<p>This idea is known as the <a href="https://en.wikipedia.org/wiki/Fisher_separation_theorem" rel="nofollow noreferrer">Fisher separation theorem</a>.</p>
<p>Without the investment opportunity to transfer <span class="math-container">$h$</span> units of present day value into <span class="math-container">$w(h)$</span> units of future value, the perfect credit market gives us the intertemporal budget constraint of
<span class="math-container">$$
c_1 + \frac{c_2}{1+r} = y,
$$</span>
which can be represented by a straight line.</p>
<p><a href="https://i.sstatic.net/jVk2F.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/jVk2F.png" alt="enter image description here" /></a></p>
<p>Without knowledge of the consumer's preferences, it is not possible to say what the optimal <span class="math-container">$(c_1,c_2)$</span> is here. But we can say with certainty, that the higher the budget line is, the more options the consumer has, and assuming monotonic preferences she will be better off.</p>
<p>Now the yield curve <span class="math-container">$w$</span> of the investment opportunity allows us to shift the basic credit market budget line. Starting from the lower right point of <span class="math-container">$(y,0)$</span> (i.e. not putting money into the credit market at all) we can reach cash flows <span class="math-container">$(y-h, w(h))$</span>.</p>
<p><a href="https://i.sstatic.net/DfafT.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/DfafT.png" alt="enter image description here" /></a></p>
<p>But by also putting some money into the credit market (or borrowing), one can reach other cash flows as well. These always transfer <span class="math-container">$x$</span> units of present value into <span class="math-container">$x(1+r)$</span> units of future value (in case of borrowing <span class="math-container">$x$</span> is negative), so they are always paralel to the original budget line, but instead of starting from <span class="math-container">$(y,0)$</span>, they start from the point <span class="math-container">$(y-h, w(h))$</span> where <span class="math-container">$h$</span> is the size of the investment.</p>
<p>The optimal investment size is the one that reaches the highest budget line:</p>
<p><a href="https://i.sstatic.net/V5BES.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/V5BES.png" alt="enter image description here" /></a></p>
<hr />
<p>The idea is that given perfect credit markets one can separate consumption behavior and investment behavior. The higher the net present value of my cash flow, the better of I will be once I adjust my cash flow via the credit market to my intertemporal preferences (this is where <span class="math-container">$\beta$</span> matters). Thus I need to make investment decisions that maximize the present value of my cash flow, and I do not have to take anything else into consideration, e.g., the timing of the payments.</p>
<hr />
<p>A mathematical tidbit:</p>
<p>In the above example, given the optimal <span class="math-container">$h$</span> (assuming it is an interior point) the budget line is tangential to the orange curve depicting the possible investments, so we have
<span class="math-container">$$
w'(h) = 1 + r,
$$</span>
the marginal yield from the investment and the credit market are equal. This is equivalent to the final equation in your question.</p>

