I have a small - potentially dumb - question regarding the modeling of the Government side. In my model the Government taxes households. There are two kinds of taxes (tau): labour and capital. Accordingly, I need to specify an equation for the Government’s “problem”. My guess is : T_t = tau_h * w * h_t + tau_k * r * k_t. The variable T appears in the right-hand-side of the household’s equation as the Government is supposed to provide them with a lump-sum transfer.
My first question is how to name the variable T, do you think “Government expenditure” is correct?
Second, I have see some people model it as G_t = tau_h * w * h_t + tau_k * r * k_t + T. Could someone tell me what the difference is ? What is this G ? Is this a more appropiate way to model it ?
In your first version you can call it government expenditure but everything the government does is to combine two taxes and to transfer the sum to the household as a lump-sum. Normally when we talk about government expenditure we mean some form of consumption that is done by the government.
This is probably what is meant here. In the second version what people normally do is to specify G either as a shock or as shock times output such that government consumes a fraction of output that can stochastically change. But then you should think about the sign of T to not get confused. If you still want it to be a transfer to households and interpret it in a similar way to the above version I would write it on the left hand side as it is part of the expenditures. Not that it really matters, otherwise it would just be negative in case of a transfer.
The most appropriate way is the one that best fits whatever you are trying to model. Only you know what that is, if you need a fiscal sector, and if yes, what kind of it.
Therefore, if I want to stick with my specification, in which left-hand side should I write it ? You mean like this : 0 = T_t + tau_h * w * h_t + tau_k * r * k_t ?
Also, for completeness, do I have to specify that T in the general output equation to make it align with the national accounts standard specification. That is : y = c + i + T ?
Like I said, it does not really matter if you write it like
0 = T_t + tau_h * w * h_t + tau_k * r * k_t
where a positive lump sum transfer to the household would result in a T_t < 0, or
T_t = tau_h * w * h_t + tau_k * r * k_t
where a positive lump sum transfer leads to T_t > 0.
It just depends how you want to write it. Try to think about the signs of incomes and expenditures of the government in this case then it should be clear what I am trying to say.
My guess, but without knowing the model I can’t say for certain, is that you do not have to include it in the economy-wide resource constraint. The government in the specification, that you want to have, does not consume anything. It only transfers something from the firms to the households which they can then consume, i.e. I would say it is already included in C.
To reiterate: In G_t = \tau_h w h_t + \tau_k r k_t + T
we have G_t being government final consumption of goods and services. This is stuff the government uses directly and therefore it enters the resource constraint as these resources cannot be used for e.g. private consumption. This is very different from lump sum transfers Z_t. These are resources either transferred to or from the households. They can then be used for final use somewhere else, e.g. if there is a transfer to households, it will be used for private consumption C_t. If it’s taxes on households it will be used for government consumption G_t. Putting it into the resource constraint as well would amount to double-counting.