<p>To get general formula for optimal linear taxation, you first need to start with the following objective function:</p>
<p><span class="math-container">$$\max_{t,g} \int_N \Psi (v((1-t)n,g))f(n)dn$$</span></p>
<p>where <span class="math-container">$t$</span> is the tax rate, <span class="math-container">$g$</span> is the government transfer.  Next government wants to  maximize welfare given its social welfare function <span class="math-container">$\Psi$</span> this function can be Rawlsian, utilitarian etc, but government has to have some. You have to superimpose so government social utility otherwise there is no point in redistribution and you can maximize welfare with lump-sum tax.</p>
<p>Next <span class="math-container">$v(\cdot)$</span> is the indirect utility of people, which depends on after tax <span class="math-container">$(1-t)n$</span> and government transfers <span class="math-container">$g$</span>. You want to use indirect utility so later it is easier to get parameter estimates.</p>
<p>Next government faces the following budget constraint (assume balanced budgets - otherwise you would have to use some overlapping generations model which would be nightmare):</p>
<p><span class="math-container">$$t\int_N nl_n((1-t)n,g)f(n)dn = g+G$$</span></p>
<p>Now you can set up Lagrangian:</p>
<p><span class="math-container">$$\mathcal{L}(t,g) = \int_N \left(\Psi (v((1-t)n,g)+\lambda( t nl_n((1-t)n,g)\right) -g-G)f(n)dn$$</span></p>
<p>Note Lagrangian above is only function of <span class="math-container">$t,g$</span> everything else is exogenously given (these optimal tax formulas are all partial equilibrium analysis <span class="math-container">$\implies$</span> there are no general equilibrium effects here, although that is not realistic this is how the 'sausage is made' this is why the results from literature have to be taken with a grain of salt &lt;- but general equilibrium effects would here only reduce the tax rate so at least the outcome can be used to find the upper bound for optimal tax rate).</p>
<p>Next just find the focs:</p>
<p><span class="math-container">$$\frac{\partial \mathcal{L}}{\partial g} = \int_N \left( \Psi'(v_n) \frac{\partial v_n}{\partial g} - \lambda + \lambda t n\frac{\partial l_n}{\partial g} \right)f(n)dn = 0 $$</span></p>
<p><span class="math-container">$$\frac{\partial \mathcal{L}}{\partial t} = \int_N \left( \Psi'(v_n) \frac{\partial v_n}{\partial t} + \lambda n l_n + \lambda t n\frac{\partial l_n}{\partial t} \right)f(n)dn = 0 $$</span></p>
<p>And of course dont forget about the budget constraint itself.</p>
<p>Now, providing that all functions are well behaved so all the derivatives exist, and providing there is solution, you can clearly solve explicitly for <span class="math-container">$t$</span> as a function of the exogenously given parameters so you will end up with some:</p>
<p><span class="math-container">$$t^* = \tau (n,G)$$</span></p>
<p>However, the formula above has little practical use because it would be still too hard to find all the parameters. The whole point of the Saez (2001) work is to find better formulas for which parameters are more easier to estimate. That is the whole advantage of rewriting everything in terms of elasticities.</p>
<p><strong>Edit: Per clarification I also provide close form solution</strong></p>
<p>In order to provide closed form solution we have to provide some specific assumptions for what the societal welfare function, utility function, income distribution and labor supply is. Hence I will use: <span class="math-container">$\Psi(v) =v^2$</span> - society wants to maximize square of money metric utility, <span class="math-container">$v=(1+t)n+g$</span> money metric utility is just sum of net income and government subsidy, <span class="math-container">$l_n=l$</span> labor supply is perfectly inelastic, <span class="math-container">$n$</span> is normalized so that <span class="math-container">$n\in [0,1]$</span> and that <span class="math-container">$n$</span> is uniformly distributed  we have:</p>
<p><span class="math-container">$$t\int_N (nl) dn = g+G \implies g = t\frac{l}{2}-G \tag{1}$$</span></p>
<p><span class="math-container">$$\frac{\partial \mathcal{L}}{\partial g} = \int_N 2(1-t)n+2g - \lambda )dn = 0 =&gt; \lambda = 2g+1-t\tag{2}$$</span></p>
<p><span class="math-container">$$\frac{\partial \mathcal{L}}{\partial t} = \int_N \left( -2((1-t)n+g)n + \lambda n c  \right)dn = 0 \implies  t = \frac{3}{2}g+1 - \frac{3}{4}l\lambda \tag{3}$$</span></p>
<p>Substituting 1 and 2 into 3 yields closed form solution for the <span class="math-container">$t$</span>:</p>
<p><span class="math-container">$$ t^*= \frac{4- 6G(1-l)+3l}{(4  +   3l^2)}$$</span></p>
<p>The above is closed form for <span class="math-container">$t^*$</span> based on exogenously given <span class="math-container">$G$</span> and <span class="math-container">$l$</span> given above mentioned distribution of <span class="math-container">$n$</span>.</p>

