# What does an individual's demand function describe?

Now, this seems a super easy and straightforward question, right? From wikipedia:

The demand curve is the graph depicting the relationship between the price of a certain commodity and the amount of it that consumers are willing and able to purchase at that given price.

To me, this reads as follows: for an individual, the demand function answers, for all values of $p$, the question: "if the price is $p$ per unit, how many units $q(p)$ would the individual buy?"

As an example, let's consider my personal demand curve $q = 10-p$ for a certain good. I would interpret that, also with the above definition, as follows: if the price is 10/unit or more, I purchase nothing. If it is 7/unit, I'll purchase 3 units.

The big point here is that this seems to imply, that I'd pay 7/unit for each of the 3 units. Is that correct? Things would be different if I were faced with a situation in which I had to pay more for the first 2 units. In that case, I might decide to buy less than 3 units. And I wouldn't be contradicting my demand function. At least, not, if it were as I interpreted the definition.

Alternative: it seems less close to the text, yet much more convenient to the mathematics, to interpret the demand function in the following way. The demand function answers, for all values of $q$, the question: "if an individual already has $q-1$ units, how much would he be willing to pay for unit $q(p)$?" (or, as an integral, "if an individual already has an amount $q$, how much would $p(q)$ be, with $p(q)dq$ the amount he's willing to pay for an additional amount $dq$?").

Thanks!

The demand function does not say how much he's willing to pay for an additional amount dq. The demand function says: given the price of good is $p$, you will purchase $Q(p)$ of them. I think your confusion lies in the fact that it is possible to charge different prices for each unit of good. Notice that our theory is assuming one price, not multiple.

Moreover, the notion that consumer surplus measure the difference between what you paid and what you are willing to pay is completely false. Suppose the demand function is $Q(p)=-5p+100$. Suppose further that the price is $5$, so you would be purchasing at a quantity $75$. The consumer surplus is: $$CS=\int_0^5Q(p)dp=\int_0^5 -5p+100dp=562.5.$$ If interpreting it as the difference between what you paid and what you are willing to pay, we know that you are willing to pay $562.5+5\times75=937.5$ for $75$ units of them. In other words, you are willing to accept a price $937.5/75=12.5$. However, plugging this number into our demand function, we find that $Q(12.5)=37.5$, i.e. you are only willing to buy 37.5 units of the good, which clearly contradicts to our interpretation of consumer surplus.

For a better treatment of consumer surplus, I would recommend you to consult advanced texts on microeconomics theory such as Mas-Colell's or Varian's more advanced book.

• Hey Kun, thanks for writing! I'll check out the sources you provide. You integrate from 0 to 5 but this should be 5 to 20, I think you meant that as the number is correct. I agree with you that if the demand function describes the average acceptable cost, then the CS cannot be interpreted as the diff between willing and having to pay. My question however is, if that's what it describes in the first place. Consider the situation of perfect price discrimination (and constant marginal cost of production of 5), and answer me this: how much would I be paying, and for how many units? Thnx! – ElRudi Feb 13 '16 at 9:57
• @EIRudi Hello, your point is well taken. In fact, I very much agree with your point. I cannot understand how firm will be able to charge different prices for each additional unit consumed since the demand function only takes one price. And the demand theory only assumes one price. I asked two of the professors in my department, though they both claim this should work out, none actually were able to explain it to me satisfactory. Let me try some more sources. – Kun Feb 13 '16 at 15:50
• I have detialed my response in a pdf at space.zeo.net/g/5k4ij . Please check it out. – Kun Feb 13 '16 at 16:40
• Hey Kun, thanks for your detailed answer. I think you correctly saw that p=1/q does not provide a finite value for CS, but I think you incorrectly concluded from that, that the surface area cannot be representing the CS. I've also made a detailed response, here: wp.me/s7eZd7-59 Check it out! – ElRudi Feb 15 '16 at 21:28
• I think I agree with your point. You should post it as an answer here so other people will see that. :-) – Kun Feb 15 '16 at 22:00

In the discussions with Kun it seems we've found a satisfying answer.

TL;DR: it is as I suggested in the last paragraph of my question.

This is becoming a bit of an overkill for a question that could be looked up in a couple of minutes, but it's an interesting exercise to see if we can up with the correct answer ourselves.

What are we trying to accomplish again?

We have

• An infinitely divisible good
• A consumer with demand function $q = Q(p)$ for this good, with corresponding inverse demand function $p=P(q)$. These are described by a curve (the demand curve) in the $q,p$-plane.

