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 & 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:
$$ frac{dot{c}}{c} = left(frac{1}{theta} right) cdot left( underbrace{f(L) – L cdot f'(L)}_{phi_0} – delta – rhoright) $$
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:
$$ frac{dot{c}}{c}biggvert_{text{planner}} = left( frac{1}{theta} right) cdot (underbrace{f(L)}_{phi_1} – delta – rho) $$
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 – rhoright) frac{dot{c}}{c}biggvert_{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.