Q2
Assume a Cobb-Douglas production function in a deterministic growth model. Assume leisure is inelastically
supplied.
a) Show that the capital-output ratio and the rental rate on capital in a steady state is invariant to
changes in
• a proportional tax on labor income;
• changes in a proportional tax on consumption;
• invariant to permanent TFP changes;
• not invariant to changes in a tax on capital income
b) Who bears the burden of a tax on labor income?
Solution:
a) Let the aggregate production function be given by
F (k, n) = Ak α n1−α
α ∈ (0, 1)
Consider a TDCE with flat rate taxes on ct (τct ), on wt nt (τnt ), and rt kt (τkt )
From the HH problem, we obtain the following FOC’s and a labor supply condition:
β t Uct
= λPt (1 + τct )
1
(lt = 0)
(1)
if
wt (1 − τnt ) > 0
nt
=
Pt
= Pt+1 (1 − δ) + rt+1 Pt+1 (1 − τkt+1 )
(2)
(3)
From the firm’s problem, we have
F easibility :
Fkt
= rt
(4)
Fnt
= wt
(5)
= ct + kt+1 − (1 − δ)kt + gt
(6)
F (kt , nt )
Then, from (18), (20)-(23), and the production function, we can obtain:
Uct [Aktα − kt+1 + (1 − δ)kt − gt ]
1 + τct
α−1
α
=β
Uct+1 [Akt+1
− kt+2 + (1 − δ)kt+1 − gt+1 ][αAkt+1
(1 − τkt+1 ) + (1 − δ)]
1 + τct+1
Thus, assuming τct → τc , τkt → τk ,τn t → τn and gt → g
1
=
α−1
β[αAkss
(1 − τk ) + (1 − δ)]
⇒
α−1
βαAkss
(1 − τk ) = 1 − β(1 − δ)
1
1 − β(1 − δ) α−1
kss =
βαA(1 − τk )
⇒
(7)
and the rental rate on capital (in real terms) is
rss
α−1
= Fk (ks s, 1) = αAkss
= αA
=
1 − β(1 − δ)
β(1 − τk )
1 − β(1 − δ)
βαA(1 − τk )
(8)
1
and the capital output ratio in SS is
βα(1 − τk )
kss
kss
1−α
= A−1 kss
=
=
α
yss
Akss
1 − β(1 − δ)
As you can see from (25) and (26), rss and
( let’s call it r)
(9)
kss
are independent of the steady state level of τn , τc and
yss
A(T F P ). The same is not true for τk :
drss
dτk
dr
dτk
=
=
1 − β(1 − δ)
>0
β(1 − τk )2
βα
−
<0
1 − β(1 − δ)
Suppose that the representative HH does not value gt . (also writing F (k, n) = Ak α n1−α implicitly
assumes that gt does not affect the production in the economy.) Note from (24) τn does not affect kss ,
and since labor is inelastically supplied (assume τn ≤ 1), the aggregate output in the the economy does
α
− δkss = css + gss =
not depend on τn . However, by (23), evaluated at steady state, it follows that Akss
css + τn wss + τk rss kss + τc css , where I use the fact that the government budget must be balanced in the
present value (in steady state, with all values constant over time imposing budget balance in prevent value
⇒ budget balance period by period). Then since we just showed that τn does not affect rss , kss and wss is
given by
wss
We have that τn ↓
⇒
css ↓
⇒
∞
P
t=0
=
Fn (kss , 1)
=
α −α
(α − 1)Akss
nss
=
α
(α − 1)Akss
β t U (css ) =
U (css )
↓ Hence household bears the burden.
1−β
2