Question

Assume that demand for a commodity is represented by the equation P = 10 − .2Qd and supply by the equation P = 2 + .2Qs, where Qd and Qs are quantity demanded and quantity supplied, respectively, and P is the price. Using the equilibrium condition Qs = Qd, solve the equations to determine equilibrium price and equilibrium quantity.

Short answer

The equilibrium price will be 6.

The equilibrium quantity will be 20 units.

Step-by-step solution

Meaning of a demand function and supply function

The demand function gives a mathematical relationship between the quantity demanded of a good and the price of the good. For example: D=5-0.6P.

Here, P stands for the price, D stands for quantity demanded, 0.6 is the demand curve slope, and 5 is the intercept term.

The supply function gives a mathematical relationship between the quantity supplied of a good and the price of the good. For example: S=2+0.7P

Here, P stands for the price, S stands for quantity supplied, 0.7 is the supply curve slope, and 2 is the intercept term.

Determination of equilibrium level of quantity and price using demand and supply functions

The equilibrium quantity can be determined by equating the demand and supply functions. The first step is to convert the two equations (inverse demand and supply functions) to proper demand and supply functions. The quantity demanded and supplied are considered dependent variables, and the price is considered an independent variable.

$$\begin{gathered} P=10-0.2Q_d \\ {Q}_d=\frac{10-P}{0.2} \\ P=2+0.2Q_s \\ {Q}_s=\frac{P-2}{0.2} \end{gathered}$$

Now, equating the two, you get

$$\begin{gathered} Q_d=Q_s \\ \frac{10-P}{0.2}=\frac{P-2}{0.2} \\ 10+2=2P \\ P=6 \end{gathered}$$

Put the price value into the demand equation (or supply equation, as both are equal)

$$\begin{gathered} Q_d=\frac{10-P}{0.2} \\ {Q}_d=\frac{10-6}{0.2} \\ {Q}_d=20units \end{gathered}$$

Thus, the equilibrium price is estimated to be 6, and the quantity is 20 units.