Q: Is the bundle $(A, B) = (3, 2)$ affordable to the consumer with income $I = 12$, prices $p_A = 3$, $p_B = 2$?

No, because $3 \times 3 + 2 \times 2 = 9 + 4 = 13 > 12$, so it is not affordable.

Question 1

Which of the following binary relations is transitive? Let $A$ and $B$ be subsets of a set $X$ of all books. Consider the relation: 'A has the same number of pages as B.'

Accepted Answer

Yes, because if $A$ has the same number of pages as $B$, and $B$ as $C$, then $A$ has the same number of pages as $C$.

Question 2

Suppose a consumer holds 8 apples and 6 bananas and is willing to give up 2 bananas to receive one additional apple. Which of the following consumption bundles would the consumer prefer: (A) 9 apples and 4 bananas, (B) 8 apples and 6 bananas?

Accepted Answer

Bundle (A) is preferred because the consumer is willing to trade 2 bananas for 1 apple.

Question 3

If the bundles $(A, B) = (8, 6)$ and $(A, B) = (9, 4)$ are equally good for the consumer, what does this imply about the marginal rate of substitution (MRS) between apples and bananas?

Accepted Answer

The MRS is 2, meaning the consumer is willing to give up 2 bananas for 1 apple.

Question 4

Calculate the marginal rate of substitution (MRS) at the bundle $(A, B) = (9, 12)$ for the utility function $U(A, B) = A + 2B$. What is the MRS?

Accepted Answer

$MRS = \frac{\partial U/\partial B}{\partial U/\partial A} = 2$

Question 5

For the utility function $U(A, B) = AB$, what is the marginal rate of substitution (MRS) at the bundle $(A, B) = (9, 12)$?

Accepted Answer

$MRS = \frac{B}{A} = \frac{12}{9} = \frac{4}{3}$

Question 6

A consumer with income $I = 12$ faces prices $p_A = 3$ and $p_B = 2$ for apples and bananas, respectively. What is the equation for the consumer's budget line?

Accepted Answer

$3A + 2B = 12$

Question 7

Given the budget line $3A + 2B = 12$, what is the maximum number of apples the consumer can afford if they buy only apples?

Accepted Answer

$A = \frac{12}{3} = 4$

Question 8

Given the budget line $3A + 2B = 12$, what is the maximum number of bananas the consumer can afford if they buy only bananas?

Accepted Answer

$B = \frac{12}{2} = 6$

Question 9

Is the bundle $(A, B) = (3, 2)$ affordable to the consumer with income $I = 12$, prices $p_A = 3$, $p_B = 2$?

Accepted Answer

No, because $3 	imes 3 + 2 	imes 2 = 9 + 4 = 13 > 12$, so it is not affordable.

Question 10

Which of the following utility functions satisfies that preferences are convex?

Accepted Answer

$U(A, B) = AB$

Question 11

Suppose a consumer has income $I = 24$ and faces prices $p_A = 3$ and $p_B = 2$ for apples and bananas. If the government imposes a 2 per unit tax on bananas, what is the new price of bananas?

Accepted Answer

$p_B = 2 + 2 = 4$

Question 12

After the introduction of a 2 per unit tax on bananas, what is the new budget line for a consumer with income $I = 24$, $p_A = 3$, and $p_B = 4$?

Accepted Answer

$3A + 4B = 24$

Question 13

How many bananas can the consumer afford at most after the introduction of the tax, if they buy only bananas?

Accepted Answer

$B = \frac{24}{4} = 6$

Question 14

How has the consumer's purchasing power changed after the introduction of the tax on bananas?

Accepted Answer

Purchasing power has decreased because the consumer can now afford fewer bananas for the same income.

Question 15

For the utility function $U(A, B) = \sqrt{A} + \sqrt{B}$, what is the marginal rate of substitution (MRS) at the bundle $(A, B) = (9, 16)$?

Accepted Answer

$MRS = \frac{\partial U/\partial A}{\partial U/\partial B} = \frac{1/(2\sqrt{9})}{1/(2\sqrt{16})} = \frac{1/6}{1/8} = \frac{8}{6} = \frac{4}{3}$

Question 16

If a consumer's utility function is $U(A, B) = A + B$, what does this imply about their preferences for apples and bananas?

Accepted Answer

The consumer is indifferent between apples and bananas; they are perfect substitutes.

Question 17

If a consumer's utility function is $U(A, B) = \min\{A, B\}$, what does this imply about their preferences?

Accepted Answer

The consumer views apples and bananas as perfect complements.

Question 18

Suppose a consumer faces the budget constraint $3A + 2B = 12$. If the price of apples increases to $p_A = 4$, what is the new budget line?

Accepted Answer

$4A + 2B = 12$

Question 19

If a consumer's budget line is $3A + 2B = 12$, what is the slope of the budget line?

Accepted Answer

Slope is $-\frac{p_A}{p_B} = -\frac{3}{2}$

Microeconomics Problem Set: Consumer Choice, Utility, and Budget Constraints

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