Question

Mark consumes apples and other products. His utility function for apples x and money to spend on other goods y is given by
u(x, y) = 100x - x² + y

(a) Let p denote the price for apples and y be a numeraire whose price is set equal to 1. Then, what is his demand curve for cereals?
(b) If the price of apples is $20, how many apples will he consume? What if the price of apples is $30?
(c) Suppose that Mark has $2000 in total to spend and the price of apples has increased from $20 to $30. How much does his income have to increase to make him not complain about the price increase? (In other words, what is the compensating variation?)
(d) What is the change in consumer's surplus when the price changes from $20 to $30?
Compare this number to the answer to (c).

Answer 1

•U(X,Y) = 100X- X^2+ Y

Budget line: M= Px.X+ Py.Y

•Budget line: M= Px.X+ 1.Y

Answer a)• At optimization : MUX/Px=MUY/Py

MUX=∆U/∆X = 100-2X

MUY=∆U/∆Y= 1

Py= 1

100-2X/Px= 1

100-2X= Px

X= (100-Px)/2

Demand curve for cereals : X=(100-Px)/2

Answer b)• Px= 20

X= (100-Px)/2

X= 80/2

X= 40

He will consume 40 apple when P= 20

•Px=30

X=(100-Px)/2

X= (100-30)/2

X= 70/2

X= 35

He will Consume 35 apples when price is 30.

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Answer c) Budget line: M=Px.X+ Y

Budget line: 2000= Px.X+ Y

Old Px= 20

Old X= (100-Px)/2

Old X= 40

so, 2000= 20*40+Y

Old Y= 1200

•New Px= 30

We have to keep value of X and Y as earlier i.e 40 and 1200

To make him Consume old X i.e 40 let the new income be 2000+z

2000+z = Px.X+ Y

2000+z= 30*40 + 1200

z= 2400-2000 = 400

so, income has to increase by 400.

Answer d) Demand function : X= (100-Px)/2

X intercept ( keeping Px=0)= 50

Y intercept ( keeping X=0) =100

• At price of 20 , Consumer surplus= Area of triangle ADE

Area of triangle ADE= (1/2)*base*height

Base= 40

Height= 100-20 = 80

Old Consumer surplus= (1/2)*40*80= 1600

• At price of 30, new Consumer surplus= Area of triangle ABC

Area of triangle ABC= (1/2)*base*height

Base= 35

Height= 100-30 = 70

New Consumer surplus=(1/2)*35*70 = 1225

Change in consumer surplus= 1600-1225 = 375

Consumer surplus has decreased by 375