CC BY-SA 4.0
2020-06-29T09:52:18.907Z
2020-06-30T10:18:57.773Z
2020-06-30T10:18:57.773Z
Giskard
<p>Well i just edited my answer a lot. I made a fundamentally mistake, from <span class="math-container">$ L_ {h} $</span> arises an unique value of h, even when <span class="math-container">$ h $</span> is in other first order conditions, that doesn't change anything. Once that i have clear that there is no relationship between <span class="math-container">$ \beta $</span> and <span class="math-container">$ h $</span>, I think I know what is happening.</p>
<p>The reason why it is invested in <span class="math-container">$ h $</span> although <span class="math-container">$ \beta = 0 $</span> is because h increases the NPV, and this allows to increase consumption in the first period. What happens is <span class="math-container">$\beta=0 \ \Rightarrow \ \ c_{2}=0$</span>. But let´s see this more carefully.</p>
<p>To do this we have to add restrictions to the model <span class="math-container">$c_{1},c_{2},h &gt; 0$</span>, so the lagrangian will change:</p>
<p><span class="math-container">\begin{align}
L= L = u(c_1) + \beta u(c_2) + \lambda_{1} \left( y - h + \frac{w(h)}{1+r} - c_1 - \frac{c_2}{1+r} \right) +\lambda_{2}c_{1}+\lambda_{3}c_{2}+\lambda_{4}h  
\end{align}</span></p>
<p>The new first order conditions will be:</p>
<p><span class="math-container">\begin{align}  
\frac{\partial L}{\partial c_{1}} =  u^{ ' }(c_{1}) - \lambda_{1} + \lambda_{2} =0 \ (1)\\
\frac{\partial L}{\partial c_{2}} =  \beta u^{ '}(c_{2}) - \frac{\lambda_{1}}{1+r} + \lambda_{3}=0 \ (2)\\
\frac{\partial L}{\partial h} =(\frac{ w^{ ' }(h)}{1+r} -1)\lambda_{1} + \lambda_{4}= 0 \ (3)\\
\end{align}</span></p>
<p>But we need for constraints with inequalities the complementary slackness conditions (we can work with the Kuhn-Tucker lagrangian, but that it's just a special case of this general formulation):</p>
<p><span class="math-container">\begin{align}
\lambda_{1}(y-h + \frac{ w(h)}{1+r} -  \frac{ c_{2}}{1+r}-c_{1}) =0 \ (4) \\
\lambda_{2}(c_{1})=0 \ (5) \\
\lambda_{3}(c_{2})=0 \ (6) \\
\lambda_{4}(h)=0 \ (7) \\
\end{align}</span></p>
<p>This means that either the restriction or the <span class="math-container">$\lambda_{i}$</span> will be 0. To solve this equations we need to exhaust all posibilities, and see if we can arrive to a solution that is consistent or discard cases that lead to contradiction. Im going to show that <span class="math-container">$\beta=0 \ \Rightarrow \ c_{2}=0, \ h=constant$</span>. You can look for yourself to check if there is other posible solutions.</p>
<p>So let's see if <span class="math-container">$\beta=0 \ \Rightarrow \ c_{2}=0, \ h=constant$</span> it's a solution. if <span class="math-container">$\beta=0$</span> the logical thing would be to think that the optimal solution would be <span class="math-container">$c_{2}^{*}=0$</span>, so let's just assume that this it's the case, if it is not, at some point the first order conditions must show that this is not consistent.</p>
