Problem Set 3, IDEA-UAB

Solutions

Professor: Francesc Obiols i Homs

TA: Rodica Fazakas

1.1 Given k¯1 , a sequential markets equilibrium (SME) is allocations for households
c̄10 , {(ĉtt , ĉtt+1 , ŝtt )}∞ ∞ ∞
t=1 , allocations for the firm {(K̂t , L̂t )}t=1 and prices {(r̂t , ŵt )}t=1 such
that

• For all t ≥ 1, given (ŵt , r̂t+1 ), (ĉtt , ĉtt+1 , ŝtt )∞

t=1 ) solves

(ctt )1−σ − 1 (ct )1−σ − 1

max t + β t+1
t t
{ct ,ct+1 ,st } 1−σ 1−σ

s.t. ctt + stt ≤ ŵt

ctt+1 ≤ (1 + r̂t+1 − δ)stt

• Given k̄1 and r̂1 , cˆ10 solves

max
0
U (c01 )
{c1 }

s.t. c01 ≤ (1 + r̂1 − δ)k̄1

• For all t ≥ 1, given (r̂t , ŵt ), (K̂t , L̂t ) solves

max Ktθ Lt1−θ − r̂t Kt − ŵt Lt

{Kt ,Lt }

• For all t ≥ 1

– (Goods Market)

Ntt ĉtt + Ntt−1 ĉt−1

t + K̂t+1 − (1 − δ)K̂t = K̂tθ L̂t1−θ

– (Asset Market)
Ntt ŝtt = K̂t+1
– (Labor Market)
Ntt = L̂t

1
1.2 Substitution of first and second period consumptions from the corresponding budget
constraints into the households’ utility function yields:

(wt − stt )1−σ − 1 ((1 + rt+1 − δ)stt )1−σ − 1

max +β
t
{st } 1−σ 1−σ

The first order condition with respect to savings is given by:

1 1
(wt − stt )−σ = β(1 + rt+1 − δ)((1 + rt+1 − δ)stt )−σ ⇔ wt − stt = β − σ (1 + rt+1 − δ)1− σ stt

t
h
− σ1 1− σ1
i wt
⇔ wt = st 1 + β (1 + rt+1 − δ) ⇒ stt = − σ1 1
1 + β (1 + rt+1 − δ)1− σ
From the firm’s maximization problem we have that:
 θ
δYt Kt
wt = = (1 − θ)Ktθ L−θ
t = (1 − θ) ⇔ wt = (1 − θ)ktθ
δLt Lt
 θ−1
δYt Kt
rt = = θKtθ−1 Lt1−θ = θ θ−1
⇔ rt+1 = θkt+1
δKt Lt
Combining the households and firm’s optimality conditions we find that:

(1 − θ)ktθ
stt = − σ1 θ−1 1
1+β (1 + θkt+1 − δ)1− σ

The capital law of motion will then be given by:

t (1 − θ)ktθ
kt+1 = 1 θ−1 1 (1)
(1 + n)[1 + β − σ (1 + θkt+1 − δ)1− σ ]

To investigate monotonicity of transitional dynamics, let

1−θ ktθ
F (kt+1 , kt ) = 1 θ−1 1 − kt+1 = 0
1 + n 1 + β − σ (1 + θkt+1 − δ)1− σ

By the implicit function theorem we have that:

dkt+1 Fk
=− t
dkt Fkt+1

The numerator is:

1−θ θktθ−1
Fkt = 1 > 0
1 + n 1 + β − σ1 (1 + θkt+1
θ−1
− δ)1− σ

while the denominator is:

1−θ ktθ − σ1 1 θ−1 − σ1 θ−2
Fkt+1 = −1 − 1 θ−1 1 β (1 − )(1 + θk t+1 − δ) θ(θ − 1)kt+1
1 + n [1 + β − σ (1 + θkt+1 − δ)1− σ ]2 σ

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Then, we have that:
dkt+1 Fkt
= − σ1 θ−1 θ−2 1 (2)
dkt 1 + Fkt 1
kt
1 β (1 − σ1 )(1 + θkt+1 − δ)− σ θ(θ − 1)kt+1
1+β − σ θ−1
(1+θkt+1 −δ)1− σ

dkt+1
Note that σ ≤ 1 implies dkt
> 0, hence monotone adjustment.

1.3 In a stationary equilibrium, variables per capita are constant over time. Equation (1)
implies that the stationary capital stock kt+1 = kt = k ∗ is given by the solution to the
following equation:
1+n 1 1
(k ∗ )θ−1 = [1 + β − σ (1 + θ(k ∗ )θ−1 − δ)1− σ ]
1−θ
To study stability, we evaluate (2) at the steady state level of the capital stock:

dkt+1 ∗ Fkt (k ∗ )
(k ) = (k∗ )θ−1 1 1
dkt 1 + Fkt (k ∗ ) 1 1
θ−1
β − σ (1 − σ1 )(1 + θkt+1 − δ)− σ θ(θ − 1)
−σ
1+β (1+θ(k∗ )θ−1 −δ)1− σ

dkt+1 ∗ θ
(k ) = 1 θ−1 1
dkt 1 + θ2 (1 + n)( 1−σ
σ
)β − σ (1 + θkt+1 − δ)− σ
dkt+1
It is then easy to see that dkt
(k ∗ ) < 1 adjustment to steady state is monotone, if σ ≤ 1.

