Monetary Policy and Macroeconomic Dynamics

All variables in the model below are measured in percentage deviations from their long-run levels
(percentage deviations of gross rates in the cases of ináation and the interest rate).
Suppose that the dynamics of output (yt) are determined by the following intertemporal condition:
yt = Et (yt+1) + yt1  [it Et (t+1)] + ut
; (1)
where 0   < 1 captures the impact of past output dynamics on current output (for instance, because of habits in consumption behavior),  > 0 is the elasticity of intertemporal substitution,
it
is the nominal interest rate between t and t + 1 decided by the central bank at time t, Et (:) is
the expectation operator, t+1 is ináation at t + 1, and ut
is an exogenous output demand shock.
Ináation is determined by the following modiÖed New Keynesian Phillips curve:
t = t1 + yt + Et (t+1) + zt
; (2)
where 0   < 1 captures persistence in ináation implied, for instance, by indexation mechanisms,  > 0,  2 (0; 1) is the representative householdís subjective discount factor, and zt
is an exogenous
supply-side shock.
Finally, the central bank sets the interest rate to respond to ináation and output according to
the Taylor rule:
it =  1t +  2yt + t
; (3)
where the policy response coe¢ cients are such that  1 > 1 and  2  0, and t
is an exogenous
interest rate shock.
Assume that the shocks ut
, zt
, and t are such that:
ut = uut1 + “u;t; (4)
zt = z
zt1 + “z;t; (5)
and
t = 
t1 + “;t; (6)
where the persistence parameters u
, z
, and  are all strictly between 0 and 1, and “u;t, “z;t,
“;t are identically and independently distributed innovations with zero mean and variance 
2
“u
, 
2
“z
,
and 
2
“
, respectively.
We can guess that the solution of the system (1)-(3) can be written as:
yt = yyyt1 + yt1 + yuut + yzzt + yt
; (7)
t = yyt1 + t1 + uut + zzt + t
; (8)
and
it = iyyt1 + it1 + iuut + izzt + it
; (9)
where the ís are elasticities that can be obtained with the method of undetermined coe¢ cients.
 Explain why we make this guess based on the information I have given you.
 Now assume that the parameters  and  are such that  =  = 0. What do you
think happens to yy, y, y, , iy, and i in this case