Algorithm in the AI and Digital Economy

As artificial intelligence becomes a central engine of modern economic growth, standard macroeconomic frameworks must evolve to conceptualize algorithms not merely as decision rules, but as multidimensional technologies subject to strict computational constraints. This paper develops a novel endogenous growth model bridging theoretical computer science and macroeconomics, characterizing algorithmic innovation through the intrinsic trade-off between time complexity (compute utilization efficiency) and space complexity (memory utilization efficiency). We establish a unique asymptotic Balanced Growth Path (BGP) where, despite exploding data privacy costs, the economy sustains growth by directing innovation exclusively toward time-complexity optimization. The framework provides a formal macroeconomic foundation for "Wirth's Law,'' demonstrating that rapid exogenous hardware progress (Moore's Law) actively crowds out algorithmic R&D. Furthermore, we show that sudden algorithmic breakthroughs generate an ambiguous short-run effect on hardware demand but unambiguously increase long-run demand, explaining recent financial market dynamics as a computational Jevons Paradox. Normative analysis reveals that decentralized markets systematically under-invest in algorithmic innovation due to classic monopoly distortions and a novel "over-reaction to hardware growth.'' Finally, we demonstrate that when the economy collides with an aggregate energy bottleneck, physical hardware scaling halts, fully reversing Wirth's Law and inducing an explosive reallocation of labor into algorithmic R&D as the sole engine of long-run growth.