Interpretation of this VAT tax

In Barrdear and Kumhof (2019), page 24, the VAT tax (\tau^c_t) altogether with the consumption transaction costs (s^c_t), are set up like this:


and the authors call it the “After-tax real consumption expenditure including transaction costs”. Now what is the intuition behind this, since I would say that for the government transaction costs are not observable (and they are not for any agent, for my better understanding there’s no measure of how much money eases the economy’s transactions and neither there’s a measure of how much those transaction costs account for. Nonetheless, I’ might be rampantly wrong in this, if so please let me know). In that sense, why would government tax consumption including transaction costs?

Actually this is an abstraction that is made for the purpose of including money explicitly in the model (although those transaction costs take real resources as seen in market clearing), but at the end as this is a “shortcut” for including money, why would one like to tax also the transaction costs?

To make my point more clear, wouldn’t it be more appealing for the subsequent FOC finding and algebra manipulation, and also more intuitive, to set that term like this

c_t(1+\tau_t^c+s_t^c)\equiv c_t + \tau_t^cc_t+c_ts_t^c?

What do you think? Maybe I’m interpreting something in the wrong way?

Also, to notice some difference, market clearing with first specification would imply:

y_t = c_t(1+(1+\tau_t^c)s_t^c)+...=c_t(1+s_t^c+\tau_t^cs_t^c)+...

Which clearly implies a taxation of the transaction costs itself, but in the government side there’s no income from TC taxation, which makes VAT tax to appear in the market clearing, which seems odd to me.

With the second one

y_t = c_t(1+s_t^c)+...

If I read this correctly, then the government is NOT taxing the transaction costs. Only \tau_cc_t enters the government budget constraint (61). But for households, the transaction costs apply to the price including VAT. Say the price excluding taxes is 100€ and there is a 10% tax. Then households will have to pay 110€ at the counter. Transaction costs will apply to this price.

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