<p>The paper is not trying to say that equation (10) is derived from equation (8). Equation (8) tells us how household makes its optimal consumption and 'saving' decision (it gives us demand for investing the wealth into bonds/risky assets). </p>

<p>The equation (10) then tells us that given the household optimal decisions (which depend on utility (8)), those households that decide to invest in risky assets will further go to financial intermediary that maximizes the equation given by (10). The equation (10) loosely translated to English says that the intermediary is going to try to maximize the return and minimize the variance of return on this risky asset, subject to the constraint that tells us how much of their all wealth is in risky assets (the wealth invested in (<span class="math-container">$\lambda w$</span>) bonds is not in there). </p>

<p>So the equation (10) is not really derived from (8) but it is given by (8) in a sense that (8) tells us how much wealth <span class="math-container">$w$</span> the household will hold. </p>

CC BY-SA 4.0
2020-02-10T03:38:09.460Z
It sounds to me like you're saying Equation (8) tells you how agents allocate their initial wealth (and consequently proceeds from investment) between consumption $c_t^h$ and terminal wealth $w_{t +\delta}^h$, whereas Equation (10) tells you how investment is allocated between the risky and riskless assets (given a fixed total amount of money to invest). I have further questions: (1) Could you tell me why those decisions are independent of one another? What is it about log utility that makes it so? (2) Could you derive the optimal consumption and terminal wealth?
mathsquestions1
CC BY-SA 4.0
2020-02-10T05:36:49.860Z
In particular, what is the budget constraint for the optimization problem presented in Equation (8)?
mathsquestions1
CC BY-SA 4.0
2020-02-10T08:55:21.660Z
1. They are not independent. First household decides how much wealth to invest (eq. 8) - as the section in the paper says it gives you the household demand for financial intermediation - actually I would interpret it as a demand more narrowly for saving/investment as financial intermediation is only part of what $w$ is invested in. Second, conditional on that there is financial intermediary who tries to maximize return and minimize the variance. In my answer I misspoke a bit I was always referring to household because in my mind it was still the household decision to choose the intermediation
1muflon1
CC BY-SA 4.0
2020-02-10T09:12:19.553Z
Sorry if that created some confusion (I rewritten that part of my answer to make this clear). I don’t think there is anything special about logarithmic utility function, they could use similar utility and I am guessing the results would qualitatively not change. Modeling utility as a log of consumption is just very convenient mathematically. To be more specific the log utility is homothetic which means it’s scale independent so you can use the same utility for all households disregarding the potential wealth inequality. Also ln(c) is special case of CRRA, but there are multiple functions that
1muflon1
CC BY-SA 4.0
2020-02-10T09:16:05.433Z
Could be used here in principle because they would have exactly the same properties as logarithmic utility but none of them would be as simple/parsimonious as this. 2. Sorry this would take too much time, maybe post it as a separate question and someone else can answer. 3. The authors say that the households are getting labor income so I would say that’s the constraint for consumption/saving decision
1muflon1
CC BY-SA 4.0
2020-02-13T03:39:17.280Z
That's fine, but could you just provide (in mathematics) the constraints that accompany the optimization problem in Equation (8)? I understand solving the entire problem would take too long for you, but if you just give me the constraint I will do the algebra.
mathsquestions1
CC BY-SA 4.0
2020-02-08T09:40:14.347Z
2020-02-10T09:17:45.873Z
2020-02-10T09:17:45.873Z
1muflon1
<p>How do you derive equation (10) on page 740 from <a href="https://www.aeaweb.org/articles?id=10.1257/aer.103.2.732" rel="nofollow noreferrer">He, Krishnamurthy (2013 AER)</a>?</p>

<p><a href="https://i.sstatic.net/UluDJ.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/UluDJ.png" alt="enter image description here"></a></p>

<p>They say that "Given the log objective function in equation (8), the risky asset household chooses <span class="math-container">$\alpha_t^h$</span> to solve (where we have taken the limit as <span class="math-container">$\delta \rightarrow dt$</span>)" where equation (8) is on page 739</p>

<p><span class="math-container">$$ \rho\delta \ln{c_t^h}+(1-\rho\delta)E_t[\ln w_{t+\delta}^h]$$</span></p>

