Models with Learning by Doing and Knowledge Spillovers - Barro, Sala-i-Martin (2003)
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.