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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 $0\leq a<1$

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    $\begingroup$ 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 :) $\endgroup$
    – Armenthus
    Dec 25 '20 at 23:46
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    $\begingroup$ @Armenthus, you should pose this as an answer and collect the much earned bounty :) $\endgroup$
    – Brennan
    Dec 28 '20 at 3:45
  • $\begingroup$ @Armenthus Fixed the typo, thanks. Are there any other definitions of "Surplus"? $\endgroup$ Dec 28 '20 at 14:28
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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, @actuarialboi9 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:

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

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  • $\begingroup$ 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? $\endgroup$ Dec 30 '20 at 14:03
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    $\begingroup$ 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 $\endgroup$
    – Armenthus
    Dec 30 '20 at 17:59

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