# Quantum computation of the Cobb-Douglas utility function via the 2D-Clairaut differential equation

## Description

This paper inserts the well-known Cobb-Douglas (CD) utility model in the context of quantum computation through the Clairaut-type differential formula. The new economic-physical model is studied from the envelope theory allowing the direct relation with the quantum entanglement that defines the emergent probabilities of the linear optimal utility function for the consumption of two goods with or without the limit of the available expenditure. Characterization of the Cobb-Douglas model with quantum computation and Clairaut differential equations also allows for the elucidation of system entropy for both the envelopes and intercepts of the optimal utility. We introduce new algorithms based on 2D-Clairaut’s differential equation to quantization formulation of the CD function expressed correctly in a circuit for one and two qubits. Our calculations and those coded for an IBM-Q computer will converge on analytic predictions. This approach linked the utility and the set budget of the CD function in an explicit model based on the envelope representation of the 2D-Clairaut’s differential equation and from the canonical equation of the line, whose normalized intercepts express the probabilities. The exact expressions derived here for one- and two-qubit quantum entanglement in the econometric context perform more efficiently than the calculations of the IBM-Q simulations, requiring a large number of shots to match the exact results