<p>If you have taxes, it is important to distinguish between the price that consumers pay <span class="math-container">$p_D$</span> and the price that producers receive <span class="math-container">$p_S$</span>.</p>
<p>Assume that the demand curve and the supply curve are linear, like
<span class="math-container">$$
q_D = \alpha_D - \beta_D p_D
$$</span>
and
<span class="math-container">$$
q_S = \alpha_S + \beta_S p_S
$$</span></p>
<h3>No taxation</h3>
<p>In case that there is no taxation, the price consumers pay is the same as the price that producers receive, so <span class="math-container">$p_S = p_D$</span>. Equilibrium on the market requires demand to equal supply, so <span class="math-container">$q_D = q_S$</span>. This means that for the equilbrium, we have
<span class="math-container">$$
\alpha_D - \beta_D p_D = \alpha_S + \beta_S p_S
$$</span>
Using <span class="math-container">$p_S = p_D$</span> gives the equilibrium price
<span class="math-container">$$
p_D = p_S = \frac{\alpha_D - \alpha_S}{\beta_S + \beta_D}. \tag{1}
$$</span></p>
<h3>Taxes</h3>
<p>In case there is taxation (say an amount <span class="math-container">$t$</span>), the price that consumers pay will be an amount <span class="math-container">$t$</span> higher than the price that the producers receive, so
<span class="math-container">$$
t = p_D - p_S.
$$</span></p>
<p>There is still equilibrium on the market so <span class="math-container">$q_D = q_S$</span>, which gives the same equilibrium condition:
<span class="math-container">$$
\alpha_D - \beta_D p_D = \alpha_S + \beta_S p_S
$$</span>
Now we do not have a single equilibrium price anymore as the price that the producers receive is not equal to the price that the consumers pay (due to the tax).</p>
<p>Now we can make two choices: either we can express everything in terms of the supply price <span class="math-container">$p_S$</span> or we can express everything in terms of the demand price <span class="math-container">$p_D$</span>.</p>
<h4>option 1</h4>
<p>If we express everything in terms of <span class="math-container">$p_S$</span> we substitute <span class="math-container">$p_D = p_S + t$</span> into the equilibrium condition and get:
<span class="math-container">$$
\alpha_D - \beta_D p_S - \beta_D t= \alpha_S + \beta_S p_S
$$</span>
This gives an equilibrium supply price
<span class="math-container">$$
p_S = \frac{\alpha_D - \alpha_S}{\beta_S + \beta_D} - \frac{\beta_D}{\beta_S + \beta_D}t
$$</span>
This price is lower than the equilibrium price if there is no tax (equation (1)).</p>
<p>If we visualize the function <span class="math-container">$\alpha_D - \beta_D p_S - \beta_D t$</span>, it corresponds to a shift of the original demand curve (in the <span class="math-container">$p_D-q$</span> plane) to the left. But now it is given in the <span class="math-container">$p_S-q$</span> plane. The intersection of the two gives indeed the supply price <span class="math-container">$p_S$</span> and in order to get to the demand price <span class="math-container">$p_D$</span> we have to add the tax <span class="math-container">$t$</span>. So
<span class="math-container">$$
p_D = \frac{\alpha_D - \alpha_S}{\beta_S + \beta_D} + \frac{\beta_S}{\beta_S + \beta_D}t.
$$</span></p>
<h4>option 2</h4>
<p>On the other hand, if we express everything in terms of demand prices <span class="math-container">$p_D$</span> we substitute <span class="math-container">$p_S = p_D - t$</span> and get:
<span class="math-container">$$
\alpha_D - \beta_D p_D = \alpha_S + \beta_S p_D - \beta_S t
$$</span>
This gives the equilibrium demand price:
<span class="math-container">$$
p_D = \frac{\alpha_D - \alpha_S}{\beta_S + \beta_D} + \frac{\beta_S t}{\beta_S + \beta_D} 
$$</span>
This new demand price will be higher than the equilibrium demand price when there is no taxation (equation (1)).</p>
<p>The function <span class="math-container">$q_S = \alpha_S + \beta_S p_D - \beta_S t$</span> (in the <span class="math-container">$p_D-q$</span> plane) corresponds to a shift of the original supply curve (in the <span class="math-container">$p_S-q$</span> plane) to the left. The intersection of these two curves gives the demand price <span class="math-container">$p_D$</span>. In order to get to the equilibrium supply curve, we need to subtract the tax <span class="math-container">$t$</span>, so
<span class="math-container">$$
p_S = \frac{\alpha_D - \alpha_S}{\beta_S + \beta_D} - \frac{\beta_D}{\beta_S + \beta_D}t.
$$</span></p>
<h3>Conclusion</h3>
<p>Once you introduce taxes, there is no such thing as an &quot;equilibrium price&quot; as the price that the supplier gets is not the same as the price that the consumer pays.</p>
<p>As such, you either choose to express everything in consumer prices or in producer prices.</p>
<p>For the first choice, you can find the consumer price by looking at the intersection of the demand curve with the shifted supply curve. This price will be higher than the one found under no taxation.</p>
<p>For the second choice, yu can find the producer price by looking at the intersection of the supply curve with the shifted demand curve. This price is below the equilibrium price without taxation.</p>
<p>Once you have found the producer or consumer price, you can find the other one by taking into account the tax difference between the two.</p>

