I have the following social planner problem to maximize $\{c_t, k_t, n_t \}$

$\begin{gather*}E_0 \sum_{t=0}^\infty \beta^t U(c_t, 1 - n_t), 0 < \beta < 1\end{gather*}$

subject to

$\begin{gather} y_t = F(k_{t-1}, n_t) \\ c_t + x_t = F(k_{t-1}, n_t) \\ k_t = h(x_t, k_{t-1}) \end{gather}$

For the final equation for capital stock evolution, $h$ is a function $h(x_t, k_{t-1}) = k_{t-1} g(x_t/k_{t-1})$ with $g(0) = 0$, $g' > 0$, and $g'' < 0$.

Standard variable terminology applies, $k_{t-1}$ is the capital stock carried into period $t$, $n_t$ is hours worked at $t$ and $c_t$ and $x_t$ are consumption and investment at $t$.

To determine first order conditions in this type of problem, I would normally reduce the constraints down to a single equation. With standard capital adjustment, this would be trivial. However, the inclusion of $h$ is confusing me. It's easy to see that the first two constraints can be combined, $c_t + x_t = y_t$ , but is it possible to combine that and the remaining constraint without functional forms?

It's probably a simple mathematical property that I'm forgetting. Can anyone help me out?

  • $\begingroup$ Chain rule of differentiation (which works without the need for functional forms), and use of multipliers. $\endgroup$ – Alecos Papadopoulos Apr 11 '15 at 3:26

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