<p>Your claim that if there is a market described by a linear demand curve and we increase the number of people in it that necessarily pushes it to a higher intercept with the same slope is false.</p>
<p>If you want to increase the numbers without changing this scale free distribution, all you have to do is change both the intercept and slope proportionally, i.e:
<span class="math-container">$$
D(p) = a - bp \Rightarrow 2a - 2bp
$$</span>
This is equivalent to only changing the slope in the inverse demand curve, which is the linear case's version of stretching the inverse demand curve along the quantity axis, which is the goal when scaling population numbers.</p>

CC BY-SA 4.0
2024-04-10T01:46:22.990Z
I think I am confused by your example. I said that if you increase $a$ without changing $b$ then you get a higher intercept. You said that is false and then you showed an example where you changed both $a$ and $b$ proportionally, which is not the scale-dependent situation I'm discussing. In fact changing both $a$ and $b$ proportionally is precisely the "scale-free" model that has the form $D(p) = N f(p)$ for some $f(p)$ independent of the number of people.
motherboard
CC BY-SA 4.0
2024-04-10T04:31:11.653Z
"*I said that if you increase a without changing $b$ then you get a higher intercept. You said that is false*" This is not what I wrote. Please read my first sentence again. I wrote that your claim that scaling implies *higher intercept* + *same slope* is false. In the situation I wrote about $f(p)$ is indeed independent of the number of people. If this is not what you want, I am afraid I do not understand your question, as you wrote you wanted a scale independent distribution. This is it.
Giskard
CC BY-SA 4.0
2024-04-10T15:47:25.847Z
Ah what you wrote then is not an accurate representation of my question. I am saying that in the typical presentation of demand curves, when you have the linear equation defined by two independent parameters ($a$ and $b$ in your equation), then increasing the number of people who buy the product (i.e., shifting the demand curve) is often represented by an increase in $a$ without an increase in $b$. My question was about the real-life motivation for such a situation.
motherboard
CC BY-SA 4.0
2024-04-10T15:48:12.450Z
It is of course clear to me that you can increase both parameters by the same factor (which is the "scale free" situation I describe in the second paragraph), but that is not how I have seen demand curves presented typically (e.g., shifts occur without changes in the slope) Example: https://www.thebalancemoney.com/shift-in-demand-curve-when-price-doesn-t-matter-3305720#toc-demand-curve-shifts-right
motherboard
CC BY-SA 4.0
2024-04-10T17:53:46.673Z
And I will grant you that there are a lot of bad presentations explanations out there. Still, stretching the inverse demand curve along the quantity axis is the good way to go if you want scale independent distribution.
Giskard
CC BY-SA 4.0
2024-04-05T03:02:48.673Z
2024-04-05T05:46:58.823Z
2024-04-05T05:46:58.823Z
Giskard
<p><strong>Setup:</strong> Say I have a store and I have 50 bottles that I want to sell. Outside this store there are 100 people who want water bottles and each differs in the price they are willing to pay for a water bottle.</p>
<p>Generally, we say the number of people willing to buy the bottle at price <span class="math-container">$b$</span> is a monotonically decreasing function of <span class="math-container">$b$</span>, but let's further say that the distribution in this price is <strong>&quot;scale-free&quot;</strong> in that it is independent of the number of people outside. In other words, the fraction of people who are willing to pay at least 5 coins for the water bottle is independent of the number of people out there. This means that if 10 people are willing to pay at least 5 coins when there are 100 people outside, then 100 people will be willing to at least 5 coins when there are 1000 people outside. Such an assumption makes sense if you consider each person as independent and to be &quot;drawn&quot; from a distribution of people who want to buy water bottles.</p>
<p><strong>Comparison:</strong> This assumption of &quot;scale-free&quot; fraction in demand differs from typical presentations. In particular, I usually see demand represented as a linear function such as <span class="math-container">$N_{D}(b) = N_D(0) - m_D b$</span> where <span class="math-container">$N_D(0)$</span> is the total number of people in the market and <span class="math-container">$m_D$</span> is an arbitrary constant. By this linear equation, when you increase the number of people in the market, you are simply pushing the <span class="math-container">$N_{D}(b)$</span> curve to a higher intercept with the same slope. (see examples ....)</p>
<p>What this implies is that the fraction of people willing to pay a certain price,</p>
<p><span class="math-container">$$N_D(b)/N_D(0) = 1 - m_D b/N_D(0)$$</span></p>
<p>is not &quot;scale free&quot; and in particular having more people leads to a greater fraction of people willing to pay above a certain amount. For example, if you have 100 people and 50 of them are willing to pay above 2 coins, then having 1000 people means 950 of them are willing to pay above 2 coins.</p>
<hr />
<p><strong>Question:</strong> This &quot;scale-dependent&quot; pricing distribution makes much less sense to me than a scale-free version, yet the former is often how supply and demand is presented. What am I missing? Is &quot;scale-free&quot;ness not a good general assumption?</p>
<p>(I realize that what I'm missing might simply be that the linear equation is an approximation, but I'm wondering if I'm missing anything more than this.)</p>

<supply-and-demand>
Scale-Dependent Demand Curve
motherboard
# Scale-Dependent Demand Curve

(Asked by: username_0 on 2022-12-11)

**Setup:** Say I have a store and I have 50 bottles that I want to sell. Outside this store there are 100 people who want water bottles and each differs in the price they are willing to pay for a water bottle.

