Stochastic Growth with Discrete Choice

Production:

$$ Y_t = A_t K_t^\alpha L_t^{1-\alpha}. $$

TFP envolution:

$$ \log A_{t+1} = \rho \log A_t + \epsilon_{t+1},\\ \epsilon_{t}\sim\mathcal{N}(0, \sigma^2). $$

Period utility:

$$ u_t = \frac{c_t^{1-\gamma}}{1- \gamma} + B\frac{(\bar{L} - L_t)^{1-\eta}}{1- \eta}. $$

Maximization objective:

$$ \sum\limits_{t=0}^\infty \beta^t u_t. $$

Budget constraint:

$$ c_t + K_{t+1} = Y_t + (1-\delta)K_t. $$

First-order condition:

$$ c_t^{-\gamma} = \beta\mathbb{E}t\Bigg{ c{t+1}^{-\gamma} \Big[\alpha A_{t+1}(\frac{L_{t+1}}{K_{t+1}})^{1-\alpha} + 1 - \delta\Big] \Bigg},\ c_t^{-\gamma} (1-\alpha) A_t (\frac{K_t}{L_t})^{\alpha}= B(\bar{L}-L_t)^{-\eta}, \text{if L is continuous.} $$

Steady state:

$$ Y=K^{\alpha}L^{1-\alpha}\\ c+\delta K = Y\\ \alpha (\frac{L}{K})^{1-\alpha} + 1 - \delta = \frac{1}{\beta}\\ c^{-\gamma} (1-\alpha) (\frac{K}{L})^{\alpha}= B(\bar{L}-L)^{-\eta} $$