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I need help in a maximization problem(finding the optimal investment portfolio). enter image description here where $R_s$ and $\Phi$ are $n$ by $1$, with other variables being scalars.

$C^s$ is consumption (or wealth) of an investor, $R_s$ (or $R$) is the return rates of risky assets while $R_f$ is the return rate of a risk-free asset(say government bond), $\Phi$ is the quantities of risky assets, $\phi_f$ is the quantity of the risk-free asset, $\omega$ is the quantity of all the assets(so $\phi_f+\Phi' \cdot 1=\omega$), and $u(\cdot)$ is the investor's utility function.

The solution to this problem gives the following first order condition(FOC) i.e. taking derivative w.r.t. $\Phi$ (assuming integration and differentiation can be exchanged):

enter image description here

But there's actually a constraint: $\phi_f+\Phi' \cdot 1=\omega$, so I tried Lagrange but couldn't get the same result: $$L(\Phi, \lambda)=E[u(c)]+\lambda(\omega-\phi_f-\Phi' \cdot 1 )$$ with FOCs: $$\partial L/\partial \Phi =E[u'(c)(R-1\cdot R_f)]-\lambda \cdot 1=0$$ $$\partial L/\partial \lambda =\omega-\phi_f-\Phi' \cdot 1 =0$$ I couldn't get the same result from above. To have the result in the solution I must have $\lambda=0$ but I'm not sure in what cases it can hold. Please let me know which parts I did wrong.

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  • $\begingroup$ Hi, it would be helpful if you could take the picture portions and turn them into normal math markup for the site. Thanks $\endgroup$ – Kitsune Cavalry Sep 22 '18 at 20:45
  • $\begingroup$ Your constraint is not actually a constraint. You have defined that consumption is what it is because $\phi_f + \Phi' \cdot 1 = w$, so, by construction, that is true. You'd get a $\lambda = 0$ because the constraint is trivial. $\endgroup$ – rafaelc Sep 23 '18 at 22:32
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A $\lambda = 0$ means that the objetive function's derivative with respect to the restriction is zero. In more intuitive terms, one cannot change the expected utility of consumption by relaxing or tightening the budget restriction. This is a weird case, for sure, I think you're missing something here. Maybe telling us what the variables mean can help?

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  • $\begingroup$ $C^s$ is consumption (or wealth) of an investor, $R_s$ is the return rates of risky assets while $R_f$ is the return rate of a risk-free asset(say government bond), $\Phi$ is the quantities of risky assets, $\phi_f$ is the quantity of the risk-free asset, $\omega$ is the quantity of all the assets(so $\phi_f+\Phi' \cdot 1=\omega$), and $u(\cdot)$ is the investor's utility function. $\endgroup$ – Hank Sep 23 '18 at 2:03
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Hi: 2 things I noticed.

1) You're using $R$ in places where I think you mean $R_{s}$.

2) more importantly, you don't need the lagrange multiplier approach because the constraint is already implied-being used in the derivation of the objective function. Therefore, the use of the lagrange multiplier approach in order to satisfy the constraint is unnecessary.

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  • $\begingroup$ 1. is a typo $R^s$ and $R$ are same. 2. I know for this question I can directly substitute in the constraint and get the same FOC as the solution's, but I think I should get the same result if I use Lagrange multiplier. I just wonder what's wrong in my Lagrange approach. $\endgroup$ – Hank Sep 23 '18 at 17:58
  • $\begingroup$ Read the comment by RafaelC. You already have the constraint in the obective function through the use of substitution. So, if you want to use LM approach, then don't make that substitution in the objective function which is currently being made. If you don't make that substitution and use the LM approach, you should get the same answer but you also have to take the derivative with respect to the other parameter $\omega$. $\endgroup$ – mark leeds Sep 24 '18 at 2:34
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There is no problem with your approach, the correct f.o.c. is indeed

$$\partial L/\partial \Phi =E[u'(c)(R-1\cdot R_f)]-\lambda \cdot 1=0$$

They give you a constraint meaning that $\omega$ and $\phi_s$ are treated as exogenously fixed numbers (and indeed you are not asked to maximize with respect to them). So you have an exogenously given total quantity for risky assets that you have to allocate among them, equal to $\omega+\phi_s$. It follows that if this constraint is relaxed, most probably your expected consumption and utility will change since you will be able to buy more/less risky assets, so it is the correct thing to do to form a Lagrangean. This constraint operates in the same way as a "budget constraint" would, only it does so at the level of quantities and not values.

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