# Consumer surplus calculation

Suppose $$P(Q) = 10-0.5Q$$. If a firm is producing $$q_1$$ and $$q_2$$ units from its two different machines such that $$Q = q_1 + q_2$$, what is the consumer surplus at $$(q_1, q_2)$$?

I think the consumer surplus is $$\left(\frac{Q}{2}\right)^2$$.

However, when I calculate the consumer surpluses separately, the result is different. In other words, at $$q_i$$, the consumer surplus is $$(q_i/2)^2$$ but the total consumer surplus $$(q_1/2)^2 + (q_2/2)^2 \neq (Q/2)^2$$.

Why's this happenning?

• Why would you "calculate the surpluses separately" for each production machine and then add them together? They are related to how demand varies with price. You might calculate consumer surpluses separately for each consumer, but not for each producer. Commented Apr 6 at 16:06
• @Henry Why can't we do that? I can see it does not work but I don't see why. Commented Apr 6 at 17:13

If we are going to consider the separate effects on consumer surplus (CS) of $$q_1$$ and $$q_2$$, what we must do is first find the CS when $$Q=q_1$$ and then consider the total change in CS which results from increasing $$Q$$ from $$q_1$$ to $$q_1+q_2$$. In the diagram below, the CS when $$Q=q_1$$ is represented by area $$A$$ which equals $$(q_1/2)^2$$.

The increase from $$q_1$$ to $$q_1+q_2$$ has two effects on CS. It adds area C which equals $$q_2/2)^2$$. But also, by lowering the price, including the price for the original $$q_1$$ units, it increases the CS on those units. This second effect is represented by area B, a rectangle with length $$q_1$$ and height $$q_2/2$$ and therefore area $$q_1q_2/2$$. Thus the total CS obtained in this way is as below:

$$CS = \Big(\frac{q_1}{2}\Big)^2+\frac{q_1q_2}{2}+\Big(\frac{q_2}{2}\Big)^2 =\frac{q_1^2+2q_1q_2+q_2^2}{4} = \Big(\frac{q_1+q_2}{2}\Big)^2$$

which equals $$(Q/2)^2$$ when $$Q=q_1+q_2$$.

So the answer to your question "Why's this happening?" is that your calculation ignores area B, the effect of producing additional units on the consumer surplus of existing units.

• That works providing you assume machine 1 comes first providing consumer surplus A and then machine 2 comes along to add C and B. Arguably each machine provides B in addition to A or C : if machine 1 was not there then machine 2 would provide C, so machine 1 increases consumer surplus by A and B. Then the problem becomes trying to avoid overcounting B, Commented Apr 6 at 21:48
• @Henry Yes, in the interests of brevity I only considered machine 1 coming first, but the argument as you indicate is easily adapted for machine 2 coming first. Commented Apr 6 at 22:25
• @Henry Right, and if both the machines work together, then we simply think of it as the final quantity $Q$ (rather than separating them). Commented Apr 7 at 4:42