My question is from overlapping generations
$\Rightarrow$ I want to prove this part of the theorem
$\textbf{First}$ proceed by showing that the allocations in a competitive equilibrium must satisfy(12)}
Of the macroeconomic variables, $C_t=\sum_{i=1}^Ic^{i}_t$, $L_t=\sum_{i=1}^{I}\varepsilon^{i} l_{t}^{i}$ and the condition of the labor market clearing $Y_t=C_t+I_t+G_t$ (x is competitive equilibrium allocations)
$\sum_{i=1}^Ic^{i}_t+K_{t+1}-(1-\delta)K_t+G_t=F(K_t,\sum_{i=1}^{I}\varepsilon^{i} l_{t}^{i})$
$\textbf{Second}$ proceed by showing that the allocations in a competitive equilibrium must satisfy(13)
The implementability constraint for each generation is constructed by substituting the consumer first-order conditions of the consumer's problem
$U_{c}(c^{i}_{t+i},l^{i}_{t+i})=-\lambda_{t+i}(1+\eta_{t+i})$ ..........($\star$)
$U_{l}(c^{i}_{t+i},l^{i}_{t+i})=\lambda_{t+i}(1-\tau_{t+i})w_{t+i}\epsilon^i$ ........($\star\star$)
$\lambda_{t+i}=\beta \lambda_{t+i+1}(1+r_{t+1}(1-\theta_{t+1}))$
To derive the implementability constraint we have to multiply ($\star$), ($\star\star$) by their respective control variables and then add them up for all i.
$c^{i}_{t+i}U_{c}(c^{i}_{t+i},l^{i}_{t+i})=-c^{i}_{t+i}\lambda_{t+i}(1+\eta_{t+i})$ $\rightarrow \sum_{i=1}^{I}c^{i}_{t+i}U_{c}(c^{i}_{t+i},l^{i}_{t+i})=-\sum_{i=1}^{I}c^{i}_{t+i}\lambda_{t+i}(1+\eta_{t+i}) $ .....(a)
$l^{i}_{t+i}U_{l}(c^{i}_{t+i},l^{i}_{t+i})=l^{i}_{t+i}\lambda_{t+i}(1-\tau_{t+i})w_{t+i}\varepsilon^i$ $\rightarrow \sum_{i=1}^{I}c^{i}_{t+i}U_{c}(c^{i}_{t+i},l^{i}_{t+i})=\sum_{i=1}^{I}c^{i}_{t+i}\lambda_{t+i}(1+\eta_{t+i}) $........(b)
(a) and (b) into the intertemporal budget constraint
$\sum_{i=0}^{I-1}p_{t+i}(1+\eta_{t+i})c^i_{t+i}\leq \sum_{i=0}^{I-1}p_{t+i}(1-\tau_{t+i})w_{t+i}\varepsilon^il^i_{t+i} $
where $p_{t+i}$ denotes the Arrow–Debreu price for the consumption good at period $t+i$
$-\sum_{i=0}^{I-1}c^{i}_{t+i}U_{c}(c^{i}_{t+i},l^{i}_{t+i})=\sum_{i=0}^{I-1}l^{i}_{t+i}U_{l}(c^{i}_{t+i},l^{i}_{t+i})$ (where $\lambda_{t+i}=p_{t+i}$)
$-\sum_{i=0}^{I-1}\beta^{i}c^{i}_{t+i}U_{c}(c^{i}_{t+i},l^{i}_{t+i})=\sum_{i=0}^{I-1}\beta^{i}l^{i}_{t+i}U_{l}(c^{i}_{t+i},l^{i}_{t+i})$
$\sum_{i=1}^{I}\beta^{i-1}\left[c^{i}_{t+i-1}U_{c}(c^i_{t+i-1},l^{i}_{t+i-1})+l^{i}_{t+i-1}U_{l}(c^i_{t+i-1},l^{i}_{t+i-1})\right]=0$
$\textbf{Third}$ proceed by showing that the allocations in a competitive equilibrium must satisfy(14)
I don't know how I could get $U_{c^s_{0}}a^{s}_{0}$ where $s=2,3,...,I$
Any recommendations for the third part of the proposition? Thanks!
