From Pascal Michaillat and Emmanuel Saez:
We construct a microfounded, dynamic version of the IS-LM-Phillips curve model by adding two elements to the money-in-the-utility-function model of Sidrauski . First, real wealth enters the utility function. The resulting Euler equation describes consumption as a decreasing function of the interest rate in steady state—the IS curve. The demand for real money balances describes consumption as an increasing function of the interest rate in steady state—the LM curve. The intersection of the IS and LM curves defines the aggregate demand (AD) curve. Second, matching frictions in the labor market create unemployment. The aggregate supply (AS) curve describes the amount of consumption sold for a given market tightness. Tightness adjusts to equalize AD and AS curve for any price process. With a rigid price process, this steady-state equilibrium captures Keynesian intuitions. Demand and supply shocks affect tightness, unemployment, consumption, and output. Monetary policy affects aggregate demand and can be used for stabilization. Monetary policy is ineffective in a liquidity trap with zero nominal interest rate. In contrast, with a flexible price process, aggregate demand and monetary policy are irrelevant when the nominal interest rate is positive. In a liquidity trap, monetary policy can usefully increase inflation. We discuss equilibrium dynamics under a Phillips curve describing the slow adjustment of prices to their flexible level in the long run.