Question

Suppose we have linear supply and demand functions for a good. At the price of 100 euros the requested quantity is 25 units of the good while the offered quantity is 50 units. When the price changes from 100 euros and comes to equilibrium the demand elasticity is -3 and the supply elasticity is 1.

1.Explain the signs of elasticity

2.Find the supply and demand functions

3.Find the price and quantity balance

Answer 1

1) Price elasticity of demand is the percentage change in quantity demanded in response to the percentage change in the price.

Negative price elasticity of demand(-3) means the negative relation between price and quantity demanded. When price increases, the quantity demanded decreases and vice versa

Positive price elasticity of supply(1) means the positive relation between price and quantity supplied. When price increases, the quantity supplied increases and vice versa.

2) A linear supply function have the following form

Qs = a + bP

Qs = quantity supplied

a = Price when quantity supplied is 0

P = price

The given elasticity of supply = 1

Elasticity of supply can be calculated as follows

Es =

Where is the slope of the supply curve. It is indicated as 'b' in the above supply function. P and Q are initial price and demand.

When price change from 100 to equilibrium, elasticity is 1

Initial price = 100

Initial quantity =50

Therefore, the elasticity is

1 = b * (100/50)

1= b*2

b = 1/2 = 0.5

Slope of the supply curve = 0.5

Now we have to calculate' a ' which is the price when quantity supplied is 0

We can calculate it as follows with given price 100 and given supply 50

50 = a + 0.5 *100

50 = a + 50

a = 50-50 = 0

Price when quantity supplied is zero = 0

We have calculated

a = 0

b = 0.5

Supply function can be written as follows

QS = 0 + 0.5P

QS = 0.5P

A liner demand function has the following form

QD = a - b P

Qd = quantity demaned

a = price when quantity demanded is 0

b = slope of the demand curve

P = price

The given elasticity of demand = -3

Price elasticity of demand can be calculated as follows

Ed =

Where is the slope of the demand curve (b in the demand function)

P = initial price

Q = initial demand

When price change from 100 to equilibrium, elasticity is -3

Therefore

-3 = b * (100/25)

-3 = b*4

b = -3/4 = -0.75

Therefore slope of the demand curve = -0.75

Now we have to calculate the price when quantity demanded is 0(a in the equation)

We can calculate it as follows with price 100 and demand 25

25 = a - 0.75*100

25 = a - 75

a = 25+75 = 100

a = 100

We have calculated,

a = 100

b = -0.75

The demand function can be formed as follows

Qd = 100 - 0.75P

Demand function :

QD = 100 - 0.75P

SUPPLY FUNCTION

QS = 0.5P

3) Market is in equilibrium(balance) when quantity demanded and quantity supplied become equal

Qd = 100 - 0.75P

QS = 0.5P

QD = QS

100 - 0.75P= 0.5P

100 = 0.5P + 0.75P = 1.25P

P = 100/1.25 = 80

Substitute P= 80 in either demand or supply function.

0.5 * 80 = 40

Equilibrium price = 80

Equilibrium quantity = 40