The Response of Consumption to Interest Rates with Borrowing Constraints: An Analytical Approach

I derive an explicit mapping from initial assets, income and the real interest rate to consumption for an income fluctuation problem with a borrowing constraint and CARA utility. I show that there exists a threshold of initial wealth over which the partial equilibrium consumption response to a permanent increase in the real interest rate is positive, consistent with recent empirical evidence. I further show that precautionary savings reinforce the possibility of crowding in and that the possibility of a positive response extends to a more standard CRRA utility whenever the elasticity of intertemporal substitution is strictly less than 1.