What if the equilibrium quantity evaluates to a negative number?

A negative equilibrium quantity indicates that the cost of production is too high relative to consumer demand. In a physical market, this means no trade occurs, and the effective quantity is zero.

How do you handle non-linear supply and demand curves?

The same method applies. Set $Q_d = Q_s$ and solve the resulting non-linear equation (often a quadratic) for $P$. Discard any negative roots, as price must be non-negative.

What if demand is perfectly inelastic?

A perfectly inelastic demand is given by $Q_d = k$ for some constant $k$. The equilibrium quantity is exactly $k$. Substitute $Q = k$ into the supply equation and solve for $P^*$.

How to find market equilibrium from supply and demand equations

Market equilibrium occurs when the quantity supplied equals the quantity demanded. This method applies whenever you are given continuous algebraic functions for supply and demand. To find the equilibrium, set the supply and demand equations equal to each other and solve for the equilibrium price. Substitute this price back into either original equation to find the equilibrium quantity.

The setup

Identify the demand equation $Q_d(P)$ and the supply equation $Q_s(P)$ , where $P$ is the price and $Q$ is the quantity. Ensure both equations are expressed in terms of the same independent variable, typically $P$ .

The steps

Set the quantity demanded equal to the quantity supplied: $Q_d = Q_s$ . 2. Solve the resulting algebraic equation for $P$ . This value is the equilibrium price, $P^*$ . 3. Substitute $P^*$ into either the original demand equation or the original supply equation. 4. Solve for $Q$ to find the equilibrium quantity, $Q^*$ .

Checking the result

Substitute $P^*$ into both the supply and demand equations. If $Q_d(P^*) = Q_s(P^*)$ , the calculations are correct. If they differ, check your algebra in step 2.

Common errors

A frequent error is mixing up standard and inverse demand equations. If the equations are given as $P = f(Q)$ , you must set $P_d = P_s$ and solve for $Q^*$ first, then substitute back to find $P^*$ . Another error is accepting negative prices or quantities; in standard economic models, $P^* \geq 0$ and $Q^* \geq 0$ . Negative results indicate the market cannot exist under the given parameters.

Worked example

Given the demand equation $Q_d = 100 - 2P$ and the supply equation $Q_s = 20 + 3P$ , find the market equilibrium price and quantity.

Set $Q_d = Q_s$ . Therefore, $100 - 2P = 20 + 3P$ . Add $2P$ to both sides: $100 = 20 + 5P$ . Subtract 20 from both sides: $80 = 5P$ . Divide by 5: $P = 16$ . The equilibrium price is $P^* = 16$ . Substitute $P = 16$ into the demand equation: $Q_d = 100 - 2(16) = 100 - 32 = 68$ . Substitute $P = 16$ into the supply equation to verify: $Q_s = 20 + 3(16) = 20 + 48 = 68$ . The equilibrium quantity is $Q^* = 68$ .

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References: OpenStax Principles of Microeconomics Chapter 3 · Mankiw Principles of Economics Chapter 4