Beta as an exponent?
This is not a structural paper. The authors are not "deriving" a log-linear relationship from a model, which they then proceed to estimate. They rather start by postulat the relationship: 


  Let $C^J_{it}$ be the number of citations made by the patent
  applications Japanese firm i filed in year t to the cumulated stock of
  bindigenousQ U.S.-invented patents granted as of year $t$. I can then
  write the log of $C^J_{it}$ as a simple log-linear function of several
  other observables
  
  $$ c^J_{it} = \beta_0 + \beta_1 p_{it} + \beta_2 FDI_{it} + \gamma_i +
&gt; \alpha_t + \epsilon_{it}$$


which is their equation (1).