<p>Also, we expect that <span class="math-container">$c_{1}&gt;0$</span> which implies by (5) <span class="math-container">$\lambda_{2}=0$</span>. We can see from (1) that <span class="math-container">$\lambda_{1}&gt;0$</span> because the assumption <span class="math-container">$u(c_{i})&gt;0$</span>. Now, here comes the interesting part if <span class="math-container">$\frac{w^{'}(h^{*})}{1+r}-1&gt;0$</span> implies that <span class="math-container">$\lambda_{4}=0$</span> by (3). Not always this would be the case, for example suppose that <span class="math-container">$w(h)=h(1+v)$</span> where <span class="math-container">$v$</span> is the return of investment in <span class="math-container">$h$</span>. <span class="math-container">$\frac{w^{'}(h^{*})}{1+r}-1$</span> would be <span class="math-container">$ \frac{v-r}{1+r}$</span>. If <span class="math-container">$v&gt;r$</span> then <span class="math-container">$\frac{w^{'}(h^{*})}{1+r}-1&gt;0$</span> and <span class="math-container">$\lambda_{4}=0$</span>. But if <span class="math-container">$v&lt;r$</span> then <span class="math-container">$\lambda_{4}&gt;0$</span> by (3) and <span class="math-container">$h=0$</span> by (7).</p>
<p>This means that the optimal choice of h depends on this condition. If the return of h is bigger that his cost of oportunity (including r), then <span class="math-container">$h&gt;0$</span> and would be a constant.</p>
<p>Now let's rule out the posibility that <span class="math-container">$c_{2}&gt;0$</span>. If this is true,  <span class="math-container">$\lambda_{3}=0$</span>, which by (2) implies that <span class="math-container">$\lambda_{1}=0$</span>, which implies by (1) that <span class="math-container">$u^{'}(c_{1})=0$</span> that generally it's not true. So our assumption that <span class="math-container">$c_{2}^{*}=0$</span> it's in general correct.</p>
<p>So all this observations lead us to the following equations:</p>
<p><span class="math-container">\begin{align}
 u^{ ' }(c_{1}) = \lambda_{1}  \ (8)\\
\lambda_{1}=(1+r)\lambda_{3} \ (9)\\
\frac{ w^{ ' }(h)}{1+r} = 1 \ (10)\\
c_{1}^{*} = y-h + \frac{ w(h)}{1+r} - \frac{1}{1+r} (11)
\end{align}</span></p>
<p>(8) comes from (1), (9) from (2), (10) from (3), and (11) from (4). From this system of equations the solutions to all the endogenous variables are obtained for <span class="math-container">$c_{1}^{*}$</span> is (11) and for <span class="math-container">$h$</span> is (10). So <span class="math-container">$\beta=0 \ \Rightarrow \ c_{2}=0$</span>. Why <span class="math-container">$h_{*}&gt;0$</span>? This happens because investing in <span class="math-container">$h$</span> increases the NPV, so increases the consumption today, so invest in h increases utility in period 1.This does not depend on <span class="math-container">$\beta$</span>, because it's a monetary relationship. But note that if the return of <span class="math-container">$r$</span> is greater thatn <span class="math-container">$h$</span>, then it could happen that <span class="math-container">$h^{*}=0$</span>.</p>

