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?
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.