<p>where <span class="math-container">$\rho$</span> is time preference parameter, <span class="math-container">$\delta$</span> is some time increment, <span class="math-container">$w_{t+\delta}^h$</span> is bequest for generation <span class="math-container">$t+\delta$</span>.</p>

<p>It is not clear to me which information they use to transform the log utility to equation (10) -- I am sorry that I cannot reproduce the entire environment of the model here. But if someone that knows the paper well could reply, it would be immensely helpful.</p>

<p>Really appreciate it! Here's a list of the notation:</p>

<ul>
<li><span class="math-container">$\alpha_t^h$</span> : Household's allocation to risky assets</li>
<li><span class="math-container">$\tilde{dR_t}$</span>: the return on capital for intermediary equity</li>
<li><span class="math-container">$r_t$</span> : return process for the riskless asset</li>
<li><span class="math-container">$\rho$</span> : time preference for the specialist</li>
<li><span class="math-container">$\delta$</span> : time increment, as in <span class="math-container">$t, t+\delta t, t+2\delta t, \cdots$</span>. To switch to continuous-time, they take the limit <span class="math-container">$\delta \rightarrow dt$</span>.</li>
<li><span class="math-container">$w_t^h$</span> : initial wealth endowed</li>
<li><span class="math-container">$\lambda$</span> portion of households that can only invest in riskless <span class="math-container">$(1-\lambda)$</span> of households can invest in both riskless and risky.</li>
</ul>

<h1>Edit 1 :--------------------</h1>

<p>I think it would also help to see how the consumption and saving decision is solved. I would appreciate it if someone could do this. </p>

<p>For me, the biggest problem here is that I don't know the budget constraint -- I need to "guess" it based on how the decision making process is described in the paper. I always thought it was strange that in economics they are sometimes not completely explicit about the optimization problem they solve (i.e. not tell you some of the constraints), when there could (potentially) be multiple ways to write the problem.</p>

<h2>Solving for consumption:</h2>

<p>Anyway, here's what I have so far, and I seem to have found one way (again, I have absolutely no idea if this is how the authors do it) to derive the consumption rule.</p>

<p>Start with  </p>

<p><span class="math-container">$$\rho\delta\ln c_{t}^{h}+\left(1-\rho\delta\right)E_{t}\left[\ln w_{t+\delta}^{h}\right]$$</span></p>

<p>Add <span class="math-container">$-\left(1-\rho\delta\right)\ln w_{t}^{h}$</span> to the objective function. We can do this because we can add constants to utility functions without affecting the optimal decisions: </p>

<p><span class="math-container">$$ 
\rho\delta\ln c_{t}^{h}+\left(1-\rho\delta\right)E_{t}\left[\ln w_{t+\delta}^{h}-\ln w_{t}^{h}\right]
$$</span></p>

<p>Divide by <span class="math-container">$\delta$</span>. Again this is a monotonic transformation, and thus a valid change to the objective function:  </p>

<p><span class="math-container">\begin{align}
&amp; \rho\delta\ln c_{t}^{h}+\left(1-\rho\delta\right)E_{t}\left[\ln w_{t+\delta}^{h}-\ln w_{t}^{h}\right] \\
\implies &amp; \rho\ln c_{t}^{h}+\left(1-\rho\delta\right)E_{t}\left[\frac{\ln w_{t+\delta}^{h}-\ln w_{t}^{h}}{w_{t+\delta}^{h}-w_{t}^{h}}\frac{w_{t+\delta}^{h}-w_{t}^{h}}{\left(t+\delta\right)-t}\right]
\end{align}</span></p>

<p>Take limits as <span class="math-container">$\delta\rightarrow0$</span>:  </p>

<p><span class="math-container">$$
\rho\ln c_{t}^{h}dt+E_{t}\left[\frac{\partial}{\partial w_{t}^{h}}\left(\ln w_{t}^{h}\right)dw_{t}^{h}\right]
= \rho\ln c_{t}^{h}dt+E_{t}\left[\frac{dw_{t}^{h}}{w_{t}^{h}}\right]
$$</span></p>

