# NaN in theoretical moment and how to specify tax rate

Dear all,

I am using second order approximation to solve a RBC model with tax and government debt. However, my APROXIMATED THEORETICAL MOMENTS for capital(k), output(y), consumption© and investment(i) end up being NaN.

Another question is how to specify tax rate. In my model, all the level variables are express as Y=exp(y), where y = log(Y). However, tax rates, which are between 0 and 1, show up in some equations. Do I need to transform tax rate into log form?

Does anyone know what potentially cause this problem? Thanks.
example_RBC_gtcb.mod (2.85 KB)

1. Your model has a unit root in debt and taxes. Hence, the unconditional second moment does not exist.
2. You can put your tax rates in exp(), but typically that is not desired. Performing a log substitution for variables serves the purpose of expressing the in percentage terms. Tax rates are already in percent.

[quote=“jpfeifer”]1. Your model has a unit root in debt and taxes. Hence, the unconditional second moment does not exist.
2. You can put your tax rates in exp(), but typically that is not desired. Performing a log substitution for variables serves the purpose of expressing the in percentage terms. Tax rates are already in percent.[/quote]

Hi Dr. Pfeifer,

Thank you so much for your answer. Is there anyway that I can solve this unit root problem?

Thanks.

You were the model builder. You need to understand why you introduced a unit root and how you can eliminate it. There must be an economic reason for it.

Hi Dr. Pfeifer,

Thanks for your comment. I guess the reason why there exists a unit root in my model is that I introduce government into the rbc model.

My model is based on the simple RBC model I post here. Then, I introduce consumption tax and government debt into the model. Is it common that the introduction of government causes unit root?

Thanks.
example_RBC.mod (2.05 KB)

No, it is not common. It rather depends on your specification. If government expenditure is financed by lump-sum taxation, there will be no unit root. If in contrast there is perfect tax smoothing, i.e. additional debt is financed by just taxing to pay the resulting annuity and never repaying debt, then there will be a unit root. Thus, it depends very much on the specified fiscal rules.