Question

1/Laura's preferences over commodities x₁ and x₂ can be represented by U(x₁,x₂)=min{3x₁, x₂}. She maximizes her utility subject to her budget constraint. Suppose there is an increase in p1.  

		There is an income effect but not a substitution effect of this price change.
		There are both income and substitution effects of this price change.
		There is a substitution effect but not an income effect of this price change.
		It is unclear whether the consumer will buy more or less x1 as a result of the increase in p1. 2/ As price falls along a downward sloping compensated demand curve (in the x1, p1 plane), consumer utility will    rise if the income effect is LESS than the substitution effect. rise if the income effect is LESS than the substitution effect. rise. stay the same. rise if the income effect is GREATER than the substitution effect. fall if the income effect is GREATER than the substitution effect. fall. 3/ You consume goods X, Y and Z. If an increase in the price of x, (holding constant the price of Y, the price of Z, and income) decreases the quantity of good Y that you consume then goods X and Y are net complements. goods X and Y are net substitutes. goods X and Y are gross complements. goods X and Y are gross substitutes. 4/ Matt gets utility from commodities x1 and x2 and his preferences can be represented by the utility function, U(x1, x2)=10(x1)(x2). Matt minimizes his expenditures subject to achieving utility level U0. Matt's compensated demand function for x2, x2c(p1, p2, U0), is ? I/(2p1) (U0/10)1/2(p2/p1)2/3 ((U0/10)1/2(p2/p1)1/2 (U0/10)1/2(p1/p2)1/2 (U0/10)1/3(p1/p2)1/3 (10)(U0)2/3(p2/p1)2/3

Answer 1

1. The correct answer is option A) because utility function is perfect complement. Under perfect complement the substitution effect is zero.
2. The correct answer is option C) because the compensated demand curve shows only the substitution effect.
3. The correct answer is option A) because net complements takes prices only into account I.e if the increase in the price of one good decreases the quantity demanded of other goods then they are net complements. (Only substitution effect is considered not the income effect)
4. For compensated demand we will minimize the expenditure function for a given level of utility.

Expenditure = p1x1+p2x2

Setting up lagrangean

L = p1x1 +p2x2 + m[U0- 10x1x2]

m is the lagrangean multiplier

FOC:

δL/δx1 = p1-10mx1 = 0------------------------------1)

δL/δx2 = p2-10mx2 = 0 -----------------------------------2)

δL/δm = U0- 10x1x2 = 0 -----------------------------------------3)

Divide equation 1) by 2) we get

P1/p2 = x1/x2

X1 = p1/p2 *x2

Put in equation 3)

U0 = 10(p1/p2)*(x2)2

x2 = [(U0*p2)/10p1]1/2

X1 = [(U0*p1)/10p2]1/2

The correct answer is option B)