CC BY-SA 4.0
2023-08-04T13:28:40.337Z
2023-08-04T13:28:40.337Z
tdm
<p>Let <span class="math-container">$p_d$</span> be the consumer price and let <span class="math-container">$p_s$</span> be the producer price.</p>
<ul>
<li><p>If there is a specific tax <span class="math-container">$t$</span> on consumption, then a consumer pays
<span class="math-container">$t$</span> on top of the producer price <span class="math-container">$p_s$</span>, so that <span class="math-container">$p_d=p_s+t$</span>.</p>
</li>
<li><p>If there is specific tax <span class="math-container">$t$</span> on production, then a producer receives
<span class="math-container">$p_d$</span> but pays <span class="math-container">$t$</span> to the government, so that <span class="math-container">$p_s=p_d-t$</span>.</p>
</li>
</ul>
<p>Since</p>
<p><span class="math-container">$$ p_d=p_s+t \iff p_s=p_d-t,$$</span></p>
<p>it doesn't matter here whether the tax is levied on producers or consumers.</p>
<hr />
<p>Now, let <span class="math-container">$D$</span> denote the demand function and <span class="math-container">$S$</span> the supply function. The quantity demanded will be <span class="math-container">$D(p_d)$</span> and the quantity supplied will be <span class="math-container">$S(p_s)$</span>. The equilibrium is found where <span class="math-container">$D(p_d)=S(p_s)$</span>. Without a tax <span class="math-container">$p_d=p_s$</span> and with a specific tax <span class="math-container">$p_d-p_s=t$</span>.</p>
<p>There are three ways we could draw the diagram to show the effect of imposing a production or consumption tax:</p>
<ol>
<li><p>Think of the vertical axis as measuring the producer price <span class="math-container">$p_s$</span>, i.e. think of both supply and demand as being functions of <span class="math-container">$p_s$</span>. Then, the demand function with the tax <span class="math-container">$D^t$</span> is given by <span class="math-container">$$D^t(p_s)=D(p_d)=D(p_s+t).$$</span> The demand curve <span class="math-container">$D^t$</span> is <span class="math-container">$D$</span> shifted down by <span class="math-container">$t$</span> and the equilibrium is where this <span class="math-container">$D^t$</span> curve crosses the <span class="math-container">$S$</span> curve. <a href="https://i.sstatic.net/1FBEC.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/1FBEC.png" alt="enter image description here" /></a></p>
</li>
<li><p>Think of the vertical axis as measuring the consumer price <span class="math-container">$p_d$</span> (i.e. think of both supply and demand as being functions of <span class="math-container">$p_d$</span>). Then, the supply function with the tax <span class="math-container">$S^t$</span> is given by <span class="math-container">$$S^t(p_d)=S(p_s)=S(p_d-t).$$</span> The supply curve <span class="math-container">$S^t$</span> is <span class="math-container">$S$</span> shifted up by <span class="math-container">$t$</span> and the equilibrium is where this <span class="math-container">$S^t$</span> curve crosses the <span class="math-container">$D$</span> curve. <a href="https://i.sstatic.net/gqGcS.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/gqGcS.png" alt="enter image description here" /></a></p>
</li>
<li><p>Think of the vertical axis as measuring both <span class="math-container">$p_s$</span> and <span class="math-container">$p_t$</span>. Then neither curve moves. Instead we find the equilibrium where the demand curve is above the supply curve and the vertical distance between the two curves is <span class="math-container">$t$</span>. (This simply reflects the fact that in equilibrium <span class="math-container">$D(p_d)=S(p_s)$</span> and <span class="math-container">$p_d-p_s=t$</span>.) <a href="https://i.sstatic.net/6YKpy.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/6YKpy.png" alt="enter image description here" /></a></p>
</li>
</ol>
<p>1 is more common for a consumption tax, while 2 is more common for a production tax. I like 3 because it makes it clear that it doesn't matter whether it is a production or consumption tax and because it doesn't require adding an extra curve to the diagram. (If looking at an ad valorem tax, then diagram 1 or 2 is better.)</p>
<hr />
<p>The imposition of a tax does not affect the demand or supply function as long as we consider the demand function as a function of the consumer price and the supply function as a function of the producer price. These functions/curves can be used to find/represent consumer and producer surplus with or without the tax.
The area under the demand curve represents (total) benefit from consumption while the area under the supply represents variable production costs. The tax does not affect the benefits from consumption or the costs of production. We can think of a consumption tax as affecting the optimal choices of consumption by increasing the marginal cost of consumption, while we can think of a production tax as affecting the optimal choices of production by decreasing the producer's marginal revenue.</p>
<hr />
<p>Examples of consumption taxes are: value added tax (in the UK), goods and services tax (in Australia and New Zealand) and sales tax (in the US). Other examples include excise taxes on alcohol and cigarettes.</p>

