Quantum computation of the Cobb-Douglas utility function via the 2D-Clairaut differential equation
Description
This paper introduces the integration of the Cobb-Douglas (CD) utility model with quantum computation, utilizing the Clairaut-type differential formula. We propose a novel economic-physical model using envelope theory to establish a direct link with quantum entanglement, thereby defining emergent probabilities in the optimal utility function for two goods within a given expenditure limit. The study illuminates the interaction between the CD model and quantum computation, emphasizing the elucidation of system entropy and the role of Clairaut differential equations in understanding the utility's optimal envelopes and intercepts. Innovative algorithms utilizing the 2D-Clairaut differential equation are introduced for the quantum formulation of the CD function, showcasing accurate representation in quantum circuits for one and two qubits. Our empirical findings, validated through IBM-Q computer simulations, align with analytical predictions, demonstrating the robustness of our approach. This methodology articulates the utility-budget relationship within the CD function through a clear model based on envelope representation and canonical line equations, where normalized intercepts signify probabilities. The efficiency and precision of our results, especially in modeling one- and two-qubit quantum entanglement within econometrics, surpass those of IBM-Q simulations, which require extensive iterations for comparable accuracy. This underscores our method's effectiveness in merging economic models with quantum computation.