Help with word problem

Question

At a unit price of $16,000 the demand of a product is 300units and at a unit price of $48,000 the demand is 100units. At a unit price of $30,000 the supply is 550units and at a unit price of $50,000 the supply is 650units.

i) determine the demand equation

ii) determine the supply equation

iii) determine the equilibrium price and quantity

iv)illustrate the demand and supply equation on a graph

Answer 1

Demand:

Units Price

300 16000

100 48000

Assuming a linear function, the slope (m) is the change in price divided by the change in quantity (ΔP/ΔQ):

(48000 - 16000)/(100 - 300) = 32000/-200 = -$160/unit

We can write a linear equation using the point-slope form which, for some point (x₁, y₁) the equation is:

y - y₁ = m(x - x₁) in this case, Q (quantity) is the "x" and P (Price) is the "y" so it would be better to write the point-slope form as:

P - P₁ = m(Q - Q₁) for a point (Q₁, P₁)

Let's let out point (Q₁, P₁) be (300, 16000) so the equation would be:

P - 16000 = -160(Q - 300)

P - 16000 = -160Q + 48000

P = -160Q + 64000 ← This is a version of the Demand Function

Since this is Demand, lets call it Q_d and, despite the fact that in math we usually write a function in terms of the dependent variable, with supply/demand they usually flip it around:

P - 64000 = -160Q_d

Q_d = -(1/160)P + 400

Demand:

Units Supply

550 30000

650 50000

Slope (m) = ΔP/ΔQ = (50000 - 30000)/(650 - 550) = 20000/100 = $200/unit

P - P₁ = m(Q - Q₁) for a point (Q₁, P₁)

Let's let out point (Q₁, P₁) be (550, 30000) so the equation would be:

P - 30000 = 200(Q - 550)

P - 30000 = 200Q - 110000

P = 200Q - 80000

Since this is Supply, lets call it Q_s and, again despite the fact that in math we usually write a function in terms of the dependent variable, with supply/demand they usually flip it around:

P + 80000 = 200Q_s

Q_s = (1/200)P + 400

The equilibrium price is when the supply quantity equals the demand quantity

So, since Q_s = Q_d then:

-(1/160)P + 400 = (1/200)P + 400

-(1/160)P = (1/200)P

0 = (1/200)P + (1/160)P

0 = (9/800)P

P = $0

The associated quantity is 400 units

Crazy, put it appears you get to give away your product if you make 400 units.

Graph: