<p>The household side of this model is pretty standard. Denote $K(t)$ be household's assets at time $t$. Then the transversality condition (which is an optimizing condition, not a constraint), is </p>

<p>$$\lim_{t \to \infty} [e^{-\rho t}\lambda(t) K(t)] = 0$$</p>

<p>where $\lambda(t)$ is the current value multiplier on assets in the Hamiltonian. Given the assumed form of the utility function we have the f.o.c.</p>

<p>$$\lambda(t) = \frac {1}{c(t)^{\theta}}$$</p>

<p>So we want </p>

<p>$$\lim_{t \to \infty} [e^{-\rho t}\frac {1}{c(t)^{\theta}} K(t)] = 0$$</p>

<p>and since in the model we have a constant consumption growth rate, $g_c$ we have</p>

<p>$$c(t)^{\theta} = c(0)^{\theta}\cdot e^{g_c\theta t}$$</p>

<p>Ignoring inconsequential constants, we require</p>

<p>$$\lim_{t \to \infty} [\frac {1}{e^{(\rho+g_c\theta) t}} K(t)] = 0$$</p>

<p>Use the relation between household assets  and the number of goods $N$, the relation between income (from labor and assets) to aggregate output, as well as the expression of aggregate output as a function of $N$, to obtain a first-order linear differential equation in $N$, and proceed  as in pages 208-209 of the book.</p>

CC BY-SA 3.0
2017-03-16T00:08:09.710Z
2017-03-16T00:08:09.710Z
Alecos Papadopoulos
<p>I'm studying the Lab Equipment Model (Barro-Sala-i-Martin, Chapter 6). I'm having trouble when trying to prove that every variables grows at the same rate as consumption. </p>

<p>I was able to prove that $\dot{C}/C = g_{c}$, which is constant. Also, I have that</p>

<p>$b \dot{N} = Y - X - C$ where $b$ is a parameter,</p>

<p>$Y/N$ is a constant related to the interest rate and</p>

<p>$X = a^{2}Y$, where $a$ is a parameter from the production function.</p>

<p>From the second and third equations above, it's clear that $Y,X$ and $N$ grow at the same rate. However, it's not clear that this rate should be $g_{c}$ or even constant at all. I'm trying to use the first equation to show it, but I couldn't do it.</p>

<p>The book forgoes this proof. It says that it's quite similar to the AK model since there's no transitional dynamics. Is there a simple way to show this result? I think it should follow from a simple manipulation of the first equation. Any ideas? thanks a lot in advance!</p>

CC BY-SA 3.0
2017-03-12T22:10:57.193Z
To what book are you referring?
Alecos Papadopoulos
CC BY-SA 3.0
2017-03-12T22:35:10.870Z
@AlecosPapadopoulos it's this one: https://www.amazon.com/Economic-Growth-Barro-Sala-I-Martin/dp/8120325516
Raul Guarini Riva
CC BY-SA 3.0
2017-03-12T23:15:51.660Z
In their footnote 15 the authors say that a proof that no other equilibrium exists than where all growth rates are equal and constant, can be constructed "along the lines of chapter 4". They refer to pages 208-209.
Alecos Papadopoulos
CC BY-SA 3.0
2017-03-14T02:06:35.223Z
@AlecosPapadopoulos I took a serious look on your references and got the feeling of it. But on pages 208 and 209 the author strongly uses the transversality condition. My problem is that I cannot establish the correct TC here or order to replicate the proof. Could you please provide more details?
Raul Guarini Riva
<macroeconomics><economic-growth><consumption><endogenous-growth><proof-reference>
How to prove that GDP grows at the same rate as consumption on Lab Equipment Model?
Raul Guarini Riva
# How to prove that GDP grows at the same rate as consumption on Lab Equipment Model?

(Asked by: username_0 on 2017-03-11)

I'm studying the Lab Equipment Model (Barro-Sala-i-Martin, Chapter 6). I'm having trouble when trying to prove that every variables grows at the same rate as consumption.

I was able to prove that $\dot{C}/C = g\_{c}$, which is constant. Also, I have that

$b \dot{N} = Y - X - C$ where $b$ is a parameter,

$Y/N$ is a constant related to the interest rate and

$X = a^{2}Y$, where $a$ is a parameter from the production function.

From the second and third equations above, it's clear that $Y,X$ and $N$ grow at the same rate. However, it's not clear that this rate should be $g\_{c}$ or even constant at all. I'm trying to use the first equation to show it, but I couldn't do it.

The book forgoes this proof. It says that it's quite similar to the AK model since there's no transitional dynamics. Is there a simple way to show this result? I think it should follow from a simple manipulation of the first equation. Any ideas? thanks a lot in advance!

**Tags:** macroeconomics, economic-growth, consumption, endogenous-growth, proof-reference
> **Comments:**
> **username_1**: To what book are you referring?  
> **username_0**: @AlecosPapadopoulos it's this one: https://www.amazon.com/Economic-Growth-Barro-Sala-I-Martin/dp/8120325516  
> **username_1**: In their footnote 15 the authors say that a proof that no other equilibrium exists than where all growth rates are equal and constant, can be constructed "along the lines of chapter 4". They refer to pages 208-209.  
> **username_0**: @AlecosPapadopoulos I took a serious look on your references and got the feeling of it. But on pages 208 and 209 the author strongly uses the transversality condition. My problem is that I cannot establish the correct TC here or order to replicate the proof. Could you please provide more details?  

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## 1 answers

### Answer by username_1 on 2017-03-16. Score: 1

The household side of this model is pretty standard. Denote $K(t)$ be household's assets at time $t$. Then the transversality condition (which is an optimizing condition, not a constraint), is

$$\lim\_{t \to \infty} [e^{-\rho t}\lambda(t) K(t)] = 0$$

where $\lambda(t)$ is the current value multiplier on assets in the Hamiltonian. Given the assumed form of the utility function we have the f.o.c.

$$\lambda(t) = \frac {1}{c(t)^{\theta}}$$

So we want

$$\lim\_{t \to \infty} [e^{-\rho t}\frac {1}{c(t)^{\theta}} K(t)] = 0$$

and since in the model we have a constant consumption growth rate, $g\_c$ we have

$$c(t)^{\theta} = c(0)^{\theta}\cdot e^{g\_c\theta t}$$

Ignoring inconsequential constants, we require

$$\lim\_{t \to \infty} [\frac {1}{e^{(\rho+g\_c\theta) t}} K(t)] = 0$$

Use the relation between household assets and the number of goods $N$, the relation between income (from labor and assets) to aggregate output, as well as the expression of aggregate output as a function of $N$, to obtain a first-order linear differential equation in $N$, and proceed as in pages 208-209 of the book.


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