This paper develops a search-theoretic model to study the interaction between banking and monetary policy and how this interaction affects the allocation and welfare. Regarding how banking affects the welfare costs of inflation: First, we find that, with banking, inflation generates smaller welfare costs. Second, we show that, lowering inflation improves welfare not just by reducing consumption/production distortions, but also by avoiding intermediation costs. Therefore, understanding the nature of intermediation cost is critical for accurately assessing the welfare gain of lowering the inflation target. Regarding how monetary policy affects the welfare effects of banking: First, banking always improves efficiency of production, but the banking technology has to be efficient to improve welfare (especially in low inflation economy). Second, welfare effects of banking depend on monetary policy. For low inflation, banking is not active. For high inflation, banking is active and improves welfare. For moderate inflation, banking is active but reduces welfare. Owing to general equilibrium feedback, banking is supported in equilibrium even though welfare is higher without banking.