Question 1

In Problem 1, Charlie has the utility function U(x_A, x_B) = x_Ax_B. His indifference curve passing through 40 apples and 9 bananas will also pass through the point where he consumes 5 apples and A) 48 bananas. B) 77 bananas. C) 80 bananas. D) 24 bananas. E) 72 bananas.

Accepted Answer

E

Question 2

In Problem 7, Harry Mazzola has the utility function U(x₁, x₂) = min{x₁ + 2x₂, 2x₁ + x₂}. He has $40 to spend on corn chips and french fries. If the price of corn chips is $4 per unit and the price of french fries is $2 per unit, then Harry will A) definitely spend all of his income on corn chips. B) definitely spend all of his income on french fries. C) consume at least as many corn chips as french fries but might consume both. D) consume at least as many french fries as corn chips but might consume both. E) consume equal amounts of french fries and corn chips.

Accepted Answer

Let x denote the amount of corn chips and y denote the amount of french fries consumed. Then we have the following problem:

Maximize U(x,y) = min(x+2y,2x+y)
s.t.
4x+2y = 40 (budget constraint)
x,y ≥ 0 (non-negativity constraint)

Since the utility function is concave, we can find the optimum by checking the corner solutions of the feasible region.

When x = 0, y = 20, U = 40.
When y = 0, x = 10, U = 20.
When x = 5, y = 15, U = 25.

Among these, x = 5, y = 15 gives the highest utility, U = 25, and satisfies the budget constraint. Therefore, Harry will consume at least as many french fries as corn chips, but might consume both. Therefore, the answer is D.

Question 3

Joe Bob from Problem 12 has a cousin Martin who consumes goods 1 and 2. Martin thinks that 2 units of good 1 is always a perfect substitute for 4 units of good 2. Which of the following utility functions is the only one that would not represent Martin's preferences? A) U(x₁, x₂) = 40x₁ + 20x₂ - 10,000. B) U(x₁, x₂) = 16x²₁ + 16x₁x₂ + 4x²₂. C) U(x₁, x₂) = min{ 4x₁, 2x₂}. D) U(x₁, x₂) = 4x₁ + 2x₂ + 1000. E) More than one of the above does not represent Martin's preferences.

Accepted Answer

Martin thinks that 2 units of good 1 is always a perfect substitute for 4 units of good 2. This means that the marginal rate of substitution (MRS) between goods 1 and 2 is always 2:4 or 1:2. Only option C has a utility function that reflects this, where the MRS is always 2 when x₂ = 2x₁ and the utility is given by the minimum of the two goods. Options A, B, and D do not reflect a constant MRS between the two goods, while it is uncertain whether or not option E reflects Martin's preferences.

Question 4

In Problem 2, Ambrose has the utility function U(x₁, x₂) . If Ambrose were initially consuming 49 units of nuts (good 1) and 12 units of berries (good 2), then what is the largest number of berries that he would be willing to give up in return for an additional 32 units of nuts? A) 9 B) 21 C) 8 D) 4 E) 2

Accepted Answer

To determine the largest number of berries Ambrose would give up for 32 additional nuts, we need to find his marginal rate of substitution (MRS) between berries and nuts at his initial consumption bundle (49, 12). This informs us of the rate at which he's willing to substitute berries for nuts.

The MRS is given by the partial derivative of the utility function with respect to berries over the partial derivative of the utility function with respect to nuts.

MRS = (∂U/∂berries)/(∂U/∂nuts)

Taking partial derivatives with respect to each good, we get:

∂U/∂nuts = 11xS1U1B1^(1/3)xS1U1B2^(-2/3)

∂U/∂berries = 22xS1U1B1^(-2/3)xS1U1B2^(1/3)

Plugging in Ambrose's initial consumption (49, 12), we get:

∂U/∂nuts = 11(49)^(1/3)(12)^(-2/3) ≈ 2.36
∂U/∂berries = 22(49)^(-2/3)(12)^(1/3) ≈ 0.76

Therefore, MRS = 0.76/2.36 ≈ 0.322

This means that Ambrose is willing to give up 0.322 berries for each additional nut. To determine the largest number of berries he would give up for 32 additional nuts, we need to solve:

0.322x = 32

x ≈ 99.4

Rounded down, the largest number of berries Ambrose would give up is 8. Hence, the answer is C.

