Macroeconomic closure with money and/or bonds

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 it’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 don’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 agents’ budget constraints or I just know “ex-ante” 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 “bonds” case implicitly here, since in such models it’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.

Without knowing more details, it’s hard to follow. If we are talking frictionless models like the classical monetary economy, then
m_{j,t}=(P_{t-1}/P_t)m_{j,t-1} \forall t
does not seem too wrong. If classical dichotomy holds, real variables will be independent of monetary conditions and prices should adjust so that the condition holds.

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In that sense, should I expect in models with short-run price stickiness and money or bonds in the budget constraint to have a macroeconomic closure with the presence of m_{j,t}, for example in the base NK model in Galí (2008) chapter 3 (assuming some form of MIU, CIA, etc. that provokes money to enter de budget constraint)?

And for example if we take the example of my question to be as in Galí (2008) chapter 2 (assuming we investment process and other small differences to my shown budget constraint), do I get correct macroeconomic closure as given in equation (15) from combining exogenous path for money supply (section 2.4.3) and budget constraint?

Thanks Professor Pfeifer!

What do you mean? Unless you have a cashless limit, the money stock should show up.
See DSGE_mod/Gali_2015_chapter_2.mod at master · JohannesPfeifer/DSGE_mod · GitHub

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To explain myself better, and contrast if I’m following: suppose Galí (2008) section 2.5, which is Classical Monetary Model with MIU, the households’ budget constraint is:

P_tC_t+Q_tB_t+M_t\le B_{t-1}+M_{t-1}+W_tN_t-T_t

Given that in the book in this specific section is never shown macroeconomic closure/market clearing (equivalent to equation 15 in the baseline MM), then as you explain me I should not expect this equation in this section to be the typical: C_t=Y_t, but instead C_t+Q_tb_t+m_t= \frac{b_{t-1}+m_{t-1}}{1+\pi_t}+Y_t?

In other words, in section 2.5 is C_t+Q_tb_t+m_t= \frac{b_{t-1}+m_{t-1}}{1+\pi_t}+Y_t the equivalent of eq. 15 from Classical Monetary Model (without money) or I’m missing some detail?

Hope my question is clear, thanks!

I am still not sure what you are after. But it seems you are having a problem clearly distinguishing the resource constraint from budget constraints and other market clearing conditions.

Equation (15) is the goods market clearing condition. The final good can only be used for consumption. The equation you posted is the budget constraint and therefore the equivalent of equation (2).

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Okay, I think I finally got it. Indeed I had that confusion, since in RBC baseline model it’s possible to derive the resource constraint from the budget constraint in a similar fashion as in my first post (with no money). The conclusion I get is, when there’s the inclusion of money in the budget constraint one can no longer obtain the resource constraint from budget constraint, but one just assume what you just said that at the end output will be either consumed or invested (or exported/imported in an open economy), is that right?

The resource constraint is not an assumption. It’s a market clearing condition.

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