Generally, we say the number of people willing to buy the bottle at price $b$ is a monotonically decreasing function of $b$, but let's further say that the distribution in this price is **"scale-free"** in that it is independent of the number of people outside. In other words, the fraction of people who are willing to pay at least 5 coins for the water bottle is independent of the number of people out there. This means that if 10 people are willing to pay at least 5 coins when there are 100 people outside, then 100 people will be willing to at least 5 coins when there are 1000 people outside. Such an assumption makes sense if you consider each person as independent and to be "drawn" from a distribution of people who want to buy water bottles.

**Comparison:** This assumption of "scale-free" fraction in demand differs from typical presentations. In particular, I usually see demand represented as a linear function such as $N\_{D}(b) = N\_D(0) - m\_D b$ where $N\_D(0)$ is the total number of people in the market and $m\_D$ is an arbitrary constant. By this linear equation, when you increase the number of people in the market, you are simply pushing the $N\_{D}(b)$ curve to a higher intercept with the same slope. (see examples ....)

What this implies is that the fraction of people willing to pay a certain price,

$$N\_D(b)/N\_D(0) = 1 - m\_D b/N\_D(0)$$

is not "scale free" and in particular having more people leads to a greater fraction of people willing to pay above a certain amount. For example, if you have 100 people and 50 of them are willing to pay above 2 coins, then having 1000 people means 950 of them are willing to pay above 2 coins.

---

**Question:** This "scale-dependent" pricing distribution makes much less sense to me than a scale-free version, yet the former is often how supply and demand is presented. What am I missing? Is "scale-free"ness not a good general assumption?

(I realize that what I'm missing might simply be that the linear equation is an approximation, but I'm wondering if I'm missing anything more than this.)

**Tags:** supply-and-demand


---

## 1 answers

### Answer by username_1 on 2024-04-05. Score: 0

Your claim that if there is a market described by a linear demand curve and we increase the number of people in it that necessarily pushes it to a higher intercept with the same slope is false.

If you want to increase the numbers without changing this scale free distribution, all you have to do is change both the intercept and slope proportionally, i.e:
$$
D(p) = a - bp \Rightarrow 2a - 2bp
$$
This is equivalent to only changing the slope in the inverse demand curve, which is the linear case's version of stretching the inverse demand curve along the quantity axis, which is the goal when scaling population numbers.

> **Comments:**
> **username_0**: I think I am confused by your example. I said that if you increase $a$ without changing $b$ then you get a higher intercept. You said that is false and then you showed an example where you changed both $a$ and $b$ proportionally, which is not the scale-dependent situation I'm discussing. In fact changing both $a$ and $b$ proportionally is precisely the "scale-free" model that has the form $D(p) = N f(p)$ for some $f(p)$ independent of the number of people.  
> **username_1**: "*I said that if you increase a without changing $b$ then you get a higher intercept. You said that is false*" This is not what I wrote. Please read my first sentence again. I wrote that your claim that scaling implies *higher intercept* + *same slope* is false. In the situation I wrote about $f(p)$ is indeed independent of the number of people. If this is not what you want, I am afraid I do not understand your question, as you wrote you wanted a scale independent distribution. This is it.  
> **username_0**: Ah what you wrote then is not an accurate representation of my question. I am saying that in the typical presentation of demand curves, when you have the linear equation defined by two independent parameters ($a$ and $b$ in your equation), then increasing the number of people who buy the product (i.e., shifting the demand curve) is often represented by an increase in $a$ without an increase in $b$. My question was about the real-life motivation for such a situation.  
> **username_0**: It is of course clear to me that you can increase both parameters by the same factor (which is the "scale free" situation I describe in the second paragraph), but that is not how I have seen demand curves presented typically (e.g., shifts occur without changes in the slope) Example: https://www.thebalancemoney.com/shift-in-demand-curve-when-price-doesn-t-matter-3305720#toc-demand-curve-shifts-right  
> **username_1**: And I will grant you that there are a lot of bad presentations explanations out there. Still, stretching the inverse demand curve along the quantity axis is the good way to go if you want scale independent distribution.  

---

