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Consider the following equations:

$$ \tag{1} E_t - E_t^* = \gamma \cdot (Y_t - Y_t^*) + \eta_t, \gamma > 0, $$ $$ \tag{2} U_t - U_t^* = \delta \cdot (E_t - E_t^*) + \mu_t, \delta < 0 $$ where $E_t$ is the log of employment, $Y_t$ is the log of output, $U_t$ is the unemployment rate ,and $^*$ indicates a long-run level.

We can then derive Okun's Law by substituting $(1)$ in $(2)$: $$ \tag{3} U_t -U_t^* = \beta \cdot (Y_t - Y_t^*) + \epsilon_t, \beta < 0 $$ where $\beta = \gamma \cdot \delta$ and $\epsilon_t = \mu_t + \delta \cdot \eta_t$.

I now quote the authors:

Past research provides guidance about the values of the parameters in equations (1)- (3). To see this, suppose first that changes in output and employment are movements along a neoclassical production function: more labor produces more output. For the United States, economists believe that the elasticity of output with respect to labor is about 2/3, based on factor shares of income. If we invert the production function, we get equation (1) with γ = 3/2 = 1.5.

How does one go about 'inverting the production function' and getting equation $(1)$? I am quite lost and would appreciate any help.

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    $\begingroup$ Could it be that $y = F(e)$ with functional form assumption $y = e^\alpha$ so inverse is $y^{1/\alpha} = e$ then around some initial point $(y^*,e^*)$ in log form you get (1). $\endgroup$
    – Jesper Hybel
    Dec 14, 2020 at 7:36
  • $\begingroup$ Hi. Could you expand a little bit more on the Taylor expansion bit (I'm assuming that's what you meant by 'then around...' ? I've tried to get equation (1) to no avail. Thanks. $\endgroup$
    – Pedro Cunha
    Dec 14, 2020 at 13:36
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    $\begingroup$ $\log y = \alpha \log e$ and evaluated at $e^*$ you have $\log y^* = \alpha \log e^*$ so deviation from some equilibrium or steady state $\log y - \log y^* = \alpha (\log e - \log e^*)$. $\endgroup$
    – Jesper Hybel
    Dec 14, 2020 at 20:38
  • $\begingroup$ Perfect. Thank you, @JesperHybel $\endgroup$
    – Pedro Cunha
    Dec 14, 2020 at 21:42

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