Q: Given the utility function $$U(X,Y) = X$$^{1/2}$$Y^{1/2}$ and the optimal bundle found in the previous question, what is the consumer’s maximum utility?

$$U = \sqrt{6 \times 4} = 4.90$$

Question 1

Suppose a consumer’s utility function is given by $$U(X,Y) = X$$^{1/2}$$Y^{1/2}. $$The consumer has $$120 to spend. The price of X$ is P_X = 10$, and the price of Y$ is P_Y = 15$. How much X$ and Y$ should the consumer purchase to maximize utility?

Accepted Answer

$$X = 6, Y = 4$$

Question 2

Given the utility function $$U(X,Y) = X$$^{1/2}$$Y^{1/2}$ and the optimal bundle found in the previous question, what is the consumer’s maximum utility?

Accepted Answer

$$U = \sqrt{6 	imes 4} = 4.90$$

Question 3

If the price of $$X$$ increases to $$20 ($$with $$P_Y = 15$$ and income $$= 120$$), what is the new optimal bundle of $$X$$ and $$Y$$ for the consumer with $$U(X,Y) = X$$^{1/2}$$Y^{1/2}$?

Accepted Answer

$$X = 3, Y = 4$$

Question 4

How much money would the consumer be willing to pay to avoid the price increase from $$P_X = 10 to$$ $$P_X = 20 ($$compensating variation)?

Accepted Answer

$$18.04$$

Question 5

How much additional money would the consumer need to reach the same utility after the price change as before the price change (equivalent variation)?

Accepted Answer

$$13.42$$

Question 6

Of the total change in the quantity demanded of $$X$$ due to the price increase, how much is due to the substitution effect?

Accepted Answer

$$-0.5$$

Question 7

True or False: For every point on the contract curve in an Edgeworth box, the MRS for consumer A is equal to the MRS for consumer B.

Accepted Answer

True

Question 8

Given $$U(X,Y) = \min\{X,3Y\}$$, $$X = 6$$, $$Y = 10$$, what is the marginal rate of substitution (MRS) at this bundle?

Accepted Answer

$$MRS is $$undefined

Question 9

Suppose two consumers, A and B, have utility functions $$U_A(X,Y) = X$$^{1/2}$$Y^{1/2}$$ and $$U_B(X,Y) = \min\{X,Y\}. $$The initial allocation is $$W^A = (4,6)$$ and $$W^B = (6,4)$. Which of the following is true about the contract curve?

Accepted Answer

It consists of allocations where $$X^A = Y^A$$

Question 10

Nicole consumes orange juice and champagne, which are perfect complements in a 3:1 ratio. If the price of orange juice is $$0.25$$ and champagne is $$1.50$$, and Nicole has $$12 to $$spend, how many units of orange juice will she buy?

Accepted Answer

$$9$$

Question 11

If apples and oranges are perfect substitutes for Nicole, and she is willing to trade 2 apples for 1 orange, with apples priced at $$0.50$$ and oranges at $$1.00$$, what bundle will she choose if she has $$10 to $$spend?

Accepted Answer

All apples

Question 12

In a two-consumer Edgeworth box with $$U_A(X,Y) = X$$^{1/2}$$Y^{1/2}$$ and $$U_B(X,Y) = X$$^{1/2}$$Y^{1/2}$$, which of the following is true at general equilibrium?

Accepted Answer

The marginal rate of substitution for A equals that for B

Question 13

If the price of good $$X$$ increases, what happens to the consumer’s budget line?

Accepted Answer

It rotates inward around the $$Y-$$intercept

Question 14

Which of the following best describes the Engel curve for a perfect complement?

Accepted Answer

It is a straight line through the origin

Question 15

If a consumer’s utility function is $$U(X,Y) = X + 2Y$$, and $$P_X = 1$$, $$P_Y = 2$$, and income is $$10$$, what is the optimal bundle?

Accepted Answer

$$X = 0, Y = 5$$

Question 16

Which of the following statements about the substitution effect is correct?

Accepted Answer

It always increases the quantity demanded of the good whose price falls

Question 17

If the marginal rate of substitution (MRS) between $$X$$ and $$Y is$$ $$2$$, and the price ratio $$P_X/P_Y = 1$$, what should the consumer do to maximize utility?

Accepted Answer

Buy more $$Y$$ and less $$X$$

Microeconomics Study Guide: Consumer Choice, Indifference Curves, Contract Curves, Engel Curves, and General Equilibrium

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