Finding optimality conditions NK in real or nominal terms

Hi. Suppose base NK model with money (let’s say MIU framework). The problem of the household is:


Where c_t=\left(\int_0^1c_t(i)^{\frac{\epsilon-1}{\epsilon}}di\right)^{\frac{\epsilon}{\epsilon-1}} . In the third sequence that the HH must choose optimally I put a dot by purpose since that is my question. One can write the sequence of budget constraints in nominal terms as:

\int_0^1 P_t(i)c_t(i)di+M_t+B_t\le W_tn_t+(1+i_{t-1})B_{t-1}+X_t

Or in real terms (given that P_tc_t\equiv\int_0^1P_t(i)c_t(i)di, and P_t^{1-\epsilon}=\theta P_{t-1}^{1-\epsilon}+(1-\theta)(P_t^\ast)^{1-\epsilon}):

c_t+m_t+b_t\le w_tn_t+\frac{(1+i_{t-1})}{1+\pi_t}b_{t-1}+x_t

Where lower case denotes real variables (y_t=Y_t/P_t).

Now here’s my question, taking into account this is a NK model (i.e. in the supply side there’s monopolistic competition and price stickiness), does it make any difference to find FOC using the nominal terms budget constraint (choosing for M_t) or using the real terms budget constraint (choosing for m_t)?


It depends. In most contexts, P_t is know at time t (contained in your information set). In that case, the transformation to real variables is just a normalization and does not alter any of the results (of course, the interpretation of the Lagrange multipliers will change). In contrast, if you are in a partial information context, such a division would alter the information set.


Thanks. In the case of Calvo pricing, that optimal price P_t^\ast turns out to be a function of expected future variables such as mc_{t+s}, Y_{t+s}, etc. As P_t depends of P_t\ast, and the latter depends on expectations of variables, is in this case P_t in the information set in t?

Also, in order to avoid any mistake regarding this topic in the set up, is it safe to just derive equilibrium conditions in nominal terms (in general)?

Thanks a lot, Prof. Pfeifer.

All variables in the model usually depend on future expectations. But that is irrelevant. It’s about the information set. P_t is decided at time t and is therefore know to the agent when making the decisions.

I think get it. Nevertheless, I’d better read more about this specifically, do you have some literature recommendation that treats this topic?

Not really.

And could you tell me as a example, which variable in the NK-Calvo model would not be in the information set at t maybe? To check if i got it. Thanks!!

Anything that is dated t+1 or later.

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My experience is that it is always easiest and worth the initial headache to write out budget constraints in nominal terms, but then convert to real terms. Typically, the objective function (the utility function, for example) is specified in real terms, and thus you will want to compare apples:apples.

It also makes the thought process of what is happening with your equations (i.e., your intuition…the most important part of research!) much more clear.

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But what would be the difference between solving in nominal terms and solving in real terms with inflation rates here and there?

Could you provide some (extreme if you want) example of a case where solving in real terms may mess up analysis later?

And following the apples:apples approach, would not be a reason for solving in real terms (for having real utility and real b.c.)? Your answer is very appreciated!

I think the most reasonable approach to this is to look at your utility function. All of the arguments are in real terms. Why would it make sense to maximize a utility function, in real terms but with a budget constraint in nominal terms. Let me pose a challenge to you: attempt to re-express your optimization problem with a value-function approach instead of a recursive formulation. It shouldn’t make sense.

The other issue is that I do not think you will be able to close your model out. Typically in baseline MIU models (think of your model, but without the price stickiness), you will need to specify a growth rule for M. This is a dynamic equation with a formulation of something to the effect of

m_t = [(1 + \theta_t)/(1 + \pi_t)]m_{t - 1},

where \theta_t is a money supply shock. This money supply rule will typically be used to cancel the m terms in the household’s budget constraint. So, what you will typically need for your budget constraint to be legitimate is an M_{t - 1} term, or (in case my LaTeX doesn’t come through: you need

\cdots + M_t + \cdots \leq \cdots M_{t - 1}

where the dots represent other terms and the inequality can be less than/equals as you specified in your O.P.

I would highly recommend that you put your budget constraint in real terms, so that you Lagrangian multiplier can actually map real values into utils to match the units of your utility function. Then, I would also look into closing out the model and see if the budget constraint you have in current form (with just the contemporaneous nominal money supply variable) does it. I don’t think I’ve seen a model like that. The usual story is that a household will enter the period with nominal balances from the previous period along with bond holdings and return to those holdings. With these nominal assets, they will choose current money holdings (to satisfy the monetary argument in the utility function) and their current bond holdings (in order to inter-temporally transfer wealth).

Your above formulation looks like the money is burned at the end of the period??? It reminds me of models where people assume that the capital stock completely depreciates at the end of the period; this assumption is almost always used in the effort to isolate analytical solutions to models, which I don’t think you’re trying to do.

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@ChrisL There is absolutely no reason whatsoever to work with a real budget constraint instead of a nominal one. The two are completely isomorphic if P_t is measurable. The

argument is not a convincing one. Utility is measured in utils and therefore always in different units than the budget constraint. It is the job of the Lagrange multiplier to transform the units of the budget constraint into utils. If the budget constraint is real, then the Lagrange multiplier transforms units of the final goods, if it’s nominal then it transforms units of the respective currency into utils. Sidenote: the Lagrange multiplier is the derivative of the value function. Expressing everything recursively would still work like a charm.

Both specifications will yield the same first order conditions and will yield exactly the same intuition.

And yes, there are models where it’s easier to start with a real budget constraint. But sometimes the opposite is true. I find working with open economy modes with different currencies and types of goods very hard to work with if one first tries to make everything real.

@dsge_modeling I agree with @ChrisL that there is a term related to M_{t-1} missing.