# Derivation of Surplus in Paul Romer's paper on “mathiness”

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 $$0\leq a<1$$

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

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 $$q_0$$).
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:
$$$$S= C(a,b,m) N^{\frac{b(a-1)}{a+b}} \; \cdot \; N \; = \; C(a,b,m)N^{\frac{a(1+b)}{a+b}}$$$$