Each token in each Pool has a utilization rate $U_t$ at each time $t$, which is defined by the ratio $U_t = L_t/A_t$where $L_t$ represents total borrows and $A_t$ represents total liquidity. This ratio regulates how much of a deposited token is used in the market.

The kinked model is based on several parameters and is described as follows. Firstly, the ratio optimal utilization rate $U_\text{optimal}$ is a parameter that determines the best-fit utilization rate for the protocol. It varies from token to token and depends on the risk and volatility of the token. For example, it could be 80% for stable tokens like DAI, or just 65% for unstable coins like LINK.

Next, the interest rate is a function that varies based on utilization rate. Three parameters are used to define this function: (1) Initial rate $R_0$, (2) the constant $R_\text{slope1}$ is used when $U_t < U_\text{optimal}$, and (3) the constant $R_\text{slope2}$ is used when $U_t \geq U_\text{optimal}$.

Specifically, the interest rate $R_t$ is defined by:

If $U_t <U_\text{optimal}$ then $R_t = R_0 + \frac{U_t}{U_\text{optimal}} \times R_\text{slope1}$

If $U_t \geq U_\text{optimal}$ then $R_t= R_0 + R_\text{slope1} + \frac{ U_t - U_\text{optimal}}{1 -U_\text{optimal}} \times R_\text{slope2}$

The interest rates have a kink in them: they sharply change at a certain point. A variety of protocols use interest rates like this, including Aave and Compound. As users mint, redeem, borrow, repay, or liquidate tokens, the interest rate changes with the utilization rate $U_t$ . As a result, it's also known as a variable interest rate.** **

The borrow rate is calculated using the current interest rate$R_t$(per year) and is updated every second. That is, borrowers will be charged interest that changes every second. He must pay more interest if the Utilisation Rate is higher, and vice versa. Mathematically, at each time t, the Borrow rate (per second) is $B_t = R_t/Y$ where $Y= 31536000$ is the number of seconds in a year. If the user is currently borrowing $T_t$ tokens then the interest payable is $T_t \times B_t$ . The interest payable is accrued to the loan in the next second, that is $T_{t+1}=T_t(1+B_t)$ .

Deposit rate and Borrow rate are closely linked. Precisely, at each time $t$, deposit rate $D_t$ is determined based on Utilization Rate $U_t$ and borrow rate $B_t$ and reserve factor $r$. Precisely, deposit rate is defined by $D_t = U_t \times B_t\times (1-r)$.

In terms of probability, $U_t$ is probability that each token is borrowed, thus $D_t$ is the average value of interest earned from each token deposited into the Pool.

For deposit, the calculation of interest is updated after each operation deposit, withdraw, borrow, repay. In particular, if a user is currently depositing $T_t$ tokens, after a period of time $\Delta{t}$ (in seconds) he will receive an interest of $T_t \times D_t \times \Delta{t},$ and this interest is accrued to the deposited token, i.e. after time $\Delta{t}$ he have $T_{t+\Delta{t}} = T_t \times (1 + D_t \times \Delta{t}).$