# Stochastic Dynamic Programming: Deriving the Steady-State for a Lottery

I am working through the basic examples of the stochastic RBC models in the book by McCandless (2008): The ABCs of RBCs, pp. 71 - 75

### A Standard Stochastic Dynamic Programming Problem

Here is a formulation of a basic stochastic dynamic programming model: \begin{equation} y_t = A^t f(k_t) \end{equation}

\begin{equation} A^t = \cases{A_1 \text{ with probability } p \\ A_2 \text{ with probability } (1 - p) } \end{equation}

\begin{equation} k_{t+1} = A^tf(k_t) + (1 - \delta)k_t - c_t \end{equation}

With an agent maximizing the expected utility function: \begin{equation} E_t \sum_{t}^\infty \beta^t u(c_t) \end{equation}

Substituting consumption from the previous equation and using the recursive formulation of the problem gives the following problem: \begin{equation} V(k_t, A^t) = \max_{k_{t+1}} \left[u(A^tf(k_t) + (1 - \delta)k_t - k_{t+1}) + \beta E_t V(k_{t+1},A^{t+1})\right] \end{equation}

Then McCandless proceeds saying that the algorithm to solve to the problem is almost identical to the deterministic case. One finds the first-order conditions (a derivative of the value function with respect to $k_{t+1}$) for the control variables, then does the same for $k_t$ and applies the Envelope theorem to get an analytical solution. Steady-states found, model written, paper submitted. Profit.

### A Lottery Augmented Version

Now I want to investigate a little different case. Take the very same model but introduce another control variable. Call it $s_t$ for a security: \begin{equation} l_{t+1} = l_t + s_t \end{equation}

And the $l_t$ enters the problem through the variable $A^t$: \begin{equation} A^t = \cases{A_1 \text{ with probability p} \\ A_2l_{t} \text{ with probability (1 - p)}} \end{equation}

The main difference is easy to see if one writes the equation for income in a period $t$ explicitly opening up the expectation sign: \begin{equation} y_t = pA_1f(k_t) + (1-p)A_2l_tf(k_t) \end{equation}

In this case we have a deterministic control variable $l$ "turning on" when a certain event happens (as if you would win in a lottery, which increases your income by the factor of how much you invested in it - yes the example makes little sense but I am interested in the principle itself). The Question: Does the following lottery augmentation changes the process of how to solve the model? If yes, what is the idea behind and what does change? If no, why is that?

P.S. If someone could point me a paper with an example model, which is very close to the one I described, that would be brilliant.