Question

Given: ( x is number of items) Demand function: d(x)=485.1−0.6x² Supply function: s(x)=0.5x²

Find the equilibrium quantity: Find the producers surplus at the equilibrium quantity:

Answer 1

The equilibrium quantity is 21. The producer's surplus at the equilibrium quantity is 2320.25.

Step-by-step explanation:

The equilibrium quantity can be found by setting the demand function equal to the supply function:

d(x) = s(x)

485.1 - 0.6x² = 0.5x²

Combining like terms:

1.1x² = 485.1

Dividing both sides by 1.1:

x² = 441

Taking the square root of both sides:

x = ±21

Since we are looking for a positive quantity, the equilibrium quantity is 21.

To find the producer's surplus at the equilibrium quantity, we need to find the price at the equilibrium quantity and then calculate the area between the supply curve and the price at the equilibrium quantity.

First, substitute the equilibrium quantity (x = 21) into the supply function:

s(21) = 0.5(21)²

s(21) = 0.5(441)

s(21) = 220.5

The price at the equilibrium quantity is 220.5.

Next, calculate the producer's surplus by finding the area between the supply curve and the price at the equilibrium quantity:

Producer's Surplus = 0.5 * (Price - Marginal Cost) * Quantity

Producer's Surplus = 0.5 * (220.5 - 0) * 21

Producer's Surplus = 0.5 * 220.5 * 21

Producer's Surplus = 2320.25