In this paper, we investigate the bifurcations of a multiplier-accelerator model with nonlinear investment function in an anticyclical fiscal policy rule. First, we give the conditions that the model produces supercritical flip bifurcation and subcritical one, respectively. Second, we prove that the model undergoes a generalized flip bifurcation and present a parameter region such that the model possesses two two-periodic orbits. Third, it is proved that the model undergoes supercritical Neimark–Sacker bifurcation and produces an attracting invariant circle surrounding a fixed point. Fourth, we present the Arnold tongues such that the model has periodic orbits on the invariant circle produced from the Neimark–Sacker bifurcation. Finally, to verify the correctness of our results, we numerically simulate an attracting two-periodic orbit, a stable invariant circle, an Arnold tongue with rotation number 1/7 and an attracting seven-periodic orbit on the invariant circle.