Question 1

If someone has a utility function U = 2 min{x, y}, then x and y are perfect complements for that person.

Accepted Answer

In a utility function of the form U = 2 min{x, y}, the person derives satisfaction from the smaller of the two goods. This indicates that the goods x and y are perfect complements, as they must be used together in fixed proportions to maximize utility.

Question 2

A person with the utility function U(x₁, x₂) = min{x₁ + 2x₂, 2x₁ + x₂} has convex but not strictly convex preferences.

Accepted Answer

False

Question 3

If one utility function is a monotonic transformation of another, then the former must assign a higher utility number to every bundle than the latter.

Accepted Answer

A monotonic transformation of a utility function preserves the order of preferences but does not necessarily assign a higher utility number to every bundle. It means if one bundle is preferred over another in the original utility function, it will also be preferred in the transformed utility function, but the actual utility numbers can be higher, lower, or the same, as long as the order of preference is maintained.

Question 4

Ike's utility function is U(x, y) = 25xy. He has 12 units of good x and 8 units of good y. Ben's utility function for the same two goods is U(x, y) = 4x + 4y. Ben has 9 units of x and 13 units of y.
A) Ike prefers Ben's bundle to his own bundle, but Ben prefers his own bundle to Ike's.
B) Ben prefers Ike's bundle to his own, but Ike prefers his own bundle to Ben's.
C) Each prefers the other's bundle to his own.
D) Neither prefers the other's bundle to his own.
E) Since they have different preferences, there is not enough information to determine who envies whom.

Accepted Answer

To compare the two bundles, we need to compute the utility values for each of them: 
Ike's bundle: U(12,8) = 25(12)(8) = 2400 
Ben's bundle: U(9,13) = 4(9) + 4(13) = 92

Ike's utility from Ben's bundle: U(9,13) = 25(9)(13) = 2925 
Ben's utility from Ike's bundle: U(12,8) = 4(12) + 4(8) = 80

Therefore, Ike prefers Ben's bundle (2925 > 2400) but Ben prefers his own bundle (92 > 80). So the correct answer is A.

Question 5

Fiery Demon is a rotgut whisky made in Kentucky. Smoothy is an unblended malt whisky imported from Scotland. Ed regards these brands as perfect substitutes. When he goes into a bar, he sometimes buys only Fiery Demon. Other times he buys only Smoothy. This shows that Ed has unstable preferences.

Accepted Answer

Ed's preference for Fiery Demon or Smoothy is dependent on his mood or situation. This does not necessarily mean that he has unstable preferences. It could simply mean that he has a preference for variety or a willingness to try different things.

Question 6

Mr. Surly consumes only two goods and hates them both. His utility function is U(x, y) = 2max{x, y}. Mr. Surly has (weakly) convex preferences.

Accepted Answer

Since Mr. Surly has (weakly) convex preferences, it means that his indifference curves are (weakly) convex. This is consistent with his utility function, as the max function generates a kinked indifference curve, which is (weakly) convex.

Question 7

A person with the utility function U(x, y) = 10 + y² + x has convex preferences.

Accepted Answer

The utility function is quasi-linear in y and linear in x. Quasi-linear utility functions have convex preferences, so this person has convex preferences.

Question 8

A person with the utility function U(x, y) = y + x²has convex preferences.

Accepted Answer

A utility function is convex if its second derivative is non-negative. The second derivative of U(x, y) with respect to x is 2, which is positive, indicating convexity in x, but the function is linear in y, which does not satisfy the convexity condition for y. Therefore, the preferences are not convex overall.

Question 9

Alice's utility function is U(x, y) = x²y. Steve's utility function is U(x, y) = x²y + 2x. Alice and Steve have the same preferences since Steve's utility function is a monotonic transformation of Alice's.

Accepted Answer

Steve's utility function is not a monotonic transformation of Alice's because it includes an additional term, 2x, which changes the preferences. Monotonic transformations involve multiplying by a positive constant or adding a constant, not adding a term dependent on the variables.

Question 10

Angela's utility function is U(x₁, x₂) = (x₁ + x₂)³. Her indifference curves are downward-sloping, parallel straight lines.

Accepted Answer

The fact that Angela's indifference curves are parallel straight lines indicates that her marginal rate of substitution is constant, which is a characteristic of perfect substitutes. Therefore, as she increases the quantity of one good, she must decrease the quantity of the other good by a fixed proportion to remain indifferent.

Question 11

Quasilinear preferences are homothetic when the optimal amount of good 1 is not affordable.

Accepted Answer

Quasilinear preferences are not homothetic when the optimal amount of good 1 is not affordable. Homothetic preferences are only valid when the consumer can afford to buy the optimal bundle of goods.

