Define consumers surplus using mathematics

Consumer surplus is a measure of the economic benefit or gain that consumers receive when they are able to purchase a good or service at a price lower than the maximum price they are willing to pay. Mathematically, consumer surplus can be represented using the concept of the demand curve.

Let's denote the demand curve as \(D(p)\), where \(p\) is the price of the good or service. The demand curve represents the quantity of the good or service that consumers are willing to buy at different prices. The consumer surplus (\(CS\)) is the area between the demand curve and the price axis up to the actual market price (\(P\)).

The mathematical representation of consumer surplus (\(CS\)) is given by the integral of the demand curve from zero to the market price:

\[CS = \int_{0}^{P} D(p) \, dp\]

In simpler terms, consumer surplus is the area under the demand curve up to the market price. The integral is used to calculate this area. This formula assumes a continuous demand curve, and the interpretation is that consumer surplus represents the difference between the total amount consumers are willing to pay (the area under the demand curve) and the total amount they actually pay at the market price.