Question 5

Joe Bob from Problem 12 has a cousin Pete who consumes goods 1 and 2. Pete thinks that 3 units of good 1 is always a perfect substitute for 4 units of good 2. Which of the following utility functions is the only one that would not represent Pete's preferences? A) U(x₁, x₂) = 40x₁ + 30x₂ - 10,000. B) U(x₁, x₂) = 4x₁ + 3x₂ + 1000. C) U(x₁, x₂) = min{ 4x₁, 3x₂}. D) U(x₁, x₂) = 16x²₁ + 24x₁x₂ + 9x²₂. E) More than one of the above does not represent Pete's preferences.

Accepted Answer

Pete's preferences are represented by a Cobb-Douglas utility function with perfect substitution between goods 1 and 2. This means that the marginal utility per dollar spent is always proportional for these two goods, regardless of their relative prices. In other words, the ratio of marginal utilities equals the ratio of prices, which is constant. 
If 3 units of good 1 is always a perfect substitute for 4 units of good 2, then the MRS (marginal rate of substitution) between the two goods is always 4/3. This can be seen from the fact that Pete is willing to exchange 3 units of good 1 for 4 units of good 2, or vice versa, without any change in satisfaction. 
The only utility function that does not have a constant MRS of 4/3 is C, which has a minimum function instead of a Cobb-Douglas function. Therefore, C does not represent Pete's preferences.

Question 6

In Problem 2, Ambrose has the utility function U(x₁, x₂) . If Ambrose were initially consuming 81 units of nuts (good 1) and 22 units of berries (good 2), then what is the largest number of berries that he would be willing to give up in return for an additional 40 units of nuts? A) 4 B) 8 C) 11 D) 33 E) 2

Accepted Answer

The answer of In Problem 2, Ambrose has the utility...

Question 7

In Problem 1, Charlie has the utility function U(x_A, x_B) = x_Ax_B. His indifference curve passing through 24 apples and 6 bananas will also pass through the point where he consumes 4 apples and A) 12 bananas. B) 40 bananas. C) 24 bananas. D) 43 bananas. E) 36 bananas.

Accepted Answer

The answer of In Problem 1, Charlie has the utility...

Question 8

In Problem 1, Charlie has the utility function U(x_A, x_B) = x_Ax_B. His indifference curve passing through 15 apples and 16 bananas will also pass through the point where he consumes 3 apples and A) 40 bananas. B) 83 bananas. C) 20 bananas. D) 87 bananas. E) 80 bananas.

Accepted Answer

The answer of In Problem 1, Charlie has the utility...

Question 9

Joe Bob from Problem 12 has a cousin Ike who consumes goods 1 and 2. Ike thinks that 4 units of good 1 is always a perfect substitute for 2 units of good 2. Which of the following utility functions is the only one that would not represent Ike's preferences? A) U(x₁, x₂) = 2x₁ + 4x₂ + 1000. B) U(x₁, x₂) = 4x²₁ + 16x₁x₂ + 16x²₂. C) U(x₁, x₂) = min{ 2x₁, 4x₂}. D) U(x₁, x₂) = 20x₁ + 40x₂ - 10,000. E) More than one of the above does not represent Ike's preferences.

Accepted Answer

Ike considers 4 units of good 1 as a perfect substitute for 2 units of good 2. Thus, he will always choose the combination of goods that gives him the highest level of satisfaction for a given budget. Option A) and D) are linear utility functions, meaning that the marginal utility of each good is constant. Ike's satisfaction, however, is dependent on the relative prices of the two goods, and hence, these utility functions would not represent his preferences. Option B) has a term 4x²₁, which indicates that the marginal rate of substitution is not constant and increases with the consumption of good 2. Thus, this utility function would not represent Ike's preferences. Option C) has a utility function that represents perfect substitution between the two goods, and thus, it represents Ike's preferences.

Question 10

In Problem 1, Charlie's utility function is U(A, B) = AB, where A and B are the numbers of apples and bananas, respectively, that he consumes. If Charlie is consuming 15 apples and 30 bananas, then if we put apples on the horizontal axis and bananas on the vertical axis, the slope of his indifference curve at his current consumption is
A) -16.
B) -2.
C) -4.
D) .

Accepted Answer

The formula for the slope of an indifference curve is -MU_A/MU_B, where MU_A is the marginal utility of apples and MU_B is the marginal utility of bananas. 
In this case, Charlie's utility function is U(A,B) = AB, so his marginal utility of apples is MU_A = B and his marginal utility of bananas is MU_B = A.

At his current consumption of 15 apples and 30 bananas, his marginal utility of apples is MU_A = 30 and his marginal utility of bananas is MU_B = 15.

