Consider a three period version of a model with uncertainty. There are three periods t = 0,1,2. The shock z_1 has a density p(z_1). The shock z_2 is independent of z_1 and has the same density p(z_2). The optimization problem is: \begin{gather} \max_{c_0, k_1, \{c_1(z_1), k_2(z_1)\}, \{c_2(z_1, z_2)\}} \quad U(c_0) + \beta \int U(c_1(z_1)) p(z_1) d z_1 + \beta^2 \int U(c_2(z_1, z_2))p(z_1)p(z_2)dz_1 dz_2 otag \\ \text{s.t.} \quad f(k_0) = c_0 + k_1, otag \\ \exp(z_1)f(k_1) = c_1(z_1) + k_2(z_1) ~ \text{for all $z_1$}, otag \\ \exp(z_2)f(k_2(z_1)) = c_2(z_1, z_2) ~ \text{for all $(z_1, z_2)$}. otag \end{gather}. Notice that k_2(z_1) denotes the capital shock chosen in period 1 after the realization of z_1 which will be used in production in period 2. Write down the Lagrangian and derive the first order conditions