linear equation

Question

1. It is estimated that if a price of $15 per unit is charged for a product, five units per period will be sold. If the price is lowered to $13, sales will be six units per period.

The supply curve for the product is given by P = 3Q + 5

                                                                                   = 3+0

                                                                                   = 3#

a) Assuming that the demand relationship is linear, find the demand equation.

b) Find the equilibrium price and quantity (i.e. the price where quantity demanded = quantity supplied).

Answer 1

(a) Assuming that the demand relationship is linear, the demand D(p) as a function of the price p is D(p) = ap+b, where a and b are constants. Substitute the given points (15,5) and (13,6) into this function to yield two linear equations in terms of a and b:

5 = 15a+b

6 = 13a+b

Solving these two equations via elimination or substitution yields a = -1/2 and b = 25/2, so that the demand equation is D(p) = -1/2p + 25/2

(b) Let D(p) = Q(p), where D(p) = -1/2p + 25/2 and Q(p) = 1/3p-5/3. Then

-1/2p+25/2 = 1/3p-5/3

=> p = 17

Hence, the equilibrium price is $17 per unit, and the quantity produced is D(17) = 4 units.