# Unified growth theory optimization problem

I'm trying to understand the standard unified growth theory model as summarized on page 60 here: https://www.econstor.eu/bitstream/10419/80210/1/481894578.pdf;Unified

The basic household optimization problem is

$$max_{n_{t},e_{t+1}} c_{t}^{1-\gamma}(n_{t}h_{t+1})^{\gamma}$$

subject to

$$z_{t}(1-n_{t}(\tau+e_{t+1}))\geq \tilde{c}$$,

$$n_{t} \geq 0$$,

and

$$e_{t+1} \geq 0$$,

where $$h_{t+1}=h(e_{t+1},g_{t+1})$$.

I see how $$n_{t}(\tau+e_{t+1})=\gamma$$ is obtained, however, I don't understand how $$n_{t}(\tau+e_{t+1})=1-\frac{\tilde{c}}{z_{t}}$$ is obtained. Any help on this is appreciated.

Seems like the $$n_{t}(\tau+e_{t+1})=1-\frac{\tilde{c}}{z_{t}}$$ equation you don't understand is just a simple rearrangement of the first condition $$z_{t}(1-n_{t}(\tau+e_{t+1}))\geq \tilde{c}$$ if it holds as an equality.
• This is helpful. But how does simply rearranging the budget constraint and plugging in $\tilde{c}$ give the optimal value for $n_{t}(\tau+e_{t+1})$ when $z_{t}$ is below a certain threshold? In doing this, we're ignoring the utility function entirely which is, after all, our objective function. May 4, 2022 at 16:24
• I think I may see how this works. When $z_{t}(1-n_{t}(\tau+e_{t+1})) \geq \tilde{c}$ is not binding ($z_{t} \geq \tilde{z}_{t}$), consumption can be derived from the FOCs as $c_{t}=\frac{1-\gamma}{\gamma}n_{t}z_{t}(\tau+e_{t+1})$. When the constraint is binding ($z_{t} \leq \tilde{z}_{t}$), however, $\tilde{c}>c_{t}=z_{t}(\frac{1-\gamma}{\gamma})n_{t}(\tau+e_{t+1})$. Therefore we insert $c_{t}=\tilde{c}$ into the budget constraint and solve for $n_{t}(\tau+e_{t+1})$ obtaining $n_{t}(\tau+e_{t+1})=1-\frac{\tilde{c}}{z_{t}}$. Is that right? May 4, 2022 at 22:36