Macroeconomics Theory II
Francesco Franco
Novasbe
February 2016
Francesco Franco (Novasbe)
Macroeconomics Theory II
February 2016
1 / 20
The classical model w/o capital
In the standard business RBC model capital is central. Here we simplify
and abstract from capital (no investment) but it is easily reintroduced. We
are going to follow Gali 2015 and specify an “RBC” without capital
starting by the households:
•
max E0
 bt U (Ct , Nt )
t =0
subject to
Pt Ct + Qt Bt  Bt
1
+ W t N t + Dt
and a no-Ponzi condition. Notice that in the previous slides we had
Pt Ct + Pt Bt  (1 + it 1 )Pt Bt 1 + Wt Nt + Dt , we are just redifining the
Bonds as an asset that deliver one unit next period in nominal terms.
Francesco Franco (Novasbe)
Macroeconomics Theory II
February 2016
2 / 20
FOC
The first order conditions are the familiar
Un,t
Wt
=
Uc,t
Pt
Qt = bE
Francesco Franco (Novasbe)
⇢
Uc,t +1 Pt
Uc,t Pt +1
Macroeconomics Theory II
February 2016
3 / 20
Specific functional forms
We are going to focus on a specific functional form (simplify the algebra +
implications we have discussed)
U (Ct , Nt ) =
Ct1
1
s
1
s
1+ j
Nt
1+ j
which fives the following FOC
Wt
j
= Cts Nt
Pt
(✓
)
◆ s
Ct +1
Pt
Qt = bEt
Ct
Pt + 1
Francesco Franco (Novasbe)
Macroeconomics Theory II
February 2016
4 / 20
Log-Linear model
We want to focus on dynamics (business cycles), we know that we are
going to work with linear approximations.The advantage of the above
specification is that in log-linear form the foc are
pt = sct + jnt
wt
our labor supply equation. And
c t = Et { c t + 1 }
1
( it
s
Et { p t + 1 }
r)
where pt ⌘ pt pt 1 , it ⌘ log Qt , r ⌘ log b. If you had a money in
the utility (separable and isoelastic) you would also get
mt
Francesco Franco (Novasbe)
p t = ct
hit
Macroeconomics Theory II
February 2016
5 / 20
Firms
We are going to work with a technology that only depends on labor and
technology
Yt = At Nt1
a
with at ⌘ log At the TFP that follows an AR(1)
at = r a at
1
+ #at
they maximize profits
max Pt Yt
Francesco Franco (Novasbe)
W t Nt
Macroeconomics Theory II
February 2016
6 / 20
FOC
the FOC is
Wt
= (1
Pt
a ) A t Nt
a
which in log-linear terms is
wt
pt = at
ant + log(1
a)
which is the labor demand.
Francesco Franco (Novasbe)
Macroeconomics Theory II
February 2016
7 / 20
Equilibrium
Goods market clearing
yt = c t
Labor market clearing
sct + jnt = at
ant + log(1
a)
Asset market clearing (zero net supply)
Bt = 0
and
it
Et {pt +1 } = r + sEt {Dct +1 }
Aggregate output
yt = a t + ( 1
Francesco Franco (Novasbe)
a ) nt
Macroeconomics Theory II
February 2016
8 / 20
Solution
Use the undetermined coefficient method by postulating policy functions
for quantities:
nt = yna at + yn
yt = yya at + yy
and prices:
rt = r
w t ⌘ wt
syya (1
r a ) at
pt = ywa at + yw
(here you have a constant because the approximation is only for the Euler
equation). The values of the coefficients are in Gali.
Francesco Franco (Novasbe)
Macroeconomics Theory II
February 2016
9 / 20
Solution
As expected the allocations do not depend on the fluctuations of
technology, even if you had a monetary authority controling m or i it
would not a↵ect the allocations: neutrality. The only thing that monetary
policy does is potentially to determine the nominal variables as in the last
class. Potentially because not every interest rate monetary policy rule
determines the nominal variables. We will do exercise on this. Let us play
with the computer. (class5.mod)
Francesco Franco (Novasbe)
Macroeconomics Theory II
February 2016
10 / 20
Neo Classical Ingredients
Last class we have found that in this class of models, P m = P1 the price of
money (purchasing power in terms of goods), was really another asset
price, where the dividends where liquidity/utils services:
⇣
⌘
P
m
•
=
Âb
j =1
j
µt +j
Pt + j
lt
.
