Market demand function

Question

The market  for good X consists  of 2 consumers. consumer  1',s demand  for good X is:

X₁ :  15 - 3P_x + 0.5P_Y + .02I₁

I₁ and I₂ are incomes of consumer 1 and 2, respectively.  P_x and P_y are the prices of goods X and Y, respectively.

a. What is the equation  for the market  demand  function  for X? Graph the two individual demand curves and the market  demand  curve  for the case which  I₁ : $2000, I₂: $3000, and P_y:$ 10.

b. Suppose P_x rises  from $5 to $5.05. What is the market price elasticity of demand?

c. Suppose  income  is redistributed so that each consumer  has $2500. If P_x: 5 and P_y: 10, how much does the quantity of X demanded  change because  of the redistribution?

Answer 1

a) Equation for consumer 1: X₁= 15-3P_x + 0.5 P_y +0.2I₁

Equation for consumer2:  X₂= 15-3P_x + 0.5 P_y + 0.2 I₂

Market demand curve is  calculated by aggregating the individual demand curves.

So, By adding the two demand curves we get:  X*=30-6Px +Py + 0.2I1+ 0.2I2

Put the value of P_y and I₁ and I₂.

X*= 30 -6P_x + 10 + 0.2(2000) + 0.2(3000) is the market demand curve for the good X

Individual Demand curves will be:

X₁= 15-3P_x +5 + 400 or X₁= 420-3P_x
X₂= 15-3P_x+ 5 +600 or X₂= 620-3P_x
X*= 30-6P_x + 10+ 1000 or X*= 1040-6P_x

b. For market   price  elasticity we use market demand curve:

X*= 1040- 6P_x

Elasticity:
dx/dp(p/x)=
dx/dp= -6
-6(5/1010)=-0.029

P= original price-which is 5(that is price before the price change)
X= orginal  quantity: quantity demanded at original price of 5= 1040-6(5)=1010
And dx/dP=slope of market demand curve

c. Now each consumer has 2500. So, Put the values In the market demand curve:

X*= 2040-6P_x
If P_x=6
Then X* demanded will be 2004
And Earlier it would be: X*= 1040-36= 1004
So the change in quantity demanded will be: 1000