# is surplus just an approximation?

let's say we have a consumer with a demand curve qd=10-p and the good it's price is 5. So the consumer will consume 5 units. The surplus of the consumer would then be equal to (10-5)+(9-5)+(8-5)+(7-5)+(6-5) = 5+4+3+2+1 = 15. However, when solving this in the conventional manner it gives me (10-5)*5/2 = 12.5

What am I doing wrong? Or is surplus just an approximation?

Your second answer is correct. The problem your first approach has, is that you assume that you may only buy a discrete amount of goods. However, your demand curve is continuous and the usual assumption in microeconomics is that you can consume any amount of this good, for instance $1.478926574$ units. Thus, your first calculation for the surplus is not correct, the correct way to do it is
$$CS = \int_{5}^{10} (10-p) \mathcal{d}p = 10p-\dfrac{p^{2}}{2}|^{10}_{5} = 100-50-50+12.5=12.5$$
• Just draw the graphs, for discrete consumption and for continuous consumption. The demand function in your case always cuts off 0.5 of your CS below the bars you constructed. Note: through your calculations you already assumed how the discrete demand looks like. The first bar is equal to $p=10$ for values between $0$ and $1$ on your $q$-axis, $p = 9$ for $q$ between $1$ and $2$, and so on. – saguru May 5 '18 at 14:46
• so for discrete demand you're getting $0.5$ CS more per unit, which equals $2.5$ for $q=5$. – saguru May 5 '18 at 14:48