CC BY-SA 4.0
2020-06-28T07:09:36.087Z
You seem to be talking about *savings*, not investment.
Giskard
CC BY-SA 4.0
2020-06-28T07:20:58.847Z
Yes, because you save to invest in h or to consume in period two.
juarez_Samuel
CC BY-SA 4.0
2020-06-28T07:24:46.567Z
If it's not clear that the dependence exists. You can just substitute the restriction on the utility function for some $c_{i}$ and derivate for $h$ and $c_{2}$, and it's more clear that a dependence exists. Or just choose some utility function and solve it.
juarez_Samuel
CC BY-SA 4.0
2020-06-28T07:26:29.637Z
$h$ appears in two first order conditions, you need to solve them to see of what depends $h$
juarez_Samuel
CC BY-SA 4.0
2020-06-28T09:43:23.130Z
Unless I misunderstand the model, "*investing*" is not the only way to make money in period 2, you can also put your money into the "*perfect credit market*". Otherwise the discount factor $1/(1+r)$ makes little sense. Thus savings would equal the invested sum $h$ plus the amount of money earning interest in this market.
Giskard
CC BY-SA 4.0
2020-06-28T14:11:56.700Z
Thanks for your answer! To clarify, @Giskard is right that money in period 1 can either be spent on c_1, saved with interest rate (1+r), or invested with return w(h). Also, I'm a little confused by the last two equations in your answer. I get that beta and h both appear in the first equation, but only h appears in the second. So it still seems that changing beta doesn't change h? It only changes c?
azvecon
CC BY-SA 4.0
2020-06-28T17:10:13.333Z
Yes, you are right @Giskard, you can save at r competitive. But that does not change that to invest you have to save; h is "invested" or "saved" (you have to reduce $ c_ {1} $ to invest in $ h $ as it is a two-period model only) in competitive markets otherwise the payment function would be $ w (h, w) $, since the size of your investment would affect the return of $ h $. Almost nothing changes from what I said in the answer. But i forgot that you can invest in two assets.
juarez_Samuel
CC BY-SA 4.0
2020-06-28T17:14:37.507Z
Now, if $ h \neq r $ this means that there is no perfect mobility between the two types of investment or the two uses for your savings. If there were perfect mobility, nobody would invest in the market that returns $ r $ or $ h $, depending on which one was less. So I assume that this agent has access to a possibly international market that is not integrated into international markets. In addition, the rate that is higher is what all your savings will go to.
juarez_Samuel
CC BY-SA 4.0
2020-06-28T17:22:51.363Z
And aswering your question @asvecon, you are right, $\beta$ appear only in the first equation, But $h$ appear in both, remember that $c_{1}=R_{c_{1}}= y-h + \frac{ w(h)}{1+r} -  \frac{ c_{2}}{1+r}$, so $R_{c_{1}}$ is a function of h. So $\beta$ affect $h$.
juarez_Samuel
CC BY-SA 4.0
2020-06-28T20:46:21.470Z
@Giskard could you briefly explain what is incorrect? we have an optimization problem with 3 endogenous variables and 1 restriction, so we have 4 first order condition. I only substituted one in the other, until I reached a system of 2 equations with 2 endogenous variables
juarez_Samuel
CC BY-SA 4.0
2020-06-28T21:40:48.810Z
@Giskard i just realized what it's wrong. I thought that the implicit theorem function will show a different relationship implied by the third first order condition but it does'nt
juarez_Samuel
CC BY-SA 4.0
2020-06-29T09:52:57.437Z
@SamuelCuevas I posted an answer.
Giskard
CC BY-SA 4.0
2020-06-28T05:54:50.107Z
2020-06-29T03:26:57.387Z
2020-06-29T03:26:57.387Z
juarez_Samuel
<p>In lecture, my professor defined the following 2-period consumption model:</p>
<p><span class="math-container">$c_i = $</span> consumption in period <span class="math-container">$i$</span>.</p>
<p><span class="math-container">$y =$</span> endowed income in period 1.</p>
<p><span class="math-container">$r = $</span> interest rate in perfect credit markets.</p>
<p><span class="math-container">$h = $</span> money invested in period 1.</p>
<p><span class="math-container">$w(h) = $</span> period 2 return on money invested in period 1.</p>
<p><span class="math-container">$U(c_1, c_2) = u(c_1) + \beta u(c_2)$</span> an additively separable utility function with discount factor <span class="math-container">$\beta$</span>.</p>
<p>Then the budget constraint is <span class="math-container">$$c_1 + \frac{c_2}{1+r} = y - h + \frac{w(h)}{1+r}$$</span> so the Lagrangian for utility maximization is <span class="math-container">$$L = u(c_1) + \beta u(c_2) + \lambda \left( y - h + \frac{w(h)}{1+r} - c_1 - \frac{c_2}{1+r} \right).$$</span> But then <span class="math-container">$h^*$</span> is determined entirely by one of the first order conditions, <span class="math-container">$$L_h = \frac{w'(h)}{1+r} - 1 = 0,$$</span> which is independent of <span class="math-container">$\beta$</span>.</p>
<p>My professor said that this means &quot;patient and impatient people invest the same amount <span class="math-container">$h^*$</span>, maximizing the net present value of income.&quot; Mathematically, I understand why this has to be true, but intuitively it doesn't make sense. If <span class="math-container">$\beta = 0$</span>, for example, shouldn't we invest <span class="math-container">$h=0$</span> because all consumption in the future provides 0 utility? Why doesn't similar reasoning apply to very small <span class="math-container">$\beta$</span>? And why would we be trying to maximize the NPV of income if income in the first period provides more utility than income in the second period?</p>

