<p>We can show this by adding some public good to the model that will be financed by lump-sum taxes (which is also discussed in Barro &amp; Sala-i-Martin (2004). Economic Growth 2nd ed. ch 4.4.1). So suppose Cobb-Douglas is given like in <a href="https://dash.harvard.edu/bitstream/handle/1/3451296/Barro_GovernmentSpending.pdf?sequence=4" rel="nofollow noreferrer">Barro 1990</a> as:</p>
<p><span class="math-container">$$Y_i=AL_i^{1-\alpha} K_i^{\alpha}G^{1-\alpha} \tag{1}$$</span></p>
<p>Now for any given <span class="math-container">$G$</span> profit maximizing firms will equate the marginal product of capital to the rental price <span class="math-container">$r+\delta$</span> and this will give us:</p>
<p><span class="math-container">$$\alpha A k_i ^{ -(1-\alpha)}G^{1-\alpha}= r+\delta. \tag{2}$$</span></p>
<p>Since firms are homogenous they will all choose some optimal <span class="math-container">$k_i=k$</span> and hence we get:</p>
<p><span class="math-container">$$Y= AL^{1-\alpha}K^{\alpha}G^{1-\alpha} \implies G = \left(\frac{G}{Y}\right)^{\frac{1}{\alpha}} (AL)^{\frac{1}{\alpha}}k \tag{3}$$</span></p>
<p>Now we have to assume government will pick some constant <span class="math-container">$G/Y$</span> and using this combining (3) and (4) gives us:</p>
<p><span class="math-container">$$\alpha A^{(1/\alpha)}(G/Y)^{(1-\alpha)/\alpha}L^{(1-\alpha)/\alpha}   = r+ \delta \tag{4}$$</span></p>
<p>Now because <span class="math-container">$G/Y$</span> and <span class="math-container">$L$</span> are constant also marginal product of capital will be constant with respect to time. As a result:</p>
<p><span class="math-container">$$\frac{\dot{c}}{c} = \frac{1}{\theta} \left( \alpha A^{(1/\alpha)}(G/Y)^{(1-\alpha)/\alpha}L^{(1-\alpha)/\alpha}   -  \delta - \rho \right) \tag{5} $$</span></p>
<p>Now the above also happens to be optimum chosen by a benevolent social planner in this case because the planner would choose <span class="math-container">$c$</span>, <span class="math-container">$k$</span> and <span class="math-container">$G$</span> to max:</p>
<p><span class="math-container">$$\int^\infty_0 e^{-\rho t}\frac{c^{1-\theta}-1}{1-\theta} dt \tag{6}$$</span></p>
<p>which is the household's utility, subject to the constraint:</p>
<p><span class="math-container">$$ \dot{k} = A K^{\alpha}G^{1-\alpha}-c-\delta k - G/L \tag{7}$$</span></p>
<p>we combine (6) and (7) by setting Hamiltonian:</p>
<p><span class="math-container">$$H =  e^{-\rho t}\frac{c^{1-\theta}-1}{1-\theta} + v\left(  A K^{\alpha}G^{1-\alpha}-c-\delta k - G/L \right) \tag{8}$$</span></p>
<p>which will give us the following FOC's:</p>
<p><span class="math-container">$$ e^{-\rho t} c^{-\theta} = v \tag{9}$$</span></p>
<p><span class="math-container">$$  A(1-\alpha)k^{\alpha}G^{-\alpha} = \frac{1}{L} \implies \partial Y/ \partial G =1 \tag{10} $$</span></p>
<p><span class="math-container">$$ - \dot{v} = v \left(A \alpha K^{\alpha-1}G^{ 1-\alpha} - \delta \right) \tag{11}$$</span></p>
<p>and we also have to impose the transversality condition.</p>
<p>Now actually the FOC given in equation (10) by implying that <span class="math-container">$ \implies \partial Y/ \partial G =1$</span> tells us that in the optimum <span class="math-container">$G/Y=1-\alpha$</span> (which is where the value of investment tax credit comes from).</p>
<p>Finally we find that when <span class="math-container">$G/Y=1-\alpha$</span> social planner would choose:</p>
<p><span class="math-container">$$\frac{\dot{c}}{c} \bigg\vert_{\text{social planner}}= \frac{1}{\theta} \left( \alpha A^{(1/\alpha)}(G/Y)^{(1-\alpha)/\alpha}L^{(1-\alpha)/\alpha}   -  \delta - \rho \right) \tag{12} $$</span></p>
<p>which is exactly the same as a decentralized equilibrium given by 5. However, the assumption of lump-sum taxes is important for this result and generally using some distortionary tax we won't get the same result.</p>

