Ross derives utility from only two goods, chocolates (x) and donuts (y). His uti

Question

Ross derives utility from only two goods, chocolates (x) and donuts (y). His utility function is as follows: U(x,y) = 2xy. His marginal utility from chocolates (x) and donuts (y) are given as follows: MUx = 2y and MUy = 2x. Ross has an income of $100 and the price of chocolates (Px) and donuts (Py) are both $0.50. Suppose price of donuts (Py) increase to $1.00, price of chocolates and income remain unchanged. How much is the total effect of this price change on Ross's consumption of donuts? How much of this total effect is due to income effect and how much is due to substitution effect?

Explanation / Answer

U = 2xy

Before price rise, Budget line: 100 = 0.5x + 0.5y, or

200 = x + y

Consumption is optimal when MUx / MUy = Px / Py = 0.5 / 0.5 = 1

2y / 2x = 1

y = x

Substituting in budget line,

200 = x + y = x + x = 2x

x = 200/2 = 100

y = x = 100

After price rise, Py = 1. New budget line: 100 = 0.5x + y, or

200 = x + 2y

Price ratio = Px / py = 0.5 / 1 = 1 / 2

MUx / Muy = 2y / 2x = 1 / 2

y / x = 1 / 2

x = 2y

Substituting in new budget line: 200 = x + 2y = x + x = 2x

x = 200/2 = 100

y = x/2 = 100/2 = 50

So, total effect (TE) = Decrease in consumption of y = 100 - 50 = 50

With previous (x, y) bundle, U = 2xy = 2 x 100 x 100 = 20,000

Keeping utility level the same & substituting x = 2y in utility function:

20,000 = 2. 2y. y

20,000 = 4. y2

y2 = 20,000/4 = 5,000

y = 70.71

x = 2y = 2 x 70.71 = 141.42

Substitution effect (SE) = 100 - 70.71 = 29.29

Income effect = TE - SE = 50 - 29.29 = 20.71

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