1. Government Spending Shocks in the Basic Real Business Cycle Model

Consider an economy with identical infinitely-lived agents with pref- erences given by

E0  t βt  ┌ Ct1_σ _ Nϕ ┐

where Ct  is consumption and Nt  are hours worked. Each consumer seeks to maximise utility subject to a sequence of budget constraints:

Ct + Qt Bt = Wt Nt + Bt_1 + Dt _ Tt

for all t and states where Wt  denotes the real wage, Bt  stands for the stock of one period real bond (which is in zero net supply), Qt  represents the real price of the bond, Dt  are firm dividends and Tt  are lump sum taxes.  The representative firm has access to the following production technology:

Yt = At Nt1_α

with Yt  and At  being output and an exogenous technology shock respectively.   Moreover, at   =  log At  follows a stationary AR(1) process:

at = ρaat_1 + εt(a)

with ρa  e [0, 1) and {εt(a)( is a white noise.

Finally, assume that the government wishes to purchase an exoge- nous stream of government spending, {Gt ( and in each period it finances the purchase Gt  by collecting lump-sum taxes from the representative household.  Put differently, in each period output is produced in the private sector, and the government purchases an exogenous amount Gt of this output, with the remainder consumed by the representative household.  In addition, t  = log Gt _ log G follows a stationary AR(1) process:

t = ρg t_1 + εt(g)

with ρg  e [0, 1) and {εt(g)( is a white noise.

(a) Obtain the first-order conditions to the household and the firm

(b) Write down the government budget constraints.                        [7]

(c) List all the market clearing conditions and provide a definition      of competitive equilibrium for this economy.                             [8]

(d) Suppose χ =  where G and Y are the values of Gt  and Yt at the deterministic steady state. Express output, consumption, hours worked, government spending, the real interest rate and the real wage at the deterministic steady state as a function of χ and the exogenous parameters of the model.

Hint:  recall that the goods market clearing condition at the

deterministic steady state can be written as Y = C + χY .        [10]

(e) Assume β = 0.99, α = 0.25, ϕ = 5, σ = 1, ρg  = 0.9 and χ =       0.25. Plot the dynamic responses of output, consumption, hours       worked, the real interest rate and the real wage to a 1% increase       in government spending. Explain the transmission mechanism of       this shock. Is this a good candidate to reconcile the predictions       of the real business cycle model with the business cycle statistics       we observe in the data? Explain why or why not.                      [17]

2. Monetary  Policy  shocks  under  an  Exogenous  Monetary Supply in the Basic New Keynesian Model

Consider a version of the basic New Keynesian model with no pref-erence shocks such that:

πt = κy˜t + βEt {πt+1(

y˜t = Et {y˜t+1( _  [it _ Et {πt+1( _ rt(n)]

rt(n) = ρ + Et {at+1 _ at (

with κ = (σ + ) and the notation being as in class.  Assume as usual that at  is an exogenous TFP shock which follows a stationary AR(1) process at  = ρaat_1 + εt(a) .  where ρa  e [0, 1) and {εt(a)( is a white noise. Moreover, assume that the money supply grows according to the following law of motion:

∆mt = ρm ∆mt_1 + εt(m)

where ρm  e [0, 1) and {εt(m)( is a white noise.  In this context it is useful to rewrite money demand condition in terms of the output gap as follows:

lt = y˜t _ ηit + yt(n)

where lt  = mt  _ pt  and η  > 0.  It also helps to recall that real

balances are related to inflation and money growth through the fol-

lowing identity:

t_1 = t + πt _ ∆mt

(a) Assume β  =  0.99,  α  =  0.25,  ϕ  =  5,  σ  =  1, ρm   =  0.5       ε  =  9, η  =  4 and θ  =  0.75.   Plot the dynamic responses       of output, output gap, consumption, hours worked, inflation,       the nominal interest rate, the real interest rate, the real wage       and the money supply to a 0.25 increase in εt(m)  (See the files       Example Basic NK Model ExMonRules in the Matlab folder) .   [20]

(b) Discuss intuitively the effects of an expansionary monetary policy       shock found in point (a). Compare these effects with the VAR       evidence on monetary policy shocks discussed in class.               [14]