Question

In Problems 5- 7, you are given the utility function u(x, y), income I and tweo sets of prices: initial prices pr,py and final prices p, p,. For each problem, you are to fin: (a) the optimal choice at the initial prices (b) the optimal choice at the final prices (c) the change-optimal choice at final prices-optimal choice at initial prices (d) the income effect and the substitution effect

Answer 1

Solution:

U(X,Y) = X + 3Y

Income, I = 12, px = 1, py = 2, px' = 1, py' = 4

Budget line: px*X + py*Y = I

Note that the given utility function is that of perfect substitutes. Since, the goods are perfect substitutes, a consumer can earn on the extremes of the budget line, that is he/she might buy only one good. In case of perfect substitutes we have the following:

If MRSxy > price ratio, consume only good X and 0 good Y (one of the extremes)

If MRSxy = price ratio, consume anywhere on the budget line (including the extremes)

If MRSxy < price ratio, consume only good Y, and 0 units of good X (another extreme)

Where MRSxy is the marginal rate of substitution of good X for good Y, and

MRSxy = Marginal utility of X (MUx)/Marginal utility of Y (MUy)

And price ratio is px/py

(a) At initial prices, price ratio = px/py = 1/2

MUx = $\partial U(X,Y)/ \partial X$ = 1

MUy = $\partial U(X,Y)/ \partial Y$ = 3

So, MRSxy = MUx/MUy = 1/3

Clearly, MUx/MUy < price ratio (that is 1/3 < 1/2), so the optimal consumption would be to spend entire income on good Y and nothing on good X.

So, optimal bundle is X* = 0, and Y* = I/py = 12/2 = 6 units. Optimal bundle : (X*, Y*) = (0, 6)

(b) At final prices: price ratio = px'/py' = 1/4

Comparing it with the MRSxy = 1/3, since 1/3 > 1/4 meaning MRSxy is greater than the price ratio, the consumer will buy only good X and none of good Y.

Optimal consumption (at final prices) : Y* = 0, X* = I/px = 12/1 = 12 units

So new optimal bundle (X*, Y*) = (12, 0)

(c) The change (ot total price effect) = new optimal bundle - old optimal bundle

So, change in consumption of good X = 12 - 0 = 12 units

Change in consumption of good Y = 0 - 6 = 6 units less

(d) Substitution effect = Bundle at new prices and new (compensated) income - old income

Income required to make the old bundle affordable at new prices = 1*0 + 4*6 = $24

But at this new income and with new prices, optimal bundle that consumer would choose : X* = 24/1 = 24 units and Y* = 0 (since MRSxy > px'/py')

So substitution effect gives : change in X = 24 - 0 = 24 units

Change in Y = 0 - 6 = 6 units less

Income effect = optimal bundle at final prices - bundle at new prices and compensated income

Bundle at new prices and compensated income we already know is (X*, Y*) = (24, 0)

So, income effect gives change in X = 12 - 24 = 12 units less

Change in Y = 0 - 0 = 0 units

Finally breaking the total price effect into income effect and substitution effect.

Total effect = substitution effect + income effect

So, for good Y,

Substitution effect = -6, income effect = 0 and - 6 + 0 = -6 = total price effect (from part (c)). So, entire effect is substitution effect for good Y.

For good X,

Substitution effect = 24 units, income effect = -12 units and

24 + (-12) = 12 units = total effect (again see the total effect in part (c)).