Question

Given market demand Qd = 40 -3 P, and market supply P =1/2 Qs + 10
a. Find the market equilibrium price and quantity?
b. What would be the state of the market if market price was fixed at Birr 10 per unit?
c. Calculate and interpret price elasticity of demand at the equilibrium point.

Answer 1

$a)$ Step $1$ : Rewrite the supply curve as function of price

$P=$  $\frac{1}{2}$ $Q_{s}+10$

$\frac{1}{2}$ $Q_{s}$ $=P-10$ .........$($ divide both sides by $\frac{1}{2}$ $)$

$Q_{s}$$=2P-20$ ......... $($ supply function).....$(1)$

Step $2$ : Leave the demand function untouched

$Q_{d}$$=40-3P$ .........$($ demand function$)$ .....$(2)$

Step $3$ : At equilibrium;

Quantity demanded= Quantity supplied i.e $($ $Q_{d} =$ $Q_{s}$$)$

$40-3P=2P-20$ $($ substitute for $Q_{d}$ and $Q_{s}$ $)$

$-3P-2P=-20-40$ $($ group like terms together$)$

$-5P=-60$

$P=12$

Step $4$ : Substitute P in either of the functions

$Q_{s}$ $=2P-20$  $\implies$ $2$ $\times$$12-20=4$

$Q_{d}$ $=40-3P$  $\implies$ $40-3$$\times$$12 =4$

Therefore,

$Q_{d}$$=$  $Q_{s}$ $=$ $4$

Step 5: Answer

Equilibrium price $(P_{e})$  $=12$

Equilibrium Quantity $(Q_{e})$ $=4$

$b)$ When the price is set at $10$ Birr per unit; then $P=10$

Substitute P in both supply and demand functions

 $Q_{s}=2P-20$ $\implies$  $2$ $\times10-20=0$

$Q_{d} =40-3P\implies$$40-3\times10=$ $10$

$\therefore$ When price is set at $P=10$ ; then quantity supplied drops from $4$ units to (zero) units whereas quantity demanded increased from $4$ units to $10$ units.

$c)$ The price elasticity of demand at equilibrium point $($ $e)$ is the point elasticity at that point

$E_{D}=-\frac{\Delta Q}{\Delta P}\times\frac{P}{Q}$

Substitute for values at the equilibrium point where $P=12$ and $Q_{s}=4$ to get the

Slope $=-\frac{\Delta Q}{\Delta P}=-\frac{4}{12}=-\frac{1}{3}$

Get the reciprocal of the slope and multiply it with equilibrium price and quantity to obtain $E_{D}.$

Reciprocal $(-\frac{1}{3})=-3$

$E_{D}=-3\times\frac{12}{4}= -9$

The price of elasticity of demand at equilibrium is less than $1$ , meaning the quantity demanded is inelastic. Any change in price disproportionately affects the quantity demanded.