CC BY-SA 4.0
2023-08-04T08:17:48.973Z
2023-08-04T13:51:39.970Z
2023-08-04T13:51:39.970Z
smcc
<p>I've just started learning Economics a few days ago.
There's a question here that I can't understand. If a tax is imposed on consumers, the demand curve should shift to the left, and a new market equilibrium will form, with a lower market price. This is clearly not the case, but why?</p>
<p>Also, no matter whether supply or demand changes due to taxation, a new supply/demand curve will come out. When we are working on consumer/producer surplus, why are we still working on the 'old' supply and demand curve? Or tax will not alter demand/supply curves? Any clarification?</p>
<p>And is there really a tax that is imposed on consumers? Could you please give an example?</p>
<p>My current understanding is that even if a tax is imposed on consumers, the demand curve won't change. There will only be a movement along the current demand curve. So in this context, only the supply curve will shift to the left if a tax is imposed on producers, whereas a tax on consumers will cause a movement along the demand?</p>

<microeconomics><supply-and-demand><taxation>
tax imposition on supply and demand curve
Nonenicht
# tax imposition on supply and demand curve

(Asked by: username_0 on 2023-08-04)

I've just started learning Economics a few days ago.
There's a question here that I can't understand. If a tax is imposed on consumers, the demand curve should shift to the left, and a new market equilibrium will form, with a lower market price. This is clearly not the case, but why?

Also, no matter whether supply or demand changes due to taxation, a new supply/demand curve will come out. When we are working on consumer/producer surplus, why are we still working on the 'old' supply and demand curve? Or tax will not alter demand/supply curves? Any clarification?

And is there really a tax that is imposed on consumers? Could you please give an example?

My current understanding is that even if a tax is imposed on consumers, the demand curve won't change. There will only be a movement along the current demand curve. So in this context, only the supply curve will shift to the left if a tax is imposed on producers, whereas a tax on consumers will cause a movement along the demand?

