Assume I have the following objective, which consists of choosing the bundle of goods that will maximise the intertemporal utility between date 0 and date T:
$$ \max_{C1(t), C2(t), C3(t)} U(C(t))= \int_{t=0}^T e^{-\rho.t} . ln [C{1}^{a}(t) . C{2}^{b}(t) .C{3}^{c}(t)] . dt$$
$C1$ to $C3$ are consumptions of 3 heterogenous goods. $a$,$b$ and $c$ are coefficients, the shares of the goods in the utility function U(C(t)).
$a+b+c=1$
And this budget constraint: $$ \int_{t=0}^{T} ( \sum_{j} R(t) ) . e^{-r.t} . dt = \int_{t=0}^{T} [p1 (t) . C1(t) + p2 (t) . C2(t)+p3 (t) . C3(t)] . e^{-r.t} . dt $$
The goal is to select the bundle of goods $C1$ to $C3$ that maximises the intertemporal utility. The consumer picks a bundle for each point in time. Consumption could rise or drop over time, or it could remain constant from 0 to T. So I assume $C_1$ to $C_3$ are function of time. Should I use the maximum principle to solve this or just the Lagrangian? The integrals don't make it easy. Any help would be appreciated.