Where Supply Meets Demand

Buyers want less of a product as its price rises, and sellers offer more. Each relationship is a line in the price–quantity plane, and the market settles where they cross: the solution of two equations in two unknowns. A tax shifts one line and moves the solution.

How it works

Systems of two linear equations

In one sentence

A system of two linear equations in two unknowns has exactly one solution when the lines have different slopes, none when they are parallel and distinct, and infinitely many when they coincide. Solve by substitution or elimination: here $a - bq = c + dq$ gives $q^* = (a - c)/(b + d)$ .

General formula

\begin{cases} {\htmlData{sym=0}{p}} = {\htmlData{sym=2}{a}} - {\htmlData{sym=3}{b}}{\htmlData{sym=1}{q}} & \text{(demand)} \\ {\htmlData{sym=0}{p}} = {\htmlData{sym=4}{c}} + {\htmlData{sym=5}{d}}{\htmlData{sym=1}{q}} & \text{(supply)} \end{cases} \implies {\htmlData{sym=6}{q^*}} = \frac{{\htmlData{sym=2}{a}} - {\htmlData{sym=4}{c}}}{{\htmlData{sym=3}{b}} + {\htmlData{sym=5}{d}}}, \quad {\htmlData{sym=7}{p^*}} = {\htmlData{sym=2}{a}} - {\htmlData{sym=3}{b}}{\htmlData{sym=6}{q^*}}

Each line lists every price–quantity pair one side of the market will accept. Only the crossing point satisfies both sides.

What the symbols mean

$p$
Price
Dollars per unit.
$q$
Quantity
Thousands of units bought and sold.
$a$
Demand intercept
The price at which nobody buys.
$b$
Demand slope
How much the price must fall to sell one more thousand units.
$c$
Supply intercept
The lowest price at which anyone sells.
$d$
Supply slope
How much more price it takes to bring out one more thousand units.
$q^*$
Equilibrium quantity
The quantity where buyers and sellers agree.
$p^*$
Equilibrium price
The price where buyers and sellers agree.

The demand line $p = a - bq$ slopes down: higher prices, fewer buyers. The supply line $p = c + dq$ slopes up: higher prices, more sellers.
The market price and quantity must satisfy both equations at once. Setting the right-hand sides equal eliminates $p$ : $a - bq = c + dq$ , so $q^* = \frac{a - c}{b + d}$ .
Substituting back gives the price, $p^* = a - bq^*$ . On the graph it's where the lines cross.
A tax of $u$ per unit raises the supply line by $u$ . The new crossing has a smaller quantity, and buyers pay only part of the tax: the price rises by $\frac{b}{b+d}u$ .

Learn the math

Applications of Linear EquationsAlgebra · Lesson 9, Solving Equations and Inequalities Linear Systems with Two VariablesAlgebra · Lesson 44, Systems of Equations