<p>I think you have a typo in your  <span class="math-container">$q_0$</span>: the exponent of <span class="math-container">$N$</span> should be  <span class="math-container">$-\frac{b}{a+b}$</span>. I did the whole calculus with this corrected type of <span class="math-container">$q_0$</span> and I was able to replicate your results (this is why assume <span class="math-container">$q_0$</span> has only a typo and you actually did the algebra with the correct <span class="math-container">$q_0$</span>).</p>
<p>I suggest that the discrepancies to the paper are indeed connected to the Surplus function. But i cannot come up with any idea why this surplus function might be wrong.</p>
<hr />
<p><strong>Edit:</strong> Outside from the typo there was no error, @actuarialboi9 and my humble self calculated the individual surplus. If one multiplies that Surplus function with <span class="math-container">$N$</span> the number of people in the market results will be equal:</p>
<p><span class="math-container">$\begin{equation} S= C(a,b,m) N^{\frac{b(a-1)}{a+b}} \;  \cdot \; N \;  = \; C(a,b,m)N^{\frac{a(1+b)}{a+b}} \end{equation}$</span></p>

CC BY-SA 4.0
2020-12-30T14:03:33.597Z
I think I found what it is. If I multiply by N the Surplus I found it leads to the correct formula. My suspicion is that I actually found the individual Surplus and the total Surplus would just be N times the individual surplus (since all consumers share the same demand function). So the problem boils down to this: Is it sound to add individual surpluses to find the total surplus?
actuarialboi9
CC BY-SA 4.0
2020-12-30T17:59:35.963Z
I just reread the Romer article and i found following sentence: " Let q stand for individual consumption of mobile phone services ". So you were right we were calculating with indiviudal quantities. I cant see any problem adding indiviudal surpluses
Levittstyle
CC BY-SA 4.0
2020-12-28T17:49:47.777Z
2020-12-30T23:03:15.613Z
2020-12-30T23:03:15.613Z
Levittstyle
<p>In this paper by P. Romer <a href="https://pubs.aeaweb.org/doi/pdfplus/10.1257/aer.p20151066" rel="nofollow noreferrer">https://pubs.aeaweb.org/doi/pdfplus/10.1257/aer.p20151066</a>
I'm wondering the Surplus <span class="math-container">$S$</span> was derived.</p>
<p>By using the given condition I found that <span class="math-container">$$q_0=m^{-\tfrac{1}{a+b}}N^{-\tfrac{b}{a+b}}$$</span>
By Surplus I assume he means total Surplus (Consumer Surplus + Producer Surplus) which can be calculated by the integral <span class="math-container">$$\displaystyle{S=\int_{0}^{q_0}[D(q)-S(q)]dq=\int_{0}^{q_0}[q^{-a}-N^{b}q^{b}]dq}$$</span>
With a little bit of algebra I find <span class="math-container">$$S=C(a,b,m)N^{\tfrac{b(a-1)}{a+b}}$$</span> where <span class="math-container">$C(a,b,m)=\tfrac{1}{1-a}m^{\tfrac{a-1}{a+b}}-\tfrac{1}{b+1}m^{-\tfrac{b+1}{a+b}}$</span></p>
<p>So I'm probably wrong, is my idea of <span class="math-container">$S$</span> correct? I need to know because if it is then it's likely a mistake in my algebra. I also assumed that <span class="math-container">$b&gt;0$</span> and <span class="math-container">$0\leq a&lt;1$</span></p>

