← Research

Depositing more should not earn less

Supplier income monotonicity and rate curve design in pooled lending markets

By Ahzam Khan and Cavey Cool
Read full paper · September 2026

A supplier can add capital to a lending pool and end up earning less interest than before. The deposit lowers utilization, the borrow rate falls, and that lower rate applies to the supplier's entire balance.

The problem

This work began with an observation from Cavey about supplier income in utilization dependent lending markets.

Most pooled lending markets use a kinked interest rate curve. Borrow rates rise gradually below an optimal utilization level, then much faster above it. The steep branch is useful: expensive borrowing encourages repayment and attracts new supply when available liquidity is scarce.

It also creates price impact for suppliers. A deposit increases total supply and lowers utilization. That moves the whole market down the borrow rate curve, so the resulting lower supply rate applies not only to the new deposit but to every token the supplier already had in the pool.

Suppose a supplier deposits $10,000 at a 20% annualized supply rate. That position earns about $2,000 per year. If another $10,000 deposit pushes the rate down to 8%, the larger $20,000 position earns only $1,600.

The numbers are illustrative. The underlying problem is that supplier income need not increase with supplied capital. Steep utilization dependent interest rate models can make withholding liquidity the better choice.

Supplier income

Let $T$ be total supply after the deposit, $B$ total borrowings, and $b$ the supplier's resulting balance. Utilization is:

U = \frac{B}{T}

If $R_{b} (U)$ is the borrow rate, aggregate borrowers pay approximately $B R_{b} (U)$ in annualized interest. Ignoring fees, distributing this across all supplied tokens gives the supply rate:

R_{s} (U) = \frac{B R_{b} (U)}{T} = U R_{b} (U)

The supplier's annualized interest income is therefore:

I = b U R_{b} (U)

Now add a marginal amount to the same supplier's deposit while holding borrowings fixed. Define the supplier's share of final pool supply as:

α = \frac{b}{T}

This is the supplier's share after the marginal deposit is added. That distinction matters because both the supplier balance and total pool supply change during the transaction.

Let $δ$ be additional supply from this supplier. Borrowings are held fixed during the transaction, while $\frac{d b}{d δ} = 1$ and $\frac{d T}{d δ} = 1$ . The resulting change in utilization is:

\frac{d U}{d δ} = \frac{d}{d δ} (\frac{B}{T}) = - \frac{B}{T^{2}} = - \frac{U}{T}

Differentiating income shows the two effects of the deposit:

\frac{d I}{d δ} = U R_{b} (U) + b [R_{b} (U) + U R_{b}^{'} (U)] \frac{d U}{d δ}

The first term is what the marginal token earns. The second is the rate change applied to the supplier's full resulting balance. Substituting $\frac{d U}{d δ} = - \frac{U}{T}$ and $α = \frac{b}{T}$ gives:

\frac{d I}{d δ} = U R_{b} (U) [1 - α (1 + η)]

Here $η$ is borrow rate elasticity with respect to utilization:

η (U) = \frac{U R_{b}^{'} (U)}{R_{b} (U)}

The condition

For positive utilization and a positive borrow rate, $U R_{b} (U)$ cannot change the sign of the income derivative. Everything reduces to the bracketed term.

The requirement is that another marginal deposit does not reduce supplier income:

\frac{d I}{d δ} \geq 0

Strictly increasing income uses $\frac{d I}{d δ} > 0$ . For monotone income, the condition is equivalent to:

α (1 + η) \leq 1

The condition balances position size against rate sensitivity. A small supplier has little existing income to reprice, so the marginal deposit is more likely to help. A large supplier moves the rate against a larger balance. Likewise, a curve with low elasticity tolerates larger suppliers than one whose rate responds sharply to utilization.

The largest supplier share protected at a given utilization is therefore:

α_{safe} (U) = \frac{1}{1 + η (U)}

Borrow rate elasticity	Safe supplier share
1	50%
9	10%
19	5%
99	1%

This gives “too steep” a precise meaning. An elasticity of 19 protects suppliers up to 5% of final pool supply. Above that share, another deposit may reduce their total interest income.

A fixed supplier share guarantee

Suppose a market wants to protect every supplier with a final pool share up to $\bar{α}$ . The corresponding maximum elasticity is:

K = \frac{1 - \bar{α}}{\bar{α}}

The design rule is simply $η (U) \leq K$ . The economic choice of supplier protection becomes a constraint that can be applied to an existing IRM.

