Handbook of the Economics of International Migration

Appendix Solution of the Model in Section 5.1.2

We now prove the claims in Section 5.1.2 by fully solving the vertical innovation model. Combining p(i) = r/α with

$\mathit{p}\left(\mathit{i}\right)=\mathit{\alpha B}{\left(\frac{\mathit{A}\left(\mathit{i})\right)}{\mathit{x}\left(\mathit{i}\right)}\right)}^{1-\mathit{\alpha }}{\left({\mathit{L}}^{\mathit{Y}}\right)}^{\mathit{\beta }}{\mathit{Z}}^{1-\mathit{\alpha }-\mathit{\beta }}\left(=\frac{\partial \mathit{Y}}{\partial \mathit{x}\left(\mathit{i}\right)}\right)$

  (18.67)

and solving for x(i) implies

$\mathit{x}\left(\mathit{i}\right)=\mathit{A}\left(\mathit{i}\right){\left(\mathit{B}\frac{{\mathit{\alpha }}^2}{\mathit{r}}{\left({\mathit{L}}^{\mathit{Y}}\right)}^{\mathit{\beta }}{\mathit{Z}}^{1-\mathit{\alpha }-\mathit{\beta }}\right)}^{\frac{1}{1-\mathit{\alpha }}}$

  (18.68)

Taking wage rate w as given, ex ante of production, producer i chooses R&D labor input in period t to maximize

${\Pi }_{\mathit{t}+1}\left(\mathit{i}\right)\equiv \frac{{\mathit{\pi }}_{\mathit{t}+1}\left(\mathit{i}\right)}{1+\mathit{r}}-{\mathit{w}}_{\mathit{t}}{\mathit{l}}_{\mathit{t}}\left(\mathit{i}\right)-{\mathit{w}}_{\mathit{t}}\mathit{f}$

  (18.69)

where future profits π_t + 1(i) ≡ ( ...