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2021-12-12T18:20:55.153Z
Comments are not for extended discussion; this conversation has been [moved to chat](https://chat.stackexchange.com/rooms/132271/discussion-on-answer-by-1muflon1-closed-form-solution-for-the-optimal-flat-tax).
1muflon1
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2021-12-13T13:51:11.587Z
1. I don't see why the FOC should hold with a quadratic social welfare function (don't you get a corner solution?) 2. Can you please use the notation from my question? (As well as the specific assumptions I make in my question, not your own assumptions.)
afreelunch
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2021-12-13T14:30:15.930Z
@afreelunch 1.Quadratic social welfare does not give corner solution, clearly the expression above is not necessarily corner solution with t=1 or t=0. For example for G=0, and l= 1 we have t=0.54. The FOCs will hold for any social welfare function that is twice differentiable. 2. I use v because government cannot observe u here you need to use money metric utility function. Other than that I follow all your assumptions. 3. I already use notation you use except I use g instead of Z and no instead of z, I think using the same letter for two different things is confusing. I actually had to derive
1muflon1
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2021-12-13T15:40:19.820Z
The solution if all parameters and variables have the same name just one is capitalized and other not especially if one of them is government choice variable would make it intractable.
1muflon1
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2021-12-11T16:35:35.143Z
2021-12-12T22:16:55.643Z
2021-12-12T22:16:55.643Z
1muflon1
<p>I used to think that <a href="https://eml.berkeley.edu/%7Esaez/derive.pdf" rel="nofollow noreferrer">Saez (2001)</a> provides an explicit formula for the flat tax (it's framed as the optimal top marginal tax rate, but this is the same thing if the economy has a linear tax schedule). As has been clarified by a <a href="https://economics.stackexchange.com/questions/48669/the-optimal-tax-rate">previous question</a>, however, this formula only determines the optimal flat tax <em>implicitly</em>. After all: (1) the formula uses the elasticity of income with respect to the net of tax rate, which may depend somewhat on the tax rate (2) the formula uses the 'average social marginal welfare weight', which depends on individuals' marginal utilities, which depend on their consumption, which in turn depend on the chosen tax rate.</p>
<p><strong>Question:</strong> Are there any <em>closed form solutions</em> for the optimal flat tax?</p>
<p>To elaborate, I am considering the following kind of model:</p>
<ul>
<li>There is a single tax rate <span class="math-container">$t \in [0, 1]$</span></li>
<li>If <span class="math-container">$Z(t)$</span> denotes total income given that the tax rate is <span class="math-container">$t \in [0, 1]$</span>, then total tax paid is <span class="math-container">$tZ(t)$</span>.</li>
<li>There is a continuum of individuals, indexed by￼<span class="math-container">$i \in [0, 1]$</span>, and with total mass of one.</li>
<li>Each individual gets utility <span class="math-container">$u_i[c_i, z_i(t)]$</span> where <span class="math-container">$c_i$</span> is their consumption and <span class="math-container">$z_i(t)$</span> is their income given the tax rate <span class="math-container">$t$</span>. We assume this is increasing and concave in both arguments.</li>
<li>The tax revenue is shared between all, so each individual consumes <span class="math-container">$c_i = z_i(1-t) + tZ(t)$</span></li>
</ul>
<p>The problem, then, is to find the tax rate <span class="math-container">$t \in [0, 1]$</span> that maximises total utility</p>
<p><span class="math-container">$$ \int_0^1 u_i[c_i(t), z_i] di = \int_0^1 u_i[z_i(t) + tZ(t), z_i] di $$</span></p>
<p>or indeed a weighted sum of total utility (if we want to relax the utilitarian assumption).</p>
<p>If required, I'm happy to make simplifying assumptions -- but of course the more general the formula the better.</p>

