# Balanced Growth Path Calculation

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 = $$w_tl^h_t$$ + $$r_t\bar{k}$$ - $$c^h_t$$

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$$w_tL_t$$ = $$(τA(1-α)\bar{K})^{1/α}l_t^{1-α/α}$$

$$w_t$$= $${A(1-α)}^{1/α}(τ\bar{K})^{1-α/α}L_t^{1-2α/α}$$

$$r_t$$= $${A}^{1/α}(τα(1-α)\bar{K})^{1-α/α}L_t^{1-α/α}$$

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?

References

Alesina rodrik paper

Lecture notes