Question 1

They are thinking of buying a sofa.Bob's utility function is U_B(S, M_B)= (1 + S)M_B and Ray's utility function is U_R(S, M_R)= (4 + S)M_R, where S = 0 if they don't get the sofa and S = 1 if they do and where M_B and M_R are the amounts of money they have respectively to spend on their private consumptions.Bob has a total of $800 to spend on the sofa and other stuff.Ray has a total of $4,000 to spend on the sofa and other stuff.The maximum amount that they could pay for the sofa and still arrange to both be better off than without it is

A)$1,000.
B)$1,200.
C)$1,800.
D)$650.
E)$2,400.

$1,200.

Accepted Answer

$2,000.&#10;

Question 2

Just north of the town of Muskrat, Ontario,is the town of Brass Monkey, population 12,800.Brass Monkey, like Muskrat, has a single public good, the town skating rink and a single private good, Labatt Ale.Everyone's utility function is U_i(X_i, Y)= X_i - 121/Y, where X_i is the number of bottles of ale consumed by i and Y is the size of the skating rink in square meters.The price of ale is $1 per bottle.The cost of the skating rink to the city is $8 per square meter.Everyone has an income of at least $5,000.What is the Pareto efficient size for the town skating rink?

A)560 square meters.
B)885 square meters.
C)280 square meters.
D)440 square meters.
E)None of the above.

440 square meters.

Accepted Answer

440 square meters.&#10;

Question 3

They are thinking of buying a sofa.Bob's utility function is U_B(S, M_B)= (1 + S)M_B and Ray's utility function is U_R(S, M_R)= (4 + S)M_R, where S = 0 if they don't get the sofa and S = 1 if they do and where M_B and M_R are the amounts of money they have respectively to spend on their private consumptions.Bob has a total of $800 to spend on the sofa and other stuff.Ray has a total of $4,000 to spend on the sofa and other stuff.The maximum amount that they could pay for the sofa and still arrange to both be better off than without it is

A)$1,000.
B)$1,200.
C)$1,800.
D)$650.
E)$2,400.

$1,200.

Accepted Answer

$1,200.&#10;

Question 4

They are thinking of buying a sofa.Bob's utility function is U_B(S, M_B)= (1 + S)M_B and Ray's utility function is U_R(S, M_R)= (4 + S)M_R, where S = 0 if they don't get the sofa and S = 1 if they do and where M_B and M_R are the amounts of money they have respectively to spend on their private consumptions.Bob has a total of $800 to spend on the sofa and other stuff.Ray has a total of $4,000 to spend on the sofa and other stuff.The maximum amount that they could pay for the sofa and still arrange to both be better off than without it is

A)$1,000.
B)$1,200.
C)$1,800.
D)$650.
E)$2,400.

$1,200.

Accepted Answer

To solve this problem, we need to find the maximum amount that both Bob and Ray are willing to pay for the sofa and still be better off than without it. We can start by finding Bob and Ray's willingness to pay (WTP) for the sofa. Bob's utility function is: $U_{B}(S, M) = 10(1 + S)M$ Where $S$ is the sofa and $M$ is the amount of money Bob has left after buying the sofa. We can rewrite this function as: $U_{B}(S, M) = 10M + 10MS$ We can think of $10M$ as the utility Bob gets from spending money on other things besides the sofa. We can think of $10MS$ as the additional utility Bob gets from buying the sofa. Similarly, Ray's utility function is: $U_{R}(S, M) = 30(1 + S)M$ Which we can rewrite as: $U_{R}(S, M) = 30M + 30MS$ Now, let's assume that Bob and Ray will split the cost of the sofa evenly. Let's call the maximum amount they are willing to pay for the sofa $x$. Then Bob's WTP for the sofa is: $WTP_{B} = \frac{1}{2}(1200 - x)$ Why $1200-x$? Because Bob has a total of $1200$ to spend on both the sofa and other things, and he's already spent $\frac{1}{2}x$ on the sofa. So he has $\frac{1}{2}(1200-x)$ remaining to spend on other things. Similarly, Ray's WTP for the sofa is: $WTP_{R} = \frac{1}{2}(4000 - x)$ Now we need to find the maximum value of $x$ such that both Bob and Ray are better off with the sofa than without it. In other words, we need to find the maximum value of $x$ such that: $U_{B}(x, \frac{1}{2}(1200 - x)) + U_{R}(x, \frac{1}{2}(4000 - x)) > U_{B}(0, 1200) + U_{R}(0, 4000)$ The left-hand side of this equation represents the total utility that Bob and Ray would get if they each spent $\frac{1}{2}x$ on the sofa. The right-hand side represents the total utility they would get if they didn't buy the sofa at all. We can plug in the utility functions we derived earlier: $10(\frac{1}{2}(1200 - x)) + 10(\frac{1}{2}(1200 - x))S + 30(\frac{1}{2}(4000 - x)) + 30(\frac{1}{2}(4000 - x))S > 10(1200) + 30(4000)$ Simplifying: $12000 - 5x + 20Sx > 122000$ $20Sx - 5x > 110000$ $x(20S - 5) > 110000$ $x < \frac{110000}{20S - 5}$ Now we need to find the largest value of $x$ that satisfies this inequality. We can start by setting $20S - 5 = 1$ (the lowest possible value of $S$). $x < \frac{110000}{1}$ $x < 110000$ So the maximum amount they could pay for the sofa is $110000$. However, we also need to check that this value satisfies the inequality for all values of $S$. If we plug in $S = \frac{1}{10}$ (the highest possible value of $S$), we get: $x < \frac{110000}{2} = 55000$ Since this value is lower than our previous result, we know that the maximum amount they could pay for the sofa is between $55000$ and $110000$. To get a more precise answer, we could try plugging in other values of $S$, but we can also use common sense to eliminate some of the choices. We know that Bob's budget is much smaller than Ray's budget ($1200$ vs $4000$), and that Bob's utility function is less sensitive to changes in $S$ than Ray's utility function (the coefficient of $S$ is $1$ for Bob and $3$ for Ray). So it's likely that the maximum amount Bob is willing to pay will be much lower than the maximum amount Ray is willing to pay. Eliminating choices A, C, and E (which are all higher than $1600$), we are left with choice B as the most reasonable answer. Of course, this is not a rigorous proof, but it's a way to arrive at the correct answer quickly using intuition and common sense.

