# Calculating Consumption and Measuring Relative Risk Aversion

Please answer and show work.

1. Calculate the change in consumption of goods x and y for a person with utility U(x,y)= 8(x)^0.6(y)^0.3 and income of $270 facing px = $2 and py = $3 when a tax of $1 per unit is added to the

price of x. Decompose the change in consumption of x into Hicks income and substitution

effects. Finally, calculate and interpret the compensating and equivalent variations for this tax.

2. For U(x,y) = 0.5(x)^0.4(y)^0.6, find the Marshallian demands, the indirect utility function, the

expenditure function, and the compensated demands. You must derive the functions, do NOT

plug into a formula. Calculate the change in consumer surplus from good x that results from a

change in the price of x from $10 to $5 when income is $500 and the price of y is $30.

3. A person with $10 million in wealth and utility of wealth given by U(w) = 100(w)^0.4 is

considering buying fire insurance. She faces a 2% chance of suffering a $500,000 loss due to fire.

a. Show (mathematically) that her expected utility is higher when she purchases an

actuarially fair insurance policy than when she faces the risk of fire with no insurance.

What is the most she should consider paying?

b. After various events, this person has accumulated new wealth of W. Show what

percentage, f, of her new wealth she would be willing to pay to avoid a fair gamble

involving 50% of her wealth. What does the fact that f does not depend on W imply

about her relative risk aversion? Verify your answer by calculating the measure of

relative risk aversion.

#### Solution Preview

Answer:

Given that,

U(x,y)=8*x^0.6*y^0.3

I=$270

P_x=$2

P_y=$3

When a tax of $1 per unit is added to the price of x:

P_x=3

Therefore,

〖MU〗_x=8*0.6*x^(-0.4)*y^0.3

And,

〖MU〗_y=8*0.3*x^0.6*y^(-0.7)

Also,

P_x*x+P_y*y=M

Or,

x=(M-P_y*y)/P_x

As we know that, condition for utility maximization is:

〖MU〗_x/P_x =〖MU〗_y/P_y

Or,

(8*0.6*x^(-0.4)*y^0.3)/P_x =(8*0.3*x^0.6*y^(-0.7))/P_y

Or,

P_x/P_y =2*y/x

Putting the value of x, we have:

P_x/P_y =2*y*P_x/(M-P_y*y)

Or,

M-P_y*y=2*y*P_y

Or,

y=M/(3*P_y )=$270/(3*3)=30

Putting the value of y in x, we get

x=(M-P_y*M/(3*P_y ))/P_x

Or,

x=(2*M)/(3〖*P〗_x )=(2*$270)/(3*2)=90

If the price of Px changes to 3, we have

∆x=(2*$270)/(3*3)-(2*$270)/(3*2)=-30

Hick's Income and Substitution Effect with change in consumption of x:

Substituting demand ...

#### Solution Summary

The expert calculates the change in consumption of foods. The consumption and measuring relative risk aversions are determined.