<p>Assume that <span class="math-container">$A_{t+1} = (1 + g)A_t$</span>, where <span class="math-container">$g$</span> is constant and <span class="math-container">$g \in \mathbb{R}$</span>.</p>
<p><span class="math-container">\begin{align}
S_t &amp;= I_t, \quad \forall t \tag{1} \\
S_t &amp;= sY_t, \quad s \in (0, 1) \tag{2} \\
K_{t+1} &amp;= (1 - \delta)K_t + I_t, \quad \forall t \tag{3} \\
L_{t+1} &amp;= (1 + n)L_t, \quad \forall t \tag{4} \\
Y_t &amp;= F(K_t, L_t) \tag{5}
\end{align}</span></p>
<p>We have the properties of Neoclassical production function.</p>
<p>Let ((Y_t, K_t, I_t, S_t)\geq 0) be a BGP (Balance Growth Path) if:</p>
<p>(i) They are a solution to the model.</p>
<p>(ii) There exists positive numbers (Y, K, I, S) and (g_Y, g_K, g_I, g_S \in \mathbb{R}) such that
<span class="math-container">$$
Y_t = Y(1 + g_Y)^t, \quad \forall t &gt; 0
$$</span>
<span class="math-container">$$
K_t = K(1 + g_K)^t, \quad \forall t &gt; 0
$$</span>
<span class="math-container">$$
I_t = I(1 + g_I)^t, \quad \forall t &gt; 0
$$</span>
<span class="math-container">$$
S_t = S(1 + g_S)^t, \quad \forall t &gt; 0
$$</span></p>
<p>In other words, there exists a solution that grows at a constant rate, and the set ((Y, K, I, S)) will be called steady-state. Indeed, if the values at ( t = 0 ) of these variables are equal to the steady state, then the solution to the model will remain on the BGP.</p>
<p><span class="math-container">$$
\frac{S_{t+1} - S_t}{S_t} = \frac{I_{t+1} - I_t}{I_t} \implies g_S = g_I \tag{8}
$$</span>
<span class="math-container">$$
\frac{K_{t+1}}{K_t} = (1 - \delta) + \frac{I_t}{K_t} \tag{9}
$$</span>
Since LHS of equation (9) is constant, i.e., equal to ((1+g_K)), RHS must also be constant, hence ( \frac{I_t}{K_t} ) must be constant, i.e.,
<span class="math-container">$$
g_I = g_K \tag{10}
$$</span>
<span class="math-container">$$
g_I = g_K = g_S \tag{11}
$$</span>
<span class="math-container">$$
[S = sY] \land g = g_Y = g_I = g_K = g_S \tag{12}
$$</span>
From this and the relevant equations from the model, we can create the below table (The table environment is not working here, and I am new, I do not know another method to create tables, so I put my LaTeX output screenshot), feel free to ask if you have any further question: <a href="https://i.sstatic.net/ElkdFQZP.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/ElkdFQZP.png" alt="enter image description here" /></a></p>

CC BY-SA 4.0
2024-05-09T07:29:48.297Z
@Nonenicht if you think that the solution is true, accept the answer so other users see it as answered.
bruno
CC BY-SA 4.0
2024-05-08T07:41:28.973Z
2024-05-08T07:57:41.853Z
2024-05-08T07:57:41.853Z
bruno
<p>We say the economy converges to a balanced growth path if all variables are growing at a constant rate. But what does it mean? For example in Solow model with production function <span class="math-container">$F(K_t,A_tL_t)$</span> where <span class="math-container">$A_t$</span> and <span class="math-container">$L_t$</span> are exogenous. We say it follows a balanced growth path if <span class="math-container">$K_t$</span> and <span class="math-container">$Y_t$</span> is growing at constant rate?</p>
<p>And suppose we are considering the Cobb-Douglas production function in RBC model. I read from some notes that the growth rate of wage rate (which is the marginal product of labour here) and consumption should be the same. Using this fact, we can prove KPR preference satisfies balanced growth. But why must the <span class="math-container">$C_t$</span> and <span class="math-container">$w_t$</span> have the same growth rate?</p>

