# Summary

Steepen rebase curve and tighten equilibrium bands to promote faster equilibrium discovery

## Abstract

This proposes an update to the growth, upper asymptote, and lower asymptote configuration parameters of the Sigmoid reaction lag function. Additionally, it proposes tightening the equilibrium band within which no rebases occur.

## Motivation

The Sigmoid function (implemented in AIP-5) replaced a linear function with an S-shaped function, allowing for sophisticated supply reaction tuning. However, to ease this transition, AIP-5 kept the steepest segment of the new curve as similar as possible to that of the original linear function. For commentary on the original proposal see: Rebase curve upgrade proposal

There is now sufficient reason to believe that a steeper slope would result in a tighter price distribution around the target, and thus better serve AMPLâ€™s unit of account function. Recognizing that the primary source of price discovery and liquidity exists on constant product AMMâ€™s, we can more precisely estimate the supply change required to bring price into equilibrium as:

*An***X% price deviation**requires an**X% supply adjustment**to restore price equilibrium.

Moreover, tightening equilibrium bands (ie: the threshold of tolerance within which no rebases occur) would allow the system to adjust granularly according to smaller changes in demand rather than waiting until price is 5% away from its target to take effect, causing AMPL To spend more time at its precise target.

## Specification

Recall that the sigmoid function curve is defined by:

- F(x) : fraction of supply added or removed (supply change %)
- x : normalized price deviation

It has shaping parameters that determine: lower asymptote, upper asymptote, and the steepness of the curve (ie: growth rate).

- l = lower asymptote
- u = upper asymptote
- g = growth rate

## Proposed Parameter Values

This update proposes changing the parameter values to:

- l = -0.077
- u = 0.05
- g = 45

The resulting reaction lag curve looks as follows:

The red line, pictured above, represents a linear curve that would offset an X% price deviation with an X% supply change.

The proposed parameter changes produce a sigmoid curve that closely resembles the slope of the red line near the origin, but asymptotes horizontally at a max supply change of +5% above and -7.7% below AMPLâ€™s price target.

This update also proposes changing the deviation ratio values:

- deviation_ratio = 0.025

## Conclusion

The Ampleforth protocolâ€™s rebase reaction curve will keep its existing structure, but will more quickly and precisely adjust supply in response to demand, such that AMPL spends a maximum amount of time near its target.