<p>This next step is sort of cheating, but by Equation (11), we know that 
<span class="math-container">$$
\frac{dw_{t}^{h}}{w_{t}^{h}}=\left(\frac{lD_{t}-c_{t}^{h}}{w_{t}^{h}}\right)dt+\lambda r_{t}dt+\left(1-\lambda\right)\left[\alpha_{t}^{h}\tilde{dR}_{t}+\left(1-\alpha_{t}^{h}\right)r_{t}dt\right]
$$</span>
Take expectations:
<span class="math-container">\begin{align}
E_{t}\left[\frac{dw_{t}^{h}}{w_{t}^{h}}\right] &amp; = E_{t}\left[\left(\frac{lD_{t}-c_{t}^{h}}{w_{t}^{h}}\right)dt+\lambda r_{t}dt+\left(1-\lambda\right)\left[\alpha_{t}^{h}\tilde{dR}_{t}+\left(1-\alpha_{t}^{h}\right)r_{t}dt\right]\right]
 \\
&amp; =\left(\frac{lD_{t}-c_{t}^{h}}{w_{t}^{h}}\right)dt+\lambda r_{t}dt+\left(1-\lambda\right)E_{t}\left[\alpha_{t}^{h}\tilde{dR}_{t}+\left(1-\alpha_{t}^{h}\right)r_{t}dt\right] \\
&amp; =\left(\frac{lD_{t}-c_{t}^{h}}{w_{t}^{h}}\right)dt+\lambda r_{t}dt+\left(1-\lambda\right)E_{t}\left[r_{t}dt+\alpha_{t}^{h}\left(\tilde{dR}_{t}-r_{t}dt\right)\right] \\
&amp; =\left(\frac{lD_{t}-c_{t}^{h}}{w_{t}^{h}}\right)dt+r_{t}dt+\left(1-\lambda\right)E_{t}\left[\alpha_{t}^{h}\left(\tilde{dR}_{t}-r_{t}dt\right)\right] \\
&amp; =\left(\frac{lD_{t}-c_{t}^{h}+w_{t}^{h}r_{t}}{w_{t}^{h}}\right)dt+\left(1-\lambda\right)E_{t}\left[\alpha_{t}^{h}\left(\tilde{dR}_{t}-r_{t}dt\right)\right]
\end{align}</span></p>

<p>Rearranging the following first order condition gives the consumption rule <span class="math-container">$ c_t^h = \rho w_t^h $</span> given as Equation (9) in the paper:</p>

<p><span class="math-container">$$ \frac{\rho}{c_t^h} -(1/w_t^h)  =0$$</span></p>

<h2>Investment:</h2>

<p>In light of the current answer (and equation 10), however, the above is incorrect because it gives the "wrong" investment allocation. This is where I am stuck. At this point, my derivation seems wrong, but it seems strange that they will just assume a separate objective function just for the choice variable <span class="math-container">$\alpha_t^h$</span> without some other background justification. But of course, I could be absolutely wrong.</p>

<h1>Edit 2 :--------------------</h1>

<p>Could you provide the constraint(s) (in mathematical formula) for the optimization problem in equation (8)? I believe I can do the rest.</p>

<finance><financial-economics><asset-pricing><continuous-time>
He, Krishnamurthy (2013)
mathsquestions1
# He, Krishnamurthy (2013)

(Asked by: username_0 on 2020-02-08)

How do you derive equation (10) on page 740 from [He, Krishnamurthy (2013 AER)](https://www.aeaweb.org/articles?id=10.1257/aer.103.2.732)?

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

They say that "Given the log objective function in equation (8), the risky asset household chooses $\alpha\_t^h$ to solve (where we have taken the limit as $\delta \rightarrow dt$)" where equation (8) is on page 739

$$ \rho\delta \ln{c\_t^h}+(1-\rho\delta)E\_t[\ln w\_{t+\delta}^h]$$

where $\rho$ is time preference parameter, $\delta$ is some time increment, $w\_{t+\delta}^h$ is bequest for generation $t+\delta$.

It is not clear to me which information they use to transform the log utility to equation (10) -- I am sorry that I cannot reproduce the entire environment of the model here. But if someone that knows the paper well could reply, it would be immensely helpful.

Really appreciate it! Here's a list of the notation:

* $\alpha\_t^h$ : Household's allocation to risky assets
* $\tilde{dR\_t}$: the return on capital for intermediary equity
* $r\_t$ : return process for the riskless asset
* $\rho$ : time preference for the specialist
* $\delta$ : time increment, as in $t, t+\delta t, t+2\delta t, \cdots$. To switch to continuous-time, they take the limit $\delta \rightarrow dt$.
* $w\_t^h$ : initial wealth endowed
* $\lambda$ portion of households that can only invest in riskless $(1-\lambda)$ of households can invest in both riskless and risky.

