Question

Lucy consumes only apples and oranges, and her utility is given by U(X,Y)=5⋅X+8⋅Y, where X denotes apples and Y denotes oranges. Lucy has a budget of 1,560, and the price of apples is initially 6 and the price of oranges is 13. Find the optimal amounts of each good that Lucy will buy.

X= 260 Y= 0

Now suppose the price of apples changes to 10, now how much of each good will she buy?

X= 0 Y= 120

What is the substitution bundle? We mostly hav whole number for answers, but occassionally it is not possible. When necessary, please give all answers as fractions rather than decimals. X=? Y=?

Focusing on apples, we will break down this change into the substitution and income effects.

The substitution effect is ________?

the income effect is________?

Finally the compensating variation is _______? (Again please use fractions, rather than decimals, if needed.)

Answer 1

Solution:

We are given the utility function as U(X,Y) = 5*X + 8*Y.

Notice that here, the apples and oranges are used as perfect substitutes.

Using the information given, we have the budget line as Px*X + Py*Y = M, where M is the income of consumer

6*X + 13*Y = 1560

Marginal utility of apples, MUx = = 5

Marginal utility of oranges, MUy = = 8

For perfect substitutes, we know that optimal consumption is depicted as follows:

If MUx/MUy > Px/Py, consume only good X, so Y = 0 (one extreme)

If MUx/MUy = Px/Py, consume anywhere on the budget line (including the extremes)

If MUx/MUy < Px/Py, consume only good Y, so, X = 0

Here we have MUx/MUy = 5/8 = 0.625 > 0.463 = 6/13 = Px/Py, so only good X will be consumed. With Y = 0, budget line gives us 6* X+ 13*0= 1560

X = 260 apples and 0 oranges.

With change in price of apples to $10, Px/Py = 10/13 = 0.769 > 0.625 = MUx/MUy, so now only good Y will be purchased, good X = 0. Using the new budget line (with updated prices), it becomes: 10*0 + 13*Y = 1560

Y = 1560/13 = 120 oranges and 0 apples.

Finding the substitution bundle (using Hicksian formulation, and not Slutsky). Substitution bundle is the one which even with changed prices, results in same utility as before.

Initial utility, U = 5*260 + 8*0 = 1300

With new prices, we know already that only good Y will be purchased, so in order to maintain the old utility, 1300 = 5*0 + 8*Y

Y = 1300/8 = 162.5 oranges. (And again 0 apples).

So (in fractions) substitution bundle is (X, Y) = (0, 325/2)

Substitution effect = substitution bundle - old bundle

S.E = (0, 162.5) - (260, 0) = (-260, +162.5)

Income effect = final bundle - substitution bundle

I.E. = (0, 120) - (0, 162.5) = (0, - 42.5) . No income effect on good X, as perfect substitutes.

Compensating variation is the extra income to be given to consume in order to make them as better off as before. In other words, it is simply the change in income required to reach the substitution bundle. With changed prices, substitution bundle costs 10*0 + 13*162.5 = $2112.5

So compensating variation = compensated income - actual income

C.V = 2112.5 - 1560 = $552.5. Or in fraction (not decimals) = 1105/2