A question about Lagrange multiplier(when $\lambda=0$)

Question

I need help in a maximization problem(finding the optimal investment portfolio). 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):

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.

Answer 1

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?

Answer 2

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.

Answer 3

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.