Government budget constraint: Forward-looking vs. regressive, differences

Papers I’ve read use to write govt. budget constraint (in non-cashless economy) in “regressive” fashion as (see for example SGU (2007)):

M_t+B_t +P_t\tau_t \ge (1+i_{t-1})B_{t-1}+M_{t-1}+P_tG_t

Nevertheless HH’s budget constraint are written in a forward-looking fashion i.e.:

E_{t}\{Q_{t,t+1}B_{t+1}\}+M_t+P_t(C_t+K_{t+1}-(1-\delta)K_t)\le B_t+M_{t-1}+r_tP_tK_t-P_tT_t+P_t \Pi_t.

Note that despite actually SDF is Q_{t,t+1}=1/(1+i_t) (where i_t is the one period nominal return of B_{t-1} from t-1 to t), if I were to algebraically try to get resource constraint, combining HH’s and govt. budget constraints, I couldn’t get rid off money and bonds, since first timing would not match. And second, bonds in one govt.'s b.c. are in terms of “nominal interest” notation and in the HH’s side are in terms of “asset pricing” notation (not sure if this terminology is ok, but you get the point). Which are two ways of seeing the same thing but strictly speaking, if both are not accordingly, it’s not algebraically possible to obtain aggregate resource constraint.

Then, is it correct if I rearrange government b.c. in this way
M_t+E_{t}\{Q_{t,t+1}B_{t+1}\}+P_t\tau_t \ge B_{t}+M_{t-1}+P_tG_t

I know at the end, independant the way you write timing the purpose is obvious, and the results are the same (something like Y_t=C_t+I_t+G_t). But I like to have my equations well justified, then I’d appreciate some comment on that, as also I’m aware that in rational expectations framework, wrong timing in equations can cause major mistakes, or lead to very different results.

Also a second thing, it’s a bit basic then I don’t see necessary open another topic on this: Is it right to define a SDF in real terms? i.e. that Q_{t,t+1} is not Q_{t,t+1}=1/(1+i_t), but instead Q_{t,t+1}=(1+\pi_{t+1})/(1+i_t)? I ask since I don’t recall seeing that, but real terms has the advantage of being directly usable for setting firms discounted-sum type problem (when it’s already in real terms, and of course terms are arranged in HH’s b.c. so prices are not counted twice).


  1. I am not sure I understand the issue. In SGU, it’s E_t d_{t,t+1} x_{t+1} for a random payment at time t+1. You cannot simply plug in B_t for x_t due to bonds being predetermined and the timing reflecting this.
  2. The nominal SDF is for transforming nominal cash flows to utils, while the real SDF is for real cash flows.
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Thanks for your answer!!

  1. In that case how to prevent all those variables x_{t+1}, B_t, etc. that are not common to resource constraint to appear in it? Would it be right to formulate govt.'s resource constraint in my own model as all those variables timing match and cancel each other?

My understanding is that with the appropriate timing convention/consistent notation they should indeed cancel.

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