Question 12

A person with utility function U(x, y) = 5 + y² + 2x has nonconvex preferences.

Accepted Answer

The existence of the term y² in the utility function indicates that the preferences are not convex. This term introduces a kink or nonlinearity in the utility curve, which violates the condition of convexity.

Question 13

Jeans utility function is U(x, y) = x + y 2 - y. If we draw her indifference curves with x on the horizontal axis and y on the vertical axis, then these indifference curves are everywhere downward sloping and get flatter as one moves from left to right.

Accepted Answer

The utility function U(x, y) = x + y^2 - y is not linear in y, which means the indifference curves are not necessarily downward sloping everywhere or getting flatter as one moves from left to right. The term y^2 suggests that the curves could have a more complex shape.

Question 14

A consumer has preferences represented by the utility function U(x₁, x₂) = 10(x²₁+ 2x₁x₂ + x²₂) - 50. For this consumer, goods 1 and 2 are perfect substitutes.

Accepted Answer

The utility function only depends on the quantity of goods 1 and goods 2 consumed, not on their specific characteristics. Therefore, goods 1 and 2 are perfect substitutes in this case.

Question 15

The utility function U(x₁, x₂) = 2lnx₁ + 3lnx₂ represents Cobb-Douglas preferences.

Accepted Answer

The utility function given represents Cobb-Douglas preferences as it is a logarithmic function of two goods with positive coefficients, which is the characteristic of the Cobb-Douglas utility function.

Question 16

With quasilinear preferences, the slope of indifference curves is constant along all rays through the origin.

Accepted Answer

With quasilinear preferences, the slope of the indifference curves (marginal rate of substitution) changes as one moves along a ray from the origin, because the marginal utility of one good remains constant while the marginal utility of the other good changes.

Question 17

Wanda Lott has the utility function U(x, y) = max{x, y}. Wanda's preferences are convex.

Accepted Answer

Wanda's utility function represents a preference for having either x or y, whichever is higher. This is not a convex preference as there is no trade-off or diminishing marginal utility between x and y. Convex preferences occur when there is a diminishing marginal utility for either x or y, making the individual more willing to give up a higher quantity of one good in exchange for a smaller quantity of the other.

Question 18

Henrietta's utility function is U(x₁, x₂) = x₁x₂. She has diminishing marginal rate of substitution between goods 1 and 2.

Accepted Answer

Henrietta's utility function is a Cobb-Douglas function, which implies that the marginal rate of substitution (MRS) is diminishing as she consumes more of one good relative to the other.

Question 19

Mark strictly prefers consumption bundle A to consumption bundle B and weakly prefers bundle B to bundle A. These preferences can be represented by a utility function.

Accepted Answer

This is a contradiction as the statement says that Mark prefers A to B and also prefers B to A, which violates the transitivity of preferences. Therefore, these preferences cannot be represented by a utility function.

Question 20

Maximilian consumes two goods, x and y. His utility function is U(x, y) = max{x, y}. Therefore x and y are perfect substitutes for Max.

Accepted Answer

In the utility function U(x, y) = max{x, y}, x and y are perfect complements, not substitutes, because the utility is determined by the larger of the two quantities, indicating that Max derives utility from having a balance between x and y rather than substituting one for the other.

Quiz 3: Utility

If someone has a utility function U = 2 min{x, y}, then x and y are perfect complements for that person.

A person with the utility function U(x1, x2) = min{x1 + 2x2, 2x1 + x2} has convex but not strictly convex preferences.

If one utility function is a monotonic transformation of another, then the former must assign a higher utility number to every bundle than the latter.

Ike's utility function is U(x, y) = 25xy. He has 12 units of good x and 8 units of good y. Ben's utility function for the same two goods is U(x, y) = 4x + 4y. Ben has 9 units of x and 13 units of y.

Fiery Demon is a rotgut whisky made in Kentucky. Smoothy is an unblended malt whisky imported from Scotland. Ed regards these brands as perfect substitutes. When he goes into a bar, he sometimes buys only Fiery Demon. Other times he buys only Smoothy. This shows that Ed has unstable preferences.

Mr. Surly consumes only two goods and hates them both. His utility function is U(x, y) = 2max{x, y}. Mr. Surly has (weakly) convex preferences.

A person with the utility function U(x, y) = 10 + y2 + x has convex preferences.

A person with the utility function U(x, y) = y + x2 has convex preferences.

Alice's utility function is U(x, y) = x2y. Steve's utility function is U(x, y) = x2y + 2x. Alice and Steve have the same preferences since Steve's utility function is a monotonic transformation of Alice's.

Angela's utility function is U(x1, x2) = (x1 + x2)3. Her indifference curves are downward-sloping, parallel straight lines.