A.1 Proof of Proposition 1
For ease of notation I shall most often suppress the time argument. First, I characterize the unique interior candidate solution to the maximization problem (2.5). This delivers the expressions stated under Regime 1. Then, I show that these expressions are indeed the interior solution to the maximization problem of a two-period lived individual if Assumption 1 holds. Second, I turn to the corner solution of Regime 2 and the continuity of the functions stated in (2.7).
Consider an interior solution,(cy, l, co, s)∈R2++× (0, 1)×R, to problem (2.5) that satisfies the budget con-straints (2.6). Since preferences are increasing in coboth per-period budget constraints will hold as equalities and can be merged. Accordingly, such a solution will be identified by the Lagrangian
L =ln cy+ln
Upon multiplication by(1−l), condition (A.3) may be written as φx
(1−φx)w(1−l) =λ. (A.6)
Using the latter to replace λ in (A.2) and (A.4) delivers, respectively, cy=
With (A.7) and (A.8) in the budget constraint (A.5) I obtain (2.8), i. e., φx is time-invariant. Using (2.8) in (A.7) and (A.8) delivers (2.11). Since the optimal plan satisfies (2.3) and (2.8) I have 1 > ν(1+β), hence, cy >0. Since 1−ν(1+β) < (1+β)(1−ν), the budget constraint when young, cy+s=w(1−l), delivers strictly positive savings.
From the definition of x with h=1−l, I have cy = xh−11−ν
ν . Replacing cywith this expression in (2.11) and solving for h delivers htas stated under Regime 1. Clearly, ht ≤1 as long as wt ≥wc. Finally, using ht
in (2.11) delivers cyt, st, and cot+1=Rt+1st.
To see that the solution identified by the Lagrangian (A.1) is indeed a global maximum onP consider the
What remains to be shown is that the solution identified by the Lagrangian satisfies condition (2.4). With φx of (2.8) this is the case if and only if
It is not difficult to show that the latter condition is satisfied if and only if Assumption 1 holds.
From the expression of htunder Regime 1 it is obvious that ht>1 if wt<wc. Hence, for a real wage below wcthe constraint, l>0 (ht61) becomes binding, and the optimal plan involves l=0 and ht=1. Moreover, consumption when young and old (and λ) are determined by conditions (A.2), (A.3), and (A.5). Savings result form the budget constraint when young. This procedure delivers
cy
which coincides with the expression stated under Regime 2 since x≡ (cy)1−νν . From (A.9) one readily verifies that cy(w), hence, s(w)and co(w, R)exists for all w > 0. Moreover, under Assumption 1, U(cy, 1, co)is strictly concave for all(cy, 1, co)∈ Psince U11(cy, 0, co) =D1(cy, 0, co) <0, U33(cy, 0, co) =−β/(co)2<0, and U13=0.
Continuity of the functions stated in (2.7) follows from Theorem of the Maximum.
Finally, observe that members of cohort 0 satisfy their budget constraint when old as equality, i. e., we have
co1=R1s0>0.
A.2 Proof of Proposition 2
Follows from a simple application of the implicit function theorem to the expressions of Regime 2 and by
inspection of those of Regime 1.
A.3 Proof of Proposition 3
Consider htof Regime 1. It holds that
∂ht
as ∂z/∂φ>0.
Finally, consider cot+1. Since cot+1=Rt+1st, the qualitative results of the comparative statics for stapply here, too.
For Regime 2 one obtains the comparative statics for φ and β from a straightforward application of the
implicit function theorem.
A.4 Proof of Proposition 4
Immediate from the aggregate demand for hours worked of (3.3) and the aggregate supply of hours worked
of (3.4).
A.5 Proof of Proposition 5
To derive (3.10) use Proposition 1 and (2.14) to express (3.7) in units of efficient hours worked in t+1. This gives, Replacing ht/ht+1using again Proposition 1 delivers
βΓ(1−γ) Finally, invoke (2.14) again to obtain
βΓ(1−γ)
The proof of statement a) - c) follow from Proposition 1, Proposition 4, the capital market equilibrium con-dition (3.7), and the production function (2.13).
As to statement d) observe that for gA>0
∂g∗
Figure A.1: The steady state in Regime 2
0 wt
ˆ w∗∗
wc
s (wt)
w3
1 3
A.7 Proof of Proposition 7
As to the existence of a unique steady state 0<wˆ∗∗ <wcconsider Figure A.1. For the indicated parameter values it depicts the function s(w)and the function w3. From Proposition 1, individual savings are
s(w) =cy(w)
1−123cy2(w)
1 3
3−23cy2(w)
1 3
, (A.10)
where s(w)satisfies s(0) =0 and s(1) =1/3.21 Moreover, from Proposition 2 I have s0(wt) >0. Figure A.1 reveals a unique intersection at ˆw∗∗>0. Approximatively, it occurs at ˆw∗∗=0.55 and s(wˆ∗∗) =0.167.
21This follows since cy(w)is implicitly given by
cy
1+1−123c2y
1 3
3−23c2y
1 3
=w
with cy(0) =0 and cy(1) =2/3. Moreover, in Regime 2 the budget when young dictates s=w−cy.
To prove the local stability of the steady state I derive from (3.14) that
Then, implicit differentiation and evaluation at the steady state delivers s0(wˆ∗∗) =.339. Hence, with (3.14) dwˆt+1
Accordingly, the steady state is locally stable.
Global stability over the domain wt∈ (0, 1)follows since s(0) =0, s(1) =1/3, s0(w) >0, and The respective first-order sufficient conditions are
cy: U1−RU3=0,
(A.11) l : U2−RwU3=0,
where U is evaluated at(cy, l, R(w(1−l)−cy)).
Total differentiation of the two equations (A.11) delivers the comparative statics dl where D, Dlw, and DlRare the following determinants
D =
all evaluated at the optimal plan(cy, l, co, s)that satisfies (A.11) and the two constraints.
Since U is strictly concave it follows that D=U11 Since the term preceding the brackets is strictly positive I obtain
DlwR0 ⇔ −1+η1U 1+
Since, D>0 the sign of dl/dw is indeed determined by condition (4.2).
As to DlRI obtain from (A.14) that
where use is made of the budget constraints and (A.11). Since U3>0 and D>0 the sign of dl/dR is indeed
determined by condition (4.3).
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