Models with Learning by Doing and Knowledge Spillovers - Barro, Sala-i-Martin (2003) 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:
$$ \frac{\dot{c}}{c} = \left(\frac{1}{\theta} \right) \cdot \left( \underbrace{f(L) - L \cdot f'(L)}_{\phi_0} - \delta - \rho\right) $$
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:
$$ Y_i = A \cdot (K_i)^\alpha \cdot (KL_i)^{1-\alpha}, \quad 0 &lt; \alpha &lt; 1 $$
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} &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*}
Since $0 &lt; \alpha &lt; 1$, 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.
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 Barro 1990 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.