An exact consumption rule with liquidity constraints and stochastic income

This model provides a closed form solution to the problem of liquidity constrained consumption with stochastic income. To keep the model tractable we employ a quadratic utility function. Income follows a geometric Brownian motion. The analytical solution exhibits a smooth, non linear, relation between consumption and income along the optimizing path even when the constraint binds. This outcome confirms the assertions in the literature that even liquidity constrained consumers may satisfy the standard Euler equation. But, in our model this result emerges from the analytical solution.