<blockquote>
<p>With a lump-sum tax it doesn't seem reasonable to ignore the impact on the supply and demand diagrams, in fact the demand curves generally shift downward from what they would be without any tax because consumers have less money to spend.</p>
</blockquote>
<p>Nobody ignores effect of lump-sum tax on supply and demand but there isn't any direct effect. Regular taxes affect supply or demand because they change trade-offs people face. For example. 10% VAT will change people's trade-off between buying product or not buying product, 10% income tax changes people's incentive to work more or try for better job. There will be some consumers on a margin that will be deterred from purchasing the product or workers on margin that will be deterred by income tax, and that is what then also affects suppliers or employers.</p>
<p>The whole point of lump-sum tax is that person pays some fixed sum of money (not a percentage but fixed sum) regardless of what they do. For example, if the lump-sum tax is \$1 and person has to pay that \$1 regardless whether they purchase product or not, regardless of whether they earn a lot or not etc, then their decision to purchase product, to put more effort into getting higher paying job etc is not affected.</p>
<p>Also if people spends less but government spends more the effect will net out. The deadweight loss <strong>is not</strong> caused by less consumer spending (this will be shown in next section), it is caused by distorting peoples trade-offs which change people's behavior. For example, in special an unrealistic case where both demand and supply are perfectly inelastic there is no deadweight even from VAT because if people are not willing to change their decisions in response to tax there won't be any deadweight loss. Lump-sum tax is independent of what you do so there is no escaping the tax by changing your behavior (except perhaps for a decision of committing suicide - but virtually for everyone demand for staying alive will be perfectly or nearly perfectly inelastic).</p>
<blockquote>
<p>Presumably the shifts depend on elasticities, but the general shift downward is inevitable. For example if the lump-sum tax is 20% of GDP then demand curves generally drop by about 20% on the vertical axis.</p>
</blockquote>
<p>Nope, this couldn't be further from truth. First, a tax of 20% of GDP <strong>is not lump-sum tax</strong> as it depends on size of GDP. A lump-sum tax has to be fixed quantity. Second, a true lump-sum tax would lead to zero drop in GDP. This can be proven mathematically.</p>
<p>GDP for closed economy is given by (I use closed economy just so we can shorten the calculation, opening economy up would change nothing just make equations longer):</p>
<p><span class="math-container">$$Y = C+ I +G $$</span></p>
<p>Where <span class="math-container">$Y$</span> is the GDP/income, <span class="math-container">$C$</span> consumer spending, <span class="math-container">$I$</span> investment, and <span class="math-container">$G$</span> government spending.</p>
<p>The consumer spending will be modeled in detail and given by:</p>
<p><span class="math-container">$C =c_0 + c_1( Y -T)$</span></p>
<p>where <span class="math-container">$c_0$</span> is autonomous spending, <span class="math-container">$c_1$</span> is marginal propensity to consume (what % of income people consume and not save), <span class="math-container">$Y$</span> is GDP/income (GDP is also sum of all incomes in economy), <span class="math-container">$T$</span> is the lump-sum tax.</p>
<p><span class="math-container">$I$</span> will be just fixed for simplicity, and we will assume balanced budget so <span class="math-container">$T=G$</span>.</p>
<p>Hence we get:</p>
<p><span class="math-container">$$Y = c_0 + c_1( Y -T)+ I +G $$</span></p>
<p>Now since <span class="math-container">$Y$</span> is on both sides of the equation we have to first solve for <span class="math-container">$Y$</span> to see how GDP is affected.</p>
<p><span class="math-container">$$Y =    \frac{1}{1-c_1} \left( c_0  - c_1T+ I +G\right) $$</span></p>
<p>Now we can see that lump-sum tax will have no effect on GDP provided government spends the money since <span class="math-container">$T=G$</span> we can see that:</p>
<p><span class="math-container">$$Y =    \frac{1}{1-c_1} \left( c_0  - c_1G+ I +G\right) = \frac{1}{1-c_1} \left( c_0  + G(1- c_1)+ I \right)  = G + \frac{1}{1-c_1} \left( c_0  + I \right) $$</span></p>
<p>So clearly the lump-sum tax doesn't decrease GDP.</p>

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1muflon1
<p>I was reading derivations of Ramsey's Optimal Tax Formula, and noticed the claim that a lump-sum tax eliminates dead weight loss. I see this claim made in multiple sources, such as the Wikipedia articles on lump-sum tax or optimal tax.</p>
<p>Dead weight loss can be explained in terms of shifting the supply curve upwards by the amount of tax - i.e. using a supply curve that accounts for the increased effective price for consumers by adding the tax to the price.</p>
<p>This can instead be explained by shifting the demand curve downwards by the amount of tax - i.e. using a demand curve that accounts for the reduced effective price for suppliers by subtracting the tax from the price.</p>
<p>In the first approach prices include tax, in the second they don't. Either way you look at it the result is typically reduced trade and a dead weight loss.</p>
<p>With a lump-sum tax it doesn't seem reasonable to ignore the impact on the supply and demand diagrams, in fact the demand curves generally shift downward from what they would be without any tax because consumers have less money to spend. Presumably the shifts depend on elasticities, but the general shift downward is inevitable. For example if the lump-sum tax is 20% of GDP then demand curves generally drop by about 20% on the vertical axis.</p>
<p>So in what sense can it be claimed a lump-sum tax eliminates dead weight loss?</p>

<optimal-taxation>
Deadweight loss of lump-sum tax
Dbl
# Deadweight loss of lump-sum tax

(Asked by: username_0 on 2022-06-19)

I was reading derivations of Ramsey's Optimal Tax Formula, and noticed the claim that a lump-sum tax eliminates dead weight loss. I see this claim made in multiple sources, such as the Wikipedia articles on lump-sum tax or optimal tax.

