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The $\gamma$ on the RHS comes from applying chain rule when differentiating the second term with respect to $a_i$. Regarding elasticity, note that with a differentiable function $f$, the ratio $f'(x)/f(x)$ can be interpreted as the percentage change in the value of $f$ around $x$. So isolating $\gamma$ from the FOC, you'll get an expression for elasticity, ...
It just comes from the derivative of profit function. I assume that $a_i$ is the choice variable here so the derivative of $\pi$ wrt $a_i$ is (step by step): $$\frac{\partial \pi}{ \partial a_i} = \frac{\partial \pi}{ \partial a_i} [ \ln R(a_i) ] + \frac{\partial \pi}{ \partial a_i} [ \ln N_i(\gamma a_i, \gamma a_{-i}) ] \\ = \frac{1}{ R(a_i)} R'(a_i) ... 3 As you say the first step is to take log of both sides after that you are just applying the rules for logarithms and rearrange. For example:$$\ln (XZ)=\ln X + \ln Z\ln X/Z= \ln X - \ln Z\ln X^a = a \ln X\ln 1 = 0$$Also an important approximations that hold close to zero are applied here as well these are: \ln(1+x) \approx x  for x ... 2 \frac{dlnQ}{dp}=\frac{dlnQ}{dQ} \frac{dQ}{dp} thus: \frac{dQ}{dp}=\frac{dlnQ}{dp} \frac{dQ}{dlnQ} Since we know that if f(x)=lnx \Rightarrow f'(x)=\frac{1}{x}\Rightarrow \frac{1}{f'(x)}=x We replace \frac{dQ}{dlnQ} by Q. You get: \frac{dQ}{dp}=\frac{dlnQ}{dp} Q It is readily found that \frac{dlnQ}{dp}= \frac{e}{p} So our expression for ... 2 The elasticity of Y with respect to X is often estimated by running a regression like this:$$ \ln Y = \alpha + \beta \ln X + error However, this isn't always applicable because it is perfectly fine to ask what is the percent change in Y to a percent change in X even if X or Y are negative. There are other issues too, like the fact that the elasticity may ...