I'm trying to solve problem 4.4 in Munk (2011). The problem is as follows:
- Assume the market is complete and $\xi = (\xi_{t})$ is the unique state-price deflator.
- Present value of any consumption process $(c_{t})_{t \in [0,T]}$ is $E[\int^{T}_{0} \xi_{t}c_{t}dt]$.
- For an agent with time-additative preferences, an initial value $W_{0}$ and no future income except from financial transactions, the problem is formulated as $$ \begin{align*} \max_{c = (c_{t})_{t \in [0,T]}} \quad &E\left[\int^{T}_{0} e^{-\delta t}u(c_{t})dt\right]\\ \text{s.t.} \qquad &E\left[\int^{T}_{0} \xi_{t}c_{t}dt\right] \leq W_{0}. \end{align*} $$
Munk asks us to use the Lagrangian technique for constrained optimisation to show that the optimal consumption process must satisfy
$$ e^{-\delta t}u'(c_{t}) = \alpha \xi_{t}, \quad t \in [0,T], $$
where $\alpha$ is a Lagrange multiplier. Finally, we need to explain why $\xi_{t} = e^{-\delta t}u'(c_{t})/u'(c_{0})$.
My approach is to start by defining the Lagrange function:
$$ \mathcal{L}(c_{t},\alpha) = E\left[\int^{T}_{0} e^{-\delta t}u(c_{t})dt\right] + \alpha\left(W_{0} - E\left[\int^{T}_{0} \xi_{t}c_{t}dt\right]\right). $$
From here we of course need to derive the FOC which is also where my problem arises. My first thought was to use Tonelli to interchange the expectation and inner integral, but I'm not sure if that is the right approach. I'm quite new to consumption-based asset pricing and would like to know how to proceed from here and if my approach so far is correct. Help with initial steps to solve this problem would be very much appreciated and hints to move on.
Edit
If I need to provide more details then feel free to ask me for that. I hope someone can guide me in the right direction.
