The optimum Quantity of Money with Borrowing Constraints

We provide an analytical characterization of the optimal anticipated monetary policy in an economy where agents have a precautionary savings motive due to random production opportunities and the presence of borrowing constraints. Non storable production makes intrinsically useless outside money valuable to insure consumption. We show that the choice of the optimal money growth rate trades off insurance vs. incentives to produce: an expansionary policy provides liquidity to borrowing constrained agents, but distorts production incentives. The joint presence of uncertainty and borrowing constraints implies that the Friedman rule leads to autarkic allocations. If the utility function satisfies Inada conditions then the optimal money growth rate is strictly positive and finite.