We are trying to establish if a point $(q,p)$ on the demand curve describes

• A. the price per unit $p$ that the consumer is willing to pay, for each unit, for a the total quantity $q$, or

• B. the price per unit $p$ that the consumer is willing to pay for an additional amount $\text{d}q$, given a possession of $q$ units.

Let our thesis be that it is the latter, and let's see if we run into a contradiction.

## Consumer surplus

Let's take the simple straight line $p+q=25$, where a constant unit-price of 5 prescribes a demand of 20:

The area between the demand curve, $p=5$ and $q=0$, is called the Consumer Surplus CS, and we can calculate it by integration: $$\int_5^\infty Q(p)~\text{d}p=\int_5^{25} (25-p)~\text{d}p$$ This is the same as $$\int_0^{20} (P(q)-5)~\text{d}q=\int_0^{20} \big((25-q)-5\big)~\text{d}q$$ The latter shows the area might be interpreted as the difference between what the consumer is willing to pay and what he is paying -- at least, if our thesis is valid and interpretation (b) holds.

We find that the CS in this case is 200.

Our thesis can help to interpret this. If the price is 5, the consumer keeps on buying until the additional quantity he could buy does not bring that additional utility of 5. That happens to be at a quantity of 20. Because he was able to buy all units at a price of 5, this represents an advantage: he would have paid more for an additional unit when he still had fewer units. E.g. when he still only had 15 units, he would have paid 10 per unit for additional quantity.

## Total Willingness To Pay

The total willingness to pay TWTP for a quantity $q$ (i.e., the maximum accepted price for that quantity) can be calculated as the sum of the willingness to pay for each subsequent unit until $q$ (i.e., the maximum accepted price per unit for additional units): $$TWTP(q) = \int_0^{q} WTP(q')~\text{d}q'$$

So, in order to calculate how much our buyer would maximally have paid (in total) for those 20 units, we must add the maximum prices for each individual unit. If our thesis is correct and interpretation (b) holds, this is exactly the inverse demand function $P$, so that $$TWTP = \int_0^{20} P(q')~\text{d}q'$$ This Total Willingness To Pay for 20 units happens to be 300 in our case.

Now, under normal circumstances the buyer does not need to pay that amount for 20 units, but rather $5\cdot20=100$. The difference between the two is the CS of 200. This is the area above the $p=5$ line, which is what we would expect.

Should we have perfect price discrimination, the seller of the good would know the buyer's demand curve, and sell him each unit of the good at exactly the maximum price he'd be willing to pay for it; gradually dropping the price with the buyer's marginal utility: $p=25-q$. That way, the seller is able to capture all of the CS, and the buyer would thus pay 300 for the 20 units.

## If our thesis is wrong

This interpretation only works if our thesis is correct and the inverse demand function $P$ describes the willingness to pay for each additional unit. If it describes the willingness to pay per unit for that and all previous units, i.e. interpretation (a), things are different. In that case, the TWTP for $q$ units is simply the multiplication of $q$ and $P(q)$: $$TWTP(q) = q\cdot P(q)$$ In order to figure out how much would be bought at non-uniform pricing, we need to find the willingness to pay for each unit. That WTP is, as can be seen from the first equation, the derivative of the TWTP, so, in this case: $$WTP(q)=\frac{\text{d}}{\text{d}q}qP(q)=\frac{\text{d}}{\text{d}q}(25q-q^2)=25-2q$$ So, the first quantity is sold at a unit price of 25, just like before. This makes sense, as the buyer does not have any units yet, so the marginal price equals the average price. Then, however, under perfect price discrimination, the price of the good should drop twice as fast as we have previously calculated. That is due to the fact that the additional unit $\text{d}q$, that the seller is trying to sell, does not have a marginal utility given by its price (as is the case in interpretation (b)) but by the increase in total price.1 Moreover, in this situation, the seller sells his last unit for a price of 5, which is when he has sold only 10 units (compared to 20 if interpretation (b) is correct). The buyer has then spent $\int_0^{10} (25-2q)~\text{d}q=150$, which is his TWTP, but for 10 units. This too makes sense: the demand curve prescribes a maximally accepted (average) unit price of 15 - which is exactly what is being paid.

