<p>I am reading Parente &amp; Prescott 1994 JPE. This is just a typical neoclassical model with the extension of a global technology diffusion function and firm investment in technology.</p>
<p>Firms have technology on production <span class="math-container">$Y_{t}=h_{t} \cdot A_{t} \cdot N_{t} \cdot K_{t}^{\theta_{k}}$</span>  and on investment <span class="math-container">$X_{A t}=\pi \int_{A_{t}}^{A_{t+1}}\left(\frac{S}{W_{t}}\right)^{\alpha} d S$</span>, where <span class="math-container">$X_{At}$</span> is the investment cost of increasing <span class="math-container">$A_t$</span> to <span class="math-container">$A_{t+1}$</span>, and <span class="math-container">$W_t$</span> is the world frontier level of technology which grows at exogenous rate <span class="math-container">$\gamma$</span>.</p>
<p>The authors show that these can be rewritten as  <span class="math-container">$Y_{t}=\mu \cdot h_{t} \cdot(1+\gamma)^{\left(1-\theta_{z}\right) t} \cdot \bar{N} \cdot K_{t}^{\theta_{k}} \cdot Z_{t}^{\theta_{z}}$</span> and <span class="math-container">$Z_{t+1}=\frac{1}{(1+\gamma)^{\left(1-\theta_{z}\right) / \theta_{z}}} \cdot Z_{t}+\frac{1}{\pi} X_{z t}$</span> where <span class="math-container">$Z_{t} \equiv \frac{A_{t}^{\alpha+1}}{W_{0}^{\alpha}(1+\gamma)^{\alpha(t-1)}(1+\alpha)}$</span>,  <span class="math-container">$X_{z t} \equiv X_{A t}$</span>, and <span class="math-container">$\theta_{z} \equiv 1 /(1+\alpha)$</span>.</p>
<p>Firm maximizes PV of its dividends, <span class="math-container">$V\left(Z_{0}\right)=\max \sum_{t=0}^{\infty} p_{t} V_{f t}$</span> where  <span class="math-container">$V_{f t}=Y_{t}-w_{t}\left(h_{t}\right) \bar{N}-r_{k t} K_{t}-X_{z t}$</span>.</p>
<p>The household side is quite textbook:  <span class="math-container">$\max \sum_{t=0}^{\infty} \beta^{t}\left[\ln \left(c_{t}\right)+\phi_{d} \ln \left(d_{t}\right)+\phi_{l} \ln \left(1-h_{t}\right)\right]$</span> except that there is another household type physical capital <span class="math-container">$d_{t}$</span>.</p>
<p>My question here is that while the technology investment <span class="math-container">$X_{zt}$</span> seems to be a choice of the dynamic problem of the firm, the authors construct the competitive equilibrium by recognizing <span class="math-container">$Z_t$</span> as a static optimization choice. In particular, they have</p>
<p><a href="https://i.sstatic.net/QtWIY.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/QtWIY.png" alt="enter image description here" /></a>.</p>
<p>I wonder how comes equation (20) and why we can simply ignore the dynamic problem of choosing <span class="math-container">$X_{zt}$</span>.</p>

CC BY-SA 4.0
2022-02-26T16:19:45.957Z
I am not sure if I understand the question about the equation 20. It gives you definition for r_zt equation 20 is not derived as far as I can see
1muflon1
CC BY-SA 4.0
2022-02-26T16:38:06.160Z
@1muflon1 Yes it is a definition in the model (the authors call it  "implicit rental price") but it should be still derived from somewhere like $r_{kt}$ and $r_{dt}$. For example I think the equation of $r_{kt}$ is from $rk-\tau(r-\delta)k - \delta k =k i$.
Alalalalaki
<macroeconomics><technology><technological-progress>
Neoclassical Growth Model with Technology Investment
Alalalalaki
# Neoclassical Growth Model with Technology Investment

(Asked by: username_0 on 2022-02-26)

I am reading Parente & Prescott 1994 JPE. This is just a typical neoclassical model with the extension of a global technology diffusion function and firm investment in technology.

Firms have technology on production $Y\_{t}=h\_{t} \cdot A\_{t} \cdot N\_{t} \cdot K\_{t}^{\theta\_{k}}$ and on investment $X\_{A t}=\pi \int\_{A\_{t}}^{A\_{t+1}}\left(\frac{S}{W\_{t}}\right)^{\alpha} d S$, where $X\_{At}$ is the investment cost of increasing $A\_t$ to $A\_{t+1}$, and $W\_t$ is the world frontier level of technology which grows at exogenous rate $\gamma$.

The authors show that these can be rewritten as $Y\_{t}=\mu \cdot h\_{t} \cdot(1+\gamma)^{\left(1-\theta\_{z}\right) t} \cdot \bar{N} \cdot K\_{t}^{\theta\_{k}} \cdot Z\_{t}^{\theta\_{z}}$ and $Z\_{t+1}=\frac{1}{(1+\gamma)^{\left(1-\theta\_{z}\right) / \theta\_{z}}} \cdot Z\_{t}+\frac{1}{\pi} X\_{z t}$ where $Z\_{t} \equiv \frac{A\_{t}^{\alpha+1}}{W\_{0}^{\alpha}(1+\gamma)^{\alpha(t-1)}(1+\alpha)}$, $X\_{z t} \equiv X\_{A t}$, and $\theta\_{z} \equiv 1 /(1+\alpha)$.

Firm maximizes PV of its dividends, $V\left(Z\_{0}\right)=\max \sum\_{t=0}^{\infty} p\_{t} V\_{f t}$ where $V\_{f t}=Y\_{t}-w\_{t}\left(h\_{t}\right) \bar{N}-r\_{k t} K\_{t}-X\_{z t}$.

The household side is quite textbook: $\max \sum\_{t=0}^{\infty} \beta^{t}\left[\ln \left(c\_{t}\right)+\phi\_{d} \ln \left(d\_{t}\right)+\phi\_{l} \ln \left(1-h\_{t}\right)\right]$ except that there is another household type physical capital $d\_{t}$.

My question here is that while the technology investment $X\_{zt}$ seems to be a choice of the dynamic problem of the firm, the authors construct the competitive equilibrium by recognizing $Z\_t$ as a static optimization choice. In particular, they have

[![enter image description here](https://i.sstatic.net/QtWIY.png)](https://i.sstatic.net/QtWIY.png).

I wonder how comes equation (20) and why we can simply ignore the dynamic problem of choosing $X\_{zt}$.

**Tags:** macroeconomics, technology, technological-progress
> **Comments:**
> **username_1**: I am not sure if I understand the question about the equation 20. It gives you definition for r_zt equation 20 is not derived as far as I can see  
> **username_0**: @1muflon1 Yes it is a definition in the model (the authors call it  "implicit rental price") but it should be still derived from somewhere like $r_{kt}$ and $r_{dt}$. For example I think the equation of $r_{kt}$ is from $rk-\tau(r-\delta)k - \delta k =k i$.  

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## 0 answers

