Consumer maximizes expected utility.

\[max_{c_t} \sum_t^{\infty} p^t E(u(c_t))\] \[s.t.\] \[c_t + k_{t+1} = f(k_{t+1}) + (1 - \gamma)k_t\]

where $t$ is time, $c$ is consumption, $u$ is utility, $k$ is capital. $\gamma$ is the depreciation rate on the the capital at the previous time step.

The lhs of the constraint $c_t + k_{t+1}$ indicates the consumer can choose between consuming today or having more capital tomorrow. And the rhs indicates the source of income (stocahstic function $f$) and the depreciation on old capital.

\[f(k_t) = A_t k_t^\alpha\]

Lets say this is our production function. $A_t$ is a random variable.

Setup the Lagrangian, which is equal to the expectation.

\[L = E\big[\sum_{t=0}^\infty p^t[u(c_t) - \lambda_t(c_t + k_{t+1} - f(k_t) - (1-\gamma)k_t)]\big]\]

First order derivatives of this lagrangian.

\[\frac{d L}{d c_t} = u'(c_t) - \lambda_t = 0\]

And then

\[\frac{d L}{d k_{t+1}} = -\lambda_t + p E\big[ \lambda_{t+1}(f'(k_{t+1} + 1 - \gamma))\big] = 0\] \[\frac{d L}{d \lambda_t} = c_t + k_{t+1} - f(k_t) - (1-\gamma)k_t = 0\]