6

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, ...


6

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 ...


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