CC BY-SA 4.0
2020-10-03T10:08:44.010Z
I really appreciate the answer, mate, but still have a couple of questions, if you don't mind:
1) Shouldn't $(3)$ be given by $G = \left( \frac{G}{Y} \right)^{\frac{1}{\alpha}} \cdot (AL)^{\frac{1}{\alpha}} \cdot K$, where $K = k \cdot L$?;
2) To clarify, the investment tax works by making it so that the optimal value chosen by the the social planner is equal to the decentralized choice and not the other way around, right?
3) What about the case of the subsidy? How would it enter the production function? In this case I assume it would change the decentralized solution.

Thanks.
Pedro Cunha
CC BY-SA 4.0
2020-10-03T10:17:58.643Z
1. the $k$ is capital -labor ratio (sorry I did not made that clear). 2. no the social planner does not care what decentralized eq. is social planner wants to maximize the lifetime household utility, in this case we merely prove that social planer (when there is tax credit present) will chose the same equilibrium as would happen in the decentralzied equilibrium (but tax credit included). 3. But G is a subsidy - its tax credit (i.e. subsidy) funded by lump sum tax. So it enters prod. F. through G
1muflon1
CC BY-SA 4.0
2020-10-03T10:26:50.740Z
It think it was clear that $k = \frac{K}{L}$, but in order for $(3)$ to be true in the way that you wrote:
\begin{align*} G = \left( \frac{G}{Y} \right)^{\frac{1}{\alpha}} \cdot (AL)^{\frac{1}{\alpha}} \cdot \frac{K}{L} \implies &Y^\frac{1}{\alpha} = G^{\frac{1}{\alpha}-1} \cdot A^\frac{1}{\alpha} \cdot L^{\frac{1}{\alpha}-1} \cdot K  \\ &Y = G^{1-\alpha} \cdot A \cdot L^{1-\alpha} \cdot K^\alpha \end{align*}
which differs from the original equation you proposed. Am I missing something? Thanks again!
Pedro Cunha
CC BY-SA 4.0
2020-10-03T10:30:11.200Z
Perhaps your original equation in $(3)$ was supposed to be $$ Y = A \cdot L^{1-\alpha} \cdot K^{\alpha} \cdot G^{1-\alpha} $$ and not $$ Y = A \cdot L \cdot K^{\alpha} \cdot G^{1-\alpha} $$?
Pedro Cunha
CC BY-SA 4.0
2020-10-03T12:43:33.520Z
@PedroCunha you are actually right I corrected it
1muflon1
CC BY-SA 4.0
2020-10-02T22:16:19.380Z
2020-10-03T12:44:09.360Z
2020-10-03T12:44:09.360Z
1muflon1
<p>Consider the model of endogenous growth with learning by doing and knowledge spillovers presented in Barro &amp; Sala-i-Martin (2003), chapter 4, section 4.3, starting in page 212. In equilibrium, the growth rate of consumption in the decentralized economy is given by:
<span class="math-container">$$ \frac{\dot{c}}{c} = \left(\frac{1}{\theta} \right) \cdot \left( \underbrace{f(L) - L \cdot f'(L)}_{\phi_0} - \delta - \rho\right) $$</span>
where <span class="math-container">$c$</span> is per capita consumption, <span class="math-container">$\theta$</span> is the inverse of intertemporal elasticity of substitution, <span class="math-container">$L$</span> is the size of the labor force (<span class="math-container">$\frac{\dot{L}}{L} = n$</span> is assumed to be equal to <span class="math-container">$0$</span>), <span class="math-container">$\delta$</span> is the depreciation rate, <span class="math-container">$\rho$</span> is the subjective discount rate and <span class="math-container">$\phi_0$</span> is the marginal product of the capital.</p>
