3.2 Introducing a Means-Tested Program
3.2.3 Characterizing the Value Function
I refer to Γ(a, w) as feasibility correspondence, which is compact and continuous.
Note that Vt is a function of time indicated by the t subscript. To make the notation more compact for the subsequent analysis, define conditional on ϕt =ϕk:
Et{Vt+1(φ(k), ϕ0, µ1)}=IW=1
N
X
j=1
πk,jWVt+1(φ(k), ϕj, µ1) +IR=1
N
X
j=1
πk,jR Vt+1(φ(k), ϕj, µ1)
=Vt+1(φ(k), ϕ0, µ1),
where IW=1 and IR=1 are indicator functions that are one during working life and retirement, respectively.
3.2 Introducing a Means-Tested Program
Note the importance of definingaas total assets, includingS(wt) and definingkt as the choice excludingS(wt). Defining the state as assets excludingS(wt) would lead to a downward jump in Vt. The next Lemma establishes weak monotonicity of the pol-icykt(·, ϕ, µ1), which is a direct result from the strictly concave period utility function.
Lemma 2: kt(·, ϕ, µ1) is increasing.
I am now ready to establishes continuity of Vt(·, ϕ, µ1). Intuitively, optimal choices imply that a small change in the asset position does not lead to large changes in the value function, even though the law of motion of the endogenous state variable is not continuous.
Lemma 3: Vt(·, ϕ, µ1) is continuous ∀ a.
The following Lemma establishes that the policy function is not strictly increasing for agents with wt0 ≤ wtelig. More specific, I show that for a range of (a, ϕ0, µ01) the policy kt(a, ϕ0, µ01) is flat with choice kt = 1+ra¯ . The intuition is simple: The
Figure I: Policy Function in T −1
0 50 100 150 200 250 300 350
0 20 40 60 80 100 120 140 160 180
a kT−1(·,ϕ1,µ
1 1)
Notes: The figure displays the policy function of a household with state vector (a, ϕ1, µ11) in period T −1. Wages are such that the household is eligible to the means-tested program.
household has to weight the extra utility he gets from consumption smoothing against the income loss he incurs from choosing kt> 1+r¯a .
Lemma 4: kt(a, ϕ0, µ01) is not strictly increasing ∀w0t ≤ wtelig. More specific,
∃kt(a, ϕ0, µ01)∈B(kt0(a0, ϕ0, µ01)) with kt0 = 1+r¯a .
Figure I Panel A highlights this point graphically for periodT −1. The optimal policy is to choose kT−1 = 1+ra¯ in a range of the asset state. Note that this behavior inflicts a cost on social welfare. The social planner always prefers that each individual household equates the expected marginal utility of consumption.
The next Lemma highlights a point already apparent in the figure. It is optimal to choose kt> 1+r¯a fora large enough. The economic intuition behind the result is that the gains from consumption smoothing become larger than the income effect from the forgone income for sufficiently high asset level.
Lemma 5: ∃a˜t(ϕ0, µ01) s.th. ˜kt(˜at, ϕ0, µ01)> 1+r¯a ∀ a >˜at(ϕ0, µ01).
Equipped with these Lemmas, I can state my main theorems that are about condi-tions for first order condicondi-tions to be either necessary or sufficient. The first theorem deals with the retirement period.
Theorem 1: Let t > W and wt0 > weligt . Then ∂Vt(·,ϕ∂k0,µ01)
t exists and ∂Vt(·,ϕ∂k0,µ01)
t = 0
is sufficient for an optimum ∀ at+1(a, ϕ0, µ01)>0 and ∀ t∈ {W + 1, T}.
The result follows from the assumption that income is fixed during retirement.
Therefore,∀wW > wteligchoices cannot be disturbed in any period in the future. I need some more notations before stating my second main theorem. Let (˙as(ϕ0, µ01), ϕ0, µ01) be the state vector in period s s.th. under no possible realization of the world in t ∈ {s, T} the household wants to chooses kt ≤ 1+r¯a . Thus, from today on, the household chooses with certainty a policy that makes him never eligible to the transfer.