<macroeconomics><solow><real-business-cycles><balanced-growth>
balanced growth path clarification and why wage rate and consumption on BGP have same growth rate
Nonenicht
# balanced growth path clarification and why wage rate and consumption on BGP have same growth rate

(Asked by: username_0 on 2024-05-07)

We say the economy converges to a balanced growth path if all variables are growing at a constant rate. But what does it mean? For example in Solow model with production function $F(K\_t,A\_tL\_t)$ where $A\_t$ and $L\_t$ are exogenous. We say it follows a balanced growth path if $K\_t$ and $Y\_t$ is growing at constant rate?

And suppose we are considering the Cobb-Douglas production function in RBC model. I read from some notes that the growth rate of wage rate (which is the marginal product of labour here) and consumption should be the same. Using this fact, we can prove KPR preference satisfies balanced growth. But why must the $C\_t$ and $w\_t$ have the same growth rate?

**Tags:** macroeconomics, solow, real-business-cycles, balanced-growth

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## 1 answers

### Answer by username_1 on 2024-05-08. Score: 1

Assume that $A\_{t+1} = (1 + g)A\_t$, where $g$ is constant and $g \in \mathbb{R}$.

\begin{align}
S\_t &= I\_t, \quad \forall t \tag{1} \\
S\_t &= sY\_t, \quad s \in (0, 1) \tag{2} \\
K\_{t+1} &= (1 - \delta)K\_t + I\_t, \quad \forall t \tag{3} \\
L\_{t+1} &= (1 + n)L\_t, \quad \forall t \tag{4} \\
Y\_t &= F(K\_t, L\_t) \tag{5}
\end{align}

We have the properties of Neoclassical production function.

Let ((Y\_t, K\_t, I\_t, S\_t)\geq 0) be a BGP (Balance Growth Path) if:

(i) They are a solution to the model.

(ii) There exists positive numbers (Y, K, I, S) and (g\_Y, g\_K, g\_I, g\_S \in \mathbb{R}) such that
$$
Y\_t = Y(1 + g\_Y)^t, \quad \forall t > 0
$$
$$
K\_t = K(1 + g\_K)^t, \quad \forall t > 0
$$
$$
I\_t = I(1 + g\_I)^t, \quad \forall t > 0
$$
$$
S\_t = S(1 + g\_S)^t, \quad \forall t > 0
$$

In other words, there exists a solution that grows at a constant rate, and the set ((Y, K, I, S)) will be called steady-state. Indeed, if the values at ( t = 0 ) of these variables are equal to the steady state, then the solution to the model will remain on the BGP.

$$
\frac{S\_{t+1} - S\_t}{S\_t} = \frac{I\_{t+1} - I\_t}{I\_t} \implies g\_S = g\_I \tag{8}
$$
$$
\frac{K\_{t+1}}{K\_t} = (1 - \delta) + \frac{I\_t}{K\_t} \tag{9}
$$
Since LHS of equation (9) is constant, i.e., equal to ((1+g\_K)), RHS must also be constant, hence ( \frac{I\_t}{K\_t} ) must be constant, i.e.,
$$
g\_I = g\_K \tag{10}
$$
$$
g\_I = g\_K = g\_S \tag{11}
$$
$$
[S = sY] \land g = g\_Y = g\_I = g\_K = g\_S \tag{12}
$$
From this and the relevant equations from the model, we can create the below table (The table environment is not working here, and I am new, I do not know another method to create tables, so I put my LaTeX output screenshot), feel free to ask if you have any further question: [![enter image description here](https://i.sstatic.net/ElkdFQZP.png)](https://i.sstatic.net/ElkdFQZP.png)

> **Comments:**
> **username_1**: @Nonenicht if you think that the solution is true, accept the answer so other users see it as answered.  

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