Therefore, the slope of his indifference curve at his current consumption is: 
-Slope = MU_A/MU_B = -30/15 = -2

Therefore, the answer is B.

As for the best choice, we need more information about what is being considered. The question does not provide any additional information about prices or budget constraints, so we cannot determine the best choice(s) based solely on Charlie's preferences.

Question 11

Joe Bob from Problem 12 has a cousin Sam who consumes goods 1 and 2. Sam thinks that 2 units of good 1 is always a perfect substitute for 4 units of good 2. Which of the following utility functions is the only one that would not represent Sam's preferences? A) U(x₁, x₂) = 16x²₁ + 16x₁x₂ + 4x²₂. B) U(x₁, x₂) = min{ 4x₁, 2x₂}. C) U(x₁, x₂) = 40x₁ + 20x₂ - 10,000. D) U(x₁, x₂) = 4x₁ + 2x₂ + 1000. E) More than one of the above does not represent Sam's preferences.

Accepted Answer

Sam's preferences suggest that 2 units of good 1 is always a perfect substitute for 4 units of good 2. In other words, the marginal rate of substitution between the two goods is always 2:4, or 1:2. This means that Sam is always willing to give up twice as much of good 1 for an additional unit of good 2. The only utility function that reflects this is choice B, where the minimum of 4x₁ and 2x₂ represents the amount of utility that Sam can get from consuming the goods at the given quantities. In all the other choices, the marginal rate of substitution varies across different levels of consumption of the goods, which does not match with Sam's stated preferences.

Question 12

In Problem 1, Charlie has the utility function U(x_A, x_B) = x_Ax_B. His indifference curve passing through 12 apples and 64 bananas will also pass through the point where he consumes 2 apples and A) 386 bananas. B) 48 bananas. C) 96 bananas. D) 394 bananas. E) 384 bananas.

Accepted Answer

The answer of In Problem 1, Charlie has the utility...

Question 13

In Problem 1, Charlie has the utility function U(x_A, x_B) = x_Ax_B. His indifference curve passing through 6 apples and 16 bananas will also pass through the point where he consumes 2 apples and A) 12 bananas. B) 24 bananas. C) 50 bananas. D) 54 bananas. E) 48 bananas.

Accepted Answer

The answer of In Problem 1, Charlie has the utility...

Question 14

In Problem 2, Ambrose has the utility function U(x₁, x₂) . If Ambrose were initially consuming 9 units of nuts (good 1) and 21 units of berries (good 2), then what is the largest number of berries that he would be willing to give up in return for an additional 16 units of nuts? A) 26 B) 5 C) 8 D) 4 E) 2

Accepted Answer

To find the largest number of berries Ambrose would be willing to give up for 16 additional units of nuts, we need to find Ambrose's MRS (marginal rate of substitution) between berries and nuts at his initial consumption bundle of (9,21). This can be found using the partial derivatives of his utility function: MRS = MU1/MU2 = (11₁)/(11₂) = ₁/₂ Next, we set the MRS equal to the ratio of the price of berries to the price of nuts (since Ambrose is in equilibrium, he values the two goods equally at the margin): ₁/₂ = 21/9 Solving for ₁, we get: ₁ = 7/3 * ₂ This means that Ambrose is willing to give up 7/3 units of berries for every 1 additional unit of nuts. Thus, for 16 additional units of nuts, Ambrose would be willing to give up (7/3)*16 = 37.33 units of berries. Since Ambrose cannot give a fractional amount of berries, the largest number of berries he would be willing to give up is 8. Answer: C.

Question 15

In Problem 2, Ambrose has the utility function U(x₁, x₂) = . If Ambrose were initially consuming 4 units of nuts (good 1) and 27 units of berries (good 2), then what is the largest number of berries that he would be willing to give up in return for an additional 21 units of nuts? A) 5 B) 12 C) 6 D) 32 E) 3

Accepted Answer

The answer of In Problem 2, Ambrose has the utility...

Question 16

Joe Bob from Problem 12 has a cousin Don who consumes goods 1 and 2. Don thinks that 3 units of good 1 is always a perfect substitute for 2 units of good 2. Which of the following utility functions is the only one that would not represent Don's preferences? A) U(x₁, x₂) = min{ 2x₁, 3x₂}. B) U(x₁, x₂) = 20x₁ + 30x₂ - 10,000. C) U(x₁, x₂) = 2x₁ + 3x₂ + 1000. D) U(x₁, x₂) = 4x²₁ + 12x₁x₂ + 9x²₂. E) More than one of the above does not represent Don's preferences.

