We investigate the individual optimal consumption and investment problem under the timevarying liquidity constraint (TVLC), which is motivated by the U.S. data. When the agent has a constant relative risk aversion utility function, we provide the closed-form solution by a duality approach and characterize the optimal policies. The implications of the stochastic borrowing constraints for the optimal policies and implied wealth dynamics dier considerably from what has been shown in the previous literature on the xed borrowing limit such as the non-negative wealth constraint. We nd that the investment in the risky asset can be greater than the investment without any borrowing constraints due to the eect of the liquidity hedging demand. Remarkably, a risk-averse agent can rationally exhibit a locally risk-loving behavior when his/her wealth level approaches the borrowing limit. Moreover, as the income level increases, the impact of TVLC on the risky investment is not negligible, and in fact, becomes larger, even though the agent's credit limit under TVLC is higher for a higher level of income.