Explicit consumption functions with borrowing constraints: A continuous-time approach
What this paper finds — and why it matters
Layer 1 — Overview
Research question. The paper asks whether an explicit, global, closed-form solution exists for the consumption function in the standard income fluctuation problem with a borrowing constraint and constant income, a problem that has resisted closed-form solution since at least Schechtman (1976). All prior continuous-time work (Park 2006, Holm 2018, Fischer 2024) produced only implicit expressions; Achdou et al. (2022) produced explicit expressions valid only locally, near zero assets or as assets diverge to infinity, and only for r > 0.
Model. A single agent with CRRA utility (coefficient of relative risk aversion γ > 0) maximizes discounted utility over an infinite horizon, subject to the flow budget constraint da/dt = ra + y − c, with a borrowing constraint a(t) ≥ 0. The agent receives a constant, deterministic income stream y ≥ 0 and discounts at rate ρ, with the impatience condition ρ > r maintained throughout. The paper takes a continuous-time formulation arrived at by letting the discrete period length Δ → 0, nesting Helpman (1981)’s discrete-time analysis as a special case.
Key analytical device. A one-to-one mapping exists between initial assets a and the time T it takes for the consumer to fully run down her assets. This map, denoted T = h(a; y), is well-defined, strictly increasing, and concave in a (established in Proposition 1 via the Hadamard-Lévy theorem). Expressing the optimal consumption function as c*(a; y) = y · exp(ρh(a;y)/γ) evaluated at t = 0 reduces the problem to explicitly inverting the transcendental equation relating a to T.
Main result (r = 0). For the case of a zero net real interest rate, the transcendental equation can be solved explicitly using the second branch W₋₁(·) of the Lambert W function. The closed-form consumption function is (Theorem 2 and Corollary 2.1):
c*(a; y) = y · exp(ρ h(a;y) / γ), where h(a; y) = −(a/y + γ/ρ) − (γ/ρ) W₋₁(f(a;y)), and f(a;y) = −exp(−b(a + γy/ρ)/y), b := ρ/γ.
This is a global solution (valid for all a ≥ 0), in contrast to the local solutions in prior work. The paper notes that for the illustrative parameter values r = 0.01, γ = 0.5, ρ = 0.08, y = 3 (broadly consistent with average U.S. real interest rates in 2025), there is a visually sizable gap between the constrained and unconstrained consumption functions except as a → ∞, where the two converge (in line with the asymptotic linearity result of Benhabib et al. 2015).
Main result (r > 0). For positive interest rates, the Lambert W function cannot invert a sum of exponentials with different exponents (an open mathematical problem). The paper instead derives a global closed-form approximation valid for r ∼ 0, by expanding e^(−rT) ≈ 1 − rT to first order and applying the same Lambert W inversion. The approximating consumption function has the same structural form but with modified coefficients b_r, c_r, d_r that collapse to their r = 0 counterparts as r → 0 (Proposition 2). Numerical comparison against the implicit-expression solution of Park (2006) confirms the approximation is close for small r.
Characterization of the MPC and supermodularity (Section 3). Leveraging the explicit expression, the paper derives the full Jacobian vector and Hessian matrix of c*(a; y) in closed form (Propositions 3 and 4). Key findings, all proved formally and holding under the impatience condition ρ > r:
Consumption is increasing in both assets and permanent income (both entries of the Jacobian are strictly positive — Corollary 2.2). The second result (∂c*/∂y > 0 for all a) is new for the borrowing-constrained setting; Achdou et al. (2022) provided only suggestive evidence for the limiting case a ∼ 0.
Consumption is strictly concave in both assets and permanent income (both diagonal entries of the Hessian are strictly negative — Corollary 2.3). Concavity in assets was known (Carroll and Kimball 1996); concavity in permanent income under borrowing constraints is new.
The consumption function is supermodular: the cross-derivative ∂²c*/∂a∂y is strictly positive (Corollary 2.3). This means assets and permanent income are complements in generating consumption. Equivalently, the MPC out of permanent income is strictly increasing in the level of initial assets — a counter-intuitive result, since high MPCs are usually associated with poor (low-asset) agents. An identical result was obtained by Commault (2025) for a life-cycle model without borrowing constraints; the current paper confirms it holds in the presence of a borrowing constraint. By symmetry of the Hessian, the MPC out of assets is also strictly increasing in permanent income.
Intuition for supermodularity. When assets are low, an increase in permanent income produces little additional consumption because the risk of hitting the borrowing constraint is high. When assets are higher, the agent has buffer savings, faces a lower constraint-risk, and can smooth the higher future income stream into current consumption.