The boundary curve that uses the full elasticity budget satisfies $η (U) = K$ . Written as a differential equation:

\frac{U}{R_{b}} \frac{d R_{b}}{d U} = K

Separating variables and integrating gives:

\frac{d R_{b}}{R_{b}} = K \frac{d U}{U} \Rightarrow \ln R_{b} = K \ln U + C

Anchoring the constrained branch at optimal utilization $O$ and borrow rate $R_{o}$ determines the constant and yields:

R_{K} (U) = R_{o} {(\frac{U}{O})}^{K}

The power law is not an arbitrary replacement curve. It is the exact boundary implied by the elasticity ceiling, and its elasticity is equal to $K$ everywhere on that branch.

The constant $K$ is not a universal restriction on an interest rate model. It follows from one specific policy choice: protecting every supplier whose final share is below a chosen limit. A different guarantee produces a different constraint.

A guarantee for fresh depositors

Now consider a supplier who begins with no position and deposits $x$ into a pool with initial supply $S_{0}$ and fixed borrowings $B$ . Utilization before and after the deposit is:

u_{0} = \frac{B}{S_{0}} u = \frac{B}{S_{0} + x}

The supplier's final share can be written using those two utilizations:

α = \frac{x}{S_{0} + x} = 1 - \frac{u}{u_{0}}

Substituting this into the general condition gives the exact rule for a deposit path beginning at $u_{0}$ :

(u_{0} - u) R_{b}^{'} (u) \leq R_{b} (u)

For a nondecreasing borrow rate, the most demanding starting point is full utilization. The uniform rule that protects a fresh depositor from every feasible initial utilization is therefore:

(1 - u) R_{b}^{'} (u) \leq R_{b} (u)

In elasticity form:

η (u) \leq \frac{u}{1 - u}

This bound becomes looser as utilization approaches one, so it permits a steep emergency region near full utilization. It protects a different path from the fixed supplier share rule above.

Piecewise linear form

For a nondecreasing segment from $(u_{L}, R_{L})$ to $(u_{H}, R_{H})$ , the slope is:

m = \frac{R_{H} - R_{L}}{u_{H} - u_{L}}

The condition is tight at the lower endpoint and reduces to:

0 \leq m \leq \frac{R_{L}}{1 - u_{L}}

Equivalently, adjacent rates must satisfy:

\frac{R_{H}}{R_{L}} \leq 1 + \frac{u_{H} - u_{L}}{1 - u_{L}}

If every segment satisfies this inequality and the curve is continuous, a deposit can cross any number of segments while income remains nondecreasing. No extra condition between segments is needed.

Segment	Borrow APR	Rate ratio	Maximum
0% to 80%	5% to 9%	1.800	1.800
80% to 90%	9% to 13.5%	1.500	1.500
90% to 95%	13.5% to 20%	1.481	1.500
95% to 98%	20% to 32%	1.600	1.600
98% to 99.5%	32% to 56%	1.750	1.750
99.5% to 99.99%	56% to 110%	1.964	1.980

The example curve becomes sharply responsive near full utilization while satisfying the adjacent rate rule on every segment. A flat clamp after the final point also preserves monotonicity.

What the constrained curve changes

The figures below use round illustrative inputs: 80% optimal utilization, a 5% rate at the kink, a 100% maximum rate, and a 5% supplier protection target. They show the mechanism only and do not use market data. $O$ marks the kink and $U_{c}$ the point where the constrained curve rejoins the standard curve.

Borrow rate. The constrained curve spreads the increase over a wider utilization interval, then rejoins the standard curve before full utilization.

Supply rate, given by

R_{s} (U) = U R_{b} (U)

. The lower rate through the constrained interval prevents a supplier's own deposit from collapsing income across their full balance.

Largest supplier share for which another deposit does not reduce total interest income. The constrained curve maintains the 5% target through the replaced interval.

The endpoints impose a limit

A curve still needs to connect the rate at optimal utilization, $R_{o}$ , to the maximum rate $R_{max}$ at full utilization. Integrating elasticity between those endpoints gives:

\ln (\frac{R_{max}}{R_{o}}) = \int_{O}^{1} η (U) d \ln U

If elasticity is at most $K$ everywhere on the interval, then:

\ln (\frac{R_{max}}{R_{o}}) \leq K \ln (\frac{1}{O})

Rearranging puts a lower bound on the worst elasticity any continuous differentiable curve can achieve:

K^{*} = \frac{\ln (R_{max} / R_{o})}{\ln (1 / O)}

An elasticity ceiling below $K^{*}$ is impossible without changing the endpoints. The bound is tight: the constant elasticity curve with $K = K^{*}$ reaches $R_{max}$ exactly at full utilization.

