Question

Esther consumes goods X and Y, and her utility function is

     U(X,Y)=XY+Y

For this utility function,

     MU_X=Y

     MU_Y=X+1

a. What is Esther's MRS_XY?

	Y/(X + 1)
	X/Y
	(X + 1)/Y
	X/(Y + 1)

b. Suppose her daily income is $20, the price of X is $4 per unit, and the price of Y is $1 per unit. What is her best choice?

     Instructions: Enter your answers as whole numbers.

     X =

     Y =

     What is Esther's utility when her daily income is $20, the price of X is $4 per unit, and the price of Y is $1 per unit?

     Instructions: Enter your answer as a whole number.

     

     At what price for good Y is Esther's expenditure on good Y largest?

	When Y is a free good
	When the price of Y is $1
	At all positive prices of Y
	When the price of Y is $2

c. Suppose the price of good Y rises to $4 per unit. What is her new consumption bundle?

     Instructions: Enter your answers as whole numbers.

     X =

     Y =

     What is the total uncompensated change on Y?

     Instructions: Enter the absolute value.

     $

     What is Esther's utility when her daily income is $20, the price of X is $4 per unit, and the price of Y is $4 per unit?

     Instructions: Enter your answer as a whole number.

     

     Suppose Esther is given an additional $24 such that her new income is $44, the price of X is $4, and the price of Y is $4.

     What is her new consumption bundle?

     Instructions: Enter your answers as whole numbers.

     X =

     Y =

     What is Esther's utility when her daily income is $44, the price of X is $4 per unit, and the price of Y is $4 per unit?

     Instructions: Enter your answer as a whole number.

     

     When Esther's income is $20, the price of X is $4, and the price of Y increases from $1 to $4, what is the uncompensated total change in Y?

     Instructions: Enter the absolute value.

     

     What is the substitution effect on Y when the price of Y increases from $1 to $4?

    Instructions: Enter the absolute value.

     

     What is the income effect on Y when the price of Y increases from $1 to $4?

     Instructions: Enter the absolute value.

     
     What is Esther's compensating variation for the price change?

     Instructions: Enter your answer as a whole number.

Answer 1

a) What is Esther's MRS_XY?

Answer) Since,

and we have been given values of MUx=Y and MUy=X+1 in the question. Therefore, Esther's MRSxy :

-------------------------------------------------------

b) Suppose her daily income is $20, the price of X is $4 per unit, and the price of Y is $1 per unit. What is her best choice?

Answer) Esther's best choice will be when the bundle of goods satisfies the following condition of-

The Slope of IC= Slope of the BL

Step 1) Writing the Budget Line (BL)-

BL is written as where

I= Income

= Price of X

= Price of Y

= Quantity of X

= Quantity of Y

So in Esther's case, the BL is-

20 = 4X+1Y or

20= 4X+Y ------------------ (1)

Step 2) Slope of the Indifference Curve (IC)-

The slope of IC is given by the Marginal utilities (MU) of goods X and Y

and we have been given values of MUx=Y and MUy=X+1.Therefore,

Step 3) Slope of Budget Line (BL)-

The slope of the BL is given by

Therefore, putting the values of prices in the above equation, we get

Step 4) Finding the best choice-

For finding the optimal bundle the following condition needs to be satisfied-

The Slope of IC= Slope of the BL

Therefore,

Negative signs will cancel on both the sides and by cross multiplication we get

------------------ (2)

Solving (1) and (2) by putting the value of Y from (2) in (1)-

20= 4X+Y

20= 4X+4X+4

16=8X

X=2

Putting the value of X in (2), we get

Y=4(2)+4

Y=12

Therefore Esther's best choice is

(X, Y) = (2,12)

c) What is Esther's utility when her daily income is $20, the price of X is $4 per unit, and the price of Y is $1 per unit?

Answer) Esther's utility function is given as

U(X,Y)=XY+Y

To find the utility, we need to substitute the values of X and Y found in (b) in the given utility function, we get-

U(X, Y)=XY+Y

= (2.12)+12

= 24 +12

=36

Therefore, Esther's utility is 36 units when her daily income is $20, the price of X is $4 per unit, and the price of Y is $1 per unit.

d) At what price for good Y is Esther's expenditure on good Y largest?

	When Y is a free good
	When the price of Y is $1
	At all positive prices of Y
	When the price of Y is $2

Answer)

• When Y is a free good, this means Esther does not need to pay for good Y. she can get it for free. So her expenditure is 0 on Y in this case. So this option is not correct.
• When the price of Y is $1, then Esther's best choice is X=2 units and Y=12 units, so the total expenditure is

20= 4X+Y

20= (4.2)+12

20=20 which is the exact income and expenditure on Y is $12

• For all positive prices of Y, as the price of Y will keep on increasing, Esther's expenditure on Y will keep on decreasing. So this option is not right.

If the price of Y becomes $2, the new units of X and Y will be X= 3 and Y= 8. So the total expenditure on the goods is

20= 4X+2Y

(4.3) + 2(8)= 12+16=28

But Esther's income is $20 and expenditure here is $28. Thus this case is not possible.

Therefore Esther's highest expenditure on good Y is when she opts the best choice, i.e Y=12 units at Y=$1.