Monetary Economics Discussion Assignment

Monetary Economics Discussion Assignment

Monetary Economics HW # 2

Part 1:

I- A. Consider a Money-in-the-Utility model of the kind we studied in the class with no population growth, i.e., n = 0: A-1) Assume that the aggregate production function is Yt = F(Kt-1,Nt) =Kt-1αNt1-α; Kt-1 is total capital stock at the beginning of t carried over from t-1; Nt is total physical labor; and 0 <α < 1. From the above aggregate production function, we can derive the per capita output yt = (Yt/Nt) in terms of kt-1=(Kt-1/Nt-1) and n, population or labor growth rate. A-2) Assume population growth, n = 0 A-3) The representative agent can hold money, nominal debt, and capital for asset, and receives per capita real money transfer = τt at the beginning of each period t. A-4) Suppose the representative agent chooses the time path of ct (consumption), kt (capital), bt (real per capita nominal bonds), and mt (real pre capita money holdings) for all t>=0 to maximize lifetime discounted utility given by (1) Σt=0 βt u (ct, mt) subject to the following per capita budget constraint (2) f(kt-1) + (1-δ)kt-1 + [(1+it-1) bt-1] /(1+πt) + τt + mt-1/(1+πt) = ct + kt + bt + mt for all t >=0 where β =individual subjective discount rate, it-1= nominal interest rate at time t-1, δ=a constant capital depreciation rate, and πt= inflation rate The first order conditions of the optimization problems are given by the following equations:

Monetary Economics Discussion Assignment

(3) ),( ttc mcu = β ),( 11 ++ ttc mcu ( δ−+1)(‘ tkf ) for all t >=0

(4) ),( ttc mcu = β ),( 11 ++ ttc mcu )1( )1(

1++ +

t

ti π

for all t >=0

(5) ),( ttc mcu = ),( ttm mcu + β )1( )),((

1

11

+

++

+ t ttc mcu

π for all t >=0

Monetary Economics Discussion Assignment

Q I-A-1) Show that the above production function is a constant return to scale production function jointly with respect to capital (= Kt-1) and labor (Nt), and derive the per capita

output yt = (Yt/Nt) in terms of kt-1=(Kt-1/Nt-1) and n (population or labor growth rate) using the constant return to scale property of the production function. Q I-A-2) Explain what each of the first order conditions (equation (3)-(5)) says and why they are necessary conditions for intertemporal utility maximization. I-B This time, let’s consider the following variation of Sidrauski’s MIU model, augmented with variable labor supply. B-1) Per capita production function: 𝑦𝑦𝑡𝑡 = 𝑓𝑓(𝑘𝑘𝑡𝑡−1,

