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I am currently studying the lecture note from Jesus. In page 49, he provides 4 equilibrium conditions:

$$\frac1{c_t} = \beta E_t\left\{\frac1{c_{t+1}}\left(1 + \alpha e^{z_{t+1}}k_{t+1}^{\alpha -1}l_{t+1}^{1-\alpha} - \eta\right)\right\}$$ $$\frac{1-\xi}{1-l_t} = \frac{\xi}{c_t}(1-\alpha) e^{z_t} k_t^{\alpha} l_{t}^{-\alpha}$$ $$c_t + k_{t+1} = e^{z_t}k_t^{\alpha}l_t^{1-\alpha} + (1-\eta) k_t$$ $$z_t = \rho z_{t-1} + \varepsilon_t$$

Then, in page 50, he says that with $\eta=1$, we have $$l_t = l = \frac{(1-\alpha)\xi}{(1-\alpha)\xi + (1-\xi)(1-\alpha\beta)}$$ $$k_{t+1} = \alpha \beta e^{z_t}k_t^{\alpha} l^{1-\alpha}$$

I am not sure how we derive these. With $\eta = 1$, the Euler is reduced to $$\frac1{c_t} = \alpha \beta E_t \left\{\frac{e^{z_{t+1}}k_{t+1}^\alpha l_{t+1}^{1-\alpha} }{k_{t+1} c_{t+1}}\right\}$$ and the budget constraint is reduced to $$c_t + k_{t+1} = e^{z_t}k_t^\alpha l_t^{1-\alpha}$$ I am playing with these two equations, but can't figure it out.

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