<microeconomics><consumer-theory><optimization><saving><intertemporal-choice>
In an intertemporal (2-period) consumption model, why is the investment rate independent of discount factor?
azvecon
# In an intertemporal (2-period) consumption model, why is the investment rate independent of discount factor?

(Asked by: username_0 on 2020-06-27)

In lecture, my professor defined the following 2-period consumption model:

$c\_i = $ consumption in period $i$.

$y =$ endowed income in period 1.

$r = $ interest rate in perfect credit markets.

$h = $ money invested in period 1.

$w(h) = $ period 2 return on money invested in period 1.

$U(c\_1, c\_2) = u(c\_1) + \beta u(c\_2)$ an additively separable utility function with discount factor $\beta$.

Then the budget constraint is $$c\_1 + \frac{c\_2}{1+r} = y - h + \frac{w(h)}{1+r}$$ so the Lagrangian for utility maximization is $$L = u(c\_1) + \beta u(c\_2) + \lambda \left( y - h + \frac{w(h)}{1+r} - c\_1 - \frac{c\_2}{1+r} \right).$$ But then $h^\*$ is determined entirely by one of the first order conditions, $$L\_h = \frac{w'(h)}{1+r} - 1 = 0,$$ which is independent of $\beta$.

My professor said that this means "patient and impatient people invest the same amount $h^\*$, maximizing the net present value of income." Mathematically, I understand why this has to be true, but intuitively it doesn't make sense. If $\beta = 0$, for example, shouldn't we invest $h=0$ because all consumption in the future provides 0 utility? Why doesn't similar reasoning apply to very small $\beta$? And why would we be trying to maximize the NPV of income if income in the first period provides more utility than income in the second period?

**Tags:** microeconomics, consumer-theory, optimization, saving, intertemporal-choice

---

## 2 answers

### Answer by username_1 on 2020-06-29. Score: 4

This idea is known as the [Fisher separation theorem](https://en.wikipedia.org/wiki/Fisher_separation_theorem).

Without the investment opportunity to transfer $h$ units of present day value into $w(h)$ units of future value, the perfect credit market gives us the intertemporal budget constraint of
$$
c\_1 + \frac{c\_2}{1+r} = y,
$$
which can be represented by a straight line.

[![enter image description here](https://i.sstatic.net/jVk2F.png)](https://i.sstatic.net/jVk2F.png)

Without knowledge of the consumer's preferences, it is not possible to say what the optimal $(c\_1,c\_2)$ is here. But we can say with certainty, that the higher the budget line is, the more options the consumer has, and assuming monotonic preferences she will be better off.

Now the yield curve $w$ of the investment opportunity allows us to shift the basic credit market budget line. Starting from the lower right point of $(y,0)$ (i.e. not putting money into the credit market at all) we can reach cash flows $(y-h, w(h))$.

[![enter image description here](https://i.sstatic.net/DfafT.png)](https://i.sstatic.net/DfafT.png)

But by also putting some money into the credit market (or borrowing), one can reach other cash flows as well. These always transfer $x$ units of present value into $x(1+r)$ units of future value (in case of borrowing $x$ is negative), so they are always paralel to the original budget line, but instead of starting from $(y,0)$, they start from the point $(y-h, w(h))$ where $h$ is the size of the investment.

The optimal investment size is the one that reaches the highest budget line:

[![enter image description here](https://i.sstatic.net/V5BES.png)](https://i.sstatic.net/V5BES.png)

---

The idea is that given perfect credit markets one can separate consumption behavior and investment behavior. The higher the net present value of my cash flow, the better of I will be once I adjust my cash flow via the credit market to my intertemporal preferences (this is where $\beta$ matters). Thus I need to make investment decisions that maximize the present value of my cash flow, and I do not have to take anything else into consideration, e.g., the timing of the payments.

---

A mathematical tidbit:

In the above example, given the optimal $h$ (assuming it is an interior point) the budget line is tangential to the orange curve depicting the possible investments, so we have
$$
w'(h) = 1 + r,
$$
the marginal yield from the investment and the credit market are equal. This is equivalent to the final equation in your question.


