Question

a retail chain will buy 900 cordless phones if the price is $40 each and 700 if the price is $50. a wholesaler will supply 550 phones at $30 each and 1950 at $70 each. assuming that the supply and demand functions are linear, find the market equilibrium point and explain what it means. market equilibrium occurs when the number of units is and the price is $ per unit.

Explanation:

Step1: Find the demand function

Let the demand function be $q = mp + b$, where $q$ is the quantity and $p$ is the price. We have two points: $(p_1,q_1)=(40,900)$ and $(p_2,q_2)=(50,700)$. The slope $m=\frac{q_2 - q_1}{p_2 - p_1}=\frac{700 - 900}{50 - 40}=\frac{- 200}{10}=-20$. Using the point - slope form with $(p_1,q_1)=(40,900)$, we get $q-900=-20(p - 40)$, which simplifies to $q=-20p+1700$.

Step2: Find the supply function

Let the supply function be $q = np + c$. We have two points: $(p_3,q_3)=(30,550)$ and $(p_4,q_4)=(70,1950)$. The slope $n=\frac{q_4 - q_3}{p_4 - p_3}=\frac{1950 - 550}{70 - 30}=\frac{1400}{40}=35$. Using the point - slope form with $(p_3,q_3)=(30,550)$, we get $q - 550=35(p - 30)$, which simplifies to $q=35p - 500$.

Step3: Find the equilibrium point

Set the demand equal to the supply: $-20p+1700=35p - 500$. Add $20p$ to both sides: $1700 = 55p-500$. Add 500 to both sides: $2200 = 55p$. Solve for $p$: $p = 40$. Substitute $p = 40$ into the demand function $q=-20\times40 + 1700=900$.

Answer:

Market equilibrium occurs when the number of units is $900$ and the price is $\$40$ per unit.