Closed-form Solution of the Ak Endogenous Growth Model with Time-Consistent Hyperbolic Discounting

Abstract This paper analyzes the dynamics of the Ak endogenous growth model with CRRA utility and time-consistent hyperbolic discounting. We derive a closed-form solution for the equilibrium paths of consumption and capital over time, showing that the model features transitional dynamics. In contrast, under exponential or time-inconsistent hyperbolic discounting—whether agents are naíve or sophisticated—the economy jumps immediately to its balanced growth path, displaying no transition. We then analytically compare the trajectories of consumption and capital across discounting specifications, when overall impatience is held constant and utility is logarithmic. In particular, long-run growth is higher under time-consistent hyperbolic discounting than under the alternative specifications.