<p>If we now consider a central planner, the following expression from the growth rate of <span class="math-container">$c$</span> can be obtained:
<span class="math-container">$$ \frac{\dot{c}}{c}\bigg\vert_{\text{planner}} = \left( \frac{1}{\theta} \right) \cdot (\underbrace{f(L)}_{\phi_1} - \delta - \rho) $$</span>
where <span class="math-container">$\phi_1 = f(L)$</span> is the average product of the capital.</p>
<p>The authors then go on to present an example using a Cobb-Douglas production given by:
<span class="math-container">$$ Y_i = A \cdot (K_i)^\alpha \cdot (KL_i)^{1-\alpha}, \quad 0 &lt; \alpha &lt; 1 $$</span>
where the index <span class="math-container">$i$</span> corresponds to the firm <span class="math-container">$i$</span>, i.e., <span class="math-container">$Y_i$</span> is the output of firm <span class="math-container">$i$</span>.</p>
<p>Under the assumptions of the model, if we substitute <span class="math-container">$y_i = \frac{Y_i}{L_i}, k_i = \frac{K_i}{L_i}$</span> and <span class="math-container">$k = \frac{K}{L}$</span>, and then set <span class="math-container">$y_i = y$</span> and <span class="math-container">$k_i = k$</span>, the average product of the capital is:
<span class="math-container">$$ \frac{y}{k} = f(L) = A \cdot L^{1-\alpha} $$</span>
and the marginal product of capital is
<span class="math-container">$$ f(L) - L \cdot f'(L) = A \cdot \alpha \cdot L^{1-\alpha} $$</span>
and, therefore,
<span class="math-container">\begin{align*} \frac{\dot{c}}{c} &amp;= \left(\frac{1}{\theta} \right) \cdot \left(A \cdot \alpha \cdot L^{1-\alpha} - \delta - \rho\right) \\ \frac{\dot{c}}{c}\bigg\vert_{\text{planner}} &amp;= \left( \frac{1}{\theta} \right) \cdot (A \cdot L^{1-\alpha} - \delta - \rho) \end{align*}</span>
Since <span class="math-container">$0 &lt; \alpha &lt; 1$</span>, the growth rate of the decentralized economy is lower than that of the economy with a central planner.</p>
<p>Continuing, the authors write the following and I quote:</p>
<blockquote>
<p>The social optimum can be attained in the decentralized economy by introducing an
investment-tax credit at the rate <span class="math-container">$(1-\alpha)$</span> and financing it with a lump-sum tax. If buyers
of capital pay only the fraction <span class="math-container">$\alpha$</span> of the cost, the private return on capital corresponds
to the social return. We can then show that the decentralized choices coincide with those
of the social planner. Alternatively, the government could generate the same outcome by
subsidizing production at the rate <span class="math-container">$ \frac{(1-\alpha)}{\alpha} $</span></p>
</blockquote>
<p>I can't understand how the tax and subsidy could make the decentralized economy achieve a socially optimum level of growth rate. Intuitively, I can see how that might work but I cannot see how the math behind it would work in this case.</p>
<p>If anyone could explain that to me or point me in the right direction I'd appreciate it. Let me know if I need to add more details to the question.</p>

