that you don’t just look at with wide eyes and consume all of.

Think of salt on a restaurant table. It’s free, right? So why don’t you just unscrew the top of it and pour it all over your food? Not to mention all the rest of the condiments on the table? #maklerlifehack ftw, amirite?

Clearly i am not rite, and if you actually did that you would instantly regret it. Luckily, we can very easily come up with a utility function that captures this, but it’s one that has to be nonmonotonic: more cannot always be better.

Think about preferences which can be represented by the quasilinear utility function of the form $u(x_1,x_2) = ax_1 - \tfrac{1}{2}x_1^2 + x_2$ Note that the marginal utility from good 1 in this is initially positive, but then goes negative: $MU_1 = a - x_1$ In other words, these preferences are nonmonotonic: beyond $a$ units of good 1, getting more good 1 actually decreases your utility!

The indifference curves of a utility function like this are parabolas: that is, the utility function corresponding to some utility level $U$ would have the equation $ax_1 - \tfrac{1}{2}x_1^2 + x_2 = U$ or $x_2 = \tfrac{1}{2}x_1^2 -ax_1 + U$ which is just the familiar equation of a parabola $y = ax^2 + bx + c$ with different letters.

So, if we look at a kinked budget constraint with this kind of utility function, we can see that the optimal bundle might very well be on the horizontal portion of the budget constraint: that is, with this kind of utility function, if your recipient doesn’t like lattes very much at all, they wouldn’t even spend the entire $50 gift card you gave them!

For example, in the situation above, with the price of a latte at $p_1 = 5$, the tangency condition will give us $a - x_1 = 5$ or $x_1^\star = a - 5$ Recall that with a €50 gift card, your recipient could buy 10 lattes. If they would do this anyway – if $a > 15$ – then they’ll spend the entire gift card as well as some of their own money. (For example, if $a = 20$, they would take the 10 free lattes from the gift card and spend another €25 on five more lattes, for a total of $x_1^\star = 20 - 5 = 15$ lattes.)

However, if they don’t like lattes very much – if $a < 10$ – then they will only use the gift card to buy $a$ free lattes, leaving the rest of the value of the card unspent. For example, if $a = 5$, they will use the gift card to only buy 5 lattes, leaving the remaining €25 on the card.

What happens if $a$ lies between 10 and 15? In this case, the marginal utility of the 10th latte is $a - 10$, which is some number between 0 and 5. In other words, it’s worth getting the 10th latte for free, but it’s not worth an additional €5 to get an eleventh one. Therefore, the solution lies at the kink.

You can see these three scenarios by dragging the parameter $a$ in the diagram below. An indifference map shows the family of indifference curves so you can see how $a$ changes the indifference curves for this function:

See interactive graph online here.

Note that this kind of behavior might also occur with a good that has a zero price generally…this utility function is a favorite for exams!

Summary and next steps

Now that we’ve looked at all the reasons Lagrange might fail – i.e., reasons that the solution to a consumer’s optimization problem might not occur at a point along their budget constraint where their indifference curve is tangent to the constraint – we can establish some conditions that guarantee an optimum at a tangency condition along the budget constraint. In particular, if the following conditions all hold, then Lagrange will work:

The consumer’s preferences are strictly monotonic. This guarantees that the solution will be on the budget line; otherwise, it might be at an interior point.
The MRS is infinite whenever $x_1 = 0$. This guarantees that the MRS is greater than the price ratio at the vertical intercept of the budget line; so there won’t be a corner solution in which the consumer only buys good 2.
The MRS is zero whenever $x_2 = 0$. This guarantees that the MRS is less than the price ratio at the horizontal intercept of the budget line; so there won’t be a corner solution in which the consumer only buys good 1.
The consumer’s preferences are strictly convex, and the MRS is continuous in both $x_1$ and $x_2$. This means the MRS is smoothly decreasing as you move from the upper-left to the bottom-right end of the budget constraint.
The budget line is a simple, straight line connecting the two axes. This ensures that there are no kinks in the budget line at which the price ratio is undefined, and that the price ratio lies strictly between zero and infinity.

Putting these all together means that the solution lies along the budget line; that the MRS is greater than the price ratio at the vertical intercept, smoothly decreases along the budget line, and is less than the price ratio at the horizontal intercept. Therefore, by a continuity argument (Implicit Function Theorem), there must be a single point at which the MRS equals the price ratio. Perhaps more intuitively and succinctly: monotonicity pulls the consumer up and to the right, convexity pulls them toward an interior solution, and conditions on the endpoints of the budget line ensure that the solution lies in the first quadrant.

When one or more of the above conditions fail, the solution may not be characterized by a tangency condition:

If the consumer’s preferences are not strictly monotonic, their optimal point might involve not spending all their money on these two goods.
If the consumer’s MRS is greater than the price ratio when $x_2 = 0$, their optimal point will be a corner solution in which they spend all their money on good 1. (Likewise, if their MRS is less than the price ratio when $x_1 = 0$, they’ll spend all their money on good 2.)
If the consumer’s MRS is not continuous, their optimal solution might occur at a point where the MRS is not defined. An example we’ve seen with this before is perfect complements.
Finally, if the budget constraint itself has a kink, the consumer’s optimal point might occur at that kink, at which point the price ratio is undefined.

This is the end of Module 2; in the third and final module of consumer theory, we’ll see how the optimal bundle changes when prices or income change. In other words, we’re ready to derive a consumer’s demand curve.

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