# Inverting the production function - Okun's Law: Fit at 50? (BALL; LEIGH; LOUNGANI, 2012)

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.

• 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). Dec 14, 2020 at 7:36
• 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. Dec 14, 2020 at 13:36
• $\log y = \alpha \log e$ and evaluated at $e^*$ you have $\log y^* = \alpha \log e^*$ so deviation from some equilibrium or steady state $\log y - \log y^* = \alpha (\log e - \log e^*)$. Dec 14, 2020 at 20:38
• Perfect. Thank you, @JesperHybel Dec 14, 2020 at 21:42