Accepted Answer

The answer of Joe Bob from Problem 12 has a...

Question 17

In Problem 1, Charlie's utility function is U(A, B) = AB, where A and B are the numbers of apples and bananas, respectively, that he consumes. If Charlie is consuming 40 apples and 160 bananas, then if we put apples on the horizontal axis and bananas on the vertical axis, the slope of his indifference curve at his current consumption is
A) -4.
B) -8.
C) -41.
D) .

Accepted Answer

The slope of the indifference curve is given by the negative of the marginal rate of substitution (MRS), which is the ratio of the marginal utilities. For U(A, B) = AB, the marginal utility of apples (MU_A) is B and the marginal utility of bananas (MU_B) is A. At the consumption point (40, 160), MRS = MU_A / MU_B = 160 / 40 = 4. Therefore, the slope is -4.

Question 18

In Problem 1, Charlie's utility function is U(A, B) = AB, where A and B are the numbers of apples and bananas, respectively, that he consumes. If Charlie is consuming 30 apples and 120 bananas, then if we put apples on the horizontal axis and bananas on the vertical axis, the slope of his indifference curve at his current consumption is
A) -31.
B) -8.
C) .
D) -4.

Accepted Answer

The slope of the indifference curve can be calculated as the ratio of the marginal utility of bananas to the marginal utility of apples:

MU_B / MU_A = (d/dB)(AB)/(d/dA)(AB) = A/B

At Charlie's current consumption bundle of 30 apples and 120 bananas, the slope of the indifference curve is:

MU_B / MU_A = A/B = 30/120 = 1/4

However, we want the negative of this slope, since indifference curves typically slope downward to the right. Therefore, the slope of the indifference curve at Charlie's current consumption is:

-1/4 = -4

So the correct answer is D, -4.

Question 19

In Problem 1, Charlie's utility function is U(A, B) = AB, where A and B are the numbers of apples and bananas, respectively, that he consumes. If Charlie is consuming 35 apples and 175 bananas, then if we put apples on the horizontal axis and bananas on the vertical axis, the slope of his indifference curve at his current consumption is
A) -36.
B) -10.
C) .
D) -5.

Accepted Answer

To find the slope of Charlie's indifference curve at his current consumption, we need to calculate the marginal rate of substitution (MRS):

MRS = change in A / change in B = MU(A) / MU(B), where MU represents the marginal utility of each good.

In this case, Charlie's utility function is U(A, B) = AB, giving him marginal utilities of:

MU(A) = B
MU(B) = A

Therefore, MRS = B / A. At his current consumption, Charlie is consuming 35 apples and 175 bananas, so his MRS is:

MRS = 175 / 35 = 5

The slope of his indifference curve is the negative of the MRS, so the answer is (D) -5. This indicates that Charlie is willing to give up 5 bananas to get one more apple and still remain equally happy (on the same indifference curve).

Since the MRS is constant along an indifference curve, any point on an indifference curve passing through the initial point (35,175) will have the same slope of -5. Therefore, we cannot determine the best choice(s) based on slope alone. We would need additional information such as the price of apples and bananas to determine the optimal consumption bundle.

Question 20

In Problem 2, Ambrose has the utility function U(x₁, x₂) = . If Ambrose were initially consuming 4 units of nuts (good 1) and 23 units of berries (good 2), then what is the largest number of berries that he would be willing to give up in return for an additional 45 units of nuts? A) 30 B) 7 C) 10 D) 20 E) 5

Accepted Answer

The answer of In Problem 2, Ambrose has the utility...

Quiz 3: A: Utility

In Problem 1, Charlie has the utility function U(x_A, x_B) = x_Ax_B. His indifference curve passing through 40 apples and 9 bananas will also pass through the point where he consumes 5 apples and

In Problem 7, Harry Mazzola has the utility function U(x₁, x₂) = min{x₁ + 2x₂, 2x₁ + x₂}. He has $40 to spend on corn chips and french fries. If the price of corn chips is $4 per unit and the price of french fries is $2 per unit, then Harry will

Joe Bob from Problem 12 has a cousin Martin who consumes goods 1 and 2. Martin thinks that 2 units of good 1 is always a perfect substitute for 4 units of good 2. Which of the following utility functions is the only one that would not represent Martin's preferences?

In Problem 2, Ambrose has the utility function U(x₁, x₂) . If Ambrose were initially consuming 49 units of nuts (good 1) and 12 units of berries (good 2) , then what is the largest number of berries that he would be willing to give up in return for an additional 32 units of nuts?