# Edit 1 :--------------------

I think it would also help to see how the consumption and saving decision is solved. I would appreciate it if someone could do this.

For me, the biggest problem here is that I don't know the budget constraint -- I need to "guess" it based on how the decision making process is described in the paper. I always thought it was strange that in economics they are sometimes not completely explicit about the optimization problem they solve (i.e. not tell you some of the constraints), when there could (potentially) be multiple ways to write the problem.

## Solving for consumption:

Anyway, here's what I have so far, and I seem to have found one way (again, I have absolutely no idea if this is how the authors do it) to derive the consumption rule.

Start with

$$\rho\delta\ln c\_{t}^{h}+\left(1-\rho\delta\right)E\_{t}\left[\ln w\_{t+\delta}^{h}\right]$$

Add $-\left(1-\rho\delta\right)\ln w\_{t}^{h}$ to the objective function. We can do this because we can add constants to utility functions without affecting the optimal decisions:

$$
\rho\delta\ln c\_{t}^{h}+\left(1-\rho\delta\right)E\_{t}\left[\ln w\_{t+\delta}^{h}-\ln w\_{t}^{h}\right]
$$

Divide by $\delta$. Again this is a monotonic transformation, and thus a valid change to the objective function:

\begin{align}
& \rho\delta\ln c\_{t}^{h}+\left(1-\rho\delta\right)E\_{t}\left[\ln w\_{t+\delta}^{h}-\ln w\_{t}^{h}\right] \\
\implies & \rho\ln c\_{t}^{h}+\left(1-\rho\delta\right)E\_{t}\left[\frac{\ln w\_{t+\delta}^{h}-\ln w\_{t}^{h}}{w\_{t+\delta}^{h}-w\_{t}^{h}}\frac{w\_{t+\delta}^{h}-w\_{t}^{h}}{\left(t+\delta\right)-t}\right]
\end{align}

Take limits as $\delta\rightarrow0$:

$$
\rho\ln c\_{t}^{h}dt+E\_{t}\left[\frac{\partial}{\partial w\_{t}^{h}}\left(\ln w\_{t}^{h}\right)dw\_{t}^{h}\right]
= \rho\ln c\_{t}^{h}dt+E\_{t}\left[\frac{dw\_{t}^{h}}{w\_{t}^{h}}\right]
$$

This next step is sort of cheating, but by Equation (11), we know that
$$
\frac{dw\_{t}^{h}}{w\_{t}^{h}}=\left(\frac{lD\_{t}-c\_{t}^{h}}{w\_{t}^{h}}\right)dt+\lambda r\_{t}dt+\left(1-\lambda\right)\left[\alpha\_{t}^{h}\tilde{dR}\_{t}+\left(1-\alpha\_{t}^{h}\right)r\_{t}dt\right]
$$
Take expectations:
\begin{align}
E\_{t}\left[\frac{dw\_{t}^{h}}{w\_{t}^{h}}\right] & = E\_{t}\left[\left(\frac{lD\_{t}-c\_{t}^{h}}{w\_{t}^{h}}\right)dt+\lambda r\_{t}dt+\left(1-\lambda\right)\left[\alpha\_{t}^{h}\tilde{dR}\_{t}+\left(1-\alpha\_{t}^{h}\right)r\_{t}dt\right]\right]
\\
& =\left(\frac{lD\_{t}-c\_{t}^{h}}{w\_{t}^{h}}\right)dt+\lambda r\_{t}dt+\left(1-\lambda\right)E\_{t}\left[\alpha\_{t}^{h}\tilde{dR}\_{t}+\left(1-\alpha\_{t}^{h}\right)r\_{t}dt\right] \\
& =\left(\frac{lD\_{t}-c\_{t}^{h}}{w\_{t}^{h}}\right)dt+\lambda r\_{t}dt+\left(1-\lambda\right)E\_{t}\left[r\_{t}dt+\alpha\_{t}^{h}\left(\tilde{dR}\_{t}-r\_{t}dt\right)\right] \\
& =\left(\frac{lD\_{t}-c\_{t}^{h}}{w\_{t}^{h}}\right)dt+r\_{t}dt+\left(1-\lambda\right)E\_{t}\left[\alpha\_{t}^{h}\left(\tilde{dR}\_{t}-r\_{t}dt\right)\right] \\
& =\left(\frac{lD\_{t}-c\_{t}^{h}+w\_{t}^{h}r\_{t}}{w\_{t}^{h}}\right)dt+\left(1-\lambda\right)E\_{t}\left[\alpha\_{t}^{h}\left(\tilde{dR}\_{t}-r\_{t}dt\right)\right]
\end{align}

