## Introduction to CES Functions (15 minute read)

##### Learning Outcome

After completing this lesson, you will understand the economic behavior that is described by a constant elasticity of substitution (CES) function, and the role of the CES elasticity in describing the flexibility of consumers and producers to substitute among options.

##### Introduction

Imagine that you want to buy 3 pounds of fruit to make a fruit tart.  You hope to buy more blueberries than raspberries, because you like blueberries better.  At the grocery store, you discover that blueberries have increased in price relative to raspberries. How willing are you to switch toward more raspberries than blueberries in your tart?

In a CGE model, constant elasticity of substitution (CES) functions describe this type of decision.  A CES function explains how willing you are to switch the mix of berries within your 3-pound basket when their relative prices change.

This type of function can be applied to many decisions in a CGE  model. In the UNI-CGE model, for example, consumers have to decide on the mix of imported versus domestic varieties of a good, such as domestic versus imported cheese, in a given basket.  And producers have to decide on the mix of labor and capital to use in their production process.

In this course, you will also learn about Constant Elasticity of Transformation (CET) functions.  Both CES and CET describe how easily consumers and producers can shift between their options as relative prices change. The important thing to know about the functions is that they describe opposite price-quantity relationships. A CES function describes agents as substituting toward more of the good whose price falls, holding the total quantity constant.  A CET function describes agents as substituting toward more of a good as its price rises, holding the total quantity constant.

##### Constant Elasticity of Substitution Function

Let's examine this concept graphically. As an example, we describe a consumer who is buying the commodity "fruit."  Fruit is a composite commodity that is made up of two varieties of fruit - blueberries and raspberries.  Figure 1 describes the quantities of each kind of fruit on its axes. Line C in the figure is a budget line that shows all possible combinations of blueberries and raspberries that cost the same total amount. The further is line C from the origin, the larger the consumer's budget. The slope of C shows the relative prices of the two fruits, Pand PB.  Curve Q is an indifference curve that describes all quantities of the two fruits that yield the same utility. Like the budget line, utility is greater as curves are plotted further from the origin.

The consumer maximizes utility at the the tangency of their budget line and their utility curve. At this point, they achieve the highest attainable utility given their budget, with a basket of quantity QR of raspberries and quantity QB of blueberries.

Figure 1.  CES Aggregation Function

Larger image available HERE.

Why do we assume the consumer remains on the same utility curve? The CES function  describes only the "substitution effect" of the price shift on the consumer basket, holding utility constant.  It does not measure the income effect of the price change on the consumer's purchasing power and achievable level of utility.

##### Elasticity of Substitution

The curvature of Q shows the willingness of consumers to substitute between the two fruit varieties as their relative prices change. The measurement of its curvature is the elasticity of substitution (σc)This elasticity parameter expresses the percent change in the quantity ratio given a percent change in the price ratio, when the consumer stays on the original utility curve:

The higher the value of the elasticity of substitution parameter, the more willing are consumers to substitute between quantities of blueberries and raspberries, and the more curved is Q. The constant elasticity of substitution function is so-named because curve Q has the same elasticity value at all points on the curve and at all levels of utility.

You will notice that some of the elasticity parameters in the UNI CGE model are named ESUB*** or  ETRA***.  The "E" in the parameter names stands for "elasticity."  The term "SUB"  signals that the the elasticity is a parameter used in a CES (SUBstitution) function.  The term "TRA" signals that it is a parameter used in a CET (TRAnsformation) function.

##### CES Function with a Price Change

Figure 2 describes how consumers respond to a relative price change. In the figure, an increase in the price of raspberries relative to blueberries causes the budget line C to rotate, as shown by the new, red line labeled C'.  The consumer moves along their utility curve to its new tangency with the consumer's budget line. The consumer now purchases more blueberries, QB', and fewer raspberries, QR',in their basket.  You might imagine that if the elasticity parameter was smaller, and the utility function was more sharply curved, the consumer would make a smaller adjustment in their fruit basket given the same change in prices. But if the curves flattens, reflecting a higher elasticity parameter, then the quantity adjustment will be quite large for the same change in relative prices.

Figure 2.  CES Function with Price Change

Larger image available HERE.

##### Why the CES Parameter Value Matters

It should be apparent that the value of the CES elasticity parameter has an important effect on the quantity results from a model experiment.  Higher elasticity parameter values will result in larger quantity results than a lower value. In your research, it is a good practice to, first, evaluate the source of the elasticities used in your model. Then, test the sensitivity of your model results to parameter values by running your model experiment with higher and lower elasticity values. If your model results are robust over alternative values, you can be more confident about your findings.

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