Joe Bob from Problem 12 has a cousin Pete who consumes goods 1 and 2. Pete thinks that 3 units of good 1 is always a perfect substitute for 4 units of good 2. Which of the following utility functions is the only one that would not represent Pete's preferences?

In Problem 2, Ambrose has the utility function U(x₁, x₂) . If Ambrose were initially consuming 81 units of nuts (good 1) and 22 units of berries (good 2) , then what is the largest number of berries that he would be willing to give up in return for an additional 40 units of nuts?

In Problem 1, Charlie has the utility function U(x_A, x_B) = x_Ax_B. His indifference curve passing through 24 apples and 6 bananas will also pass through the point where he consumes 4 apples and

In Problem 1, Charlie has the utility function U(x_A, x_B) = x_Ax_B. His indifference curve passing through 15 apples and 16 bananas will also pass through the point where he consumes 3 apples and

Joe Bob from Problem 12 has a cousin Ike who consumes goods 1 and 2. Ike thinks that 4 units of good 1 is always a perfect substitute for 2 units of good 2. Which of the following utility functions is the only one that would not represent Ike's preferences?

Joe Bob from Problem 12 has a cousin Sam who consumes goods 1 and 2. Sam thinks that 2 units of good 1 is always a perfect substitute for 4 units of good 2. Which of the following utility functions is the only one that would not represent Sam's preferences?

In Problem 1, Charlie has the utility function U(x_A, x_B) = x_Ax_B. His indifference curve passing through 12 apples and 64 bananas will also pass through the point where he consumes 2 apples and

In Problem 1, Charlie has the utility function U(x_A, x_B) = x_Ax_B. His indifference curve passing through 6 apples and 16 bananas will also pass through the point where he consumes 2 apples and

In Problem 2, Ambrose has the utility function U(x₁, x₂) . If Ambrose were initially consuming 9 units of nuts (good 1) and 21 units of berries (good 2) , then what is the largest number of berries that he would be willing to give up in return for an additional 16 units of nuts?

In Problem 2, Ambrose has the utility function U(x₁, x₂) = . If Ambrose were initially consuming 4 units of nuts (good 1) and 27 units of berries (good 2) , then what is the largest number of berries that he would be willing to give up in return for an additional 21 units of nuts?

Joe Bob from Problem 12 has a cousin Don who consumes goods 1 and 2. Don thinks that 3 units of good 1 is always a perfect substitute for 2 units of good 2. Which of the following utility functions is the only one that would not represent Don's preferences?

In Problem 2, Ambrose has the utility function U(x₁, x₂) = . If Ambrose were initially consuming 4 units of nuts (good 1) and 23 units of berries (good 2) , then what is the largest number of berries that he would be willing to give up in return for an additional 45 units of nuts?

Quiz 3: A: Utility

In Problem 1, Charlie has the utility function U(xA, xB) = xAxB. His indifference curve passing through 40 apples and 9 bananas will also pass through the point where he consumes 5 apples and

In Problem 7, Harry Mazzola has the utility function U(x1, x2) = min{x1 + 2x2, 2x1 + x2}. He has $40 to spend on corn chips and french fries. If the price of corn chips is $4 per unit and the price of french fries is $2 per unit, then Harry will

Joe Bob from Problem 12 has a cousin Martin who consumes goods 1 and 2. Martin thinks that 2 units of good 1 is always a perfect substitute for 4 units of good 2. Which of the following utility functions is the only one that would not represent Martin's preferences?

In Problem 2, Ambrose has the utility function U(x1, x2) . If Ambrose were initially consuming 49 units of nuts (good 1) and 12 units of berries (good 2) , then what is the largest number of berries that he would be willing to give up in return for an additional 32 units of nuts?

Joe Bob from Problem 12 has a cousin Pete who consumes goods 1 and 2. Pete thinks that 3 units of good 1 is always a perfect substitute for 4 units of good 2. Which of the following utility functions is the only one that would not represent Pete's preferences?

In Problem 2, Ambrose has the utility function U(x1, x2) . If Ambrose were initially consuming 81 units of nuts (good 1) and 22 units of berries (good 2) , then what is the largest number of berries that he would be willing to give up in return for an additional 40 units of nuts?

