Let $\mathbb T$ denoted the the time index set in our dynamic model. We consider the following cases:
Let's consider an utility flow $u(t)$ for $t$ in $\mathbb{T}$ of an agent.
When $u(t)$ for $t$ in $\mathbb{T}$ is the flow of profits for a firm:
Consider discrete time and an utility flow $u(t)$ for $t$ in $\mathbb{T}$ of an agent. With discounting preferences, an agent evaluates utility flows, $(u(0),u(1),u(2),\ldots)$, accordingly to the discounted value of the flow: $$U(u(0),u(1),u(2),\ldots)=\sum_{t=0}^{+\infty}d(t)\cdot u(t).$$
Here, $d(t)$ is the discount factor of period $t$, and $d(t)\cdot u(t)$ is the present utility (the utility in terms of period $t=0$ utility) of receiving $u(t)$ utils in period $t$. Since we assume agents display impatient, they prefer to receive utility sooner than later, it follows that $0<d(t)<1$.
Exponential Discounting is the standard discounting model. Under exponential discounting $d(t)=\delta^{t}$ where $0<\delta<1$. $$U(u(0),u(1),u(2),\ldots)=\sum_{t=0}^{+\infty}\delta^{t}\cdot u(t).$$ The continuous time version is: $$U(u(t)_{t\in \mathbb T})=\int_{0}^{+\infty}\exp(-\rho\cdot t)\cdot u(t),$$ where $\rho=-\log(\delta)>0$.
Consider the standard exponential discounting model with discrete time and finite horizon. We shall re-normalize the discounted utility by multiplying the utility by a factor of $1-\delta$, so the the discounted utility at period zero can be expressed as: $$U_{0}=(1-\delta)\sum_{t=0}^{T}\delta^{t}u_t.$$ Remember that $u_{t}$ is the instantaneous utility-- the utility that is actually received at period $t$. By the same token the discounted utility at period $\mathbf k$ can be written as: $$U_{k}=(1-\delta)\sum_{t=k}^{T}\delta^{t-k}u_t.$$
Notice that we can write the discounted utility at period $\mathbf t$ in a recursive manner:
$$U_{t}=(1-\delta)u_t+\delta U_{t+1} \tag{R}$$Using the fact that with a finite horizon, $U_{T+1}=0$ by definition, we can use the recursive equation (R) as a way of defining the discounted utility at a given period.
We saw that exponential discounting preferences can be stated in terms of recursive utility: $$U_{t}=(1-\delta)u_t+\delta U_{t+1},$$ The above formulation is a particular case of the following recursive preferences: $$U_{t}=\left((1-\delta)u_t^{\theta}+\delta U_{t+1}^{\theta}\right)^{\frac1\theta}$$ where $0<\theta\leq 1$ and $0<\delta < 1$.
Consider $\theta=\frac12$ and $\delta=\frac12$. Suppose $u_0=4$, $u_1=16$, and from $t=2$ onwards utility is constant: $u_2=u_3=\cdots=4$. Then: $$U_2=4, \quad U_1=\left(\tfrac12\sqrt{16}+\tfrac12\sqrt{4}\right)^2=9, \quad U_0=\left(\tfrac12\sqrt{4}+\tfrac12\sqrt{9}\right)^2=6.25.$$
Note: We assume that if the utility flow is constant from some period onwards, $u_t=\overline{u}$, then $U_t$ is also constant from that period onwards, $U_t=\overline{u}$. This assumption is needed because we cannot compute $U_\infty$ directly for infinite sequences.
Assuming exponential discounting, at period $t$, what is the maximum amount the agent would be willing to sacrifice of future utility $U_{t+1}$ in order to increase the instantaneous utility $u_{t}$ by one util?
The answer is the marginal rate of intertemporal substitution of utility, MRIS. $$ (1-\delta)\Delta u_t+\delta \Delta U_{t+1}=0 \Rightarrow \mathrm{MRIS}=\left|\dfrac{ \Delta U_{t+1}}{ \Delta u_{t}}\right|=\dfrac{\delta}{1-\delta}. $$
So, in the case of exponential discounting the MRIS is a constant.
In order to allow for preferences with non-constant MRIS, the following family of recursive utility functions is often used: $$U_{t}=\left((1-\delta)u_t^{\theta}+\delta U_{t+1}^{\theta}\right)^{\frac1\theta}$$ where $0<\theta\leq 1$.
$$\mathrm{MRIS}=\left|\dfrac{ \Delta U_{t+1}}{ \Delta u_{t}}\right|=\dfrac{\delta}{1-\delta}\left(\dfrac{U_{t+1}}{u_{t}}\right)^{1-\theta}. $$