---

### Answer by username_2 on 2020-06-28. Score: 0

Well i just edited my answer a lot. I made a fundamentally mistake, from $ L\_ {h} $ arises an unique value of h, even when $ h $ is in other first order conditions, that doesn't change anything. Once that i have clear that there is no relationship between $ \beta $ and $ h $, I think I know what is happening.

The reason why it is invested in $ h $ although $ \beta = 0 $ is because h increases the NPV, and this allows to increase consumption in the first period. What happens is $\beta=0 \ \Rightarrow \ \ c\_{2}=0$. But let´s see this more carefully.

To do this we have to add restrictions to the model $c\_{1},c\_{2},h > 0$, so the lagrangian will change:

\begin{align}
L= L = u(c\_1) + \beta u(c\_2) + \lambda\_{1} \left( y - h + \frac{w(h)}{1+r} - c\_1 - \frac{c\_2}{1+r} \right) +\lambda\_{2}c\_{1}+\lambda\_{3}c\_{2}+\lambda\_{4}h
\end{align}

The new first order conditions will be:

\begin{align}
\frac{\partial L}{\partial c\_{1}} = u^{ ' }(c\_{1}) - \lambda\_{1} + \lambda\_{2} =0 \ (1)\\
\frac{\partial L}{\partial c\_{2}} = \beta u^{ '}(c\_{2}) - \frac{\lambda\_{1}}{1+r} + \lambda\_{3}=0 \ (2)\\
\frac{\partial L}{\partial h} =(\frac{ w^{ ' }(h)}{1+r} -1)\lambda\_{1} + \lambda\_{4}= 0 \ (3)\\
\end{align}

But we need for constraints with inequalities the complementary slackness conditions (we can work with the Kuhn-Tucker lagrangian, but that it's just a special case of this general formulation):

\begin{align}
\lambda\_{1}(y-h + \frac{ w(h)}{1+r} - \frac{ c\_{2}}{1+r}-c\_{1}) =0 \ (4) \\
\lambda\_{2}(c\_{1})=0 \ (5) \\
\lambda\_{3}(c\_{2})=0 \ (6) \\
\lambda\_{4}(h)=0 \ (7) \\
\end{align}

This means that either the restriction or the $\lambda\_{i}$ will be 0. To solve this equations we need to exhaust all posibilities, and see if we can arrive to a solution that is consistent or discard cases that lead to contradiction. Im going to show that $\beta=0 \ \Rightarrow \ c\_{2}=0, \ h=constant$. You can look for yourself to check if there is other posible solutions.

So let's see if $\beta=0 \ \Rightarrow \ c\_{2}=0, \ h=constant$ it's a solution. if $\beta=0$ the logical thing would be to think that the optimal solution would be $c\_{2}^{\*}=0$, so let's just assume that this it's the case, if it is not, at some point the first order conditions must show that this is not consistent.

Also, we expect that $c\_{1}>0$ which implies by (5) $\lambda\_{2}=0$. We can see from (1) that $\lambda\_{1}>0$ because the assumption $u(c\_{i})>0$. Now, here comes the interesting part if $\frac{w^{'}(h^{\*})}{1+r}-1>0$ implies that $\lambda\_{4}=0$ by (3). Not always this would be the case, for example suppose that $w(h)=h(1+v)$ where $v$ is the return of investment in $h$. $\frac{w^{'}(h^{\*})}{1+r}-1$ would be $ \frac{v-r}{1+r}$. If $v>r$ then $\frac{w^{'}(h^{\*})}{1+r}-1>0$ and $\lambda\_{4}=0$. But if $v then $\lambda\_{4}>0$ by (3) and $h=0$ by (7).

This means that the optimal choice of h depends on this condition. If the return of h is bigger that his cost of oportunity (including r), then $h>0$ and would be a constant.

Now let's rule out the posibility that $c\_{2}>0$. If this is true, $\lambda\_{3}=0$, which by (2) implies that $\lambda\_{1}=0$, which implies by (1) that $u^{'}(c\_{1})=0$ that generally it's not true. So our assumption that $c\_{2}^{\*}=0$ it's in general correct.