<economic-growth><endogenous-growth>
Models with Learning by Doing and Knowledge Spillovers - Barro, Sala-i-Martin (2003)
Pedro Cunha
# Models with Learning by Doing and Knowledge Spillovers - Barro, Sala-i-Martin (2003)

(Asked by: username_0 on 2020-10-02)

Consider the model of endogenous growth with learning by doing and knowledge spillovers presented in Barro & Sala-i-Martin (2003), chapter 4, section 4.3, starting in page 212. In equilibrium, the growth rate of consumption in the decentralized economy is given by:
$$ \frac{\dot{c}}{c} = \left(\frac{1}{\theta} \right) \cdot \left( \underbrace{f(L) - L \cdot f'(L)}\_{\phi\_0} - \delta - \rho\right) $$
where $c$ is per capita consumption, $\theta$ is the inverse of intertemporal elasticity of substitution, $L$ is the size of the labor force ($\frac{\dot{L}}{L} = n$ is assumed to be equal to $0$), $\delta$ is the depreciation rate, $\rho$ is the subjective discount rate and $\phi\_0$ is the marginal product of the capital.

If we now consider a central planner, the following expression from the growth rate of $c$ can be obtained:
$$ \frac{\dot{c}}{c}\bigg\vert\_{\text{planner}} = \left( \frac{1}{\theta} \right) \cdot (\underbrace{f(L)}\_{\phi\_1} - \delta - \rho) $$
where $\phi\_1 = f(L)$ is the average product of the capital.

The authors then go on to present an example using a Cobb-Douglas production given by:
$$ Y\_i = A \cdot (K\_i)^\alpha \cdot (KL\_i)^{1-\alpha}, \quad 0 < \alpha < 1 $$
where the index $i$ corresponds to the firm $i$, i.e., $Y\_i$ is the output of firm $i$.

Under the assumptions of the model, if we substitute $y\_i = \frac{Y\_i}{L\_i}, k\_i = \frac{K\_i}{L\_i}$ and $k = \frac{K}{L}$, and then set $y\_i = y$ and $k\_i = k$, the average product of the capital is:
$$ \frac{y}{k} = f(L) = A \cdot L^{1-\alpha} $$
and the marginal product of capital is
$$ f(L) - L \cdot f'(L) = A \cdot \alpha \cdot L^{1-\alpha} $$
and, therefore,
\begin{align\*} \frac{\dot{c}}{c} &= \left(\frac{1}{\theta} \right) \cdot \left(A \cdot \alpha \cdot L^{1-\alpha} - \delta - \rho\right) \\ \frac{\dot{c}}{c}\bigg\vert\_{\text{planner}} &= \left( \frac{1}{\theta} \right) \cdot (A \cdot L^{1-\alpha} - \delta - \rho) \end{align\*}
Since $0 < \alpha < 1$, the growth rate of the decentralized economy is lower than that of the economy with a central planner.

Continuing, the authors write the following and I quote:

> The social optimum can be attained in the decentralized economy by introducing an
> investment-tax credit at the rate $(1-\alpha)$ and financing it with a lump-sum tax. If buyers
> of capital pay only the fraction $\alpha$ of the cost, the private return on capital corresponds
> to the social return. We can then show that the decentralized choices coincide with those
> of the social planner. Alternatively, the government could generate the same outcome by
> subsidizing production at the rate $ \frac{(1-\alpha)}{\alpha} $

I can't understand how the tax and subsidy could make the decentralized economy achieve a socially optimum level of growth rate. Intuitively, I can see how that might work but I cannot see how the math behind it would work in this case.

If anyone could explain that to me or point me in the right direction I'd appreciate it. Let me know if I need to add more details to the question.

**Tags:** economic-growth, endogenous-growth

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## 1 answers

### Answer by username_1 on 2020-10-02. Score: 2 | Accepted answer

We can show this by adding some public good to the model that will be financed by lump-sum taxes (which is also discussed in Barro & Sala-i-Martin (2004). Economic Growth 2nd ed. ch 4.4.1). So suppose Cobb-Douglas is given like in [Barro 1990](https://dash.harvard.edu/bitstream/handle/1/3451296/Barro_GovernmentSpending.pdf?sequence=4) as:

$$Y\_i=AL\_i^{1-\alpha} K\_i^{\alpha}G^{1-\alpha} \tag{1}$$

Now for any given $G$ profit maximizing firms will equate the marginal product of capital to the rental price $r+\delta$ and this will give us:

$$\alpha A k\_i ^{ -(1-\alpha)}G^{1-\alpha}= r+\delta. \tag{2}$$

Since firms are homogenous they will all choose some optimal $k\_i=k$ and hence we get:

$$Y= AL^{1-\alpha}K^{\alpha}G^{1-\alpha} \implies G = \left(\frac{G}{Y}\right)^{\frac{1}{\alpha}} (AL)^{\frac{1}{\alpha}}k \tag{3}$$

Now we have to assume government will pick some constant $G/Y$ and using this combining (3) and (4) gives us:

$$\alpha A^{(1/\alpha)}(G/Y)^{(1-\alpha)/\alpha}L^{(1-\alpha)/\alpha} = r+ \delta \tag{4}$$

Now because $G/Y$ and $L$ are constant also marginal product of capital will be constant with respect to time. As a result:

$$\frac{\dot{c}}{c} = \frac{1}{\theta} \left( \alpha A^{(1/\alpha)}(G/Y)^{(1-\alpha)/\alpha}L^{(1-\alpha)/\alpha} - \delta - \rho \right) \tag{5} $$