1.4
(wt − stt − dt )1−σ − 1 ((1 + rt+1 − δ)stt + dt (1 + n))1−σ − 1
max +β
t
{st } 1−σ 1−σ
The first order condition with respect to savings is given by:

(wt − stt − dt )−σ = β(1 + rt+1 − δ)((1 + rt+1 − δ)stt + dt (1 + n))−σ

1 1
⇔ wt − stt − dt = β − σ (1 + rt+1 − δ)− σ [(1 + rt+1 − δ)stt + dt (1 + n)]
1 1
h 1 1
i
⇔ wt − dt − dt (1 + n)β − σ (1 + rt+1 − δ)− σ = stt 1 + β − σ (1 + rt+1 − δ)1− σ
1 1
wt − dt − dt (1 + n)β − σ (1 + rt+1 − δ)− σ
⇒ stt = 1 1
1 + β − σ (1 + rt+1 − δ)1− σ
δst 1 − σ1 −1
= 1 1 [−1 − (1 + n)β (1 + rt+1 − δ) σ ] < 0
δdt 1 + β − σ (1 + rt+1 − δ)1− σ
1 1
wt − dt − dt (1 + n)β − σ (1 + rt+1 − δ)− σ δkt+1
kt+1 = − σ1 1− σ1
⇒ <0
(1 + n)[1 + β (1 + rt+1 − δ) ] δdt
The introduction of the PAYG system decreases savings and the capital in the steady
state. This is welfare improving if we have that in the model without PAYG

f 0 (k ∗ ) − δ < n

3
holds, meaning that the steady state capital labor ratio in the model without PAYG is
inefficiently high.

2.1 Given k¯1 , a sequential markets equilibrium (SME) is allocations for households
c̄10 , {(ĉtt , ĉtt+1 , ŝtt )}∞ ∞ ∞
t=1 , allocations for the firm {(K̂t , L̂t )}t=1 and prices {(r̂t , ŵt )}t=1 such
that

• For all t ≥ 1, given (ŵt , r̂t+1 ), (ĉtt , ĉtt+1 , ŝtt )∞

t=1 ) solves

max log(ctt ) + βlog(ctt+1 )

{ctt ,ctt+1 ,stt }

s.t. ctt + stt ≤ ŵt

ctt+1 ≤ (1 + r̂t+1 )stt

• Given k̄1 and r̂1 , cˆ10 solves

max
0
U (c01 )
{c1 }

s.t. c01 ≤ (1 + r̂1 )k̄1

• For all t ≥ 1, given (r̂t , ŵt ), (K̂t , L̂t ) solves

max Ktα Lt1−α − δKt − r̂t Kt − ŵt Lt

{Kt ,Lt }

• For all t ≥ 1

– (Goods Market)

Ntt ĉtt + Ntt−1 ĉt−1

t + K̂t+1 − K̂t = K̂tα L̂t1−α − δ K̂t

– (Asset Market)
Ntt ŝtt = K̂t+1
– (Labor Market)
Ntt = L̂t

2.2 Substitution of first and second period consumptions from the corresponding budget
constraints into the households’ utility function yields:

max
t
log(wt − stt ) + βlog((1 + rt+1 )stt )
{st }

The first order condition with respect to savings is given by:

1 1 + rt+1 1 β β
− t
+β t
=0 ⇔ t
= t ⇒ st = wt
wt − st (1 + rt+1 )st w t − st st 1+β

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Note that, given the log form of the utility function, savings are independent function of
the interest rate, rt+1 . The competitive firm pays labor its marginal product, so we have:
δYt
wt = = (1 − α)Ktα L−α
t = (1 − α)ktα
δLt
The difference equation characterizing the dynamic equilibrium will then be given by:

1−α β
kt+1 = kα
1+n1+β t

Monotone adjustment to the steady state is indeed a feature of this economy as:
δkt+1 1−α β
=α k α−1 > 0 (3)
δkt 1+n1+β t
The steady state level of the capital stock is then:
 1
 1−α
∗ 1−α β
k =
1+n1+β

A necessary and sufficient condition for stability is that (3) evaluated at the steady state
capital stock is less than 1:
 α−1
 1−α
δkt+1 ∗ 1−α β 1−α β
(k ) = α =α<1
δkt 1+n1+β 1+n1+β

2.3 The production function is constant returns to scale as:

2Kt 2Lt Kt
log(1 + ) = 2Lt log(1 + ) = 2Yt
2Lt Lt
Yt Kt
This means we can write it in the intensive form f (kt ) = Lt
= log(1 + Lt
) = log(1 + kt ).
The wage is then given by:
kt
wt = log(1 + kt ) −
1 + kt
The saving function is the same as before:
β
st = wt
1+β
So, the (kt+1 , kt ) locus can be described by the following equation:
 
β kt
kt+1 = log(1 + kt ) −
(1 + n)(1 + β) 1 + kt
   
δkt+1 β 1 1 + kt − kt β kt
⇒0< = − = <1
δkt (1 + n)(1 + β) 1 + kt (1 + kt )2 (1 + n)(1 + β) (1 + kt )2
for any kt . Hence, the equilibrium is stable.