<p>In the paper of Mankiw, Romer and Weil (1992: 422-423) aka MRW there is a transfer from theoretical dynamical model to empirical regression approach as follows.</p>
<blockquote>
<p>The model suggest a natural regression to study the rate of convergence. Equation (13)
<span class="math-container">$$ \frac{d \log y(t)}{dt} =  \lambda [\log y^* - \log y(t)]$$</span>
implies that (14)
<span class="math-container">$$\log y(t) = (1 - e^{-\lambda t}) \log y^* + e^{-\lambda t} \log y(0)$$</span></p>
</blockquote>
<p>In the above expression <span class="math-container">$y^*$</span> is a constant steady state level of production and <span class="math-container">$y(0)$</span> is some initial level. This can also be expressed as:</p>
<p><span class="math-container">$$
\log y(t) - \log y(0) = (1 - e^{-\lambda t}) [\log y^* - \log y(0)]
$$</span></p>
<p>What are intermediate steps between the (13) and (14)? How does <span class="math-container">$y(0)$</span> appear in the model?</p>
<p>PS. This is, perhaps, something simple but a tip is needed.
Direct solution for <span class="math-container">$y$</span> returns
<span class="math-container">$$
y = y^* \cdot e^{-e^{-\lambda \cdot t - C_1}}
$$</span></p>

CC BY-SA 4.0
2021-12-16T23:34:18.043Z
Hint:  13 is ODE, start by taking the integral of both sides
1muflon1
CC BY-SA 4.0
2021-12-17T07:36:03.133Z
@1muflon1, I've got a solution in general form (below - last line). Though I'm not quite sure how do we convert $C_1$ into $log(y(0))$.
garej
CC BY-SA 4.0
2021-12-17T08:52:20.137Z
In ODE you can always find out what C is by solving the problem for initial value that is why you have $y_0$ there
1muflon1
CC BY-SA 4.0
2021-12-17T09:54:29.563Z
@1muflon1, thank you, I've figured out finally that they differentiated for $\log y$, not for $y$ (shame on me). Hence, $C_1$ is substituted with $(\log y_0 - \log y^*)$. Is such choice of constant just for convenience?
garej
<economic-growth><regression>
Natural regression to study rate of convergence from MRW (1992)
garej
# Natural regression to study rate of convergence from MRW (1992)

(Asked by: username_0 on 2021-12-16)

In the paper of Mankiw, Romer and Weil (1992: 422-423) aka MRW there is a transfer from theoretical dynamical model to empirical regression approach as follows.

> The model suggest a natural regression to study the rate of convergence. Equation (13)
> $$ \frac{d \log y(t)}{dt} = \lambda [\log y^\* - \log y(t)]$$
> implies that (14)
> $$\log y(t) = (1 - e^{-\lambda t}) \log y^\* + e^{-\lambda t} \log y(0)$$

In the above expression $y^\*$ is a constant steady state level of production and $y(0)$ is some initial level. This can also be expressed as:

$$
\log y(t) - \log y(0) = (1 - e^{-\lambda t}) [\log y^\* - \log y(0)]
$$

What are intermediate steps between the (13) and (14)? How does $y(0)$ appear in the model?

PS. This is, perhaps, something simple but a tip is needed.
Direct solution for $y$ returns
$$
y = y^\* \cdot e^{-e^{-\lambda \cdot t - C\_1}}
$$

**Tags:** economic-growth, regression

> **Comments:**
> **username_1**: Hint:  13 is ODE, start by taking the integral of both sides  
> **username_0**: @username_1, I've got a solution in general form (below - last line). Though I'm not quite sure how do we convert $C_1$ into $log(y(0))$.  
> **username_1**: In ODE you can always find out what C is by solving the problem for initial value that is why you have $y_0$ there  
> **username_0**: @username_1, thank you, I've figured out finally that they differentiated for $\log y$, not for $y$ (shame on me). Hence, $C_1$ is substituted with $(\log y_0 - \log y^*)$. Is such choice of constant just for convenience?  

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## 0 answers

