Consider an extension of Alesina - Rodrik paper 'Distributive politics and economic growth' (1994).
Instead of wealth taxation, impose a labor income taxation with a slight change in the model specification. Suppose households differ in terms of their innate abilities. In particular, assume that capital holding across households are identical and constant, given by $\bar{k}$, but households can augment their abilities over time by investing in education such that for a household h with some given innate ability $l^h_o$,
$\dot{l^h_t}$ = d$l^h_t$/dt = $w_t$$l^h_t$ + $r_t$$\bar{k}$ - $c^h_t$
My Working: Since capital holding of household doesn't change over time. So, total capital stock, which is summation of each household's capital holding, must be constant for each period, denoted by $\bar{K}$.
There are S firms in the economy and each have access to same production technology, rental rate and wage rate. Thus, each should employ same amount of capital and labor.
Thus, $\bar{K}$ is equally distributed among each firm $\bar{K}$/S = $\hat{k}$
Then labor- capital ratio, for each firm, will be given by -
$l_{it}$/$\hat{k}$ = $l_{jt}$/$\hat{k}$ = $l_t$ = $L_t$/$\bar{K}$
Labor Income Taxation $g_t$= τ$w_t$$L_t$ = $(τA(1-α)\bar{K})^{1/α}$$l_t^{1-α/α}$
$w_t$= ${A(1-α)}^{1/α}$$(τ\bar{K})^{1-α/α}$$L_t^{1-2α/α}$
$r_t$= ${A}^{1/α}$$(τα(1-α)\bar{K})^{1-α/α}$$L_t^{1-α/α}$
Maximizing utility function for the household gave:
Growth in consumption is given by -
${c^h_{t+1} - c^h_t}/c^h_t$ = (1-τ)$w_t$ + (1 - β)/β
Growth in ability is given by -
${l^h_{t+1} - l^h_t}/l^h_t$ = (1-τ)$w_t$ + ${A}^{1/α}$$(τα(1-α)\bar{K})^{1-α/α}$$L_t^{1-2α/α}/σ^h_o$ - $c^h_t/l^h_t$
For growth in ability to be constant $c^h_t/l^h_t$ and $L_t/l^h_t$ should be constant, that is,
${l^h_{t+1} - l^h_t}/l^h_t$ = ${c^h_{t+1} - c^h_t}/c^h_t$
My problem is how to compute growth in balanced growth path? Since growth of both consumption and ability is time varying, how should I proceed to compute a constant growth rate, that is growth to ensure a balanced growth path?
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