Question

Let U=X^1/2Y², dU/dX=(1/2)X^-1/2Y², dU/dY=2X^1/2Y

Px=$15, Py=$3 and I=$300

1.(2 pts)_______________________ What is the level of happiness at X=16, Y=6?

2. (2 pts)_______________________What is the marginal utility of X at this point?

3.(2 pts)________________________ What is the slope of the indifference curve at this point?

4.(2 pts)_______________________ At this point, which is larger: the marginal utility of the last dollar spent on X or the marginal utility of the last dollar spent on Y? (You must show both marginal utilities per dollar spent)

5.(2 pts)_______________________ To increase your utility (for the same dollars spent), should you trade in some X to get more Y or trade in some Y to get more X?

6.(2 pts)_____________________ What would it cost to buy this point? (X=16, Y=6)

7.(2 pts) ________________If you bought X=8, what is the most Y you could afford to buy with an income of $300?

8.(2 pts)___________________ What is Y if X=64 on the same indifference curve (as #1)?

9.(2 pts)_____________________ What is the value of Y on the income consumption curve when X=6?

10.(5 pts)______ __________________________________What are the values of X and Y which give you the maximum satisfaction given the above income constraint? How much happiness do you have?

For the rest of this problem, suppose Px has decreased to $6.

11.(2 pts) _________________________What is the value of Y on the new income consumption curve when X=6?

12. (5 pts) _______________________________________ Find the new values of X, Y and U which maximize happiness.

13.(4 pts)________________________________________What is the equation for the PCC for X (X as a function of Y or Y as a function of X) for Py=$3 and I=$300?

14.(4 pts)______________________________________ What is the equation for the demand curve (X as a function of Px alone) for Py=$3 and I=$300.

15. (4 pts)______________________________________Calculate the intermediate point - both X & Y. (The amount of X and Y you would consume to have the same happiness of #10 when you are paying $6 to buy X).

16. (2 pts)______________________________________What is the equivalent increase in income caused by the decrease in the price of X?

17.(6 pts) Draw a rough diagram clearly showing the TWO budget constraints, TWO indifference curves and TWO ICC curves of your answers to #10, #12 & #15. Carefully show (in your diagram) the income effect and the substitution effect of the decrease in the price of X.                                                                                                                            

Answer 1

1.

U=X^1/2Y²

When X=16, Y=6:

U= (16)^1/2 (6)² = 4 x 36= 144 (level of happiness)

2.

Marginal utility of X at this point= dU/dX=(1/2)X^-1/2Y² = (1/2)(16)^-1/2 (6)² = (1/2)(1/4)(36)= 4.5

3.

Slope of indifference curve= [(dU/dX)/(dU/dY)]= [(1/2)X^-1/2Y²] / [2X^1/2Y]= (1/4)(Y/X)= (1/4)(6/16)= 0.09375

4.

The marginal utility of the last dollar spent on X= Marginal utility of X/Px= 4.5/15= 0.3

The marginal utility of the last dollar spent on Y= Marginal utility of Y/Py= 48/3= 16

The marginal utility of the last dollar spent on Y is larger than the marginal utility of the last dollar spent on X.

5.

As marginal utility of the last dollar spent on Y is larger than X it means you should trade in some X to get more Y