Theorem 2: Let a˙s(ϕ0, µ01) be defined as above. Then ∂Vs(·,ϕ∂k0,µ01)
s exists and
∂Vs(·,ϕ0,µ01)
∂ks = 0 is sufficient for an optimum ∀ as+1(a, ϕ0, µ01)>0 and as>a˙s(ϕ0, µ01).
3.2 Introducing a Means-Tested Program
Theorem 2 is a powerful result because it implies that the means-tested program has no impact on optimal choices for sufficiently rich households. The next theorem argues that for non-binding choices the same holds true when the asset state leads to choices strictly less than 1+r¯a .
Theorem 3: Let (¨as(ϕ0, µ01), ϕ0, µ01) be the state vector in period s s.th. under no possible realization of the world in t ∈ {s, T} the household chooses kt ≥ 1+r¯a . Consider all a s.th. as+1(a, ϕ0, µ01) > 0 and a ≤ ¨as(ϕ0, µ01). Then ∂Vs(·,ϕ∂k0,µ01)
s exists and ∂Vs(·,ϕ∂k0,µ01)
s = 0 is a sufficient condition for a maximum.
My last theorem is concerned with choices that are to the right ofa≥˜at(ϕ0, µ01) but to the left of ˙at(ϕ0, µ01). I argue that first order conditions are still necessary for an optimum. The main issue in proving the result is that Vt(·, ϕ, µ1) is not differentiable at all points in this range. I show the result by demonstrating that these points must be downward kinks. Because these cannot be optimal choices, it follows that the function is differentiable at all optimal choices. Standard variation arguments then lead to the necessity of first order conditions.
Theorem 4: ∂Vt(·,ϕ∂k0,µ01)
t = 0 is a necessary condition for kt(a, ϕ0, µ01) to solve (3.2)
∀ a ≥a˜t(ϕ0, µ01) and at+1(a, ϕ0, µ01)>0.
Figure IIPanel A shows the value and policy function of an eligible household in period T −1. One can see how an upward jump in the policy function translates into a downward kink in the value function. To provide a better understanding for the trade-off the household faces between the income effect and the consumption smoothing effect, let me define the following function at (˜aT−1, ϕ0, µ01):
WT−1(KT−1,˜aT−1, ϕ0, µ01) =U(˜aT−1+w0t−KT−1)+βVT((1+r)KT−1+F(KT−1, w0t), ϕ0, µ01) KT−1 ≤˜aT−1+wt0,
the return function from different admissible strategies this period and following
optimal policy next period. Obviously,
WT−1(KT−1,a˜T−1, ϕ0, µ01)≤VT−1(˜aT−1, ϕ0, µ01)
with equality at KT−1 = kT−1(˜aT−1, ϕ0, µ01). The function is depicted in Figure II, Panel B. The first local maximum is the choice KT−1 = 1+r¯a . Choices just above this point lead to lower returns because the negative income effect dominates the addi-tional consumption smoothing effect. Larger choices lead to addiaddi-tional consumption smoothing gains, which are largest at the second local maximum, where (3.2) satisfies the first order conditions.
The fact that households satisfy first order conditions to the right of a≥˜at(ϕ0, µ01) does not imply that their choices are not affected by the means-tested regime.
To see that point, note that the life-cycle dimension and stochastic income imply that households possibly attach positive probability to a state where they want to participate in the means-tested regime in the future. They adjust their savings decisions already today to fulfill the asset requirements in that case.
Let me first elaborate on the role of the life-cycle dimension. Consider a household in period T −2 that has income wt0 ≤ wtelig. Lemma 4 and Lemma 5 establish
Figure II: Value Function and Return Function (A) Value Function inT −1
0 50 100 150 200 250 300 350
−0.5
−0.4
−0.3
VT−1(·,ϕ1,µ
1 1)
a
0 50 100 150 200 250 300 350
0 50 100 150
kT−1(·,ϕ1,µ
1 1)
Value Policy
(B) Return Function inT−1
10 20 30 40 50 60 70
−0.44
−0.435
−0.43
−0.425
−0.42
KT−1
W(·,a,ϕ1,µ
1 1)
Notes: Panel Adisplays the policy and value function of a household with state vector (a, ϕ1, µ11) in period T −1. Wages are such that the household is eligible to the means-tested program. Panel B depicts the return function inT−1, i.e., the return from different admissible strategies inT−1,KT−1, and following optimal policy inT, for the same type of household.
