Suppose that I = 100, px = 1 and py = 2. How much of good x and good y will Amy optimally choose? 5. Now the price of good x rises to px = 2, while income (I = 100) and the price of good y, py = 1, remain unchanged. What quantities does Amy buy and what is her resulting utility? Illustrate graphically 6. Find the income and substitution effect for good x due to this price change. That is, which change in the consumption of x is due to pure substitution and which change in the consumption of x is due to the income effect? Illustrate your answer graphically.
PART 4 TO 6
Amy chooses between two goods, x and y, with prices px and py, respectively. She has an income I and her preferences are represented by the utility function U(x, y) = √x + y.
1. Assuming that an interior solution exists to the constrained utility maximization problem, derive Amy ’s Marshallian demand function for each of the two goods. Are both goods
normal? Explain
2. Find the indirect utility function, V (px, py, I).
3. Derive Amy’s Hicksian demand function for each of the two goods and the expenditure function. Compare the Marshallian demand for good x and the Hicksian demand for good x. Are
these different functions? If so, why? If not, why not?
4. Suppose that I = 100, px = 1 and py = 2. How much of good x and good y will Amy optimally choose?
5. Now the price of good x rises to px = 2, while income (I = 100) and the price of good y, py = 1, remain unchanged. What quantities does Amy buy and what is her resulting utility?
Illustrate graphically
6. Find the income and substitution effect for good x due to this price change. That is, which change in the consumption of x is due to pure substitution and which change in the consumption of x is due to the income effect? Illustrate your answer graphically.
7. If the government wants to compensate Amy for the price increase of good x from one to two dollars, by giving her some extra income, how much extra income would be needed to
bring her back to the old utility level? In other words, find the compensating variation for the price change. Show all your calculations and illustrate.
4) utility function is given by
U (X, Y) = X^0.5 + Y
I = 100, Px = 1, Py = 2
So budget line would be
Px X + Py Y = 100
Or, X + 2Y = 100
Setting the utility maximization problem
L = X^0.5 + Y –λ (X + 2Y – 100)
dL/dX = 0.5X^-0.5 –λ
DL/dY = 1 -2λ
dL/dλ = X + 2Y – 100
putting the all first order differentials equals to the 0
λ = 0.5
½ X^-0.5 = 0.5
X^-0.5 = 1
X*= 1
And 2Y = 99
Y* = 49.5
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