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There are two interrelated maximisation problems. The first is the infinite horizon maximisation problem: \begin{align*} v(k) = &\max_{a_1, a_2, \ldots} \sum_{t = 0}^\infty \delta^t F(k_t, c_t),\\ \text{ subject to } & k_{t+1} = g(k_t, a_t),\\ & a_t = \Gamma(k_t),\\ & k_0 = k \end{align*} Here we call $a_t$ the decision variables, $k_t$ ...