Carburetor engines with an inertia governor sometimes exhibit a self-excited oscillation in rotational speed, called hunting. The mechanism of hunting has been extensively studied for many years. It is our general understanding that the dynamic characteristic of the inertia governor is the primary factor to explain the phenomenon. However, the existence of another kind of hunting, called lean hunting, where lag in air-fuel ratio plays a more important role than the dynamic characteristic of the inertia governor, was reported by Tanaka(1,2). His previous papers, showing various experimental measurements of oscillating engine speed, air-fuel ratio and peak cylinder pressure, explain the nature of the phenomenon. To provide the foregoing work with a theoretical ground, we developed a mathematical formulation for the mechanism of lean hunting. Our new paper demonstrates, by numerical calculations, the process in which engine speed becomes oscillatory reaching a limit cycle, as air-fuel ratio becomes leaner.