3.2 Introducing a Means-Tested Program
Figure III: Savings Behavior in T −2 (A) Return Function in T−2
65 70 75 80 85 90 95 100 105
−0.599
−0.5985
−0.598
−0.5975
−0.597
KT−2
W(·,a,ϕ1,µ
1 1)
(B) Value and Policy Function in T −2
0 50 100 150 200 250 300 350
−0.7
−0.6
−0.5
VT−2(·,ϕ1,µ
1 1)
a
0 50 100 150 200 250 300 350
0 100 200
kT−2(·,ϕ1,µ
1 1)
Value Policy
Notes: Panel Adisplays the return function inT−2, i.e., the return from different admissible strategies in T −2,KT−2, and following optimal policy inT−1 for an agent choosing between the left and right of a non-differentiable point. Panel B displays the resulting value and policy function.
that the policy function has a flat part and ∃a˜T−2(ϕ0, µ01) s.th. ˜kT−2(˜aT−2, ϕ0, µ01)>
¯ a
1+r ∀ a > ˜aT−2(ϕ0, µ01). Moreover, Theorem 4 establishes that the value function has a downward kink at ˜aT−2(ϕ0, µ01). Note that VT−1 has a downward kink at
˜
aT−1(ϕ0, µ01), implying that the value function becomes steeper to the right of the non-differentiability. Consider the point ˜˜aT−2(ϕ0, µ01) s.th. aT−1(˜˜aT−2(ϕ0, µ01)) >
˜
aT−1(ϕ0, µ01). To understand the decision the household has to make, let me define the following return function:
WT−2(KT−2,a˜˜T−2(ϕ0, µ01), ϕ0, µ01) =U(˜˜aT−2(ϕ0, µ01) +wt0−KT−2)
+βVT−1((1 +r)KT−2+F(KT−2, ϕ0, µ01), ϕ0, µ01). Figure III Panel A shows the two local maxima of the return function. The first implies that the household satisfies the first order conditions by choosing to the left of ˜aT−1(ϕ0, µ01) and receives means-tested transfers at end of period T −1. The second local maximum satisfies the first order conditions by choosing to the right of ˜aT−1(ϕ0, µ01) and the household never participates in the means-tested program8. The policy function makes a second jump at ˜˜aT−2(ϕ0, µ01), and the value function
8The figure highlights that non-uniqueness inktcan arise when the household is exactly indifferent between choosing to the left and the right of a non-differentiability.
Figure IV: Consumption Behavior
0 50 100 150 200 250 300 350
60 80 100 120 140 160 180 200 220
a cT−3(·,ϕ1,µ
1 1)
w<w
elig
w>w
elig
Notes: The graph compares two consumption functions inT −3 where in the one state the agent has wage income such that he is eligible for the means-tested program (straight line) and in the other he is not eligible (dashed line).
becomes non-differentiable at this point, which I highlight graphically inPanel B. Uncertain income has a very similar effect, but households with w > wtelig also become affected. These households place positive probability on becoming eligible for means-testing in the future. Consequently, they adjust their savings behavior today to have in expectations the most possible intertemporal consumption smoothing.
Note, this leads to a rapid increase in the number of non-differentiabilities in the value function because any path of the state variables that makes the household at any point in the future eligible to means-testing has to be considered.
Finally, Figure IV shows how the non-differentiabilities in the value function translate into optimal consumption behavior. The calibration is such that agents are not eligible for the program in the high income state, but they are eligible in the low income state. Concerning the latter, the following behavior arises: In the left most asset section, workers are borrowing constrained leading to a relatively steep consumption rise (Not visible in the figure). Afterwards, decisions are characterized by Theorem 3 implying that first order conditions hold, and the consumption profile becomes flatter. Lemma 4 characterizes the following section where savings are constant and all additional assets are consumed. The following three discontinuities are induced by downward kinks in the value function. Therefore, the slope of the
3.3 Data Description, Sample Selection and Calibration