Ha and Jeon's borrowing floor is the result I would use. An agent can switch among N jobs before a fixed retirement date. Wealth must remain strictly above minus the largest discounted wage stream available across destination jobs, after deducting the cheapest route into each job. The paper proves that this floor is exact.
The model uses a complete Black-Scholes market with one risky asset and constant r, μ and σ. Job i offers a constant wage ε_i and imposes a constant disutility L_i. A move from i to j has a positive cost; returning from j to i can cost a different amount. Labour income is fully spanned. A martingale dual transformation therefore separates the common consumption utility from an N-mode switching problem. Its only continuous state is y, the dual variable measuring marginal utility of wealth. The resulting N coupled variational inequalities assign an obstacle to each job through the alternatives available elsewhere.
The dual values are convex, and the authors prove 0 ≤ Q_yy ≤ C(1+y)/y² almost everywhere. Q_y is jointly continuous and locally 1/2-Hölder in t. Spatial smooth fit holds at every contact, even where destinations tie. Strong duality then gives consumption, the portfolio and the switching rule from the dual value. But the paper's second half dismantles the boundary picture readers might expect from that regularity. As the authors put it, "The boundary results distinguish value regularity from geometric regularity."
Cheap routes change borrowing capacity
The authors close switching costs under shortest paths: d_ij is the cheapest chain from i to j, with all moves executed at one instant. The borrowing constraint is wealth > −h_i(t). Here h_i(t) is the maximum across destinations j of ε_j a_r(T−t) − d_ij, with a_r the annuity factor at rate r.
The five-job illustration uses T=30 and r=0.02. It makes the case for this closure concrete. Going directly from job 1 to job 4 costs 6.5, whereas the route 1→5→4 costs 2.6. That raises the job-4 term in job 1's envelope by 3.9 units. The floor can move by that much.
Job 5 would disappear under a simple ranking. Its wage is 0.9 versus job 3's 1.1, and its disutility is 1.8 versus 1.4. Yet entry from job 1 costs only 0.6, compared with 2.0 for direct entry to job 3. Keeping it preserves a cheap routing node. The authors' warning is explicit: "A wage-disutility comparison alone is insufficient for deleting its state."
This convention has a limit. A model that forbids simultaneous successive switches may leave the cheapest route unattainable. Shrinking the delays can approximate its value; that alone does not prove attainment.
Can wealth alone supply a threshold?
In general, no. The time-line and disconnected-contact examples come from explicit formulas, without PDE approximation.
When jobs share a common disutility, the dual value is linear in y and all boundaries are time lines. A four-job example uses wages (0,1,2,3), r=1/10, β=1/5, θ=3/10 and T=5. The contact set where switching from job 1 to job 2 is among the optimal actions has two components. For the component with t ≤ 1.433, job 2 serves only as a stop along an equal-cost route to job 4 (d14 = d12 + d24 = 6). The boundary lines occur at t ≈ 1.433, 2.769 and 3.946.
A three-job example imposes strict triangle costs, with smallest slack 1/2. Job 1 then has a vertical switching boundary at t_c = 5 + 10 log(9/10). Job 2 is the unique destination on that boundary. The switching gain, 0.9·y0, is positive. Q1 nevertheless has a time kink despite spatial smooth fit. Under CRRA γ=2, the boundary becomes {t = t_c, w > 0}. A wealth threshold w = b(t) cannot represent it: the vertical line remains intact in wealth coordinates.
Ordering wages and disutilities does not restore a connected switching set. For job 2, switches down to job 1 and up to job 3 lie on opposite sides of the continuation band (100/101, 100/33). Additive costs also allow a corner within a single boundary component for almost every entry cost k in (0, 10(1−e^{−1/2})). The two tangents have determinant −1. Under CRRA with γ = 2, that corner remains in wealth coordinates.
The positive boundary result has a narrower reach. A connected piece is a C^∞ graph in time if four conditions hold: the active destination is unique, it lies in its own continuation region, the gain is positive, and there is a spatial half-space blow-up. This last condition describes the local boundary shape after rescaling. The authors assume it rather than prove it generally, though they verify it at almost every time in one example. Whether strict triangle costs rule out corners remains open.
A desk rule of the form "switch when wealth crosses b(t)" can fail even in the paper's simplest exact examples.
Household inputs behind the portfolio formula
Under CRRA, the risky dollar position is π* = (θ/σ)(A(t)y^{−1/γ}/γ + y·Q_yy). The Merton-type weight comes first; occupational curvature supplies the second term. For θ > 0, the latter is nonnegative at a fixed y. The authors caution that this does not rank portfolios at equal financial wealth, since y changes too. Q depends on the household's job data (ε, L, d), alongside r, β, θ and T.
We did not backtest this. Market data cannot supply the household inputs: job-level wages, disutilities and a directed switching-cost matrix. A run using assumed values would measure only those assumptions. Continuous trading in a complete market also underpins the paper's financing and borrowing guarantees. Any daily-bar implementation would require its own feasibility and financing assumptions, so those guarantees would not transfer exactly to a version we could run.
What the five-job grid establishes
The main grid has 1,200 time intervals and 1,201 log-state nodes on [−8,6]. Against the identical-jobs benchmark, the error is 1.67×10⁻¹⁰. The worst intervention gap, −1.59×10⁻¹², is at roundoff.
At four probe points (y = 0.05, 0.2, 1, 3), the maximum value difference falls from 5.91×10⁻² between coarse and medium grids to 2.97×10⁻² between medium and fine grids. Each refinement roughly halves the difference. Widening the domain to [−10,8] changes values at those four points by no more than 3.56×10⁻⁴. The recursion requires O(N²) comparisons per spatial node. The theorem establishes monotone convergence of grid values up to Q.
The authors spell out the limit: "Refinement reduces the observed value differences; it does not furnish a certified error bound for the continuous-time solution or its switching boundaries." Their theorem addresses values, and it holds. The boundaries examined in the paper's second half can have corners and time lines, as the counterexamples demonstrate. There is no rate for switching curves. The colour maps in the five-job figure should therefore be read qualitatively.
The paper earns its place as a model of household financing capacity. Its shortest-path floor is the part I would use. I would hold off treating the portfolio formula as a prescription until the model accommodates unhedgeable income risk, an extension the authors leave for further work.