**Tags:** microeconomics, supply-and-demand, taxation


---

## 2 answers

### Answer by username_2 on 2023-08-04. Score: 2 | Accepted answer

Let $p\_d$ be the consumer price and let $p\_s$ be the producer price.

* If there is a specific tax $t$ on consumption, then a consumer pays
  $t$ on top of the producer price $p\_s$, so that $p\_d=p\_s+t$.
* If there is specific tax $t$ on production, then a producer receives
  $p\_d$ but pays $t$ to the government, so that $p\_s=p\_d-t$.

Since

$$ p\_d=p\_s+t \iff p\_s=p\_d-t,$$

it doesn't matter here whether the tax is levied on producers or consumers.

---

Now, let $D$ denote the demand function and $S$ the supply function. The quantity demanded will be $D(p\_d)$ and the quantity supplied will be $S(p\_s)$. The equilibrium is found where $D(p\_d)=S(p\_s)$. Without a tax $p\_d=p\_s$ and with a specific tax $p\_d-p\_s=t$.

There are three ways we could draw the diagram to show the effect of imposing a production or consumption tax:

1. Think of the vertical axis as measuring the producer price $p\_s$, i.e. think of both supply and demand as being functions of $p\_s$. Then, the demand function with the tax $D^t$ is given by $$D^t(p\_s)=D(p\_d)=D(p\_s+t).$$ The demand curve $D^t$ is $D$ shifted down by $t$ and the equilibrium is where this $D^t$ curve crosses the $S$ curve. [![enter image description here](https://i.sstatic.net/1FBEC.png)](https://i.sstatic.net/1FBEC.png)
2. Think of the vertical axis as measuring the consumer price $p\_d$ (i.e. think of both supply and demand as being functions of $p\_d$). Then, the supply function with the tax $S^t$ is given by $$S^t(p\_d)=S(p\_s)=S(p\_d-t).$$ The supply curve $S^t$ is $S$ shifted up by $t$ and the equilibrium is where this $S^t$ curve crosses the $D$ curve. [![enter image description here](https://i.sstatic.net/gqGcS.png)](https://i.sstatic.net/gqGcS.png)
3. Think of the vertical axis as measuring both $p\_s$ and $p\_t$. Then neither curve moves. Instead we find the equilibrium where the demand curve is above the supply curve and the vertical distance between the two curves is $t$. (This simply reflects the fact that in equilibrium $D(p\_d)=S(p\_s)$ and $p\_d-p\_s=t$.) [![enter image description here](https://i.sstatic.net/6YKpy.png)](https://i.sstatic.net/6YKpy.png)

1 is more common for a consumption tax, while 2 is more common for a production tax. I like 3 because it makes it clear that it doesn't matter whether it is a production or consumption tax and because it doesn't require adding an extra curve to the diagram. (If looking at an ad valorem tax, then diagram 1 or 2 is better.)

---

The imposition of a tax does not affect the demand or supply function as long as we consider the demand function as a function of the consumer price and the supply function as a function of the producer price. These functions/curves can be used to find/represent consumer and producer surplus with or without the tax.
The area under the demand curve represents (total) benefit from consumption while the area under the supply represents variable production costs. The tax does not affect the benefits from consumption or the costs of production. We can think of a consumption tax as affecting the optimal choices of consumption by increasing the marginal cost of consumption, while we can think of a production tax as affecting the optimal choices of production by decreasing the producer's marginal revenue.

---

Examples of consumption taxes are: value added tax (in the UK), goods and services tax (in Australia and New Zealand) and sales tax (in the US). Other examples include excise taxes on alcohol and cigarettes.


---

### Answer by username_1 on 2023-08-04. Score: 2

If you have taxes, it is important to distinguish between the price that consumers pay $p\_D$ and the price that producers receive $p\_S$.

