OLG_SG

OLG model

• Dirk Kruger and Felix Kobler (2004) "Computing Equilibrium in OLG models with stochastic production"
• Agents live for A periods. Over their life cycle, agents born in period $t$, consume $$c_{1,t}, c{2,t+1}, \ldots, c_{A, t+A-1}
• Following Dynare's timing convention, we use stocks on an end of period basis: firms use the stock of capital at the end of last period in the production function. Financial assets are already on an end of period basis in the original paper.
• Because, we are representing the equilibrium of the entire economy at period $t$, the subscript of some variables are different from the original paper.
• Euler equation:

$$u'(c_{i,t}) = \beta \mathbb{E}_t u'(c_{i+1, t+1}r_{t+1})$$

• Agent budget constraint: $c_{i, t} + a_{i, t} = r_t a_{i-1, t-1} + w_t l{i, t}$
• Agents die with zero wealth: $a_{A, t} = 0$
• Representative firm production: $y_t = \eta_tF(k_{t-1}, l_t)$
• Wage rate: $w_t =\eta_t F_l(k_{t-1}, l_t)$
• Net return on capital: $r_t = \eta_t F_k(k_{t-1}, l_t) - \delta_t k_{t-1}$
• Market clearing: $$k_t = \sum^A_{i=1} a_{i,t}\mbox{ and }l_t = \sum_{i=1}^A l_{i,t}$$

Specification

• $u(c_t) = log(c_t)$
• $F(k_{t-1}, l_t) = k_{t-1}^\alpha l_t^ {(1-\alpha)}$
• $l_{1,t} = 1$
• $l_{i, t} = 0$ $for 2 \le i \le A$

Calibration

Parameter	Value
$\alpha$	0.3
$\beta$	0.7

Model

• First generation:
  • $$c_{1,t} + a_{1,t} = w_t$$
  • $$\frac{1}{c_{1,t}} = \beta\mathbb{E}\frac{r_{t+1}}{c_{2, t+1}}$$
• Generations $i = 2$ to $A-1$:
  • $$c_{i,t} + a_{i,t} = r_{t+1} a_{i-1, t-1}$$
  • $$\frac{1}{c_{i,t}} = \beta \mathbb{E} \frac{r_{t+1}}{c_{i+1, t+1}}$$
• Last generation:
  • $$c_{A,t} = r_{t+1} a_{A-1, t-1}$$
• Firm:
  • $$r_t = \eta_t \alpha k_{t-1}^{\alpha-1} - \delta_t k_{t-1}$$
  • $$w_t = \eta_t (\alpha - 1) k_{t-1}^{\alpha}$$
• Market clearing:
  • $$k_t = \sum_{i=1}^{A-1} a_{i,t}$$

Stochastic processes

• The model has two exogenous stochastic processes, $\eta_t$ and $\delta Productivity $\eta_t$ follows a Markov chain
• We need to add a new integration formula to compute the conditional expectation and we need two new options for sparsegridapproximation() function. One for the possible state of each process and one for the transition matrices:
• STATEeta = [0.95, 1.05]
• STATEdelta = [0.5, 0.9]
• Teta = [0.5 0.5; 0.5 0.5]
• Tdelta = [0.5 0.5; 0.5 0.5]
• `sparsegridapproximation(..., MCstates=[STATEeat, STATEdelta], MCtransition = [Teta, Tdelta])
• the order of the stochastic variables must be consistent with the varexodeclaration
• varexi eta delta;