`𝑛𝑛𝑠𝑠) = 𝑓𝑓(𝑘𝑘𝑡𝑡−1,1 − 𝑙𝑙), where total available individual labor supply time is normalized to 1, and 𝑛𝑛𝑠𝑠 𝑎𝑎𝑛𝑛𝑎𝑎 𝑙𝑙, are per capita labor supply and leisure, respectively. So, 𝑛𝑛𝑠𝑠 = 1 − 𝑙𝑙 is the variable labor supply, which is another choice variable. B-2) Other than money (m: per capita real money), and capital (k: per capita capital), there exists 1-period risk-free nominal bond (bt : per capita (real) bond holding) which pays risk-free interest rate it. B-3) Individual utility function: 𝑢𝑢𝑡𝑡 = 𝑢𝑢(𝑐𝑐𝑡𝑡, 𝑚𝑚𝑡𝑡, 𝑙𝑙𝑡𝑡), 𝑢𝑢𝑐𝑐, 𝑢𝑢𝑚𝑚, 𝑢𝑢𝑙𝑙 > 0, 𝑢𝑢𝑐𝑐𝑐𝑐, 𝑢𝑢𝑚𝑚𝑚𝑚, 𝑢𝑢𝑙𝑙𝑙𝑙 < 0 B-4) Individual household’s objective: Max ∑ 𝛽𝛽𝑡𝑡+𝑖𝑖𝑢𝑢(𝑐𝑐𝑡𝑡+𝑖𝑖, 𝑚𝑚𝑡𝑡+𝑖𝑖, 𝑙𝑙𝑡𝑡+𝑖𝑖),∞𝑖𝑖=0 where 0 < 𝛽𝛽 < 1 is subjective discount rate. B-5) Assume no population growth: n=0 B-6) In each period, gov’t makes 𝜏𝜏𝑡𝑡 amount of real per capita money transfer and no tax. Q I-B-1) The first order conditions for the above intertemporal optimization problem is virtually the same as the conditions we derived for the Sidrauski model except we have an additional first order condition for 𝒏𝒏𝒔𝒔 𝒐𝒐𝒐𝒐 𝒍𝒍. The first order condition for 𝒍𝒍 is given by 𝒇𝒇𝒏𝒏�𝒌𝒌𝒕𝒕−𝟏𝟏, `𝒏𝒏𝒔𝒔�𝒖𝒖𝒄𝒄 = 𝒖𝒖𝒍𝒍 where 𝒇𝒇𝒏𝒏�𝒌𝒌𝒕𝒕−𝟏𝟏, `𝒏𝒏𝒔𝒔� = 𝝏𝝏𝒇𝒇 𝝏𝝏𝒏𝒏 , 𝒖𝒖𝒄𝒄 = 𝝏𝝏𝒖𝒖 𝝏𝝏𝒄𝒄 , 𝒖𝒖𝒍𝒍 = 𝝏𝝏𝒖𝒖 𝝏𝝏𝒍𝒍 , for all t Interpret the above first order condition. What does the condition say and why is the above condition a necessary condition for intertemporal utility maximization? Q I-B-2) Suppose utility function takes the following form: 𝒖𝒖𝒕𝒕 = 𝒖𝒖(𝒄𝒄𝒕𝒕, 𝒎𝒎𝒕𝒕, 𝒍𝒍𝒕𝒕) = (𝒄𝒄𝒕𝒕𝒎𝒎𝒕𝒕)𝒃𝒃𝒍𝒍𝒕𝒕𝒅𝒅. Derive individual household’s money demand function in terms of consumption and interest rate. (Hint: read page 46 and 47 of Walsh and page 29 of the lecture note) Q II. For each ARMA model below, show how an unit shock (i.e.,1) to X at t (i.e., et =1) affects Xt+i, for i=0,1,2,3,…. manually and draw a plot for it. The plot would show the impact of an unit shock at t on Xt+i (y axis) against different i (x axis). You can do this by converting ARMA to MA representation and see how an unit value (i.e.,1) of white noise at i-period before (i.e., et-i ) affects Xt. We call the plot impulse response function graph. Do this manually (i.e., do not use a software). You confirm your results with Matlab. (First specify each model using ARIMA command. Then use Impulse command. For the second ARMA model, Matlab will complain that the model is unstable and will not give you the impulse response. So, use the command to check just 1st and 3rd ARMA models. Do not submit the Matlab results.) Xt = 0.7Xt-1 + et , where et ~ iid. N(0.1) Xt =Xt-1 + et , where et ~ iid. N(0.1) Xt = et + 0.3et-1 + 0.1et-2, where et ~ iid. N(0.1) Part 2: III. Consider the following Stochastic Euler Equation discussed in the class: 𝐸𝐸𝑡𝑡[𝛽𝛽 𝑢𝑢𝑐𝑐𝑡𝑡+1 𝑢𝑢𝑐𝑐𝑡𝑡 (1 + �̃�𝑟𝑡𝑡+1 𝑖𝑖 ) ] = 1 and 𝐸𝐸𝑡𝑡[𝛽𝛽 𝑢𝑢𝑐𝑐𝑡𝑡+1 𝑢𝑢𝑐𝑐𝑡𝑡 (1 + 𝑟𝑟𝑡𝑡+1 𝑓𝑓 ) ] = 1   where 𝑟𝑟𝑡𝑡+1 𝑓𝑓 , �̃�𝑟𝑡𝑡+1 𝑖𝑖   are returns on risk-free and risky asset i, between t and t+1, respectively. 𝑟𝑟𝑡𝑡+1 𝑓𝑓 is known at t (i.e., risk free) at t, but �̃�𝑟𝑡𝑡+1 𝑖𝑖 is unknown until at t+1. Let 𝑚𝑚𝑡𝑡,𝑡𝑡+1 = 𝛽𝛽 𝑢𝑢𝑐𝑐𝑡𝑡+1 𝑢𝑢𝑐𝑐𝑡𝑡 , the Marginal Rate of Substitution between t and t+1, be the stochastic discount factor (á la Campbell, JF 2000). A-1) Utility takes the following constant relative risk aversion utility function: 𝑢𝑢 = 𝑐𝑐 1−𝛾𝛾−1 1−𝛾𝛾 , 𝛾𝛾 > 0

A-2) 1 1

+ + = t t

t x c

Monetary Economics Discussion Assignment

c , (gross) growth rate of consumption, is log-normally distributed

with 𝑙𝑙𝑛𝑛( 𝑐𝑐𝑡𝑡+1 𝑐𝑐𝑡𝑡  )~𝑁𝑁( 𝜇𝜇𝑥𝑥, 𝜎𝜎𝑥𝑥2).

Q III-1) What is the mean and variance of 𝑐𝑐𝑡𝑡+1

𝑐𝑐𝑡𝑡 ? (You can find the formula for the expected

value and variance of random variable with log-normal distribution from Wikipedia or any statistics textbook.) For variance, express it in terms of mean and 𝜎𝜎𝑥𝑥2 .