Now the above also happens to be optimum chosen by a benevolent social planner in this case because the planner would choose $c$, $k$ and $G$ to max:

$$\int^\infty\_0 e^{-\rho t}\frac{c^{1-\theta}-1}{1-\theta} dt \tag{6}$$

which is the household's utility, subject to the constraint:

$$ \dot{k} = A K^{\alpha}G^{1-\alpha}-c-\delta k - G/L \tag{7}$$

we combine (6) and (7) by setting Hamiltonian:

$$H = e^{-\rho t}\frac{c^{1-\theta}-1}{1-\theta} + v\left( A K^{\alpha}G^{1-\alpha}-c-\delta k - G/L \right) \tag{8}$$

which will give us the following FOC's:

$$ e^{-\rho t} c^{-\theta} = v \tag{9}$$

$$ A(1-\alpha)k^{\alpha}G^{-\alpha} = \frac{1}{L} \implies \partial Y/ \partial G =1 \tag{10} $$

$$ - \dot{v} = v \left(A \alpha K^{\alpha-1}G^{ 1-\alpha} - \delta \right) \tag{11}$$

and we also have to impose the transversality condition.

Now actually the FOC given in equation (10) by implying that $ \implies \partial Y/ \partial G =1$ tells us that in the optimum $G/Y=1-\alpha$ (which is where the value of investment tax credit comes from).

Finally we find that when $G/Y=1-\alpha$ social planner would choose:

$$\frac{\dot{c}}{c} \bigg\vert\_{\text{social planner}}= \frac{1}{\theta} \left( \alpha A^{(1/\alpha)}(G/Y)^{(1-\alpha)/\alpha}L^{(1-\alpha)/\alpha} - \delta - \rho \right) \tag{12} $$

which is exactly the same as a decentralized equilibrium given by 5. However, the assumption of lump-sum taxes is important for this result and generally using some distortionary tax we won't get the same result.

> **Comments:**
> **username_0**: I really appreciate the answer, mate, but still have a couple of questions, if you don't mind: 1) Shouldn't $(3)$ be given by $G = \left( \frac{G}{Y} \right)^{\frac{1}{\alpha}} \cdot (AL)^{\frac{1}{\alpha}} \cdot K$, where $K = k \cdot L$?; 2) To clarify, the investment tax works by making it so that the optimal value chosen by the the social planner is equal to the decentralized choice and not the other way around, right? 3) What about the case of the subsidy? How would it enter the production function? In this case I assume it would change the decentralized solution.  Thanks.  
> **username_1**: 1. the $k$ is capital -labor ratio (sorry I did not made that clear). 2. no the social planner does not care what decentralized eq. is social planner wants to maximize the lifetime household utility, in this case we merely prove that social planer (when there is tax credit present) will chose the same equilibrium as would happen in the decentralzied equilibrium (but tax credit included). 3. But G is a subsidy - its tax credit (i.e. subsidy) funded by lump sum tax. So it enters prod. F. through G  
> **username_0**: It think it was clear that $k = \frac{K}{L}$, but in order for $(3)$ to be true in the way that you wrote: \begin{align*} G = \left( \frac{G}{Y} \right)^{\frac{1}{\alpha}} \cdot (AL)^{\frac{1}{\alpha}} \cdot \frac{K}{L} \implies &Y^\frac{1}{\alpha} = G^{\frac{1}{\alpha}-1} \cdot A^\frac{1}{\alpha} \cdot L^{\frac{1}{\alpha}-1} \cdot K  \\ &Y = G^{1-\alpha} \cdot A \cdot L^{1-\alpha} \cdot K^\alpha \end{align*} which differs from the original equation you proposed. Am I missing something? Thanks again!  
> **username_0**: Perhaps your original equation in $(3)$ was supposed to be $$ Y = A \cdot L^{1-\alpha} \cdot K^{\alpha} \cdot G^{1-\alpha} $$ and not $$ Y = A \cdot L \cdot K^{\alpha} \cdot G^{1-\alpha} $$?  
> **username_1**: @PedroCunha you are actually right I corrected it  

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