Competitive equilibrium and surplus

A market clears when buyers and sellers face the same price and wish to trade the same quantity. Before any algebra, the idea is simple: at a high price sellers offer more than buyers want; at a low price buyers want more than sellers offer; somewhere in between, plans are consistent. This note makes that point exact for the linear case used throughout the course, and then measures how much the market accomplishes.

The linear market

Inverse demand and inverse supply are

$$p = a - bq \qquad \text{and} \qquad p = c + dq,$$

with $a > c \ge 0$, $b > 0$, $d > 0$. The intercept $a$ is the most any buyer is willing to pay; $c$ is the lowest price at which any seller is willing to produce.

Equilibrium

Setting the two prices equal,

$$a - bq = c + dq \quad\Longrightarrow\quad q^* = \frac{a - c}{b + d}, \qquad p^* = a - bq^*.$$

Since $a > c$, we have $q^* > 0$: there are mutually beneficial trades, and the market finds them. (When $a = c$ the curves meet on the price axis and $q^* = 0$ — the degenerate case shown as $0.00$ in Tool 2.)

Consumer surplus

A buyer who would have paid $p = a - bq$ for the $q$-th unit, but pays only $p^*$, gains the difference. Adding these gains over all units traded,

$$CS = \int_0^{q^*} \big[(a - bq) - p^*\big]\, \mathrm{d}q = \tfrac{1}{2}\,(a - p^*)\,q^*,$$

the area of the triangle between the demand curve and the price line.

Producer surplus

Symmetrically, the seller of the $q$-th unit would have accepted $c + dq$ but receives $p^*$:

$$PS = \int_0^{q^*} \big[p^* - (c + dq)\big]\, \mathrm{d}q = \tfrac{1}{2}\,(p^* - c)\,q^*.$$

Total surplus

$$\begin{aligned} TS &= CS + PS \\ &= \tfrac{1}{2}(a - p^*)q^* + \tfrac{1}{2}(p^* - c)q^* \\ &= \tfrac{1}{2}(a - c)\,q^*. \end{aligned}$$

Notice that $p^*$ drops out: total surplus depends only on the gap between the curves and the quantity traded. The price decides how the pie is divided; the quantity decides how large the pie is.

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Try it yourself. Open Tool 2, set $a = 80$, $b = 0.8$, $c = 20$, $d = 1.2$, and verify: $q^* = 30$, $p^* = 56$, $CS = 360$, $PS = 540$, $TS = 900$.