## 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

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

(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 πt + 1(i) ≡ ( ...

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