The price of a unit of money is the discounted sum of all periods expected
direct marginal utility of money divided by the present MU of wealth. The
other side of the medal is that the price level $P$ the price of goods in
terms of money is also an asset price.
Francesco Franco (Novasbe)
Macroeconomics Theory II
February 2016
11 / 20
New Keynesian Ingredients
The price level is the aggregate of millions of individual prices. It is
unlikely to behave like an asset price. We have to think in terms of price
setters and individual price setters might react at di↵erent times.
WhenP reacts slowly the adjustment within the model is di↵erent.Take the
foc wrt to M/P in the MIU with Umc > 0
Um (Ct , Mt /Pt )
it
=
Uc (Ct , Mt /Pt )
If Mt " andPt ! it # This will decrease rt , and a↵ect C and I .
Francesco Franco (Novasbe)
Macroeconomics Theory II
February 2016
12 / 20
The New Keynesian Model
monopolistic competition: price setters
sticky prices (staggered price setting):real e↵ects of nominal interest rate
(real)
competitive labor markets, no capital accumulation: two previous
ingredients nested on previous classical model
Francesco Franco (Novasbe)
Macroeconomics Theory II
February 2016
13 / 20
Households
Representative agent solves:
•
max E0
 bt U (Ct , Nt ; Zt )
t =0
where
Ct ⌘
✓ˆ
1
Ct (i )
1
1
e
di
0
◆ e e1
subject to
ˆ
0
1
Pt (i )Ct (i )di + Qt Bt  Bt
1
+ W t N t + Dt
plus no Ponzi condition.
Francesco Franco (Novasbe)
Macroeconomics Theory II
February 2016
14 / 20
The demand for goods
Each household faces a demand curve for its product, which we shall have
to derive (the demand for the good by all other consumers.) Think of the
total expenditure of a household
ˆ 1
Xt =
Pt (i )Ct (i )di
0
The allocation of expenditure in the various consumption goods is an
intratemporal decision.
Francesco Franco (Novasbe)
Macroeconomics Theory II
February 2016
15 / 20
The demand for goods
You can see it as households facing a the cost minimization problem in
buying the bundle:
ˆ 1
min Xt =
Pt (i )Ct (i )di
Ct (i )
subject to Ct ⌘
⇣´
1
1
0 Ct (i )
Francesco Franco (Novasbe)
0
1
e
di
⌘ e e1
Macroeconomics Theory II
February 2016
16 / 20
The demand for goods
The cost minimization problem is nice for the lagrange multiplier
associated with the consumption bundle constraint in the price deflator:
Pt ⌘
✓ˆ
1
Pt ( i )
1 e
di
0
◆ 11 e
Using the price deflator definition the minimization problem gives the
demand of each good
Ct (i ) =
and in the budget constraint
ˆ 1
✓
Pt ( i )
Pt
◆
e
Ct
Pt (i )Ct (i )di = Pt Ct
0
Francesco Franco (Novasbe)
Macroeconomics Theory II
February 2016
17 / 20
FOC
The other first order conditions are as before now
Un,t
Wt
=
Uc,t
Pt
Qt = bE
Francesco Franco (Novasbe)
⇢
Uc,t +1 Pt
Uc,t Pt +1
Macroeconomics Theory II
February 2016
18 / 20
Specification
U (Ct , Nt ; Zt ) =
where zt = rz zt
1
Ct1
1
s
1+ j
1
Nt
1+ j
s
!
Zt
+ etz and the log-linear foc
pt = sct + jnt
wt
c t = Et { c t + 1 }
Francesco Franco (Novasbe)
1
( it
s
Et { p t + 1 }
Macroeconomics Theory II
r) +
1
(1
s
r z ) zt
February 2016
19 / 20
Next Class the Firms
They produce the continuum of goods, one firm one good, they face a
downward sloping demand, and they cannot adjust price as they would
always like.
Francesco Franco (Novasbe)
Macroeconomics Theory II
February 2016
20 / 20