Dead weight loss can be explained in terms of shifting the supply curve upwards by the amount of tax - i.e. using a supply curve that accounts for the increased effective price for consumers by adding the tax to the price.

This can instead be explained by shifting the demand curve downwards by the amount of tax - i.e. using a demand curve that accounts for the reduced effective price for suppliers by subtracting the tax from the price.

In the first approach prices include tax, in the second they don't. Either way you look at it the result is typically reduced trade and a dead weight loss.

With a lump-sum tax it doesn't seem reasonable to ignore the impact on the supply and demand diagrams, in fact the demand curves generally shift downward from what they would be without any tax because consumers have less money to spend. Presumably the shifts depend on elasticities, but the general shift downward is inevitable. For example if the lump-sum tax is 20% of GDP then demand curves generally drop by about 20% on the vertical axis.

So in what sense can it be claimed a lump-sum tax eliminates dead weight loss?

**Tags:** optimal-taxation

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## 1 answers

### Answer by username_1 on 2022-06-19. Score: 1

> With a lump-sum tax it doesn't seem reasonable to ignore the impact on the supply and demand diagrams, in fact the demand curves generally shift downward from what they would be without any tax because consumers have less money to spend.

Nobody ignores effect of lump-sum tax on supply and demand but there isn't any direct effect. Regular taxes affect supply or demand because they change trade-offs people face. For example. 10% VAT will change people's trade-off between buying product or not buying product, 10% income tax changes people's incentive to work more or try for better job. There will be some consumers on a margin that will be deterred from purchasing the product or workers on margin that will be deterred by income tax, and that is what then also affects suppliers or employers.

The whole point of lump-sum tax is that person pays some fixed sum of money (not a percentage but fixed sum) regardless of what they do. For example, if the lump-sum tax is \$1 and person has to pay that \$1 regardless whether they purchase product or not, regardless of whether they earn a lot or not etc, then their decision to purchase product, to put more effort into getting higher paying job etc is not affected.

Also if people spends less but government spends more the effect will net out. The deadweight loss **is not** caused by less consumer spending (this will be shown in next section), it is caused by distorting peoples trade-offs which change people's behavior. For example, in special an unrealistic case where both demand and supply are perfectly inelastic there is no deadweight even from VAT because if people are not willing to change their decisions in response to tax there won't be any deadweight loss. Lump-sum tax is independent of what you do so there is no escaping the tax by changing your behavior (except perhaps for a decision of committing suicide - but virtually for everyone demand for staying alive will be perfectly or nearly perfectly inelastic).

> Presumably the shifts depend on elasticities, but the general shift downward is inevitable. For example if the lump-sum tax is 20% of GDP then demand curves generally drop by about 20% on the vertical axis.

Nope, this couldn't be further from truth. First, a tax of 20% of GDP **is not lump-sum tax** as it depends on size of GDP. A lump-sum tax has to be fixed quantity. Second, a true lump-sum tax would lead to zero drop in GDP. This can be proven mathematically.

GDP for closed economy is given by (I use closed economy just so we can shorten the calculation, opening economy up would change nothing just make equations longer):

$$Y = C+ I +G $$

Where $Y$ is the GDP/income, $C$ consumer spending, $I$ investment, and $G$ government spending.

The consumer spending will be modeled in detail and given by:

$C =c\_0 + c\_1( Y -T)$

where $c\_0$ is autonomous spending, $c\_1$ is marginal propensity to consume (what % of income people consume and not save), $Y$ is GDP/income (GDP is also sum of all incomes in economy), $T$ is the lump-sum tax.

$I$ will be just fixed for simplicity, and we will assume balanced budget so $T=G$.

Hence we get:

$$Y = c\_0 + c\_1( Y -T)+ I +G $$

Now since $Y$ is on both sides of the equation we have to first solve for $Y$ to see how GDP is affected.

$$Y = \frac{1}{1-c\_1} \left( c\_0 - c\_1T+ I +G\right) $$

Now we can see that lump-sum tax will have no effect on GDP provided government spends the money since $T=G$ we can see that:

$$Y = \frac{1}{1-c\_1} \left( c\_0 - c\_1G+ I +G\right) = \frac{1}{1-c\_1} \left( c\_0 + G(1- c\_1)+ I \right) = G + \frac{1}{1-c\_1} \left( c\_0 + I \right) $$

So clearly the lump-sum tax doesn't decrease GDP.

> **Comments:**
> **username_1**: Comments are not for extended discussion; this conversation has been [moved to chat](https://chat.stackexchange.com/rooms/137256/discussion-on-answer-by-1muflon1-deadweight-loss-of-lump-sum-tax).  

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