Modelling the inefficiency of redistribution in case of proportional taxation and lump-sum subsidy

Question

I would like to model the following situation in the simplest possible way. First, a government imposes a proportional tax on the whole population, say, x% of income for each household. But then, it decides to return the collected amount of taxes back by subsidizing each household with a lump-sum amount of money, say, X USD.

Intuitively, there should be some efficiency loss due to redistribution for any type of subsidy other than a proportional one (because of incentives distortion). But it also seems to me that it should be possible to prove it with some simple model that illustrates that individual supply curves change in a way that the market supply curve never gets back to it's previous level, thus, causing DWL.

So far I couldn't come up with a way on how to plot this, so I would be happy to hear some ideas on that, as well as any alternative ideas that could illustrate such inefficiency.

Answer 1

If you want some toy model you can use simple representative agent model with:

$$U = x^al^{1-a} \quad s.t. \quad px=(1-t)wL+g $$

where $U$ is utility, $x$ composite consumption good, $l$ leisure, $p$ price level, $t$ proportional tax, $w$ wage/income, $L$ labor supply. We can normalize daily time endowment to 1 to get $1=L+l \implies L=(1-l)$, and $g$ is lump sum subsidy.

To solve the model you can use Lagrangian multiplier. First substitute $L=(1-l)$ or use it as a separate constraint if you prefer doing that way. Choice variables are $x$ and $l$ (choice of $l$ implicitly gives you L).

To calculate the welfare loss as a result, compare $U_{t=0,g=0}$ with $U_{t>0,g>0}$.

Intuitively, there should be some efficiency loss due to redistribution for any type of subsidy other than a proportional one (because of incentives distortion).

This is not good intuition, proportional subsidy should create some distortions, at least generally save for some special parameters. Distortions, are created by people changing their choices when tax is introduced. Generally speaking, any non-lump sum tax and transfer will create incentive for at least some people to change their choices, unless we have some special cases where parameters are such that incentives don't change behavior (e.g. insulin demand can be thought of as being perfectly inelastic so you can tax it without welfare loss).