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2021-12-12T20:31:03.193Z
I edited my A to provide closed form solution per your clarification
1muflon1
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2021-12-12T20:38:41.220Z
also please stop reediting your question. "explicit formulae" may refer to"Analytical expression, a mathematical expression in terms of a finite or **infinite number of well-known functions**" (see [here](https://en.wikipedia.org/wiki/Explicit_formula)). In comments you complained you only want one closed form solution because you want just one clean function, so please keep your question clear, not all users read the comments. Do not purposefully introduce ambiguity by reversing the edits - purposefully making question less clear is considered vandalism on SE sites.
1muflon1
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2021-12-13T13:45:03.490Z
What edits are you referring to?
afreelunch
<public-economics>
Closed form solution for the optimal flat tax
afreelunch
# Closed form solution for the optimal flat tax

(Asked by: username_0 on 2021-12-09)

I used to think that [Saez (2001)](https://eml.berkeley.edu/%7Esaez/derive.pdf) provides an explicit formula for the flat tax (it's framed as the optimal top marginal tax rate, but this is the same thing if the economy has a linear tax schedule). As has been clarified by a [previous question](https://economics.stackexchange.com/questions/48669/the-optimal-tax-rate), however, this formula only determines the optimal flat tax *implicitly*. After all: (1) the formula uses the elasticity of income with respect to the net of tax rate, which may depend somewhat on the tax rate (2) the formula uses the 'average social marginal welfare weight', which depends on individuals' marginal utilities, which depend on their consumption, which in turn depend on the chosen tax rate.

**Question:** Are there any *closed form solutions* for the optimal flat tax?

To elaborate, I am considering the following kind of model:

* There is a single tax rate $t \in [0, 1]$
* If $Z(t)$ denotes total income given that the tax rate is $t \in [0, 1]$, then total tax paid is $tZ(t)$.
* There is a continuum of individuals, indexed by￼$i \in [0, 1]$, and with total mass of one.
* Each individual gets utility $u\_i[c\_i, z\_i(t)]$ where $c\_i$ is their consumption and $z\_i(t)$ is their income given the tax rate $t$. We assume this is increasing and concave in both arguments.
* The tax revenue is shared between all, so each individual consumes $c\_i = z\_i(1-t) + tZ(t)$

The problem, then, is to find the tax rate $t \in [0, 1]$ that maximises total utility

$$ \int\_0^1 u\_i[c\_i(t), z\_i] di = \int\_0^1 u\_i[z\_i(t) + tZ(t), z\_i] di $$

or indeed a weighted sum of total utility (if we want to relax the utilitarian assumption).

If required, I'm happy to make simplifying assumptions -- but of course the more general the formula the better.

**Tags:** public-economics
> **Comments:**
> **username_1**: I edited my A to provide closed form solution per your clarification  
> **username_1**: also please stop reediting your question. "explicit formulae" may refer to"Analytical expression, a mathematical expression in terms of a finite or **infinite number of well-known functions**" (see [here](https://en.wikipedia.org/wiki/Explicit_formula)). In comments you complained you only want one closed form solution because you want just one clean function, so please keep your question clear, not all users read the comments. Do not purposefully introduce ambiguity by reversing the edits - purposefully making question less clear is considered vandalism on SE sites.  
> **username_0**: What edits are you referring to?  

---

## 1 answers

### Answer by username_1 on 2021-12-11. Score: 1

To get general formula for optimal linear taxation, you first need to start with the following objective function:

$$\max\_{t,g} \int\_N \Psi (v((1-t)n,g))f(n)dn$$

where $t$ is the tax rate, $g$ is the government transfer. Next government wants to maximize welfare given its social welfare function $\Psi$ this function can be Rawlsian, utilitarian etc, but government has to have some. You have to superimpose so government social utility otherwise there is no point in redistribution and you can maximize welfare with lump-sum tax.

Next $v(\cdot)$ is the indirect utility of people, which depends on after tax $(1-t)n$ and government transfers $g$. You want to use indirect utility so later it is easier to get parameter estimates.

