<p>Seems like the
<span class="math-container">$$n_{t}(\tau+e_{t+1})=1-\frac{\tilde{c}}{z_{t}}$$</span>
equation you don't understand is just a simple rearrangement of the first condition
<span class="math-container">$$z_{t}(1-n_{t}(\tau+e_{t+1}))\geq \tilde{c}$$</span>
if it holds as an equality.</p>

CC BY-SA 4.0
2022-05-04T16:24:43.460Z
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.
econogist
CC BY-SA 4.0
2022-05-04T22:36:54.243Z
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?
econogist
CC BY-SA 4.0
2022-05-05T05:26:14.550Z
Hi! I don't know, I didn't read the paper and don't know the variables, I just saw what I wrote in the answer. Sorry I couldn't be of more help.
Giskard
CC BY-SA 4.0
2022-05-04T05:59:02.630Z
2022-05-04T05:59:02.630Z
Giskard
<p>I'm trying to understand the standard unified growth theory model as summarized on page 60 here: <a href="https://www.econstor.eu/bitstream/10419/80210/1/481894578.pdf;Unified" rel="nofollow noreferrer">https://www.econstor.eu/bitstream/10419/80210/1/481894578.pdf;Unified</a></p>
<p>The basic household optimization problem is</p>
<p><span class="math-container">$max_{n_{t},e_{t+1}} c_{t}^{1-\gamma}(n_{t}h_{t+1})^{\gamma}$</span></p>
<p>subject to</p>
<p><span class="math-container">$z_{t}(1-n_{t}(\tau+e_{t+1}))\geq \tilde{c}$</span>,</p>
<p><span class="math-container">$n_{t} \geq 0$</span>,</p>
<p>and</p>
<p><span class="math-container">$e_{t+1} \geq 0$</span>,</p>
<p>where <span class="math-container">$h_{t+1}=h(e_{t+1},g_{t+1})$</span>.</p>
<p>I see how <span class="math-container">$n_{t}(\tau+e_{t+1})=\gamma$</span> is obtained, however, I don't understand how <span class="math-container">$n_{t}(\tau+e_{t+1})=1-\frac{\tilde{c}}{z_{t}}$</span> is obtained. Any help on this is appreciated.</p>

<economic-growth><optimization>
Unified growth theory optimization problem
econogist
# Unified growth theory optimization problem

(Asked by: username_0 on 2022-05-03)

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.

**Tags:** economic-growth, optimization

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## 1 answers

### Answer by username_1 on 2022-05-04. Score: 2

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.

> **Comments:**
> **username_0**: 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.  
> **username_0**: 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?  
> **username_1**: Hi! I don't know, I didn't read the paper and don't know the variables, I just saw what I wrote in the answer. Sorry I couldn't be of more help.  

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