---
title: Incentives system for isolated lending markets
author: "Prevert"
date: "2024/06/06"
description: post
tags: lending
---

# Incentives system for isolated lending markets

How to split a fixed amount of tokens per period between isolated markets so that total supply is maximized?

Notations:
- $s_i$ : total deposits in market $i$
- $\rho_i$ : native APR of the market, exogenous, based on borrowers' demand
- $n_i$ : quantity of tokens distributed in market $i$ to incentivize supply $s_i$ over a given period, annualized
- $n$ : total quantity of tokens distributed over a given period, annualized
- $p$ : price of the token distributed

Market $i$'s total return is:
$$
r_i = \rho_i + \dfrac{pn_i}{s_i}
$$

## Optimality condition

The protocol is assumed to maximize total deposits in two markets (with $\lambda$ the Lagragian factor):
$$
\max s_1 \Big( \rho_1 + \dfrac{pn_1}{s_1} \Big) + s_2 \Big( \rho_1 + \dfrac{pn_2}{s_2} \Big) + \lambda (n-n_1-n_2)
$$

First-order condition requires equality of marginal increase of supply across the two markets following a marginal transfer of tokens from one market to the second:
$$
s'_1 (r_1) \dfrac{s_1 - n_1 s'_1 (r_1)}{s_1^2} = s'_2 (r_2) \dfrac{s_2 - n_2 s'_2 (r_2)}{s_2^2}
$$

Define the supply elasticity to total APR:
$$
\varepsilon_i = s'_i (r_i) \dfrac{r_i}{s_i}
$$

which is positive under normal conditions.

The optimality condition simplifies to:
$$
\dfrac{\varepsilon_1}{r_1} - n_1 \Big (\dfrac{\varepsilon_1}{r_1} \Big)^2 = \dfrac{\varepsilon_2}{r_2} - n_2 \Big (\dfrac{\varepsilon_2}{r_2} \Big)^2
$$

## Interpretation

Adding one token in market $i$ produces two effects of opposite directions with respect to supply:

- a positive effect: more APR induces more lending
- a negative (second-order) effect: more lending reduces APR due to dilution

The first effect dominates the second one for small distributed quantities $n_i$. As more tokens are directed to market $i$, the second effect increasingly reduces the marginal benefits of incentives, until a point where distributing more tokens produces a negative impact on TVL. The higher the elasticity and the smaller the APR, the sooner this happens.

The second effect is important to take into account. Suppose that market $1$ has a supply of 1$m and market $2$ a supply of 10$m. Due to the dilution effect, distributing additional tokens in market $1$ raises additional APR by a factor of $10$ compared to market $2$. Hence, unless elasticity in market $1$ is much lower than in market $2$, it is more efficient to channel the tokens into it (up to a certain level).

To a first approximation, the negative second order effect in the optimality condition could be neglected, either because APRs $r_i$ are high, or elesticities $\varepsilon_i$ are close to zero, or quantities $n_i$ are limited. We get:
$$
\dfrac{r_1}{r_2} = \dfrac{\varepsilon_1}{\varepsilon_2}
$$

that is, the higher the elasticity compared to other markets, the higher total APR in this market.

Adding the second order effect:
$$
\dfrac{\varepsilon_1}{r_1} - n_1 \Big (\dfrac{\varepsilon_1}{r_1} \Big)^2 = \dfrac{\varepsilon_2}{r_2} - n_2 \Big (\dfrac{\varepsilon_2}{r_2} \Big)^2
$$

doesn't change the general property. If $\varepsilon_1/r_1$ is close to $\varepsilon_2/r_2$, $(\varepsilon_1/r_1)^2$ is also close to $(\varepsilon_2/r_2)^2$, as long as $n_1$ is not too far from $n_2$. A gap between $n_1$ and $n_2$ only exists to align total APRs on elasticities.

To gain better understanding, suppose that optimal total APRs in markets $1$ and $2$ are $10$ and $12$\% (resp.). If native APRs are $8$ and $11$%, the distribution of tokens fills the gap in both markets to provide targetted APRs.

If native APRs in markets $1$ and $2$ are $3$ and $15$\%. Optimal APRs $10$ and $12$% should still be targeted by distributing all tokens to the first market and by applying a positive fee rate in the second market.

## Conclusion

Targetted APRs, after incentives, are, to a first approximation, proportional to supply elasticities relative to APR, which is intuitive as we want to maximize total supply. A second order effect exists however which puts a limit to how much a market can be rewarded with tokens. As the supply is increasing in a high-elasticity market, adding new tokens translates into less and less into additional APR for lenders due to the dilution effect. Hence, as a general rule, low supply markets should be prioritized if elasticities are suspected to be high enough.

A fee rate by market is not added in the model but could contribute to reaching the optimum. For instance, high native APR markets could be taxed with the proceeds redistributed to low native APR markets, to align APRs, after incentives and fees, on elasticities.

<!--

equalized by way of fees aplied to market 2.

To summarize, if elasticities are the same across markets or they are so difficult to estimate that the best guess is to suppose they are equal, then total APRs, after incentives and fees, should be equalized.

## Equal elasticities

Suppose that the supply elasticity to APR is the same across markets: $\varepsilon_i = \varepsilon$, $i=1,2$. The optimality condition simplifies to:
$$
\dfrac{1}{r_1} - n_1 \dfrac{\varepsilon}{r_1^2} = \dfrac{1}{r_2} - n_2 \dfrac{\varepsilon}{r_2^2}
$$

$$
n_1 = n_2
$$

The solution of this equation is $r_1 = r_2$.

APRs, net of incentives, should be equalized across markets.

Notice what should not ne equalized: the quantities of tokens $n_i$, the rates of distribution $n_i/s_i$ or the additional APR based on the token distribution $pn_i/s_i$.

To understand the implications,