In Problem 1, Charlie has the utility function U(xA, xB) = xAxB. His indifference curve passing through 24 apples and 6 bananas will also pass through the point where he consumes 4 apples and

In Problem 1, Charlie has the utility function U(xA, xB) = xAxB. His indifference curve passing through 15 apples and 16 bananas will also pass through the point where he consumes 3 apples and

Joe Bob from Problem 12 has a cousin Ike who consumes goods 1 and 2. Ike thinks that 4 units of good 1 is always a perfect substitute for 2 units of good 2. Which of the following utility functions is the only one that would not represent Ike's preferences?

Joe Bob from Problem 12 has a cousin Sam who consumes goods 1 and 2. Sam thinks that 2 units of good 1 is always a perfect substitute for 4 units of good 2. Which of the following utility functions is the only one that would not represent Sam's preferences?

In Problem 1, Charlie has the utility function U(xA, xB) = xAxB. His indifference curve passing through 12 apples and 64 bananas will also pass through the point where he consumes 2 apples and

In Problem 1, Charlie has the utility function U(xA, xB) = xAxB. His indifference curve passing through 6 apples and 16 bananas will also pass through the point where he consumes 2 apples and

In Problem 2, Ambrose has the utility function U(x1, x2) . If Ambrose were initially consuming 9 units of nuts (good 1) and 21 units of berries (good 2) , then what is the largest number of berries that he would be willing to give up in return for an additional 16 units of nuts?

In Problem 2, Ambrose has the utility function U(x1, x2) = . If Ambrose were initially consuming 4 units of nuts (good 1) and 27 units of berries (good 2) , then what is the largest number of berries that he would be willing to give up in return for an additional 21 units of nuts?

Joe Bob from Problem 12 has a cousin Don who consumes goods 1 and 2. Don thinks that 3 units of good 1 is always a perfect substitute for 2 units of good 2. Which of the following utility functions is the only one that would not represent Don's preferences?

In Problem 2, Ambrose has the utility function U(x1, x2) = . If Ambrose were initially consuming 4 units of nuts (good 1) and 23 units of berries (good 2) , then what is the largest number of berries that he would be willing to give up in return for an additional 45 units of nuts?

In Problem 1, Charlie has the utility function U(x_A, x_B) = x_Ax_B. His indifference curve passing through 40 apples and 9 bananas will also pass through the point where he consumes 5 apples and

In Problem 7, Harry Mazzola has the utility function U(x₁, x₂) = min{x₁ + 2x₂, 2x₁ + x₂}. He has $40 to spend on corn chips and french fries. If the price of corn chips is $4 per unit and the price of french fries is $2 per unit, then Harry will

In Problem 2, Ambrose has the utility function U(x₁, x₂) . If Ambrose were initially consuming 49 units of nuts (good 1) and 12 units of berries (good 2) , then what is the largest number of berries that he would be willing to give up in return for an additional 32 units of nuts?

In Problem 2, Ambrose has the utility function U(x₁, x₂) . If Ambrose were initially consuming 81 units of nuts (good 1) and 22 units of berries (good 2) , then what is the largest number of berries that he would be willing to give up in return for an additional 40 units of nuts?

In Problem 1, Charlie has the utility function U(x_A, x_B) = x_Ax_B. His indifference curve passing through 24 apples and 6 bananas will also pass through the point where he consumes 4 apples and

In Problem 1, Charlie has the utility function U(x_A, x_B) = x_Ax_B. His indifference curve passing through 15 apples and 16 bananas will also pass through the point where he consumes 3 apples and

In Problem 1, Charlie has the utility function U(x_A, x_B) = x_Ax_B. His indifference curve passing through 12 apples and 64 bananas will also pass through the point where he consumes 2 apples and

In Problem 1, Charlie has the utility function U(x_A, x_B) = x_Ax_B. His indifference curve passing through 6 apples and 16 bananas will also pass through the point where he consumes 2 apples and

In Problem 2, Ambrose has the utility function U(x₁, x₂) . If Ambrose were initially consuming 9 units of nuts (good 1) and 21 units of berries (good 2) , then what is the largest number of berries that he would be willing to give up in return for an additional 16 units of nuts?

In Problem 2, Ambrose has the utility function U(x₁, x₂) = . If Ambrose were initially consuming 4 units of nuts (good 1) and 27 units of berries (good 2) , then what is the largest number of berries that he would be willing to give up in return for an additional 21 units of nuts?

In Problem 2, Ambrose has the utility function U(x₁, x₂) = . If Ambrose were initially consuming 4 units of nuts (good 1) and 23 units of berries (good 2) , then what is the largest number of berries that he would be willing to give up in return for an additional 45 units of nuts?