Assume that the demand curve and the supply curve are linear, like
$$
q\_D = \alpha\_D - \beta\_D p\_D
$$
and
$$
q\_S = \alpha\_S + \beta\_S p\_S
$$

### No taxation

In case that there is no taxation, the price consumers pay is the same as the price that producers receive, so $p\_S = p\_D$. Equilibrium on the market requires demand to equal supply, so $q\_D = q\_S$. This means that for the equilbrium, we have
$$
\alpha\_D - \beta\_D p\_D = \alpha\_S + \beta\_S p\_S
$$
Using $p\_S = p\_D$ gives the equilibrium price
$$
p\_D = p\_S = \frac{\alpha\_D - \alpha\_S}{\beta\_S + \beta\_D}. \tag{1}
$$

### Taxes

In case there is taxation (say an amount $t$), the price that consumers pay will be an amount $t$ higher than the price that the producers receive, so
$$
t = p\_D - p\_S.
$$

There is still equilibrium on the market so $q\_D = q\_S$, which gives the same equilibrium condition:
$$
\alpha\_D - \beta\_D p\_D = \alpha\_S + \beta\_S p\_S
$$
Now we do not have a single equilibrium price anymore as the price that the producers receive is not equal to the price that the consumers pay (due to the tax).

Now we can make two choices: either we can express everything in terms of the supply price $p\_S$ or we can express everything in terms of the demand price $p\_D$.

#### option 1

If we express everything in terms of $p\_S$ we substitute $p\_D = p\_S + t$ into the equilibrium condition and get:
$$
\alpha\_D - \beta\_D p\_S - \beta\_D t= \alpha\_S + \beta\_S p\_S
$$
This gives an equilibrium supply price
$$
p\_S = \frac{\alpha\_D - \alpha\_S}{\beta\_S + \beta\_D} - \frac{\beta\_D}{\beta\_S + \beta\_D}t
$$
This price is lower than the equilibrium price if there is no tax (equation (1)).

If we visualize the function $\alpha\_D - \beta\_D p\_S - \beta\_D t$, it corresponds to a shift of the original demand curve (in the $p\_D-q$ plane) to the left. But now it is given in the $p\_S-q$ plane. The intersection of the two gives indeed the supply price $p\_S$ and in order to get to the demand price $p\_D$ we have to add the tax $t$. So
$$
p\_D = \frac{\alpha\_D - \alpha\_S}{\beta\_S + \beta\_D} + \frac{\beta\_S}{\beta\_S + \beta\_D}t.
$$

#### option 2

On the other hand, if we express everything in terms of demand prices $p\_D$ we substitute $p\_S = p\_D - t$ and get:
$$
\alpha\_D - \beta\_D p\_D = \alpha\_S + \beta\_S p\_D - \beta\_S t
$$
This gives the equilibrium demand price:
$$
p\_D = \frac{\alpha\_D - \alpha\_S}{\beta\_S + \beta\_D} + \frac{\beta\_S t}{\beta\_S + \beta\_D}
$$
This new demand price will be higher than the equilibrium demand price when there is no taxation (equation (1)).

The function $q\_S = \alpha\_S + \beta\_S p\_D - \beta\_S t$ (in the $p\_D-q$ plane) corresponds to a shift of the original supply curve (in the $p\_S-q$ plane) to the left. The intersection of these two curves gives the demand price $p\_D$. In order to get to the equilibrium supply curve, we need to subtract the tax $t$, so
$$
p\_S = \frac{\alpha\_D - \alpha\_S}{\beta\_S + \beta\_D} - \frac{\beta\_D}{\beta\_S + \beta\_D}t.
$$

### Conclusion

Once you introduce taxes, there is no such thing as an "equilibrium price" as the price that the supplier gets is not the same as the price that the consumer pays.

As such, you either choose to express everything in consumer prices or in producer prices.

For the first choice, you can find the consumer price by looking at the intersection of the demand curve with the shifted supply curve. This price will be higher than the one found under no taxation.

For the second choice, yu can find the producer price by looking at the intersection of the supply curve with the shifted demand curve. This price is below the equilibrium price without taxation.

Once you have found the producer or consumer price, you can find the other one by taking into account the tax difference between the two.


---

