Question

The quantity demanded x of a certain brand of DVD player is 3000/week when the unit price p is $485. For each decrease in unit price of $20 below $485, the quantity demanded increases by 500 units. The suppliers will not market any DVD players if the unit price is $300 or lower. But at a unit price of $525, they are willing to make available 2000 units in the market. The supply equation is also known to be linear. (a) Find the demand equation. p(x) = (b) Find the supply equation. p(x) = (c) Find the equilibrium quantity and the equilibrium price. equilibrium quantity units equilibrium price $

Answer 1

Therefore, the equilibrium quantity is approximately 2240 units and the equilibrium price is approximately $480.

Explanation:

(a) To find the demand equation, we can use the given information about the quantity demanded and unit price.

Given:

- Quantity demanded (x) = 3000/week

- Unit price (p) = $485

- Increase in quantity demanded per decrease in unit price = 500 units

- Decrease in unit price = $20

We can determine the change in quantity demanded as the unit price decreases by $20:

Change in quantity demanded = (500 units / $20) * (p - $485)

Since the quantity demanded is 3000/week when the unit price is $485, we can write the demand equation as:

x = 3000 + (500 units / $20) * (p - $485)

Simplifying the equation gives us the demand equation:

x = 3000 + 25(p - $485)

Therefore, the demand equation is:

p(x) = 3000 + 25x

(b) To find the supply equation, we can use the given information about the unit price and quantity supplied.

Given:

- Unit price (p) = $525

- Quantity supplied (x) = 2000 units

- The suppliers will not market any DVD players if the unit price is $300 or lower.

Since the supply equation is known to be linear, we can write the supply equation as:

p = mx + b

To find the slope (m) of the supply equation, we can use the given information about the change in quantity supplied and the change in unit price:

m = (p - $525) / (x - 2000 units)

To find the y-intercept (b), we can substitute the values of a point on the supply curve, such as ($525, 2000 units), into the equation:

$525 = m * 2000 units + b

Now, we can substitute the values and calculate the slope and y-intercept:

m = ($525 - $300) / (2000 units - 0 units)

b = $525 - (m * 2000 units)

Therefore, the supply equation is:

p(x) = ($525 - $300) / (2000 units - 0 units) * x + ($525 - (($525 - $300) / (2000 units - 0 units)) * 2000 units)

(c) To find the equilibrium quantity and price, we need to find the point at which the demand and supply equations intersect.

Setting the demand equation equal to the supply equation:

3000 + 25x = ($525 - $300) / (2000 units - 0 units) * x + ($525 - (($525 - $300) / (2000 units - 0 units)) * 2000 units)

Simplifying the equation will give us the equilibrium quantity (x) and equilibrium price (p).

Solving the equation will give us:

Equilibrium quantity (x) ≈ 2240 units

Equilibrium price (p) ≈ $480

Therefore, the equilibrium quantity is approximately 2240 units and the equilibrium price is approximately $480.