Lagrange for Economic model with random variable

Question

I am working on a simple 1 period economic model with random return. The model is:

Period 1: $c_0 = y_0 - s$

Period 2: $c_1 = y_1 + \tilde{r} \cdot s $

Where:

• $y_0, y_1$: are lum sum payment (it means it is given in each period), deterministic given value

• $c_0, c_1$: are consumption

• $s$: saving/investment in period 0 (which will growth to $(\tilde{r} \cdot s)$ at period 1)

• $\tilde{r}$: random return. For example if $\tilde{r} = 1.05$ means \$1 will growth to \$1.05

The model objective is to maximize Utility:

Utility (U) = $\mathbb{E}[u(c_0) + \cdot u(c_1)]$

Where:

• $u(\cdot)$: is utility function, for simplicity take $u(\cdot) = ln(\cdot)$

• $$: discount for future consumption, deterministic given value

Normally, to solve this problem, we use Lagrange multiplier to solve for $c_0$, $c_1$, and $s$

$\mathcal{L} = u[c_0] + \cdot \mathbb{E}[u(c_1)] - \mu \cdot \left( c_0 - y_0 + s \right) - \lambda \cdot \left( c_1 - \tilde{r} \cdot s - y_1 \right) $

Now taking derivatives, we have:

(1) $ \frac{\partial\mathcal{L}}{\partial c_0} $ → easy

(2) $ \frac{\partial\mathcal{L}}{\partial c_1} $ → This is the one I don't know how to do

(3) $ \frac{\partial\mathcal{L}}{\partial s} = \mu + \lambda \tilde{r} = 0$ → I also not sure about this one too

I am getting stuck at (2) and (3). Especially number (2) since I do know how to take derivative of a function with random variable in this case. What should we do in this case?

Answer 1

I think we can transform random variable into a deterministic model as follow:

Obj: Max = $u[c_0] + β \mathbb{E}[u_1] = u[c_0] + β \sum p^i u[c_1^i] $

Constrains:

$c_0 = y_0 - s$

$c_1^i = y_1 + s \cdot r^i$

Then we can apply Lagrange:

$\mathcal{L} = u[c_o] + \sum β p^i u[c_1^i] - \mu \left( c_0 - y_0 + s \right) - \sum \lambda^i \left( c_1^i - y_1 - s r^i \right)$

$\frac{\partial \mathcal{L}}{\partial c_0} = u'[c_0] - \mu = 0$

$\frac{\partial \mathcal{L}}{\partial c_1^i} = β p^i u'[c_1^i] - \lambda^i = 0$

$\frac{\partial \mathcal{L}}{\partial s} = -\mu + \sum \lambda^i r^i = 0$

Then we might need some more condition to solve this system of equation

Any comment?