<p>$a=0$ means implies the household does not enjoy leisure. In other words, the trade-off between consumption and leisure is switched off. It's not generally true that with $a=0$, utility is higher for a given value of $c_t$, as this depends on the previous value of $l$. Whether or not consumption rises as a consequence depends also on the parameter $b$, i.e. the utility discount factor. 
If the household does not value leisure, he might work 80 hours per week and obtain a higher income. If he is very forward-looking however, he might also save a lot for later consumption.</p>

CC BY-SA 4.0
2018-05-25T09:48:10.133Z
2018-05-25T09:48:10.133Z
E. Sommer
<p>Without labor disutility, the household will work full-time so that <span class="math-container">$e=1$</span>. The Euler equation will pin down the optimal steady state capital-to-labor ratio, which is just a function of the time preference <span class="math-container">$b$</span> and the depreciation rate <span class="math-container">$x$</span>. If the capital to labor ratio is always a fixed number regardless of <span class="math-container">$a$</span>, you can immediately see that the numerator <span class="math-container">$k$</span> must be higher if the denominator <span class="math-container">$e$</span> is at its maximum.</p>
<p>Intuitively, the production factors are q-complements: more of one factor makes the other one more productive as well. With higher labor, capital becomes more productive and it pays to invest more.</p>

CC BY-SA 4.0
2022-03-13T10:58:18.663Z
2022-03-13T10:58:18.663Z
jpfeifer
<blockquote>
<p>I have following dynamic optimization problem</p>
<p><span class="math-container">$$V(k_0)=\max \sum_{t=0}^{\infty} b^t((1-a)ln c_t+a ln(l_t))$$</span></p>
<p>Subject to <span class="math-container">$l_t+e_t=h$</span> and <span class="math-container">$y=Ak_t^pe_t^{1-p}$</span> and <span class="math-container">$k_{t+1}=i_t+(1-x)k_t$</span></p>
<p>Where c is consumption, i is inverse,ent, y is output, k is phisical capital per capita. <span class="math-container">$l$</span> is leisure time , e is working time. V is value function. x is depreciation rate. A is total factor productivity.</p>
<blockquote>
<blockquote>
<blockquote>
<p>The question I asked to you is that when a is zero, the economy would accumulate more physical capital? Or not? If yes, Why?</p>
</blockquote>
</blockquote>
</blockquote>
</blockquote>
<p>I think that</p>
<p>Capital accumulates according to <span class="math-container">$k_{t+1}=i_t+(1-x)k_t$</span></p>
<p>Since investment <span class="math-container">$i_t=y_t-c_t$</span></p>
<p>So</p>
<p><span class="math-container">$k_{t+1}=y_t+(1-x)k_t-c_t$</span></p>
<p>When a=0, the utility function becomes <span class="math-container">$u(c_t)=ln(c_t)$</span></p>
<p>That, househoulds get more utility from consumption with this new utility function in comparision to the previous utility function given. So consumption level increases, which decreases one period forward capital stock.</p>
<p>But the things that I said seems to be wrong to some extent. What do you think ? Do you have any other wise idea about this?</p>

CC BY-SA 4.0
2018-05-25T13:49:37.450Z
The reason why the model is written in mathematical terms is in order to be able to obtain definite answers on such interesting questions. Certainly an initial assessment and tentative answer is a valuable start - but you will find the answer if you solve the model under the two different assumptions.
Alecos Papadopoulos
CC BY-SA 4.0
2018-05-25T21:46:57.763Z
I solved the model in two ways by using guess verify method and also by Euler equation. But this question is the first part of the question. So I guess its answer is required without any such long solutions. @AlecosPapadopoulos
mnm123
<macroeconomics><self-study><dynamic-programming>
Capital accumulation
mnm123
# Capital accumulation

(Asked by: username_0 on 2018-05-24)

> I have following dynamic optimization problem
>
> $$V(k\_0)=\max \sum\_{t=0}^{\infty} b^t((1-a)ln c\_t+a ln(l\_t))$$
>
> Subject to $l\_t+e\_t=h$ and $y=Ak\_t^pe\_t^{1-p}$ and $k\_{t+1}=i\_t+(1-x)k\_t$
>
> Where c is consumption, i is inverse,ent, y is output, k is phisical capital per capita. $l$ is leisure time , e is working time. V is value function. x is depreciation rate. A is total factor productivity.
>
> > > > The question I asked to you is that when a is zero, the economy would accumulate more physical capital? Or not? If yes, Why?

I think that

Capital accumulates according to $k\_{t+1}=i\_t+(1-x)k\_t$

Since investment $i\_t=y\_t-c\_t$

So

$k\_{t+1}=y\_t+(1-x)k\_t-c\_t$

When a=0, the utility function becomes $u(c\_t)=ln(c\_t)$

That, househoulds get more utility from consumption with this new utility function in comparision to the previous utility function given. So consumption level increases, which decreases one period forward capital stock.

But the things that I said seems to be wrong to some extent. What do you think ? Do you have any other wise idea about this?

**Tags:** macroeconomics, self-study, dynamic-programming

> **Comments:**
> **username_1**: The reason why the model is written in mathematical terms is in order to be able to obtain definite answers on such interesting questions. Certainly an initial assessment and tentative answer is a valuable start - but you will find the answer if you solve the model under the two different assumptions.  
> **username_0**: I solved the model in two ways by using guess verify method and also by Euler equation. But this question is the first part of the question. So I guess its answer is required without any such long solutions. @username_1  

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## 2 answers

### Answer by username_2 on 2018-05-25. Score: 2

$a=0$ means implies the household does not enjoy leisure. In other words, the trade-off between consumption and leisure is switched off. It's not generally true that with $a=0$, utility is higher for a given value of $c\_t$, as this depends on the previous value of $l$. Whether or not consumption rises as a consequence depends also on the parameter $b$, i.e. the utility discount factor.
If the household does not value leisure, he might work 80 hours per week and obtain a higher income. If he is very forward-looking however, he might also save a lot for later consumption.


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### Answer by username_3 on 2022-03-13. Score: 0

Without labor disutility, the household will work full-time so that $e=1$. The Euler equation will pin down the optimal steady state capital-to-labor ratio, which is just a function of the time preference $b$ and the depreciation rate $x$. If the capital to labor ratio is always a fixed number regardless of $a$, you can immediately see that the numerator $k$ must be higher if the denominator $e$ is at its maximum.

Intuitively, the production factors are q-complements: more of one factor makes the other one more productive as well. With higher labor, capital becomes more productive and it pays to invest more.


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