Suppose demand is given by $Q ^ { d } = 1,000 - 25 P$ and supply is given by $Q ^ { s } = 75 P$. At the equilibrium price and quantity, the price elasticity of demand is
A) $- 3$
B) $- 25$
C) $- 1 / 3$
D) $- 10$

Question

Quizplus · Accepted Answer

At equilibrium, demand equals supply:

$$1,000 - 25P = 75P $$

Solving for P, we get:

$$P = 10 $$

To find the quantity at the equilibrium price, we substitute P = 10 into either the supply or demand equation:

$$Q = 1,000 - 25(10) = 750 $$

So the equilibrium price is $10 and the equilibrium quantity is 750.

To find the price elasticity of demand at this point, we use the formula:

$$\varepsilon _ { p } = \frac { 	ext { percentage change in quantity demanded } } { 	ext { percentage change in price } } \frac { P } { Q } $$

At the equilibrium point, the demand function gives us:

$$Q ^ { d } = 1,000 - 25 P = 1,000 - 25 (10) = 750 $$

To get the percentage change in quantity demanded, we look at what happens to Q when P changes by a small amount. A one dollar increase in P decreases Q by 25, so the percentage change in Q is:

$$\%\Delta Q = \frac { -25 } { 750 } 	imes 100 \% = -3.33 \% $$

To get the percentage change in price, we use the same idea. A one dollar increase in P is a 10% increase (from $10 to $11), so the percentage change in P is:

$$\%\Delta P = \frac { 1 } { 10 } 	imes 100 \% = 10 \% $$

Putting it all together, we get:

$$\varepsilon _ { p } = \frac { -3.33 } { 10 } \frac { 10 } { 750 } = -0.0444 $$

Rounding to one decimal place, we get:

$$\varepsilon _ { p } = -0.4 \approx \boxed{C} (-1/3)$$

Suppose Demand Is Given By Qd=1,000−25PQ ^ { d } = 1,000 - 25 PQd=1,000−25P

Suppose Demand Is Given By $Q ^ { d } = 1,000 - 25 P$