Question 5

They are thinking of buying a sofa.Bob's utility function is U_B(S, M_B)= (1 + S)M_B and Ray's utility function is U_R(S, M_R)= (4 + S)M_R, where S = 0 if they don't get the sofa and S = 1 if they do and where M_B and M_R are the amounts of money they have respectively to spend on their private consumptions.Bob has a total of $800 to spend on the sofa and other stuff.Ray has a total of $4,000 to spend on the sofa and other stuff.The maximum amount that they could pay for the sofa and still arrange to both be better off than without it is

A)$1,000.
B)$1,200.
C)$1,800.
D)$650.
E)$2,400.

$1,200.

Accepted Answer

Bob and Ray will be better off if their utility with the sofa is greater than without it. For Bob, (1+1)(M_B - P) > (1+0)(M_B), and for Ray, (2+1)(M_R - P) > (2+0)(M_R). Solving these inequalities with M_B = M_R = $1,200 gives P < $1,000.

Question 6

Lucy's utility function is 2X_L + G and Melvin's utility function is X_MG, where G is their expenditures on the public goods they share in their apartment and where X_L and X_M are their respective private consumption expenditures.The total amount they have to spend on private goods and public goods is $33,000.They agree on a Pareto optimal pattern of expenditures in which the amount that is spent on Lucy's private consumption is $9,000.How much do they spend on public goods?

A)$8,550
B)$4,000
C)$8,000
D)$16,000
E)There is not enough information here to be able to determine the answer.

Accepted Answer

The answer of Lucy's utility function is 2X_L +...

Question 7

Lucy's utility function is 2X_L + G and Melvin's utility function is X_MG, where G is their expenditures on the public goods they share in their apartment and where X_L and X_M are their respective private consumption expenditures.The total amount they have to spend on private goods and public goods is $33,000.They agree on a Pareto optimal pattern of expenditures in which the amount that is spent on Lucy's private consumption is $9,000.How much do they spend on public goods?

A)$8,550
B)$4,000
C)$8,000
D)$16,000
E)There is not enough information here to be able to determine the answer.

Accepted Answer

Given the total amount for spending is $24,000 and Lucy's private consumption is $9,000, the remaining amount after subtracting Lucy's private consumption is $15,000. This must be split between Melvin's private consumption and public goods. Without specific details on Melvin's private consumption, we assume the question implies that the rest is spent on public goods, making the correct answer $15,000, which is not listed. However, based on the provided options and the logic applied, it seems there might be a misunderstanding in the interpretation of the question or a mistake in the options provided.

Question 8

Lucy's utility function is 2X_L + G and Melvin's utility function is X_MG, where G is their expenditures on the public goods they share in their apartment and where X_L and X_M are their respective private consumption expenditures.The total amount they have to spend on private goods and public goods is $33,000.They agree on a Pareto optimal pattern of expenditures in which the amount that is spent on Lucy's private consumption is $9,000.How much do they spend on public goods?

A)$8,550
B)$4,000
C)$8,000
D)$16,000
E)There is not enough information here to be able to determine the answer.

Accepted Answer

In a Pareto optimal allocation, the marginal rate of substitution (MRS) between private and public goods should be equal for both individuals. Lucy's MRS is 2, and Melvin's MRS is G/X_M. Given the total budget of $24,000 and Lucy's private consumption of $6,000, the remaining $18,000 is split between Melvin's private consumption and public goods. To equalize MRS, they spend $12,000 on public goods, leaving $6,000 for Melvin's private consumption.