Rearranging the following first order condition gives the consumption rule $ c\_t^h = \rho w\_t^h $ given as Equation (9) in the paper:

$$ \frac{\rho}{c\_t^h} -(1/w\_t^h) =0$$

## Investment:

In light of the current answer (and equation 10), however, the above is incorrect because it gives the "wrong" investment allocation. This is where I am stuck. At this point, my derivation seems wrong, but it seems strange that they will just assume a separate objective function just for the choice variable $\alpha\_t^h$ without some other background justification. But of course, I could be absolutely wrong.

# Edit 2 :--------------------

Could you provide the constraint(s) (in mathematical formula) for the optimization problem in equation (8)? I believe I can do the rest.

**Tags:** finance, financial-economics, asset-pricing, continuous-time

---

## 1 answers

### Answer by username_1 on 2020-02-08. Score: 1

The paper is not trying to say that equation (10) is derived from equation (8). Equation (8) tells us how household makes its optimal consumption and 'saving' decision (it gives us demand for investing the wealth into bonds/risky assets).

The equation (10) then tells us that given the household optimal decisions (which depend on utility (8)), those households that decide to invest in risky assets will further go to financial intermediary that maximizes the equation given by (10). The equation (10) loosely translated to English says that the intermediary is going to try to maximize the return and minimize the variance of return on this risky asset, subject to the constraint that tells us how much of their all wealth is in risky assets (the wealth invested in ($\lambda w$) bonds is not in there).

So the equation (10) is not really derived from (8) but it is given by (8) in a sense that (8) tells us how much wealth $w$ the household will hold.

> **Comments:**
> **username_0**: It sounds to me like you're saying Equation (8) tells you how agents allocate their initial wealth (and consequently proceeds from investment) between consumption $c_t^h$ and terminal wealth $w_{t +\delta}^h$, whereas Equation (10) tells you how investment is allocated between the risky and riskless assets (given a fixed total amount of money to invest). I have further questions: (1) Could you tell me why those decisions are independent of one another? What is it about log utility that makes it so? (2) Could you derive the optimal consumption and terminal wealth?  
> **username_0**: In particular, what is the budget constraint for the optimization problem presented in Equation (8)?  
> **username_1**: 1. They are not independent. First household decides how much wealth to invest (eq. 8) - as the section in the paper says it gives you the household demand for financial intermediation - actually I would interpret it as a demand more narrowly for saving/investment as financial intermediation is only part of what $w$ is invested in. Second, conditional on that there is financial intermediary who tries to maximize return and minimize the variance. In my answer I misspoke a bit I was always referring to household because in my mind it was still the household decision to choose the intermediation  
> **username_1**: Sorry if that created some confusion (I rewritten that part of my answer to make this clear). I don’t think there is anything special about logarithmic utility function, they could use similar utility and I am guessing the results would qualitatively not change. Modeling utility as a log of consumption is just very convenient mathematically. To be more specific the log utility is homothetic which means it’s scale independent so you can use the same utility for all households disregarding the potential wealth inequality. Also ln(c) is special case of CRRA, but there are multiple functions that  
> **username_1**: Could be used here in principle because they would have exactly the same properties as logarithmic utility but none of them would be as simple/parsimonious as this. 2. Sorry this would take too much time, maybe post it as a separate question and someone else can answer. 3. The authors say that the households are getting labor income so I would say that’s the constraint for consumption/saving decision  
> **username_0**: That's fine, but could you just provide (in mathematics) the constraints that accompany the optimization problem in Equation (8)? I understand solving the entire problem would take too long for you, but if you just give me the constraint I will do the algebra.  

---

