Naomi\'s preference can be represented by the following utility function U = x^0

Question

Naomi's preference can be represented by the following utility function U = x^0.3y^0.7. Her wealth is $500, the price of x is $15 and the price ofy is $10. How much of each good does Naomi consume? If the price of good x increases to $25, how much of each good does she consume? What is Naomi's compensating variation for this price change? What is Naomi's equivalent variation for this price change? What is one practical issue with measuring Naomi's welfare using either compensating variation or equivalent variation?

Explanation / Answer

U = x0.3y0.7

Budget line: 500 = 15x + 10y, or

100 = 3x + 2y

(a) Consumption is optimal when MUx / MUy = Px / Py

MUx = dU / dx = 0.3 x (y / x)0.7

MUy = dU / dy = 0.7 x (x / y)0.3

So, MUx / MUy = (3/7) x (y / x) = 15 / 10 = 3 / 2

21x = 6y

7x = 2y

Substituting in budget line,

100 = 3x + 2y = 3x + 7x = 10x

x = 10

y = 7x / 2 = 70 / 2 = 35

(b) Revised budget line: 500 = 25x + 10y

100 = 5x + 2y

Revised price ratio = 25 / 10 = 5/2

MUx / MUy = (3/7) x (y / x) = 5 / 2

35x = 6y

Substituting in budget line,

100 = 5x + 2y = 5x + (35 / 3)x = 50x / 3

50x = 300

x = 6

y = 35x / 6 = 35 x 6 / 6 = 35

(c)

Utility using old prices: U0 = (10)0.3 x (35)0.7 = 2 x 12 = 24

If we keep U constant, then for new prices,

24 = x0.3. (35x / 6)0.7 = x. (35 / 6)0.7 = 3.44x

x = 7

y = 35 x 7 / 6 = 41

Required income = 7 x $25 + 41 x $10 = $(175 + 410) = $585

Compensating variation = $(585 - 500) = $85

(d)

New utility, U1 = (6)0.3 x (35)0.7 = 1.71 x 12 = 20.6

Plugging in original (case a) values of x & y:

20.6 = (x)0.3 (7x / 2)0.7 = x. (7 / 2)0.7 = 2.4x

x = 8.6

y = 7 x 8.6 / 2 = 30.1

Income required (using old prices) = 8.6 x $15 + 30.1 x $10 = $(129 + 301) = $430

Equivalent variation = $(500 - 430) = $70

Related Questions

Navigate

• Browse (All)
• Browse N
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.