So all this observations lead us to the following equations:

\begin{align}
u^{ ' }(c\_{1}) = \lambda\_{1} \ (8)\\
\lambda\_{1}=(1+r)\lambda\_{3} \ (9)\\
\frac{ w^{ ' }(h)}{1+r} = 1 \ (10)\\
c\_{1}^{\*} = y-h + \frac{ w(h)}{1+r} - \frac{1}{1+r} (11)
\end{align}

(8) comes from (1), (9) from (2), (10) from (3), and (11) from (4). From this system of equations the solutions to all the endogenous variables are obtained for $c\_{1}^{\*}$ is (11) and for $h$ is (10). So $\beta=0 \ \Rightarrow \ c\_{2}=0$. Why $h\_{\*}>0$? This happens because investing in $h$ increases the NPV, so increases the consumption today, so invest in h increases utility in period 1.This does not depend on $\beta$, because it's a monetary relationship. But note that if the return of $r$ is greater thatn $h$, then it could happen that $h^{\*}=0$.

> **Comments:**
> **username_1**: You seem to be talking about *savings*, not investment.  
> **username_2**: Yes, because you save to invest in h or to consume in period two.  
> **username_2**: If it's not clear that the dependence exists. You can just substitute the restriction on the utility function for some $c_{i}$ and derivate for $h$ and $c_{2}$, and it's more clear that a dependence exists. Or just choose some utility function and solve it.  
> **username_2**: $h$ appears in two first order conditions, you need to solve them to see of what depends $h$  
> **username_1**: Unless I misunderstand the model, "*investing*" is not the only way to make money in period 2, you can also put your money into the "*perfect credit market*". Otherwise the discount factor $1/(1+r)$ makes little sense. Thus savings would equal the invested sum $h$ plus the amount of money earning interest in this market.  
> **username_0**: Thanks for your answer! To clarify, @Giskard is right that money in period 1 can either be spent on c_1, saved with interest rate (1+r), or invested with return w(h). Also, I'm a little confused by the last two equations in your answer. I get that beta and h both appear in the first equation, but only h appears in the second. So it still seems that changing beta doesn't change h? It only changes c?  
> **username_2**: Yes, you are right @Giskard, you can save at r competitive. But that does not change that to invest you have to save; h is "invested" or "saved" (you have to reduce $ c_ {1} $ to invest in $ h $ as it is a two-period model only) in competitive markets otherwise the payment function would be $ w (h, w) $, since the size of your investment would affect the return of $ h $. Almost nothing changes from what I said in the answer. But i forgot that you can invest in two assets.  
> **username_2**: Now, if $ h \neq r $ this means that there is no perfect mobility between the two types of investment or the two uses for your savings. If there were perfect mobility, nobody would invest in the market that returns $ r $ or $ h $, depending on which one was less. So I assume that this agent has access to a possibly international market that is not integrated into international markets. In addition, the rate that is higher is what all your savings will go to.  
> **username_2**: And aswering your question @asvecon, you are right, $\beta$ appear only in the first equation, But $h$ appear in both, remember that $c_{1}=R_{c_{1}}= y-h + \frac{ w(h)}{1+r} -  \frac{ c_{2}}{1+r}$, so $R_{c_{1}}$ is a function of h. So $\beta$ affect $h$.  
> **username_2**: @Giskard could you briefly explain what is incorrect? we have an optimization problem with 3 endogenous variables and 1 restriction, so we have 4 first order condition. I only substituted one in the other, until I reached a system of 2 equations with 2 endogenous variables  
> **username_2**: @Giskard i just realized what it's wrong. I thought that the implicit theorem function will show a different relationship implied by the third first order condition but it does'nt  
> **username_1**: @SamuelCuevas I posted an answer.  

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