Interpreting the demand curve

This is becoming a bit of an overkill, but it’s an interesting exercise to see if we can up with the correct answer ourselves.

What are we trying to accomplish?

We have

• an infinitely divisible good; and
• a consumer with demand function $q = Q(p)$ for this good, with corresponding inverse demand function $p=P(q)$

Both functions described the same curve (the demand curve) in the $q,p$-plane, and we are trying to establish whether a point $(q,p)$ on the demand curve describes

(a) the average price per unit $p$ that the consumer is willing to pay when buying 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:
$$CS = \int_5^\infty Q(p)~\text{d}p=\int_5^{25} (25-p)~\text{d}p$$
This is the same as
$$CS = \int_0^{20} (P(q)-5)~\text{d}q=\int_0^{20} \big((25-q)-5\big)~\text{d}q$$
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:
$$WTP(q)=\frac{\text{d}}{\text{d}q}TWTP(q)=\frac{\text{d}}{\text{d}q}qP(q)=\frac{\text{d}}{\text{d}q}(25q-q^2)=25-2q$$
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 $25-2q$.