Question

U=(q₁q₂)^0,5

Y= 70

p₁=4$ p₂=9$

P₁ increases to 9$ calculate the compensation variation.

Answer 1

Compensating variation is the amount of compensation given to the consumer so that initial bundle remain just affordable or the level of utility to the consumer remain same.

Compensating variation= Change in price x intial optimal quantity of the good whose price changes

U=(q₁q₂)^0.5

U= q₁^0.5 q₂^0.5

Budget line: 4q₁ +9q₂ = 70

Marginal utility of q₁ (MUq₁ )= Differentiation of U wrt q₁ = 0.5q₁^-0.5 q₂^0.5  

Marginal utility of q₂ (MUq₂ )= Differentiation of U wrt q₂ = 0.5q₂^-0.5 q₁^0.5  

MRS= (MUq₁ )/(MUq₂ )= q₂ /q₁

Slope of budget line= p₁/p₂=4/9

Optimal condition:

MRS= Slope of budget line

q₂ /q₁ = 4/9

9q₂ = 4q₁ Equation 1

Use this equation in budget line:

4q₁ +4q₁ = 70

8q₁ = 70

q₁* = 70/8 (intial optimal quantity of the good whose price changes)

Now P₁ increases to 9$:

Compensating variation= Change in price x q₁*

Compensating variation= (9-4) (70/8)= 5 x (70/8)= 350/8= 175/4= 43.75

Income should be compensated by $43.75 so that initial bundle can still be just affordable.