0

I want to find the log form of this equation to simplify some of the subsequent steps. However, I am struggling with how to handle the constant, $\theta$. If anyone could explain how to log this equation, it would be much appreciated:

$$ \Pi_t^{1-\varepsilon} = \theta + (1-\theta) (P_t^{*}/P_{t-1})^{1-\varepsilon} $$

If theta were excluded, so that the equation was $\Pi_t^{1-\varepsilon} = (1-\theta) (P_t^{*}/P_{t-1})^{1-\varepsilon}$, I would understand the log form to be:

$$ (1-\varepsilon)\ln\Pi_t = \ln(1-\theta) + (1-\varepsilon)\ln(P_t^{*}/P_{t-1}) $$

ts_highbury
  • 101
  • Is $\Pi_t^{1-\varepsilon}$ a product (if so, of what?) or a constant? – Jam Mar 28 '20 at 10:41
  • Hi, it is a variable that changes over time t. It is an aggregate price index at time t – ts_highbury Mar 28 '20 at 10:43
  • I don't believe there's any way of simplifying the logarithm of the RHS in general. Ignoring subscripts, we have $\ln(\mathrm{RHS})=\ln\left(\theta+\left(1-\theta\right)\left(\frac{P^*}{P}\right)^{1-\varepsilon}\right)$ but $\log(a+b)$ does not simplify in general. – Jam Mar 28 '20 at 11:26

0 Answers0