## Homothetic, Quasi-homothetic and Non-homothetic Demand (8 minute read)

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##### Learning Objectives:

At the end of this lesson, you will understand the concept of homotheticity and the implications of homotheticity for household behavior in a CGE model.

##### What is Homotheticity?

Homotheticity is a characteristic of a utility function. It describes how consumers adjust the quantities they purchase when they have a change in income.

##### Homothetic and Non-homothetic Utility Functions

A utility function is homothetic if a consumer changes the quantity they demand by the same percentage change as a change in income - holding prices constant.  For example, suppose Lee gets a 10% raise. If Lee's demand is homothetic, Lee will increase the quantity of all goods in their consumption basket by 10%.  A homothetic utility function therefore has an income elasticity of one.

An implication of a homothetic utility function is that budget shares remain the same as income changes.  Since Lee increased the quantity purchased of every good by the same proportion as the change in income, spending on each good will still have the same share in Lee's total consumption spending.

A utility function is non-homothetic
if the percentage change in a consumer's demand for a good is more or less than the percentage change in income, at constant prices. A non-homothetic utility function allows some commodities to be luxury goods, with income elasticities of demand that are greater than one, or inferior, necessity goods, with income elasticities less than one.

For example, if Lee has a non-homothetic utility function, the composition of their basket may change after Lee's income increases by 10%.  Lee might increase their purchase of caviar by more than 10% and reduce the quantity of canned beans in their basket by more than 10%.  Because the quantity purchased of each good may go up or down, depending on the income elasticity of demand for that item, budget shares can change.

##### A Quasi-Homothetic Utility Function

A utility function is quasi-homothetic if part of the consumer's purchases is homothetic and a part is non-homothetic. The Stone-Geary utility function/LES demand system has this characteristic.

Consumers first meet their subsistence requirements by buying fixed quantities of each commodity.  Budget shares of subsistence goods in total spending can change when income changes because subsistence purchases become a smaller part of a person's over-all spending.  The subsistence part of spending is therefore non-homothetic.

After subsistence needs are met in the LES, any additional income can be allocated to discretionary spending. Budget shares in discretionary spending (called marginal budget shares) are fixed, and will remain the same if incomes change. This part of a person's spending is therefore homothetic.

What happens if all spending is discretionary?  In that case, the LES utility function is homothetic.

Homotheticity and Commonly Used Utility Functions in CGE Models

Three widely used utility functions in CGE models are the Cobb-Douglas and the Stone-Geary/LES demand system used in the UNI-CGE model, and the Constant Difference of Elasticities (CDE) used in the GTAP model.  Table 1 compares their homotheticity properties.

##### Table 1. Properties of Utility Functions Utility function Homotheticity Budget shares Cobb-Douglas Homothetic Fixed Stone Geary/LES Quasi-homothetic Flexible Constant difference of elasticities  Non-homothetic Flexible

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