# Demand, Willingness to Pay and Marginal Benefits

The market demand curve for a good originates from what individuals are willing to pay (W2P) for the good. Conceptually, it is constructed as follows: (1) start with a high price; (2) ask all potential buyers how many items they would be willing to buy at that price; (3) make a note of that price and quantity; (4) decrease the price slightly and repeat the process. The example below shows the steps in detail.

A person's willingness to pay for something shows the dollar value she attaches to it. Her willingness to pay for one more unit of a good is thus a dollar measure of the benefits the extra unit of the good gives her. As a result, the terms "willingness to pay" and "marginal benefit" are often used interchangably.

## Willingness to Pay and Individual Demand

Suppose Alice and Bob are two buyers of downloadable songs and each has a monthly W2P that can be expressed in equation form as follows:

Alice: W2Pa = 5 - Qa/2

Bob: W2Pb = 10 - Qb/5

That is, Alice is willing to pay up to $4.50 for the first song (when Qa=1),$4.00 for the second song, and so on. Bob likes music more: he's willing to pay $9.80 for the first song (when Qb=1) and$9.60 for the second song.

Given this information, we can construct each person's demand curve--the number of songs they would be willing to buy at each price. If the price were very high, say $10 per song, neither person would buy any: Alice's maximum W2P is$4.50 and Bob's is $9.80. Qa and Qb would both be 0. If the price were$9, however, Bob would be willing to buy 5 songs. To see that, look at his W2P for the first few songs:

 Qb W2P 1 9.80 2 9.60 3 9.40 4 9.20 5 9.00 6 8.80

He's willing to pay more than $9 per song for songs 1-4, and is willing to pay$9 for song 5. He wouldn't want a 6th song at that price: song 6 is only worth \$8.80 to him. Since he'll buy songs until his W2P for the last song is just equal to the price, we can use his W2P equation to find his demand curve:

W2Pb = 10 - Qb/5

W2Pb = P (for the last song he buys)

P = 10 - Qb/5 (substituting P for W2Pb)

Qb = (10-P)*5 (rearranging)

From this it's easy to calculate Bob's demand: when P=10, Qb=0; when P=9, Qb=5; etc. We can do the same thing to get Alice's demand curve:

W2Pa = 5 - Qa/2

W2Pa = P (for the last song she buys)

P = 5 - Qa/2

Qa = (5-P)*2

This reproduces the resuts we obtained above: when P=5, Qa=0; when P=4.50, Qa=1; when P=4, Qa=2.

## Deriving the Market Demand Curve

To build the market demand curve, we could go through the reasoning above for each potential price and then add up the quantities demanded by each person. If Alice and Bob are the only buyers in the market, and to keep things simple we imagine they can buy fractions of a song, the results would look like this:

 P Qa Qb Market Q 10.00 0 0 0 9.50 0 2.5 2.5 9.00 0 5 5 8.50 0 7.5 7.5 8.00 0 10 10 7.50 0 12.5 12.5 7.00 0 15 15 6.50 0 17.5 17.5 6.00 0 20 20 5.50 0 22.5 22.5 5.00 0 25 25 4.50 1 27.5 28.5 4.00 2 30 32 3.50 3 32.5 35.5

In algebra, what this says is the following, where Q is the total market demand:

P > 10:

Q = 0

P between 10 and 5:

Q = Qb

Q = (10-P)*5

P < 5:

Q = Qa + Qb

Q = (10-P)*5 + (5-P)*2

Q = 60 - 7*P

In each case, the total market demand is just the sum of the quantites demanded by the individuals.
URL: https://wilcoxen.maxwell.insightworks.com/pages/144.html
Peter J Wilcoxen, The Maxwell School, Syracuse University
Revised 08/17/2016