Constant Elasticity of Substitution (CES) for Marshallian and Hicksian Demand Functions

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This Demonstration shows the relationships of iso-utility and iso-expenditures for the goods and in the constant elasticity of substitution (CES) formulation for Marshallian and Hicksian levels of consumption. We consider both compensated and uncompensated income. Let and be the number of units of the goods.


The CES utility form is:

with scalar factor for utility , elasticity of substitution , returns to scale and preference share parameters and ; and are normalized to 1.

Marshallian consumption 1 (blue curve and blue line): maximization of with optimal consumption , subject to uncompensated income , where and are the original prices.

Hicksian consumption (blue curve and green line): minimization of compensated income with optimal consumption , subject to original utility .


Marshallian consumption 2 (red curve and red line): maximization of with optimal consumption , subject to uncompensated income , where and are the changed prices. (January 2023)
Contributed by: Carlos Angulo
Open content licensed under CC BY-NC-SA



This Demonstration is based on [1], which develops further mathematical analysis from an idea in [2]. The purpose of this Demonstration is to explore how changes in the system parameters lead to changes in the equilibrium states. Hicksian substitution and income effects are represented in the following plots: (1) blue curve and blue line (original Marshallian consumption); (2) red curve and red line (final Marshallian consumption); and (3) blue curve and green line (Hicksian consumption).


[1] C. Angulo, "Creation of Wolfram Visualisation Tools for Microeconomic Pedagogy in Comparative Statics," Undergraduate Research Support Scheme research paper, University of Warwick, UK, forthcoming.

[2] K. Sydsæter, P. Hammond, A. Strøm and A. Carvajal, Essential Mathematics for Economic Analysis, 6th ed., Hoboken, NJ: Pearson, 2021.

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