In the framework of baseline explicitly-modelled money (CIA, MIU, ST, TC* in an RBC), suppose I have typical j-indexed household budget constraint (j\in[0,1]) in nominal terms with money balances:

P_t(C_{j,t}+I_{j,t}) + M_{j,t}=W_tn_{j,t}+R_{t}K_{j,t-1}+M_{j,t-1}+\Pi_{j,t}P_t

With I_{j,t} = K_{j,t}-(1-\delta)K_{j,t-1}

According to my calculations, taking into account the constant returns to scale of the firm and consequently the zero profit condition, then the j-firm profits (\Pi_{j,t}=0) yield:

0 = P_tY_{j,t}-W_tn_{j,t}-R_{t}K_{j,t-1}\implies P_tY_{j,t}=W_tn_{j,t}+R_{t}K_{j,t-1}

Substituting in the budget constraint and translating to real terms we arrive to:

C_{j,t}+I_{j,t}+m_{j,t} = Y_{j,t}+\frac{P_{t-1}}{P_{t}}m_{j,t-1}

I expected to have the typical closed-economy macroeconomic closure (after dropping indexes because of same idiosyncratic risk between households and firms, and therefore same behaviour enabling us to use a representative agent): C_{t}+I_{t}= Y_{t}. My question is, why itâ€™s not possible to derive macroeconomic closure from household budget constraint? (unless of course m_{j,t}=(P_{t-1}/P_t)m_{j,t-1} \forall t which donâ€™t seem obviously correct for me). To a more in depth level my real question would be, in general are macroeconomic closure conditions always derived from combining agentsâ€™ budget constraints or I just know â€śex-anteâ€ť that how such equation should be and what would be the a rule-of-thumb mathematically consistent derivation of this equation?

Thanks!

*PD: Maybe we should exclude TC (transaction costs) models since in this setting the budget constraint has an additional term involving the transactions costs, and the deriving of macroeconomic closure could be different.

I include â€śbondsâ€ť case implicitly here, since in such models itâ€™s often assumed that households use bonds for intertemporal substitution and firms carry on investment process, in which case we must combine firms and households budget constraint, and we would arrive to a similar setting as the described above.