Q III-2) Using the consumption CAPM equation 𝐸𝐸𝑡𝑡��̃�𝑟𝑡𝑡+1 𝑖𝑖 − 𝑟𝑟𝑡𝑡+1

𝑓𝑓 � = −[ 𝑐𝑐𝑐𝑐𝑐𝑐𝑡𝑡( 𝑚𝑚𝑡𝑡,𝑡𝑡+1 ,�̃�𝑟𝑡𝑡+1

𝑖𝑖 −𝑟𝑟𝑡𝑡+1 𝑓𝑓

𝐸𝐸𝑡𝑡(𝑚𝑚𝑡𝑡,𝑡𝑡+1) ]

Show that 𝐸𝐸𝑡𝑡[ �̃�𝑟𝑡𝑡+1 𝑖𝑖 −𝑟𝑟𝑡𝑡+1

𝑓𝑓

𝜎𝜎𝑟𝑟𝑖𝑖 ] = −𝜌𝜌𝑚𝑚,𝑟𝑟

𝜎𝜎𝑚𝑚 𝐸𝐸𝑡𝑡(𝑚𝑚𝑡𝑡,𝑡𝑡+1)

where 𝜌𝜌𝑚𝑚,𝑟𝑟 = 𝑐𝑐𝑐𝑐𝑟𝑟𝑟𝑟𝑡𝑡(𝑚𝑚𝑡𝑡,𝑡𝑡+1, �̃�𝑟𝑡𝑡+1 𝑖𝑖 )

Q III-3) Using II-1) and II-2), Show that �𝐸𝐸𝑡𝑡[ �̃�𝑟𝑡𝑡 𝑖𝑖−𝑟𝑟𝑡𝑡

𝑓𝑓

𝜎𝜎𝑟𝑟�𝑖𝑖 ]� ≤ 𝜎𝜎𝑚𝑚

𝐸𝐸(𝑚𝑚𝑡𝑡,𝑡𝑡+1) = �𝑒𝑒𝛾𝛾2𝜎𝜎𝑥𝑥2 − 1

(The above inequality is known as Hansen- Jagannathan bound and left hand side of inequality is known as Sharpe Ratio. ) In order to show above,

a) Use 1 ),cov(

1 , ≤=≤− yx

yx yx

σσ ρ and

b) Then, using the fact that a linear transformation of normal random variable is also normally distributed (i.e., if y ~𝑁𝑁( 𝜇𝜇𝑦𝑦, 𝜎𝜎𝑦𝑦2) then a+by ~𝑁𝑁( 𝑎𝑎 + 𝑏𝑏𝜇𝜇𝑦𝑦, 𝑏𝑏2𝜎𝜎𝑦𝑦2)), show that 𝑚𝑚𝑡𝑡,𝑡𝑡+1 is also log- normally distributed. (i.e., ln(m) is normally distributed.) c) Finally, derive mean and variance of ln(m), m, and apply the formula from I-1). Q III-4) Show that

22 2 1

)ln()1ln( xx f

tr σγγµβ −+−=+

(Hint: From the Euler equation for risk free asset

�1 + 𝑟𝑟𝑡𝑡+1 𝑓𝑓 � = [

1 𝐸𝐸𝑡𝑡(𝑚𝑚𝑡𝑡,𝑡𝑡+1)

]

Q III-5) Suppose in the past 50 years, US real stock reruns have averaged about 9% with standard deviation of about 16%. On the other hand, average real return on T-Bills (risk free) is about 1%. Mean and Standard deviation of aggregate consumption growth per annum is about 1.8% and 1% respectively. What do these facts imply about the range of coefficient of Relative Risk Aversion γ? IV. Go to the following Federal Reserve Site: https://www.federalreserve.gov/monetarypolicy/bst.htm and https://www.federalreserve.gov/monetarypolicy/policytools.htm The sites provide a collection of resources for describing the Federal Reserve’s current monetary policy tools, as well as the special programs that the Federal Reserve implemented to address the financial crisis that emerged during the summer of 2007 and also in response to Covid-19 ( https://www.federalreserve.gov/publications/reports-to-congress-in-response-to-covid-19.htm ), including information about the Credit and Liquidity facilities FRB created during the crises and FRB’s historical balance sheet information.

https://www.federalreserve.gov/monetarypolicy/bst.htm
https://www.federalreserve.gov/publications/reports-to-congress-in-response-to-covid-19.htm
Compare FRB’s balance sheet trend (size and components in both assets and liabilities sides) at five different historical snapshots: January 2007, January 2009, January 2015, January 2020, January 2021 and summarize the changes in trend. You can find the weekly historical balance sheet information of the FRBs going back to 1996 at https://www.federalreserve.gov/releases/h41/about.htm ).

https://www.federalreserve.gov/releases/h41/about.htm

 

Essay writing help – Monetary Economics Discussion Assignment Online Essay Writing Agency – Grade Master-Pro.

Write my Essay. Premium essay writing services is the ideal place for homework help or essay writing service. if you are looking for affordable, high quality & non-plagiarized papers, click on the button below to place your order. Provide us with the instructions and one of our writers will deliver a unique, no plagiarism, and professional paper.

Get help with your toughest assignments and get them solved by a Reliable Custom Papers Writing Company. Save time, money and get quality papers. Buying an excellent plagiarism-free paper is a piece of cake!

All our papers are written from scratch. We can cover any assignment/essay in your field of study.