The maximum supplier share that can be guaranteed while keeping the endpoints is:

{\bar{α}}_{max} = \frac{1}{1 + K^{*}}

Optimal utilization, the rate at that point, the maximum borrow rate, and the supplier protection target cannot all be chosen independently.

The full pool edge case

For a supplier who owns the entire pool, $α = 1$ . The monotonicity condition becomes $1 + η \leq 1$ , or:

η \leq 0

A conventional utilization dependent IRM has positive elasticity, so it cannot protect a supplier who owns 100% of supply. The identity behind this is simple:

I = T U R_{b} (U) = B R_{b} (U)

With borrowings fixed, more supply lowers utilization and therefore total borrower interest. There are no other suppliers across whom that reduction can be distributed. Protecting the full pool requires a flat or decreasing borrow rate.

A minimal change to an existing curve

Consider a standard single kink curve. Below optimal utilization it rises linearly from zero. Above the kink it follows a steeper line:

R_{old} (U) = R_{o} \frac{U}{O} for U \leq O

R_{old} (U) = R_{o} + m (U - O), m = \frac{R_{max} - R_{o}}{1 - O}

The curve does not need to be replaced everywhere. Let $U_{c}$ be the first utilization after the kink where the constant elasticity boundary meets the old branch again:

R_{o} {(\frac{U_{c}}{O})}^{K} = R_{o} + m (U_{c} - O)

The modified model keeps the original ramp through the kink, follows the constant elasticity boundary only while needed, then returns to the original curve:

R_{new} (U) = {\begin{cases} R_{o} \frac{U}{O} & if U \leq O \\ R_{o} {(\frac{U}{O})}^{K} & if O < U < U_{c} \\ R_{o} + m (U - O) & if U_{c} \leq U \leq 1 \end{cases}

This construction is feasible when $K \geq K^{*}$ . It preserves the kink, maximum rate, and every part of the old curve that already satisfies the constraint. As the protected share approaches zero, $K$ approaches infinity and the construction collapses back to the original IRM.

Fixed supplier share implementation

The fixed supplier share guarantee can also be applied to piecewise linear rate curves. For a segment from $(u_{i}, r_{i})$ to $(u_{i + 1}, r_{i + 1})$ , define $Δ r = r_{i + 1} - r_{i}$ and $Δ U = u_{i + 1} - u_{i}$ .

The segment slope and rate are:

m_{i} = \frac{Δ r}{Δ U} R_{b} (U) = r_{i} + m_{i} (U - u_{i})

Its elasticity is:

η (U) = \frac{U m_{i}}{R_{b} (U)}

For a positive linear segment, elasticity is monotone within the segment. Checking both endpoints therefore bounds the whole interval. Substituting the segment slope and cross multiplying gives a division free condition:

U Δ r \leq K r Δ U

Here $(U, r)$ is each endpoint being checked. The inequality works with fixed point integer arithmetic and avoids numerical division. Implementations still need consistent scaling and sufficiently wide intermediate products.

This check belongs where a market's rate configuration is validated, not in the rate calculation hot path. Existing models can keep their piecewise linear representation and add maximum elasticity as an invariant on the configured segments.

Assumptions and limits

The derivation holds borrowings fixed while a supplier deposits. In equilibrium, a lower borrow rate may attract new borrowing and move utilization back up. That response would soften the income reduction, which makes the fixed borrowing condition conservative.

The analysis also ignores protocol fees. A constant reserve factor multiplies supplier income without changing the sign condition, but a utilization dependent fee requires its own derivative term.

The general derivative condition is local: it asks whether the next marginal token reduces income. For a finite deposit, the inequality needs to hold along the entire path from the initial state to the final state. It also prices withheld capital at zero. A supplier deciding between markets should compare against the return available elsewhere.

Finally, monotone supplier income is only one IRM objective. Borrow demand, liquidation risk, withdrawal liquidity, rate stability, and the speed at which a stressed market returns toward its target utilization still matter.

Conclusion

The general condition $α (1 + η) \leq 1$ separates the supplier's pool share from the rate curve's response to utilization. A fixed supplier share target gives a constant elasticity ceiling and a boundary described by a power law. A fresh depositor guarantee gives $(1 - u) R_{b}^{'} (u) \leq R_{b} (u)$ and the adjacent rate rule.

These are different protections derived from the same condition. Both can be checked against an existing IRM, and the useful choice depends on which supplier path the protocol intends to protect.