## Introduction to CET Functions (15 minute read)

*Learning Outcome*

After completing this lesson, you will understand the economic behavior that is described by a constant elasticity of transformation (CET) function, and the role of the CES elasticity parameter in describing the flexibility of producers and factors to substitute among their options when prices change.

*Introduction *

Imagine that you are a doctor and you learn that jobs in the field of artificial intelligence are starting to pay a much higher salary than your current job. How flexible are you to switch jobs to get higher pay? Your decision will probably depend on a number of factors, including the training requirements to enter a new field, and your personal preferences about possibly moving to a new city. In other words, your decision will depend on the transition costs that you face.

In a CGE model, including the GTAP model, a constant elasticity of transformation (CET) function describes this type of decision. A CET function depicts the flexibility of labor and capital to switch employment in response to changes in relative pay across industries.

This type of function can also be applied to other decisions in a CGE model. In the UNI-CGE model, a CET function is used to describe the producer's flexibility to change the ratio of exportable and domestically-sold varieties in their product mix. For example, the producer may make fitness trackers with distance measured in miles for the domestic market and in kilometers for the export market. Although they are the same commodity, the trackers are differentiated products, and it is not cost-free for the producer to redesign them to suit each market.

In this course, you will also learn about Constant Elasticity of Substitution (CES) functions. Both CET and CES functions describe how agents can substitute among options as relative prices change. The important thing to know about the functions is that they describe *opposite* price-quantity relationships. A CET function describes agents as substituting toward **more** of an option as its price **rises**. A CES function describes agents as substituting toward** more** of the option as its price **falls**.

*Constant Elasticity of Transformation Function*

Figure 1 shows a graph of a CET function. It is almost identical to the graph of a CES function with this important difference: The CET function is concave to the origin, whereas the CES function is convex.

Let's assume that a producer produces "fruit," which is made up of some combination of apples and oranges. The axes of Figure 1 describe quantities of the two varieties - apples and oranges. Line R is an isorevenue curve. It describes all combinations of the two types of fruit that generate the same total revenue to the producer. The further R lies from the origin, the greater the revenue earned from the apple/orange product mix. The slope of R shows the relative price of apples to oranges.

The convex curve Q is a product transformation curve. It shows all combinations of apples and oranges that can be made from a given level of resources, to produce the same quantity of fruit, Q. The further Q lies from the origin, the larger the quantity of output that can be produced, from a larger supply of resources. A producer optimizes by producing the output mix at the tangency of the product transformation curve and the highest attainable isorevenue line. It is the lowest-cost mix that achieves the highest possible revenue. In the figure, this point is the quantity mix of apples (Q_{A}) and oranges (Q_{O}). If the producer's mix was at any other point on the transformation curve Q, they would not be earning the maximum attainable revenue given the Q level of resources.

**Figure 1: Constant Elasticity of Transformation (CET) Function **** **

##### CET Elasticity

The curvature of the product transformation curve provides a measure of transition costs. It shows the technological possibilities for substituting between output of apples and oranges as their relative prices change. The flatter the curve, the smaller are transition costs.

The curvature is measured by an elasticity of transformation* *(σ^{x})*. *This elasticity parameter expresses the percent change in the quantity ratio given a percent change in the price ratio, when the producer stays on the original isorevenue line. The CET function is so-named because the transformation curve has the same, constant elasticity value at all points on the curve and at all levels of output.

Carefully compare the expressions for the CES elasticity of substitution (in the lesson on CES functions) and the CET elasticity of transformation and note the reversed positions of Q_{O} and Q_{A} in the quantity ratio. In the CET expression, if the price of oranges *rises* relative to the price of apples, then the quantity ratio of oranges to apples in the product mix *also* *rises. *The higher the value of the elasticity of transformation, the flatter is the product transformation curve, Q, the smaller are transition costs, and the more flexible is the producer in shifting among product lines as relative prices change.

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

##### CET Function with a Price Change

Figure 2 describes how the producer responds to a change in the relative prices of its product. In the figure, an increase in the relative price of oranges causes the price line to rotate, as shown by the new, red price line. The producer moves along their product transformation curve to its new tangency with the isorevenue line. As the orange price rises, the producer now makes more oranges, Q_{O}', and fewer apples, Q_{A}', in their product mix.

**Figure 2. CET Function with Price Change**

Imagine that the transformation elasticity value was low. The product transformation curve would be more sharply curved - signaling that it is difficult for the producer to switch production among the products in the output mix. In that case, even relatively large changes in prices would result in smaller quantity adjustments. If the transformation elasticity value is high, the transformation curve would be flatter. In this case, the producer could easily change the product mix, and even small changes in relative prices would lead to large changes in quantity ratios.

##### Why the Elasticity of Transformation Matters

It should be apparent that the value of the CET 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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