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程序代写案例-F 2C

时间：2021-04-14

Solutions to Trial test

1. Suppose James is endowed with 5 units of F and 1 unit of C, and has the utility function

U(F,C) = F 2C

Suppose pFpC = 4.

• James’ demand for F is equal to 3.5 (rounded to one decimal place)

- The tangency condition is 2CF = 4, and the budget condition is pFFpCC = I = pF ∗ 5 + pC ∗ 1,

and dividing by pC the budget condition is s 4F + C = 21 It is easy to solve from here.

• James’ demand for C is equal to 7 (rounded to the nearest whole number

- see above

• If pC were equal to 1, the change in income that would make James indifferent between his optimal

quantity demanded and his endowment would be -7.07 (rounded to the nearest whole number)

- James is better off with his optimal quantity demand relative to his endowment, so we have to

take away “income” so that his optimal quantity demanded yields utility 52 ∗ 1 = 25.

2. Suppose a consumer chooses how much to consume over time, with prices in each time period, interest

rates, and income given by p1, p2, i,M1,M2. Further suppose that optimal consumption in period 1, c

∗

1

satisfies c∗1 > M1/p1.

• On net, the consumer is a borrower

- M1/p1 is the zero savings point

• If interest rates go down the consumer is worse off

- If interest rates go up, the cost of borrowing goes up and the consumer is worse off. A way to

see this is to notice how the slope of the budget line pivots around the zero savings point and gets

steeper and the optimal consumption is to the right of the zero savings point because the consumer

is a borrower.

• If interest rates go up the consumer has greater consumption opportunities in period 2

- A way to see this is to notice how the slope of the budget line pivots around the zero savings point

and gets steeper and has a higher vertical intercept.

3. Refer to the figure below to answer this question. Consider an individual with preferences over two

goods—good X and good Y. At the initial prices Px1 and Py1 and income m, the individual chooses

X1 units of good X. When the price of good X decreases to Px2, (and the price of good Y remains the

same), the individual chooses X2 units of good X. Which of the following is true?

1. Good X is an inferior good but not a Giffen good.

2. Good X is a normal good.

3. Good X is a Giffen good.

4. The income elasticity of good X equals 1 since the demand for good Y stayed the same.

5. We cannot tell whether good X is a normal, inferior, or Giffen good.

4. Consider an individual who only cares about the quantity of goods she consumes this year and the

quantity of the same goods she consumes next year. Let C1 be the quantity consumed this year and

C2 be the quantity consumed next year. Her preferences can be represented by the following utility

function:

U = 2(

√

C1 +

√

C2)

Suppose she earns $1,000 this year and nothing next year. Let the price index this year be p1 = 2 and

the price index next year be p2 = 4. The interest rate is 0.05. What is her total savings (up to two

decimal places)?

- The answer is 344.27. Why? The tangency condition is

(

C2

C1

)(

1/2) = p1p2 and the budget constraint is

p1(1 + i) + p2 = M1(1 + i), solving yields c

∗

1 = 327.87 and expenditure equal to p1 ∗ c∗1 = 655.73 so that

means the remainder of income is saved.

Page 2

学霸联盟

1. Suppose James is endowed with 5 units of F and 1 unit of C, and has the utility function

U(F,C) = F 2C

Suppose pFpC = 4.

• James’ demand for F is equal to 3.5 (rounded to one decimal place)

- The tangency condition is 2CF = 4, and the budget condition is pFFpCC = I = pF ∗ 5 + pC ∗ 1,

and dividing by pC the budget condition is s 4F + C = 21 It is easy to solve from here.

• James’ demand for C is equal to 7 (rounded to the nearest whole number

- see above

• If pC were equal to 1, the change in income that would make James indifferent between his optimal

quantity demanded and his endowment would be -7.07 (rounded to the nearest whole number)

- James is better off with his optimal quantity demand relative to his endowment, so we have to

take away “income” so that his optimal quantity demanded yields utility 52 ∗ 1 = 25.

2. Suppose a consumer chooses how much to consume over time, with prices in each time period, interest

rates, and income given by p1, p2, i,M1,M2. Further suppose that optimal consumption in period 1, c

∗

1

satisfies c∗1 > M1/p1.

• On net, the consumer is a borrower

- M1/p1 is the zero savings point

• If interest rates go down the consumer is worse off

- If interest rates go up, the cost of borrowing goes up and the consumer is worse off. A way to

see this is to notice how the slope of the budget line pivots around the zero savings point and gets

steeper and the optimal consumption is to the right of the zero savings point because the consumer

is a borrower.

• If interest rates go up the consumer has greater consumption opportunities in period 2

- A way to see this is to notice how the slope of the budget line pivots around the zero savings point

and gets steeper and has a higher vertical intercept.

3. Refer to the figure below to answer this question. Consider an individual with preferences over two

goods—good X and good Y. At the initial prices Px1 and Py1 and income m, the individual chooses

X1 units of good X. When the price of good X decreases to Px2, (and the price of good Y remains the

same), the individual chooses X2 units of good X. Which of the following is true?

1. Good X is an inferior good but not a Giffen good.

2. Good X is a normal good.

3. Good X is a Giffen good.

4. The income elasticity of good X equals 1 since the demand for good Y stayed the same.

5. We cannot tell whether good X is a normal, inferior, or Giffen good.

4. Consider an individual who only cares about the quantity of goods she consumes this year and the

quantity of the same goods she consumes next year. Let C1 be the quantity consumed this year and

C2 be the quantity consumed next year. Her preferences can be represented by the following utility

function:

U = 2(

√

C1 +

√

C2)

Suppose she earns $1,000 this year and nothing next year. Let the price index this year be p1 = 2 and

the price index next year be p2 = 4. The interest rate is 0.05. What is her total savings (up to two

decimal places)?

- The answer is 344.27. Why? The tangency condition is

(

C2

C1

)(

1/2) = p1p2 and the budget constraint is

p1(1 + i) + p2 = M1(1 + i), solving yields c

∗

1 = 327.87 and expenditure equal to p1 ∗ c∗1 = 655.73 so that

means the remainder of income is saved.

Page 2

学霸联盟