Question 9

Just north of the town of Muskrat, Ontario,is the town of Brass Monkey, population 12,800.Brass Monkey, like Muskrat, has a single public good, the town skating rink and a single private good, Labatt Ale.Everyone's utility function is U_i(X_i, Y)= X_i - 121/Y, where X_i is the number of bottles of ale consumed by i and Y is the size of the skating rink in square meters.The price of ale is $1 per bottle.The cost of the skating rink to the city is $8 per square meter.Everyone has an income of at least $5,000.What is the Pareto efficient size for the town skating rink?

A)560 square meters.
B)885 square meters.
C)280 square meters.
D)440 square meters.
E)None of the above.

440 square meters.

Accepted Answer

The answer of Just north of the town of Muskrat,...

Question 10

Lucy's utility function is 2X_L + G and Melvin's utility function is X_MG, where G is their expenditures on the public goods they share in their apartment and where X_L and X_M are their respective private consumption expenditures.The total amount they have to spend on private goods and public goods is $33,000.They agree on a Pareto optimal pattern of expenditures in which the amount that is spent on Lucy's private consumption is $9,000.How much do they spend on public goods?

A)$8,550
B)$4,000
C)$8,000
D)$16,000
E)There is not enough information here to be able to determine the answer.

Accepted Answer

The answer of Lucy's utility function is 2X_L +...

Question 11

Just north of the town of Muskrat, Ontario,is the town of Brass Monkey, population 12,800.Brass Monkey, like Muskrat, has a single public good, the town skating rink and a single private good, Labatt Ale.Everyone's utility function is U_i(X_i, Y)= X_i - 121/Y, where X_i is the number of bottles of ale consumed by i and Y is the size of the skating rink in square meters.The price of ale is $1 per bottle.The cost of the skating rink to the city is $8 per square meter.Everyone has an income of at least $5,000.What is the Pareto efficient size for the town skating rink?

A)560 square meters.
B)885 square meters.
C)280 square meters.
D)440 square meters.
E)None of the above.

440 square meters.

Accepted Answer

The answer of Just north of the town of Muskrat,...

Question 12

They are thinking of buying a sofa.Bob's utility function is U_B(S, M_B)= (1 + S)M_B and Ray's utility function is U_R(S, M_R)= (4 + S)M_R, where S = 0 if they don't get the sofa and S = 1 if they do and where M_B and M_R are the amounts of money they have respectively to spend on their private consumptions.Bob has a total of $800 to spend on the sofa and other stuff.Ray has a total of $4,000 to spend on the sofa and other stuff.The maximum amount that they could pay for the sofa and still arrange to both be better off than without it is

A)$1,000.
B)$1,200.
C)$1,800.
D)$650.
E)$2,400.

$1,200.

Accepted Answer

The answer of  They are thinking of buying a...

Question 13

Just north of the town of Muskrat, Ontario,is the town of Brass Monkey, population 12,800.Brass Monkey, like Muskrat, has a single public good, the town skating rink and a single private good, Labatt Ale.Everyone's utility function is U_i(X_i, Y)= X_i - 121/Y, where X_i is the number of bottles of ale consumed by i and Y is the size of the skating rink in square meters.The price of ale is $1 per bottle.The cost of the skating rink to the city is $8 per square meter.Everyone has an income of at least $5,000.What is the Pareto efficient size for the town skating rink?

A)560 square meters.
B)885 square meters.
C)280 square meters.
D)440 square meters.
E)None of the above.

440 square meters.

Accepted Answer

The answer of Just north of the town of Muskrat,...

Question 14

Lucy's utility function is 2X_L + G and Melvin's utility function is X_MG, where G is their expenditures on the public goods they share in their apartment and where X_L and X_M are their respective private consumption expenditures.The total amount they have to spend on private goods and public goods is $39,000.They agree on a Pareto optimal pattern of expenditures in which the amount that is spent on Lucy's private consumption is $9,000.How much do they spend on public goods?

A)$9,550
B)$20,000
C)$5,000
D)$10,000
E)There is not enough information here to be able to determine the answer.

Accepted Answer

The answer of Lucy's utility function is 2X_L +...

Question 15

Just north of the town of Muskrat, Ontario,is the town of Brass Monkey, population 12,800.Brass Monkey, like Muskrat, has a single public good, the town skating rink and a single private good, Labatt Ale.Everyone's utility function is U_i(X_i, Y)= X_i - 121/Y, where X_i is the number of bottles of ale consumed by i and Y is the size of the skating rink in square meters.The price of ale is $1 per bottle.The cost of the skating rink to the city is $8 per square meter.Everyone has an income of at least $5,000.What is the Pareto efficient size for the town skating rink?

A)560 square meters.
B)885 square meters.
C)280 square meters.
D)440 square meters.
E)None of the above.

440 square meters.

Accepted Answer

The answer of Just north of the town of Muskrat,...

Deck 37: Extension: A: Public Goods