Question

I ONLY NEED HELP WITH "C". THE OTHER PARTS ARE PUT IN HERE IN CASE THE BACKGROUND INFO THERE IS NEEDED

I know that the answer for "C" is given here. What I need help with is understanding how they got the answer.

1. In the 'slutsky' equation part please explain the meaning and significance of the "h" in the hR/pR part as well as the effect it has on the equation.

2. Please explain the 'slutsky' equation to me and info on how it's broken up in an equation (both in general and specifically relating to this problem).

3. What is the substitution affect doing in this problem?

4. Please explain how the circled equation equalling "total effect" was found. Also, please explain what it means in the context of this equation and any impacts it has on the equation.

5. Why is it that when you take the derivative of root beer in respect to income the answer is 0? What does this have to do with the "no income effect!" note written close by?

6. What is the significance and meaning of "no income effect!" in the context of this equation? How does it relate to the rest of the equation?

7. How does all this info help us find the change in Claude's root beer consumption caused by the substitution effect?

8. How does all this info help us find the change in Claude's root beer consumption due to the income effect?

Also, if you're going to write please be sure to write in a way that is very legible/readable. I don't mean to be rude, but I have had people's answers in the past be completely useless because I could only half-read what they wrote down as a result of handwriting that was very difficult to read. If you're going to write your answer please be sure it's easy to read.

8. (20 points) Suppose Claude has preferences represented by the utility function: U = ln(R) + S that generate Marshallian demands for Sno-Cones (S) and Root Beer (R) given by -"-% , where lìs Claude's income and Phare prices. A. (5 points) Find Claude's indirect utility and expenditure functions. Ps B. (5 points) Find Claude's hicksian demand for Root Beer. What result from class justifies your approach?-rr. C. (6 points) Suppose the price of Root Beer increases just a little. Using the slutsky equation, find the change in Claude's root beer consumption caused by the substitution effect and change due to the income effect. dk 0

Answer 1

1.The h part here is Hicksian demand which captures the substitution effect only ( it's significance). This part explains when price of Root beer increase by 1 unit, how much does quantity demanded falls as now Sno-Cones have become relatively cheaper. The reason why 'h' is used is to show the affect of change in demand due to own price only by taking real wealth and utility constant.

2. The demand function tells us how a person when maximising his utility while making a choice respond to changes in price of the good and changes in his purchasing power.So we have to distinguish the portion of change in demand which causes movement along the indifference curve and the portion that involves moving to a different indifference curve.To understand these portions separately we use Slutsky's equation.The Slutsky's equation is decomposed to understand the substitution effects and income effects.

Essentially it means you are bring together the partial derivatives to get the total derivatives.Now, this gives the Slutsky's equation -

$\delta X/\delta P = \partial X/\partial P _{X|U=U*} - \partial I/\partial P_{X}\times \partial X /\partial I$   

The left side shows the total effect in demand due to price change.

The right side is broken in two portions as -

• The first portion shows the substitution effect i.e. when there is a unit change in price of root beer (C) how much does the demand for root beer changes(R) keeping utility constant(U = U*).
• The second portion shows the income effect ; Change in purchasing power due to price change multiplied with change in demand due to price change.

3. Claude has preferences for two goods Root beer and Sno-cone whose total utility is given in the problem.Therefore in order to maximise his utility he will consume in a way where marginal rate of substitution (MRS) is equal across goods.If MRS is > or < a constant number then he is not in equilibrium position..(Law of equi-marginal utility)Therefore we try to understand the substitution effect.

However if Claude was consuming only 1 good and spending his entire income on that good only we would not have figured out substitution effect.

4.We rearrange the R equation to get

P_r = P _s x 1 / R   

Differentiating both sides we get , Here there is change in price of root beer only.

$\Delta P_{r} = P_{s\times } -1/R^{2}\Delta R$

Again rearranging we get ,

$\Delta R/ \Delta P_{r} =-P_{s\times } / P_{r}^{2}$

or  $\partial R/ \partial P_{r} =-P_{s\times } / P_{r}^{2}$

This is total effect which is circled.

5. As per the problem we were making use of Hicksian demand which holds real wealth as constant.So derivative of a constant is 0.Further the price change does not cause a real change in income of Claude so demand does not change with respect to income.Hence there is no income effect.

6.When there is no income effect it means the price change is entirely explained by substitution effect.Here Hicksian substitution effect is sufficient to explain the effects as compared to Slutsky's equation.

7.The change in root beer's consumption due to substitution effect is explained by the first portion of Slutsky's equation.This is found by differentiating the R function only.

8.The change in root beer's consumption due to income effect is explained by the second portion of Slutsky's equation.as income effect is 0 therefore no change due to demand.