monetary econ

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-1Nt1-; Kt-1 is total capital stock at the beginning of t carried over from t-1; Nt is total physical labor; and 0 < < 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) t=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:

(3) ),( ttc mcu =  ),( 11  ttc mcu (  1)(' tkf ) for all t >=0

(4) ),( ttc mcu =  ),( 11  ttc mcu )1(

)1(

1 

t

ti

 for all t >=0

(5) ),( ttc mcu = ),( ttm mcu +  )1(

)),((

1

11





 t ttc mcu

 for all t >=0

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: 𝑦 𝑓 𝑘 ,

`𝑛 𝑓 𝑘 ,1 𝑙 , where total available individual labor supply time is normalized to 1, and 𝑛 𝑎𝑛𝑑 𝑙, are per capita labor supply and leisure, respectively. So, 𝑛 1 𝑙 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: 𝑢 𝑢 𝑐 , 𝑚 , 𝑙 , 𝑢 , 𝑢 , 𝑢 0, 𝑢 , 𝑢 , 𝑢 0 B-4) Individual household’s objective: Max ∑ 𝛽 𝑢 𝑐 , 𝑚 , 𝑙 , where 0 𝛽 1 is subjective discount rate. B-5) Assume no population growth: n=0 B-6) In each period, gov’t makes 𝜏 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 𝒏𝒔 𝒐𝒓 𝒍. The first order condition for 𝒍 is given by

𝒇𝒏 𝒌𝒕 𝟏, `𝒏𝒔 𝒖𝒄 𝒖𝒍

where 𝒇𝒏 𝒌𝒕 𝟏, `𝒏𝒔

𝝏𝒇

𝝏𝒏 , 𝒖𝒄

𝝏𝒖

𝝏𝒄 , 𝒖𝒍

𝝏𝒖

𝝏𝒍 , 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: 𝒖𝒕 𝒖 𝒄𝒕, 𝒎𝒕, 𝒍𝒕 𝒄𝒕𝒎𝒕 𝒃𝒍𝒕

𝒅. 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. 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:

𝐸 𝛽 1 �̃�   1 and 𝐸 𝛽 1 𝑟   1   where 𝑟 , �̃�   are returns on

risk-free and risky asset i, between t and t+1, respectively. 𝑟 is known at t (i.e., risk free) at t, but �̃� is unknown until at t+1. Let 𝑚 , 𝛽 , 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:

𝑢 , 𝛾 0

A-2) 1 1

   t t

t x c

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

with 𝑙𝑛   ~𝑁  𝜇 , 𝜎 .

Q III-1) What is the mean and variance of ? (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 𝜎 .

Q III-2) Using the consumption CAPM equation 𝐸 �̃� 𝑟 , , ̃

,    

Show that 𝐸 ̃

𝜌 , ,

   where 𝜌 , 𝑐𝑜𝑟𝑟 𝑚 , , �̃�    

Q III-3) Using II-1) and II-2), Show that 𝐸 ̃

  ,

𝑒 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

  and

b) Then, using the fact that a linear transformation of normal random variable is also normally distributed (i.e., if y ~𝑁  𝜇 , 𝜎 then a+by ~𝑁  𝑎 𝑏𝜇 , 𝑏 𝜎 ), show that 𝑚 , 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  

(Hint: From the Euler equation for risk free asset

1 𝑟 1

𝐸 𝑚 ,    

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.

Compare FRB’s balance sheet (both assets and liabilities sides) at five different historical snapshots: January 2007, January 2009, January 2015, January 2020, July 2020 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 ).