Why not say the buyer will *definitely* buy at price X?

Question

In supply-and-demand curve, we draw curves of indifference. Say the demand goes through (1, \$1). Then Bob the Buyer is indifferent between spending \$1 for 1 apple or just keeping the \$1. If Sally the Seller had 1 apple to sell, then the equilibrium quantity sold would be 0 or 1—we cannot say. Seen differently, Bob will buy 1 apple if the price is in the interval [\$0, \$1) (and he may buy it if the price is exactly $1).

Would it not be simpler to use closed intervals? In our example, Bob will buy 1 apple is the price is in the interval [\$0, \$1]. The equilibrium quantity sold will definitely be \$0. Seen differently, the demand curve represents the maximum price buyers will pay, and the supply curve represents the minimum sellers will sell for.

It's (literally) an infitesmal of a difference, but it seems, at least for discrete situations like the above, it would be nice to say exactly what will happen.

Why don't we do that? There must be some elegant reason.

Answer 1

Under Yatharth's approach, the set of prices at which Bob will definitely buy an apple is the closed interval $[\$0,\$1]$, which is nice. However, the set of prices at which Bob will definitely not buy an apple now becomes an open interval $(\$1,\infty]$.

In contrast, under the conventional approach, the set of prices at which Bob may buy an apple is closed: $[\$0,\$1]$. So too is the set of prices at which he may not buy an apple: $[\$1,\infty)$. So, under the conventional approach, we're dealing solely with closed intervals.

Both approaches have their merit. But I believe that all things considered, fewer inconveniences arise with the conventional approach than with your suggested approach. In general, it's "nicer" to not have to deal with any open intervals.

Example 1. Under Yatharth's approach, there is no answer to the question "What is the lowest price at which Bob may not buy an apple?"

Example 2. Under Yatharth's approach, the demand curve will be discontinuous and with jumps. In contrast, under the conventional approach, it is continuous (albeit with kinks).

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Elaborating on Example 2.

The demand function $D:\mathbb{R}_{0}^{+}\rightarrow\mathcal{P}\left(\mathbb{R}_{0}^{+}\right)$ maps each price (a non-negative number) to a set of quantities demanded (a subset of the non-negative numbers). The economic interpretation is that at each price, $D$ tells us what numbers of units of the good the consumer most prefers to buy.

We assume that the good is infinitely divisible.

Then here are the graphs of the demand function under the two approaches:

The demand function is continuous under the conventional approach, but not under Yatharth's. (Nonetheless, under Yatharth's, it is lower semi-continuous.)

Answer 2

Two arguments to complement the existing answer:

1. We need to have one price for which Bob is indifferent between buying the apple or not, otherwise his preferences are discontinuous, which does not make much sense normatively. That is, an infintesimal variation in the price would cause him to reverse his preferences.

2. Your logic is not entirely correct as Bob might also buy the apple for 1 dollar, as this action is not strictly dominated (he is indifferent). And whether he is indifferent at a price of 1 dollar or not is economically irrelevant, as it will not influence Bob's behavior in equilibrium. Suppose for instance that Sally is a monopolist who knows Bob's preferences and posts a price for the apple (a take-it-or-leave-it offer). Irrespective of whether Bob strictly or weakly prefers buying the apple for 1 dollar rather than not, the only equilibrium is the one in which Sally offers the apple for 1 dollar, and Bob accepts the offer (either because he is indifferent, or because he strictly prefers it).

Answer 3

This is essentially a request for clarification, but it is a bit long to be a comment.

Demand curves represent the amount that the consumer will definitely buy at each price (if it has the income). So the statement "the demand curve passes through the point $(1,1)$", means that the consumer will definitely buy one apple at one dollar price, (if he has one dollar). It will certainly also buy one apple (and maybe more), if the price is below one dollar (and he has the income).

It is not clear what the OP explores here, since Indifference curves are drawn in the goods space, or at least, in the "one good and income space". But the demand curve is drawn in the "good-price" space. We cannot superimpose the demand curve on the map of the indifference curves.

Answer 4

If he obtains 1 util of value from an apple that costs 1 util, then he neither gains nor loses from purchasing the apple and consuming it.

He might buy it, or might not, and at that price will be indifferent as to whether he does.

Since the difference is literally infinitessimal, presumably convenience rather than theory will justify most decisions about whether to used an open or closed bound.

Compare this to a stock trading platform which may calculate to very small, but finite, differences in prices. Being finite, and not infinitessimal, it is strictly calculable whether a gain can be had. In which case specific features of reality would dictate the answer, and not convenience.

Textbook exercises may 'arbitrarily' be designed to require one or the other -- for practical applications, the difference simply is the type of inequality operator used in the programming.