What we cannot see anywhere, is the figure of 200 which is the area above the $p=5$ line. In fact, the Consumer Surplus that was 200 in the case of interpretation (b), is actually 0 in the case of interpretation (a) -- simply because of the way the inverse demand function is defined to be the maximum average unit price: if we have a uniform price, the price is the average price, and the buyer will have an incentive to buy more as long as his willingness to pay is higher than the price. Exactly when he buys the quantity $q$ that, on his demand curve, corresponds to the offered price $p$, is the average price he's willing to pay equal to the offered price. Because the average price he's willing to pay, times the quantity, is the total price he's willing to pay for that quantity, and because that also equals the price he is paying at that point in the curve, his CS is 0.

## Conclusion

In order to have a sensible interpretation of the area between the demand curve, $p=5$ and $q=0$, called the Consumer Surplus CS, we need to interpret the inverse demand function to mean: "the price per unit $p$ that the consumer is willing to pay for an additional amount $\text{d}q$, given a possession of $q$ units" (b). The common interpretation (a) as "the price per unit $p$ that the consumer is willing to pay, for each unit, for a the total quantity $q$" is incorrect. It's easy to see, however, why it is often interpreted that way. Firstly, it is a simpler interpretation that's easier to visualise, and secondly, in everyday situations -- which all have uniform pricing -- it still predicts the correct quantity to be traded.

Footnotes:

1: This gives us another way to come to the formula. Consider the buyer, which buys a quantity $q$ when the average price per unit is $P(q)$. As he buys a quantity $q+\text{d}q$ when the price per unit is $P(q+\text{d}q)$, the total cost increases by $P(q+\text{d}q) - P(q)$, which means that is the utility of the additional unit $\text{d}q$. So, the marginal utility per unit is $\frac{(q+\text{d}q)\cdot P(q+\text{d}q)-q\cdot P(q)}{\text{d}q}$. Using the inverse demand function $P(q)=25-q$, this is turns out to be $25-2q$.

Your analysis of what demand curve represents is correct. Your confusion comes from the overly simplistic demand function that you chose. A "true" demand function would rather look like this:

Your equation is sufficient to study what happens in the middle of the curve, but not on the tails.

The upper tails has such form that you would actually purchase some units even if price goes up a lot. But the more it goes up, the less your demanded quantity changes. That is because the product has some utility to you, so you are willing to pay a relatively high price for one unit of it. But then, the lower the price, the more you could buy. But the more units you buy, the fewer utility it brings to you. While you might really enjoy eating an apple, I bet that the 100th will not taste as great as the first one, and you might be disgusted. That's why the curve does not go to infinity when the price is zero. Reaching a certain point, your demand for apple doesn't grow anymore.

• Thanks Hector for your answer. The reason I had the question was because of consumer surplus, which is the upper right corner above the horizontal line with the price. Sticking to my stylised curve, my CS would be 4.5 (at p=7/unit). The CS is supposed to be the monetary gain of the consumer. I don't yet see how that fits with my first interpretation. If 7/unit is the maximum I'm prepared to pay for each of the units when buying 3 units, my CS would be 0 as I'd be paying my maximum. If we take the second interpretation, we do get a CS of 4.5 - do you know what i mean? – ElRudi Jan 12 '16 at 9:42
• The consumer surplus represents the difference between what you effectively paid and the max of what you would have been willing to pay. So in your example you would have been willing to pay 9 for the first unit, 8 for the second and 3 for the third, but you actually pay 3 per unit, so you have a consumer surplus of 3. – Hector Jan 12 '16 at 10:09
• Thanks again. That would fit with my second interpretation then, and the demand curve point (3, 7) describes that I'd pay a maximum of 7/unit for the 3rd unit only, so, assuming I have already acquired 2 units from elsewhere. The point does not describe that I'd buy 3 units if the price where 7/unit. Thanks for the clarification! – ElRudi Jan 12 '16 at 10:31
• No, I might not have been clear. The point (3,7) means that you will pay 7 for three units. The consumer surplus is a measure of the total amount that you would have been willing to pay, but that you will save. – Hector Jan 12 '16 at 10:39
• Sorry, neither was I;) Of course, in 'normal' situations, I don't need to pay more for the 1st unit than for the 2nd or 3rd - they all have the same (i.e. 'the') price, which is great news for the consumer. But still, the demand curve is supposed to be cut loose from the supply I actually encounter, no? Consider perfect price discrimination, in which i'm charged whatever i'm willing to pay for each single unit. In that case, I'd be charged 9 for unit 1, 8 for unit 2, and 7 for unit 3, right? That 7 means what I'm willing to pay regardless of what I've paid for the others, wouldn't you say? – ElRudi Jan 12 '16 at 16:54