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My question comes out of the context of the explanation for quotas in the textbook Macroeconomics, 4th edition by Paul Krugman. The explanation is for what could've happened had a quota on cab rides not been placed. In this explanation, the price of $5.50 is apparently the minimum price consumers are willing to offer:

Looking back at Figure 4-8, you can see that starting at the quota limit of 8 million rides, New Yorkers would be willing to pay at least $5.50 per ride when 9 million rides are offered.

In the next paragraph, there is another reference to the price as being the minimum that people were willing to pay:

The same is true for the next 1 million riders: New Yorkers would be willing to pay at least $5 per ride when the quantity of rides is increased from 9 to 10 million...

Shouldn't this price be the maximum price that consumers are willing to pay for, instead of the minimum price, as the book suggests? Why would a consumer not want to pay less that $5.50$ dollars per ride, at $5.30$ dollars for example, if the same quantity of rides (9 million) is offered?

I have included a copy of the diagram that is referenced, which I redrew. The x axis is quantity of cab rides per year, in millions of rides. The y axis is the fare price per ride. The green, orange and red lines are the demand, supply and quota restriction lines respectively. Please click to enlarge image.

enter image description here

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I am perfectly willing to pay nothing to take a taxi ride in NY. If I were have to pay, I am willing to pay no more than \$5 for such ride. Therefore, I am not willing to pay \$10 for such a ride.

Add up the valuations of every other potential consumer of taxi rides in NY and you get the demand curve. Thus, this represents the maximum price individuals are willing to pay.

Regarding the apparently contradictory "at least" phrase in the second quote, this is consistent with the definition of demand. In effect, whilst the marginal consumer (the 10th million) is in fact willing to pay no more than \$5 for a ride, the other 9,999,999 consumer are willing to pay more than \$5 (for instance, the 8th million individual is willing to pay as much as \$6, whereas the 9th million consumer is willing to pay as much as \$5.5). Hence, strictly speaking it is correct to say that 10 million "New Yorkers" are willing to pay at least \$5 for a ride. This is not a contradiction, but a use of terminology that refers to different groups of people.

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A distinction must be made between the individual demand curve and the aggregate demand curve.

An individual demand curve (or demand schedule) for a product/service, represents the pairs {price, quantity demanded} that co-optimize the consumer's utility maximization problem (together with all other quantities demanded of the other products that are present in the utility function).
It may appear that then this would be the "maximum" price, but it would be confusing to characterize it like that. Because, if the price changes, what would happen to individual quantity demanded depends on whether the good is normal or not, and in general whether there are "income-effects" or not (not all utility functions allow for income effects, by the way).
Moreover, at the individual level, the concept representing the consumer is quantity demanded, not price: the consumer observes prices and responds with quantity demanded.

Let's move now to the aggregate demand curve. It is simply the sum of individual demand curves for each price. Here we have two alternative approaches:

a) The more "traditional" one is that all consumers have a non-zero demand schedule for all price levels, each lowering its demand as price increases. Here the previous discussion applies.

b) The alternative but equally "standard" approach is to argue that for each consumer there exists a (possibly different) "threshold" price, above which the individual demands zero quantity. So as price decreases, we imagine that more consumers enter the market with non-zero demand, and this is why as price decreases aggregate quantity demanded increases.
This approach implies that the price level in each price/quantity combination is the minimum ("at least") price for some consumers, while others would be willing to pay more (and so they have contributed positive quantity demanded for higher price levels).

I believe this is the approach taken by the textbook.

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The formulation quoted says:

"Looking back at Figure 4-8, you can see that starting at the quota limit of 8 million rides, New Yorkers would be willing to pay at least $5.50 per ride when 9 million rides are offered."

Note the bolded text. Assuming that a seller of rides can only charge one price regardless of which unit the seller is selling. The seller would then have to charge $5.50 if 9 million rides are to be sold. This price charged is the consumers' maximum willingness to pay for the 9th million unit.

However, buyers are willing to pay more for earlier units, that is why the demand curve is downward sloping. Hence, if the seller is able to perfectly price discriminate, that is set different prices for each ride based on the buyers' willingness to pay for that particular unit, (s)he could charge different prices for each ride. For example, if I read your graph correctly, the seller could charge $6 for the 8th million ride. The seller will still be able to sell 9 million rides, and the price will be at least \$5.50 per ride. In fact, the only ride that will be sold for \$5.50 is the 9th million, all earlier rides will be sold at a higher price becaue the seller is charging exactly what the consumer is willing to pay for each individual ride.

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  • $\begingroup$ Your answer made sense. I wish I could award the "correct answer" tag to both yours and luchonacho's $\endgroup$ – user98937 Sep 18 '17 at 20:22
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Of course everyone would be happy to pay less, but that's not exactly what is going on here. There are a few implicit assumptions going on in the supply and demand curves.

  • It's assumed that buyers will have internal utility functions determining their 'need', and the demand curve is an aggregate function representing the number of buyers that will be willing to pay at least that much. e.g. some buyers will not be willing to pay more than 5.30, some buyers will not be willing to pay more than 5.50, some buyers will be willing to pay up to 100.
  • It is assumed sellers can only offer one price. So even though some buyers would be willing to pay exorbitant prices, sellers don't get to price according to any specific buyer.

In this case, referencing the minimum is correct. The book is working a little bit counter-intuitively by increasing the supply and decreasing the price however the reverse is a bit more easy to grasp:

Each increase in price decreases the available population of potential riders as some buyers decide it is not worth it to take taxis anymore. Some buyers will still be willing to pay more (as should be obvious when the demand doesn't drop to 0 as the price is increased). So, at 5, 10 million New Yorkers are willing to pay at least 5. However, some of the people willing to pay 5 are not willing to pay 5.50, so when the price goes up to 5.50 then 1 million buyers will drop out of the market leaving 9 million willing to pay 5.50 or more.

This naturally means the converse is true: if the price decreases some people that previously were not interested in taking taxis at the higher price will be okay to take the taxi at the lower price.

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On an existing demand curve, yes. However, demand curves can move up and down laterally as well. This might be in response to an external shock such as a sudden perception that cabs are the cure for cancer.

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  • $\begingroup$ Given how the book says nothing about changes in demand (i.e. nothing about shifts), would it be correct to say that the book's writers have made a mistake? $\endgroup$ – user98937 Sep 18 '17 at 0:29

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