Scope conditions. Results are derived under CRRA utility, constant (deterministic) income, no stochastic variation, and the impatience condition ρ > r. The exact closed form applies to r = 0; the approximation is characterized as valid for r ∼ 0 and is not a local expansion in assets.
Layer 2 — Q&A
Q1. What is the longstanding gap in the literature that this paper addresses? A: Since Zeldes (1989) noted that no closed-form solution exists for the consumption function with stochastic income and CRRA utility, researchers settled for numerical solutions or local analytical approximations. In the constant-income/borrowing-constraint version studied here, Park (2006), Holm (2018), and Fischer (2024) derived only implicit continuous-time expressions. Achdou et al. (2022) gave explicit local solutions valid near a ∼ 0 or a → ∞ under r > 0. No prior work produced an explicit, global closed-form for any case.
Q2. Why does moving to continuous time enable progress that discrete time did not? A: In discrete time, the consumption function is piecewise linear (Helpman 1981), with kinks at the sequence of asset thresholds µ(T) for T = 0, Δ, 2Δ, …. As Δ → 0, the piecewise-linear function converges to a smooth function whose governing ODE can be solved analytically. This convergence to smoothness, illustrated in Figure 1, is what enables the application of the Lambert W function to invert the resulting transcendental equation.
Q3. What is the role of the Lambert W function, specifically its second branch W₋₁? A: The optimal asset-depletion time T satisfies the transcendental equation e^(bT) = yT + c (for r = 0), which cannot be solved with elementary functions. Via the change of variables z := −bT − bc/y, the equation reduces to ze^z = α, whose solution is z = W(α). The argument α lies in (−1/e, 0) for a ∈ (0, +∞), and it is precisely on this interval that the Lambert W function is double-valued; the relevant branch is W₋₁ (the second, lower branch), which is well-defined and strictly less than −1 on (−1/e, 0). It is the properties of W₋₁ on this domain — specifically that 1 + W₋₁(α) < 0 — that drive the sign conclusions for the Hessian.
Q4. Why does the Lambert W approach fail for r > 0, and what is the approximation strategy? A: For r > 0, Equation (8) contains two exponentials with different exponents — e^((ρ−r)T/γ) and e^(−rT) — and their sum cannot be inverted by the Lambert W function, which handles only a linear-plus-single-exponential structure. Inverting a sum of exponentials with different exponents is stated in the paper to be an open problem. The approximation strategy exploits the fact that for r ∼ 0, e^(−rT) ≈ 1 − rT + o(r), reducing the equation to a single-exponential transcendental form (Equation 15) with modified coefficients b_r, d_r, c_r, all of which converge to their r = 0 analogues as r → 0.
Q5. What does Proposition 1 establish, and why is it necessary before stating the main theorem? A: Proposition 1 establishes that the mapping µ(T) from depletion time T to initial assets a is smooth (infinitely differentiable), bijective (one-to-one and onto) on ℝ₊, and strictly convex. The Hadamard-Lévy theorem then guarantees that its inverse h(a;y) = µ⁻¹(a) exists, is unique, is strictly increasing, and is strictly concave in a. This is a necessary prerequisite for Theorem 2 because h(a;y) is the central object in the closed-form consumption function; without establishing its existence and uniqueness, Theorem 2 would have no well-defined object.
Q6. What does the Jacobian characterization (Proposition 3 and Corollary 2.2) contribute? A: Proposition 3 gives explicit formulas for ∂c*/∂a = (ρ/γ) · w/(1+w) and ∂c*/∂y in terms of w = W₋₁(f(a;y)). Corollary 2.2 proves both are strictly positive using the property w < −1 on (−1/e, 0), which ensures w/(1+w) > 0 and that the bracketed term in the expression for ∂c*/∂y is strictly positive. The contribution is that the positivity of ∂c*/∂y for all a was previously unproven in a borrowing-constrained setting with constant income.
Q7. What is the structure of the Hessian matrix and what signs do its entries take? A: All four entries of Hc are proportional to w/(1+w)³. Since w < −1, we have 1 + w < 0, so (1+w)³ < 0, making w/(1+w)³ > 0. The diagonal elements ∂²c*/∂a² = −(ρ²/γ²y) · w/(1+w)³ and ∂²c*/∂y² = −(ρ²a²/γ²y³) · w/(1+w)³ are both strictly negative (concavity). The off-diagonal elements ∂²c*/∂a∂y = (aρ²/γ²y²) · w/(1+w)³ are strictly positive (supermodularity/complementarity).
Q8. What is the precise counter-intuitive implication of supermodularity for MPC heterogeneity? A: Supermodularity (∂²c*/∂a∂y > 0) means the MPC out of permanent income — conventionally associated with low-wealth households — is in fact increasing in the level of initial assets. This contradicts the conventional narrative that high MPCs are a hallmark of poor agents. The paper’s intuition is that low-asset agents face high risk of hitting the constraint, suppressing their consumption response to income news, while high-asset agents can freely smooth the increased income stream. The same supermodularity implies, by the symmetry of the Hessian, that the MPC out of assets is also increasing in permanent income.
