Equivalence of producer surplus areas

I really can't understand how the sum of the rectangles leads to the sum of the triangle. My book's(Chapter 12, page 430) explanation is:

Equivalence of these areas can be shown by recognizing that each point in the supply curve in Figure 12.11d represents minimum average cost for some firm. For each such firm, P - AC represents profits per unit of output. Total longrun profits can then be computed by summing over all units of output.

The author doesn't explain how the bold part is valid and its not obvious from the graph as well.

The profit of a firm $$i$$ is given by: $$\pi_i(p) = p q_i - C_i(q_i)$$ where $$p$$ is the price, $$q$$_i is the output of firm $$i$$ and $$C_i(.)$$ is the cost function which differs across firms.

The first order condition gives: $$p = \frac{\partial C_i(q_i)}{\partial q_i} = MC_i(q_i^\ast)$$ This shows how to obtain the optimal supply of firm $$i$$, i.e. where $$MC(q_i^\ast)$$ is equal to $$p$$ (see figures a,b and c in the question).

Total profits for the firm is then: \begin{align*} \pi_i(p) &= pq_i^\ast - C_i(q_i^\ast),\\ &= \left(p - \frac{C_i(q_i^\ast)}{q_i^\ast}\right)q_i^\ast,\\ &= \left(MC(q_i^\ast) - AC(q_i^\ast)\right) q_i^\ast. \end{align*} This corresponds to the shaded areas in figures a,b and c in the question: the areas equal the difference between $$MC(q_i^\ast)$$ and $$AC(q_i^\ast)$$ multiplied by $$q_i^\ast$$.

Now, consider the first order condition: $$p = MC_i(q_i^\ast).$$ We can invert this function to obtain the supply curve of firm $$i$$: $$q_i(p) = q \text{ whenever } p = MC_i(q).$$ The total supply in the market at price $$p$$ is determined by adding the supply curves of over all firms. $$Q(p) = \sum_i q_i(p).$$ Notice that, as usual in economics, figure d in the question draws the supply and demand curves wrong, as they put $$p$$ on the vertical axis and $$q$$ on the horizontal. Mathematically, the two should be switched.

If we would do it correctly, it should look something like this:

If $$p^\ast$$ is the equilibrium price, then the producer surplus is given by: $$PS = \int_0^{p^\ast} Q(p) dp = \sum_i \int_0^{p^\ast} q_i(p) dp$$ Now let us make a change of variables $$p \to q$$, where $$p = MC_i(q)$$. Then $$dp = \frac{\partial MC_i(q)}{\partial q} dq$$ so: $$\int_0^{p^\ast} q_i(p) dp = \int_0^{q_i^\ast} q \frac{\partial MC_i(q)}{\partial q} dq.$$ Then use integration by parts to get: \begin{align*} \int_0^{q_i^\ast} q \frac{\partial MC_i(q)}{\partial q} dq &= \left[q MC_i(q)\right]^{q_i^\ast}_0 - \int_0^{q_i^\ast} MC_i(q) dq,\\ &= q_i^\ast MC_i(q_i^\ast) - C_i(q_i^\ast),\\ &= p^\ast q_i^\ast - C_i(q_i^\ast) = \pi_i^\ast. \end{align*} Here we assume $$C_i(0) = 0$$ and we used the first order condition to substitute $$MC_i(q_i^\ast) = p^\ast$$. We also used $$\pi_i^\ast$$ to denote the profit of firm $$i$$ at price $$p^\ast$$. From this: $$CS = \sum_i \int_0^{p^\ast} q_i(p) dp = \sum_i \pi_i^\ast,$$ which shows that the consumer surplus indeed equals the sum of all profits in the industry.

• It is somewhat shady to use the invertability of $MC$, as that is not necessary :) May 8 at 16:57
• Interesting. I inverted the MC as I didn't see any other way to define aggregate supply?
– tdm
May 8 at 17:05
• ? Supply is defined as inverse of MC when price is above min of AVC and 0 otherwise. May 8 at 17:44
• Agree, I would have been more correct to restrict the supply curve to the region where the MC is above the AC. I just didn't want to make the argument too complicated. Feel free to adjust the answer if you feel like.
– tdm
May 8 at 18:11
• What does this line from the book mean: "Equivalence of these areas can be shown by recognizing that each point in the supply curve in Figure 12.11d represents minimum average cost for some firm."? May 12 at 19:43