From Micro-Distributions to Macro-Regularities: A Critique and Reconstruction of the Production Function Based on the Maximum Entropy Principle

This paper aims to provide a micro-foundation for the Cobb-Douglas production function based on statistical physics, and to launch a critique of its political-economic implications. By introducing the Maximum Entropy Principle and an axiom of scale invariance, we prove that in an economic system with incomplete information, the most unbiased distribution of micro-level technical coefficients must take the form of a truncated power law. Based on this, statistical aggregation naturally leads to the emergence of a constant-returns-to-scale Cobb-Douglas function at the macro level. This result not only provides a micro-foundation for neoclassical growth models that does not rely on a representative agent or value aggregation of capital but, more importantly, reveals that the aggregate production function is essentially a lossy compression of micro-level information. In this compression process, the social-historical relations embedded in distribution parameters are 'naturalized' into seemingly eternal technical laws, which is the manifestation of Marx's critique of 'fetishism' at the level of mathematical logic. This paper further deepens the understanding of the production function as a statistical phenomenon rather than a technical law through dialogues with Marx, the Cambridge School, and Shaikh.