Depositing More Should Not Earn Less: Supplier Income Monotonicity and Rate Curve Design in Pooled Lending Markets
Ahzam Khan, Cavey Cool
q-fin.TR
Oct 27, 2026 · v1
TL;DR
Most of the paper's results on supplier income monotonicity and admissible rate curves are formalized in a Lean 4 library, released with a coverage record.
Abstract
In utilization dependent interest rate models, adding liquidity can reduce a supplier's total interest income even as the supplier's balance increases. A supplier holding a fraction $α$ of final pool supply keeps income nondecreasing with a marginal deposit exactly when $α(1+η)\leq1$, where $η$ is the elasticity of the borrow rate with respect to utilization. When the condition fails, the supplier can instead raise its income by withdrawing, which pushes utilization further up. On two public Drift markets, a lender held more of the pool than the curve protected in at least one hour in nine on SOL and one in thirteen on USDC. A protocol turns this into a design constraint by choosing, at each utilization, the largest supplier share it intends to protect. That choice caps the elasticity, and integrating the cap describes every admissible curve. Protecting the same share at every utilization gives a power law boundary, and protecting every fresh depositor gives a hyperbolic one. On a multiple kink USDT market, compliant repairs can lower or raise supplier income, because the constraint governs how income responds to a deposit and not the price level.
Problem
In utilization-dependent interest rate models for pooled lending, a supplier's extra deposit lowers utilization and therefore the rate. Because that lower rate applies to the whole balance, depositing more can reduce the supplier's total income. The paper asks when this happens and which rate curves rule it out.
Approach
The authors derive the marginal income identity dI/dδ = U R_b(U)[1 − α(1+η)], where η is the elasticity of the borrow rate with respect to utilization. A protocol-chosen protection profile caps η, and integrating that cap characterizes every admissible curve. Constant-share and fresh-depositor profiles give power-law and hyperbolic boundary curves, and an endpoint bound limits how much protection is feasible. They measure real exposure using onchain Drift market data and formalize most results in Lean 4.
Results
Income is nondecreasing under a marginal deposit exactly when α(1+η) ≤ 1. On Drift, a lender was exposed in at least one hour in nine on SOL and one in thirteen on USDC. For the illustrative endpoints, no curve can protect a supplier holding more than about 6.93% of the pool.
| Balance | U | Borrow rate | Income |
|---|
| Start | $1,000,000 | 83% | 19.25% | $159,775 |
| Deposit $375,000 | $1,375,000 | 80% | 5% | $55,000 |
| Withdraw $302,000 | $698,000 | 85.6% | 31.5% | $188,300 |
The 10% supplier at three balances, with borrowings fixed at $8.3 million (income annualized)