I'm currently trying to build a CGE, and I'm stuck at the household's problem which is about intertemporal utility maximisation. The household consumes multiple heterogeneous goods $C_i$ (I'm limiting the case to 3 goods for simplicity). I choose an intertemporal utility of substitution equal to 1, which means:

$$ U(C(t))= ln (C_{1,t}^a * C_{2,t}^b * C_{3,t}^c) $$ Here $a$,$b$,$c$ are cobb-douglas coefficients, the shares of goods in the utility function. The budget constraint is straightforward. I assume the consumer is not borrowing, he spends part of his budget on consumption $ \sum (p_{i} * C_{i}) $ and saves what remains $ S$. His income consists of wage $w$ earned from many types of labour $L$ and exogenous government transfers $TR$ (redistribution of income to lower inequality), i.e $ \sum (w_{j} * L_{j}) + TR $ (The index j indicates the many activities where the household works.

The household is supposed to optimise his consumption over time, say from an initial date $ t_0 $ to the final date $ T $. He has to pick the optimal bundle of goods that maximises his inter-temporal utility.

Now, here comes part I don't understand:

1 - In order to select the optimal bundle of goods that will maximise the intertemporal utility, is the consumer supposed to optimise/derive with respect to C1, C2 and C3?

2 - What about the intertemporal budget constraint? I understand for the case of two periods it should be: $$ \sum (p_{i,t1} * C_{i,t1}) + \frac {\sum (p_{i,t2} * C_{i,t2}) } {(1+r)} = (\sum (w_{j,t1} * L_{j,t1}) + TR) + \frac {\sum (w_{j,t2} * L_{j,t2}) + TR } {(1+r)}$$

, but what if this problems goes on from t0 to T periods? What should the budget constraint look like? (t1,t2 means period 1 and period 2)

Assuming a continuous time, I know that the objective function should be written like this ($\rho$ being the rate of time preference): $$ \max_{C_{i,t}} U(C,t) = \int ln (C_{1,t}^a * C_{2,t}^b * C_{3,t}^c) * e^{-\rho*t} $$ (the integral ranges from t0 to T, the final date) Sorry for the long text, hopefully you can at least help me find a form to the intertemporal budget constraint.



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