Question

Caleb consumes only two goods, X and Y, and faces the following utility function: U=XY. His initial budget is $800, and the prices of X and Y are $12.5 and $2.

What is the marginal utility for X?

What is the marginal utility for Y?

**Most answers should be round numbers. Answer everything to 1 decimal place, if need be**

What are the amounts of X and Y that will maximize Caleb's utility?

X =  

Y =  

How many X and Y will Caleb choose if the price of X suddenly fell to $8?

X =  

Y =   

How many X and Y are in the substitution bundle?

X =

Y =

What are the break-downs of the price change on the amount of X purchased?

Substitution effect =

Income effect =

What is the compensating variation?   

Answer 1

Solution:

Utility function: U = X*Y

Budget constraint: Px*X + Py*Y <= M ; where Px is price of good X, Py is price of good Y, and M is the income of the consumer.

So, using the given information, budget line becomes: 12.5*X + 2*Y = 800

Marginal utility of X, MUx = $\partial U/\partial X$ = Y

Marginal utility of Y, MUy = $\partial U/\partial Y$ = X

Utility maximizing bundle is the one which satisfies the optimal condition of: MUx/MUy = Px/Py

So, Y/X = 12.5/2 or in other words, Y = 6.25*X

Substituting this in the budget line, we get 12.5*X + 2*6.25*X = 800

25*X = 800

X = 800/25 = 32 units

So, Y = 6.25*32 = 200 units

With change in price of good X:

Px'= $8

Again using the optimal condition above, we now have, Y/X = 8/2, so Y = 4*X

Substituting this in the new budget line (with changed prices): 8*X + 2*Y = 800

8*X + 2*4*X = 800

X = 800/16 = 50 units

So, Y = 4*50 = 200 units

Finding the substitution bundle:

Substitution bundle is the bundle which is attained at new prices, such that the utility remains unchanged.

Initial utility, U1 = 32*200 = 6400

At new prices, we know at optimum, Y = 4*X

Unchanged utility = 6400 = X*(4*X)

X² = 6400/4 = 1600

X = 40 units

So, Y = 4*40 = 160 units

(NOTE that this is Hicksian substitution bundle, and not Slutsky substitution bundle, since nothing is specifically asked for. In case of otherwise, please let know in the comments)

Break-down of price change:

Substitution effect = Substitution bundle - old bundle

S.E. = (40, 160) - (32, 200) = (+8, -40), so with decrease in price of good X, under substitution demand for good X has increased.

Income effect = New bundle - substitution bundle

I.E. = (50, 200) - (40, 160) = (+10, +40)

So, for good Y, substitution and income effect cancel out each other, while for good X, both effects move in same direction (indicating that good X is a normal good).

Compensating variation is the amount of extra income (in case of price increase; in case of price decrease it is reduction in income) required to reach the same utility level as the old one. In other words, it is same as extra (or reduction in) income required to reach at substitution bundle from the old bundle.

Income required at substitution bundle (at new prices, of course) = 8*40 + 2*160 = $640

So, compensating variation = compensated income - actual income

C.V. = 640 - 800 = -$160