Q9. How does this result relate to Commault (2025)? A: Commault (2025) proved, in a life-cycle model with a permanent/transitory stochastic income process but without borrowing constraints, that the MPC out of permanent income is increasing in assets. The current paper obtains the same qualitative finding in the opposite environment — constant income with a borrowing constraint. The paper treats these as complementary, noting that the result thus appears robust to these different modeling choices.
Q10. What does concavity in permanent income (∂²c/∂y² < 0) add that was not previously known?* A: Carroll and Kimball (1996) established concavity of the consumption function in assets for a broad utility class. Concavity in permanent income — that the marginal consumption response to a windfall increase in y is diminishing — had been proved by Commault (2025) only in the absence of borrowing constraints. The current paper provides the first formal proof of this property in a setting with a borrowing constraint (albeit for constant, deterministic income and CRRA utility in continuous time).
Q11. What is the potential use of these closed-form results for numerical methods? A: The paper notes in the conclusion that the closed-form solutions for r = 0 and the approximation for r ∼ 0 can serve as benchmarks for assessing the reliability of continuous-time numerical methods when computing objects such as the MPC out of assets. Because the exact solution is known analytically, numerical implementations can be compared against it to detect discretization errors or convergence failures.
Q12. What parameter values are used to illustrate the consumption function, and what do they imply? A: The paper uses r = 0.01, γ = 0.5, ρ = 0.08, y = 3, where r = 0.01 is described as roughly in line with the average real interest rate in the U.S. in 2025. With these values, Figure 1 shows a visually sizable gap between the constrained and unconstrained consumption functions at low to moderate asset levels, with the two converging as a → ∞ as guaranteed by asymptotic linearity (Benhabib et al. 2015).
Key Concepts
Income fluctuation problem (with borrowing constraint): The standard infinite-horizon single-agent savings problem in which the agent faces a non-negativity constraint on assets (a(t) ≥ 0), so that the agent cannot borrow. In the paper’s formulation: maximize ∫ e^(−ρt)u(c(t))dt subject to da/dt = ra + y − c and a(t) ≥ 0, with constant income y and CRRA utility. The borrowing constraint creates the concavity of the consumption function and was the source of intractability in prior closed-form attempts.
Lambert W function (second branch W₋₁): A special transcendental function defined as the solution to we^w = x. It is double-valued on (−1/e, 0); the second branch W₋₁ takes values strictly less than −1 on this interval. In this paper, the transcendental equation linking initial assets to asset-depletion time is reduced to the form ze^z = α, enabling explicit inversion via W₋₁. The property that 1 + W₋₁(α) < 0 on (−1/e, 0) is the algebraic engine driving all sign results in the Hessian.
Asset-depletion time T = h(a; y): The time it takes for the optimal consumer to fully run down her initial assets before settling into perpetual income consumption of y. The paper establishes a bijective mapping from initial assets a to depletion time T (Proposition 1); the closed-form solution is obtained by explicitly inverting this mapping. In the paper’s formulation, h(a; y) = µ⁻¹(a) where µ(T) is derived from the ODE governing the consumption path.
Supermodularity of the consumption function: The property that the cross-derivative ∂²c*/∂a∂y is strictly positive, meaning assets a and permanent income y act as complements in generating consumption. This is an equilibrium property of the consumption function (not an assumption on the utility function), and the paper identifies it as new to the income fluctuation literature. It implies the MPC out of permanent income is increasing in a, and the MPC out of assets is increasing in y.
MPC out of permanent income (∂c/∂y):* The marginal increase in current consumption per unit increase in the constant income stream y, holding initial assets constant. This object is less studied than the MPC out of a transient asset windfall. In the paper’s setting, it is shown to be strictly positive for all a (Corollary 2.2) and, counter-intuitively, strictly increasing in a (supermodularity).
Global vs. local closed-form solution: A global solution holds for all values of the state variable (here, all a ≥ 0), while a local solution is valid only in the neighborhood of a particular value (e.g., a ∼ 0 or a → ∞). Achdou et al. (2022) produced local closed-form expressions; the current paper’s Theorem 2 (r = 0) is the first global explicit closed-form for this class of problems.
Piecewise-linear consumption function (discrete time): In Helpman (1981)’s discrete-time formulation with period length Δ = 1, the optimal consumption function is piecewise linear in assets, with slope changes at the asset thresholds µ(T) for integer T. As Δ → 0, this becomes a smooth function, enabling the passage to the continuous-time closed form derived in the paper.