CC BY-SA 4.0
2020-12-25T23:46:46.113Z
I think you have a typo in your q0, the exponent of N should be: -b/(a+b). I did the whole calculus with this corrected type of q0 and I was able to replicate your results (this is why assume q0 has only a typo and you actually did the algebra with the correct q0). I suggest that the discrepancies to the paper are indeed connected to the Surplus function. But i cannot come up with any idea why this surplus function might be wrong. In case you find anything useful keep me / this thread updatet :)
Levittstyle
CC BY-SA 4.0
2020-12-28T03:45:31.983Z
@Armenthus, you should pose this as an answer and collect the much earned bounty :)
Brennan
CC BY-SA 4.0
2020-12-28T14:28:10.877Z
@Armenthus Fixed the typo, thanks. Are there any other definitions of "Surplus"?
actuarialboi9
<mathematical-economics><supply-and-demand><economic-growth><surplus><equilibrium-price>
Derivation of Surplus in Paul Romer's paper on "mathiness"
actuarialboi9
# Derivation of Surplus in Paul Romer's paper on "mathiness"

(Asked by: username_0 on 2020-12-21)

In this paper by P. Romer <https://pubs.aeaweb.org/doi/pdfplus/10.1257/aer.p20151066>
I'm wondering the Surplus $S$ was derived.

By using the given condition I found that $$q\_0=m^{-\tfrac{1}{a+b}}N^{-\tfrac{b}{a+b}}$$
By Surplus I assume he means total Surplus (Consumer Surplus + Producer Surplus) which can be calculated by the integral $$\displaystyle{S=\int\_{0}^{q\_0}[D(q)-S(q)]dq=\int\_{0}^{q\_0}[q^{-a}-N^{b}q^{b}]dq}$$
With a little bit of algebra I find $$S=C(a,b,m)N^{\tfrac{b(a-1)}{a+b}}$$ where $C(a,b,m)=\tfrac{1}{1-a}m^{\tfrac{a-1}{a+b}}-\tfrac{1}{b+1}m^{-\tfrac{b+1}{a+b}}$

So I'm probably wrong, is my idea of $S$ correct? I need to know because if it is then it's likely a mistake in my algebra. I also assumed that $b>0$ and $0\leq a<1$

**Tags:** mathematical-economics, supply-and-demand, economic-growth, surplus, equilibrium-price

> **Comments:**
> **username_1**: I think you have a typo in your q0, the exponent of N should be: -b/(a+b). I did the whole calculus with this corrected type of q0 and I was able to replicate your results (this is why assume q0 has only a typo and you actually did the algebra with the correct q0). I suggest that the discrepancies to the paper are indeed connected to the Surplus function. But i cannot come up with any idea why this surplus function might be wrong. In case you find anything useful keep me / this thread updatet :)  
> **username_2**: @Armenthus, you should pose this as an answer and collect the much earned bounty :)  
> **username_0**: @Armenthus Fixed the typo, thanks. Are there any other definitions of "Surplus"?  

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## 1 answers

### Answer by username_1 on 2020-12-28. Score: 5 | Accepted answer

I think you have a typo in your $q\_0$: the exponent of $N$ should be $-\frac{b}{a+b}$. I did the whole calculus with this corrected type of $q\_0$ and I was able to replicate your results (this is why assume $q\_0$ has only a typo and you actually did the algebra with the correct $q\_0$).

I suggest that the discrepancies to the paper are indeed connected to the Surplus function. But i cannot come up with any idea why this surplus function might be wrong.

---

**Edit:** Outside from the typo there was no error, @username_0 and my humble self calculated the individual surplus. If one multiplies that Surplus function with $N$ the number of people in the market results will be equal:

$\begin{equation} S= C(a,b,m) N^{\frac{b(a-1)}{a+b}} \; \cdot \; N \; = \; C(a,b,m)N^{\frac{a(1+b)}{a+b}} \end{equation}$

> **Comments:**
> **username_0**: I think I found what it is. If I multiply by N the Surplus I found it leads to the correct formula. My suspicion is that I actually found the individual Surplus and the total Surplus would just be N times the individual surplus (since all consumers share the same demand function). So the problem boils down to this: Is it sound to add individual surpluses to find the total surplus?  
> **username_1**: I just reread the Romer article and i found following sentence: " Let q stand for individual consumption of mobile phone services ". So you were right we were calculating with indiviudal quantities. I cant see any problem adding indiviudal surpluses  

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