Solving the consumer’s utility-maximization problem with CES and Cobb-Douglas utility function via mathematical inequalities

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This paper presents a new, non-calculus approach to solving the consumer’s utility–maximization problem with constant elasticity of substitution (CES) utility function, as well as with Cobb-Douglas utility function in case of \(n\ge 2\) commodities. Instead of using the Lagrange multiplier method or some other method based on differential calculus of several variables which might give complicated terms and equations difficult to handle, the utility–maximization problems are solved by using Jensen’s inequality and weighted arithmetic-geometric mean (weighted AM–GM) inequality. In comparison with calculus methods, such approach does not require checking first and second order conditions.