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How do you derive equation (10) on page 740 from He, Krishnamurthy (2013 AER)?

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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.

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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.

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  • $\begingroup$ 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? $\endgroup$ – mathsquestions1 Feb 10 at 3:38
  • $\begingroup$ In particular, what is the budget constraint for the optimization problem presented in Equation (8)? $\endgroup$ – mathsquestions1 Feb 10 at 5:36
  • $\begingroup$ 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 $\endgroup$ – 1muflon1 Feb 10 at 8:55
  • $\begingroup$ 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 $\endgroup$ – 1muflon1 Feb 10 at 9:12
  • $\begingroup$ 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 $\endgroup$ – 1muflon1 Feb 10 at 9:16

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