# Kelp Protocol (KP)

The Kelp Monetary Policy framework is guided by a modified Quantity Theory of Money (QTM) which asserts that there is a direct relationship between the amount of money in circulation and the general level of prices. This modified framework is expressed as:

$P_K =\frac{CV}{R} \tag{1}$Where:

$P = P_K$

$C$ = Coin Supply

$V$ = Velocity of Coin

$R$ = Market Base

Price of the Kelp at any particular point in time is the weighted market price at which Kelp is exchanged over the many exchanges across the different pairs. This is denoted as:

$P_K = \frac{P^1_1W^1 + ... + P^n_1W^n}{P^1_KW^1 + ... + P^n_KW^n} \tag{2}$Where:

$P_K$ = Kelp price denominated in USD,

$P^1_1 ... P^n_1$ = Kelp pairwise exchange rates, while $W^1 ... W^n$ is the respective weights of the pairwise exchange rates.

Kelp’s initial target price is to reach parity with the USD. This is important for adoption as Kelp is meant to become an alternative medium of exchange but can only be considered as such if the public is able to, roughly, equate the price of goods and services in $KELP.

Once Kelp reaches parity with the USD, Kelp (PK) tracks the USD at a ratio of 1:1, but allowed to fluctuate within a band of ±2 SD (standard deviations) of the target price (PT). At this stage, Kelp is fully collateralized. This is represented as:

$P_K = P_T \pm 2 SD \tag{3}$Where:

$P_T = 1_{USD}$

A crucial assumption behind the QTM is that the velocity of money or its growth rate is constant, and money growth has no effect on real revenue growth in the long run. Studies have, however, shown that velocity is far from constant, at least in the short run. So, rather than hold the velocity or its growth rate constant, we allow the growth rate of velocity to be dictated by changes to price, market base growth and coin supply growth, as:

$v = p + r - c \tag{4}$Where:

$v$ = Velocity growth rate

$p$ = Rate of price change

$r$ = Market Base growth rate

$c$ = Coin supply growth rate

The core stability principle adjusts coin supply at the completion of each block by the percentage change determined by changes to velocity, price and market base. We assume asymmetry of information during buy and sell pressures, such that it takes longer for supply adjustment to translate to price during sell pressure, and shorter path to price during buy pressure. The coin supply equation is specified as:

$c = (p + r - v) * T_f \tag{5}$Where:

$c$ = Change in coin supply

$p$ = Change in price

$r$ = Change in market base

$v$ = Change in velocity

$T_f$ = Transmission factor which differs for contractionary and expansionary pressures.

Once the Kelp ecosystem attains maturity (maturity in this case is defined as a consistent annual activity level that is equal to, or higher than the initial reserve fund), the price of Kelp will be reflective of the size and trend of the underlying activities in its ecosystem. At this point, equation 5 above will lay emphasis on variable p, which at equilibrium is specified as:

$p = v - r + c \tag{6}$We examine the Bitcoin (BTC) and Ethereum (ETH) prices for guidance on what we might expect in the interrelationship among the variables as well as the transmission factor. Due to co-integration among our variables, a Vector Error Correction Model (VECM), which allows for short run corrections, was employed. The VECM is specified as:

$\Delta y_t = \alpha \beta y_{t-1} + \Gamma _1\Delta y_{t-1} + ... + \Gamma _{p-1} \Delta y_{t-p+1} + v_{t'} \tag{7}$Where:

$\alpha \beta y_{t-1}$ = lagged error correction term

Once enough data has been generated for the Kelp cryptocurrency, a machine learning model will handle both the Positive Price Drift and Negative Price Drift phases, guided by monetary policy and theory. The model will determine the best combination of the ecosystem variables, such that the optimal combination with minimal cost to the ecosystem, and best transmission path to price is adjusted.

At any given time, the change in coin supply is a function of the Staking Ratio(SR) and the open market operations (OMO), specified as-

$\Delta i = f\{ SR, OMO \} \tag{8}$The interest rate paid on SR and OMO instruments to stakeholders at the point of redemption is the pro-rated difference between the target price and the current price, annualized. This is specified as-

$Yield = \begin{bmatrix} \frac{P_T - P_r}{P_r} \end{bmatrix} * 100 \tag{9}$ $Pro-rated \space Yield = \begin{bmatrix} \frac{Yield}{365} \end{bmatrix} * \text{number of days to redemption} \tag{10}$When Kelp price is below target price and the Kelp Protocol makes the decision to expend part of the reserves held to purchase Kelp on exchanges, the ecosystem will buy the Kelp pair on any exchange with the highest market price, specified as-

$Buy \space P^1_1 > P^n_1 \tag{11}$When Kelp price is above target price and the Kelp Protocol makes the decision to sell Kelp on exchanges, the ecosystem sells on the exchange with the lowest priced Kelp pair, specified as-

$Sell \space P^1_1 < P^n_1 \tag{12}$