Find the equilibrium quantity and equilibrium price for the commodity whose supply and demand functions are given.
Supply: [tex]p=90 q[/tex]
Demand: [tex]p=-q^2+9,000[/tex]
(On this one, just give me the EQUILIBRIUM PRICE.)
Answer (1)
Set the supply and demand functions equal: 90 q = − q 2 + 9000 .
Rearrange into a quadratic equation: q 2 + 90 q − 9000 = 0 .
Solve for q using the quadratic formula, resulting in q = 60 (since quantity cannot be negative).
Substitute q = 60 into the supply function to find the equilibrium price: p = 90 × 60 = 5400 .
The equilibrium price is 5400 .
Explanation
Problem Analysis We are given the supply and demand functions for a commodity and asked to find the equilibrium price. The supply function is p = 90 q and the demand function is p = − q 2 + 9000 , where p is the price and q is the quantity. At equilibrium, the supply and demand are equal.
Equating Supply and Demand To find the equilibrium, we set the supply and demand functions equal to each other: 90 q = − q 2 + 9000
Forming a Quadratic Equation Rearrange the equation to form a quadratic equation: q 2 + 90 q − 9000 = 0
Using the Quadratic Formula We can solve this quadratic equation for q using the quadratic formula: q = 2 a − b ± b 2 − 4 a c where a = 1 , b = 90 , and c = − 9000 .
Calculating the Quantity Plugging in the values, we get: q = 2 ( 1 ) − 90 ± 9 0 2 − 4 ( 1 ) ( − 9000 ) q = 2 − 90 ± 8100 + 36000 q = 2 − 90 ± 44100 q = 2 − 90 ± 210
Possible Values for Quantity This gives us two possible values for q :
q 1 = 2 − 90 + 210 = 2 120 = 60 q 2 = 2 − 90 − 210 = 2 − 300 = − 150
Choosing the Positive Quantity Since quantity cannot be negative, we choose the positive value, q = 60 .
Calculating the Equilibrium Price Now, substitute the value of q into either the supply or demand function to find the equilibrium price p . Using the supply function: p = 90 q = 90 ( 60 ) = 5400
Final Answer Therefore, the equilibrium price is $5400.
Examples
Understanding equilibrium price and quantity is crucial in economics. For instance, if you're selling handmade sweaters, you need to find the price point where the number of sweaters you're willing to sell matches the number customers want to buy. If you price them too high, you'll have unsold sweaters; too low, and you might not cover your costs. Finding that equilibrium ensures you sell your sweaters efficiently and at a profitable price.
