# 辅导ECON-4330、辅导Java，Python编程

- 首页 >> Algorithm 算法 ECON-4330 Advanced Macroeconomics I Professor Nurlan Turdaliev

Homework Assignment 3 Fall 2021

Due: Friday, November 12, 2021, 6pm

Problem 1

Consider the economy with monetary expansion studied in class. Assume that

u(c1, c2) = ln(c1) + c2,

N = 200,

y = 3.

a. Suppose there is no seigniorage. For μ = 1.2, find the real value of the transfer a in the

stationary case.

b. Consider the case of seigniorage. Find function g(μ) discussed in class.

c. Plot the function g(μ) obtained in part b.

Problem 2

Consider the following economy: Individuals are endowed with y units of the consumption good

when young and nothing when old. The fiat money stock is constant. The population grows at

rate n. In each period, the government taxes each young person τ goods. The total proceeds of

the tax are then distributed equally among the old who are alive in that period.

a. Write down the first- and second-period budget constraints facing a typical individual at time

t. (Hint: Be careful; remember that more young people than old people are alive at time t .)

Combine the constraints into a lifetime budget constraint.

b. Find the rate of return on fiat money in a stationary monetary equilibrium.

c. Does the monetary equilibrium maximize the utility of future generations?

d. Does this government policy have any effect on an individual’s welfare?

e. Does your answer to part d change if the tax is larger than the real balances people would

choose to hold in the absence of the tax?

f. Suppose that tax collection and redistribution are (very) costly, so that for every unit of tax

collected from the young, only 0.5 unit is available to distribute to the old. How does your

answer to part d change?

Problem 3

Consider an overlapping generations model with the following characteristics. Each generation is

composed of 1,000 individuals. The fiat money supply changes according to Mt = 2Mt?1. The

initial old own a total of 10,000 units of fiat money (M0 = $10, 000). Each period, the newly printed

money is given to the old of that period as a lump-sum transfer (subsidy). Each person is endowed

with 20 units of the consumption good when born and nothing when old. Preferences are such that

individuals wish to save 10 units when young at the equilibrium rate of return on fiat money.

a. What is the gross real rate of return on fiat money in this economy?

b. How many goods does an individual receive as a subsidy?

c. What is the price of the consumption good in period 1, p1, in dollars?

ECON-4330 Advanced Macroeconomics I Professor Nurlan Turdaliev

Homework Assignment 4 Fall 2021

Due: Friday, December 3, 2021, 6pm

Problem 1

Consider the economy with capital studied in class. Assume the endowment vector (y, 0). Suppose

the production function is f(k) =

√

k and u(c1, c2) = ln c1 + ln c2.

a. Suppose there is no money. Find the equilibrium consumption allocation (c?1, c?2).

b. Now suppose there is money in this economy. Find the equilibrium consumption allocation

(c?1, c?2).

c. Let u?(y) = u(c?1, c?2) and u?(y) = u(c?1, c?2). Calculate the values of u?(y) and u?(y) for the

values y = 1, 5, 10. What is larger, u?(y) or u?(y)?

d. Plot u?(y) and u?(y) for values of y ∈ [0, 10] on the same graph using different colours or

thickness. You can use Maple, Excel, or any other software.

Problem 2

Consider the economy with capital and private debt but without money studied in class. Assume

the endowment vectors are (y, 0) for lenders and (0, y) for borrowers. Suppose the production

function is f(k) =

√

k and u(c1, c2) = ln c1 + ln c2.

a. Find the equilibrium gross interest rate r?. (Remember that r > 0. Note that you may obtain

more than one value of r?. Hint: use the fact that one of the roots of the equation you will

be solving is r = 1.)

b. Find the value(s) of k?.

c. Find the values of consumption in both periods for borrowers and lenders?

Homework Assignment 3 Fall 2021

Due: Friday, November 12, 2021, 6pm

Problem 1

Consider the economy with monetary expansion studied in class. Assume that

u(c1, c2) = ln(c1) + c2,

N = 200,

y = 3.

a. Suppose there is no seigniorage. For μ = 1.2, find the real value of the transfer a in the

stationary case.

b. Consider the case of seigniorage. Find function g(μ) discussed in class.

c. Plot the function g(μ) obtained in part b.

Problem 2

Consider the following economy: Individuals are endowed with y units of the consumption good

when young and nothing when old. The fiat money stock is constant. The population grows at

rate n. In each period, the government taxes each young person τ goods. The total proceeds of

the tax are then distributed equally among the old who are alive in that period.

a. Write down the first- and second-period budget constraints facing a typical individual at time

t. (Hint: Be careful; remember that more young people than old people are alive at time t .)

Combine the constraints into a lifetime budget constraint.

b. Find the rate of return on fiat money in a stationary monetary equilibrium.

c. Does the monetary equilibrium maximize the utility of future generations?

d. Does this government policy have any effect on an individual’s welfare?

e. Does your answer to part d change if the tax is larger than the real balances people would

choose to hold in the absence of the tax?

f. Suppose that tax collection and redistribution are (very) costly, so that for every unit of tax

collected from the young, only 0.5 unit is available to distribute to the old. How does your

answer to part d change?

Problem 3

Consider an overlapping generations model with the following characteristics. Each generation is

composed of 1,000 individuals. The fiat money supply changes according to Mt = 2Mt?1. The

initial old own a total of 10,000 units of fiat money (M0 = $10, 000). Each period, the newly printed

money is given to the old of that period as a lump-sum transfer (subsidy). Each person is endowed

with 20 units of the consumption good when born and nothing when old. Preferences are such that

individuals wish to save 10 units when young at the equilibrium rate of return on fiat money.

a. What is the gross real rate of return on fiat money in this economy?

b. How many goods does an individual receive as a subsidy?

c. What is the price of the consumption good in period 1, p1, in dollars?

ECON-4330 Advanced Macroeconomics I Professor Nurlan Turdaliev

Homework Assignment 4 Fall 2021

Due: Friday, December 3, 2021, 6pm

Problem 1

Consider the economy with capital studied in class. Assume the endowment vector (y, 0). Suppose

the production function is f(k) =

√

k and u(c1, c2) = ln c1 + ln c2.

a. Suppose there is no money. Find the equilibrium consumption allocation (c?1, c?2).

b. Now suppose there is money in this economy. Find the equilibrium consumption allocation

(c?1, c?2).

c. Let u?(y) = u(c?1, c?2) and u?(y) = u(c?1, c?2). Calculate the values of u?(y) and u?(y) for the

values y = 1, 5, 10. What is larger, u?(y) or u?(y)?

d. Plot u?(y) and u?(y) for values of y ∈ [0, 10] on the same graph using different colours or

thickness. You can use Maple, Excel, or any other software.

Problem 2

Consider the economy with capital and private debt but without money studied in class. Assume

the endowment vectors are (y, 0) for lenders and (0, y) for borrowers. Suppose the production

function is f(k) =

√

k and u(c1, c2) = ln c1 + ln c2.

a. Find the equilibrium gross interest rate r?. (Remember that r > 0. Note that you may obtain

more than one value of r?. Hint: use the fact that one of the roots of the equation you will

be solving is r = 1.)

b. Find the value(s) of k?.

c. Find the values of consumption in both periods for borrowers and lenders?