Next government faces the following budget constraint (assume balanced budgets - otherwise you would have to use some overlapping generations model which would be nightmare):

$$t\int\_N nl\_n((1-t)n,g)f(n)dn = g+G$$

Now you can set up Lagrangian:

$$\mathcal{L}(t,g) = \int\_N \left(\Psi (v((1-t)n,g)+\lambda( t nl\_n((1-t)n,g)\right) -g-G)f(n)dn$$

Note Lagrangian above is only function of $t,g$ everything else is exogenously given (these optimal tax formulas are all partial equilibrium analysis $\implies$ there are no general equilibrium effects here, although that is not realistic this is how the 'sausage is made' this is why the results from literature have to be taken with a grain of salt <- but general equilibrium effects would here only reduce the tax rate so at least the outcome can be used to find the upper bound for optimal tax rate).

Next just find the focs:

$$\frac{\partial \mathcal{L}}{\partial g} = \int\_N \left( \Psi'(v\_n) \frac{\partial v\_n}{\partial g} - \lambda + \lambda t n\frac{\partial l\_n}{\partial g} \right)f(n)dn = 0 $$

$$\frac{\partial \mathcal{L}}{\partial t} = \int\_N \left( \Psi'(v\_n) \frac{\partial v\_n}{\partial t} + \lambda n l\_n + \lambda t n\frac{\partial l\_n}{\partial t} \right)f(n)dn = 0 $$

And of course dont forget about the budget constraint itself.

Now, providing that all functions are well behaved so all the derivatives exist, and providing there is solution, you can clearly solve explicitly for $t$ as a function of the exogenously given parameters so you will end up with some:

$$t^\* = \tau (n,G)$$

However, the formula above has little practical use because it would be still too hard to find all the parameters. The whole point of the Saez (2001) work is to find better formulas for which parameters are more easier to estimate. That is the whole advantage of rewriting everything in terms of elasticities.

**Edit: Per clarification I also provide close form solution**

In order to provide closed form solution we have to provide some specific assumptions for what the societal welfare function, utility function, income distribution and labor supply is. Hence I will use: $\Psi(v) =v^2$ - society wants to maximize square of money metric utility, $v=(1+t)n+g$ money metric utility is just sum of net income and government subsidy, $l\_n=l$ labor supply is perfectly inelastic, $n$ is normalized so that $n\in [0,1]$ and that $n$ is uniformly distributed we have:

$$t\int\_N (nl) dn = g+G \implies g = t\frac{l}{2}-G \tag{1}$$

$$\frac{\partial \mathcal{L}}{\partial g} = \int\_N 2(1-t)n+2g - \lambda )dn = 0 => \lambda = 2g+1-t\tag{2}$$

$$\frac{\partial \mathcal{L}}{\partial t} = \int\_N \left( -2((1-t)n+g)n + \lambda n c \right)dn = 0 \implies t = \frac{3}{2}g+1 - \frac{3}{4}l\lambda \tag{3}$$

Substituting 1 and 2 into 3 yields closed form solution for the $t$:

$$ t^\*= \frac{4- 6G(1-l)+3l}{(4 + 3l^2)}$$

The above is closed form for $t^\*$ based on exogenously given $G$ and $l$ given above mentioned distribution of $n$.

> **Comments:**
> **username_1**: Comments are not for extended discussion; this conversation has been [moved to chat](https://chat.stackexchange.com/rooms/132271/discussion-on-answer-by-1muflon1-closed-form-solution-for-the-optimal-flat-tax).  
> **username_0**: 1. I don't see why the FOC should hold with a quadratic social welfare function (don't you get a corner solution?) 2. Can you please use the notation from my question? (As well as the specific assumptions I make in my question, not your own assumptions.)  
> **username_1**: @afreelunch 1.Quadratic social welfare does not give corner solution, clearly the expression above is not necessarily corner solution with t=1 or t=0. For example for G=0, and l= 1 we have t=0.54. The FOCs will hold for any social welfare function that is twice differentiable. 2. I use v because government cannot observe u here you need to use money metric utility function. Other than that I follow all your assumptions. 3. I already use notation you use except I use g instead of Z and no instead of z, I think using the same letter for two different things is confusing. I actually had to derive  
> **username_1**: The solution if all parameters and variables have the same name just one is capitalized and other not especially if one of them is government choice variable would make it intractable.  

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