## Demand for Intermediate Inputs (20 minute read)

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##### Learning outcome:
After completing this lesson, you will be able to define intermediate inputs, and understand how demand for the inputs is depicted in the UNI-CGE model.

##### Intermediate Inputs

Intermediate inputs are the goods and services that producers combine with factors (e.g., land, labor and capital) to make their final product.  For example, consider a producer that manufactures autos. The automaker requires intermediate inputs such as tires, steering wheels, fuel tanks and radios. The automaker also hires workers and buys or leases capital equipment. The workers and equipment are used to transform the intermediate inputs into a final product - an auto.

In CGE models, the producer's demand for inputs is typically divided into two separate "nests." One describes their demand for intermediate inputs - called a bundle of intermediates. The other is their demand for labor, capital and other factors - called a bundle of value added.  Input demand is divided into two (or more) parts because many CGE models describe producers as having different levels of flexibility in deciding about the mix of inputs within each bundle.

This lesson focuses on the demand for intermediate inputs.  In many CGE models, this demand is characterized by fixed input-output relationships between the quantity of intermediate inputs required per unit of output. For example, production of an auto requires a bundle of intermediate inputs that must contain 4 tires and one steering wheel. Adding a fifth tire or a second steering wheel to the input bundle will not increase the number of autos that can be produced. This fixed relationship between the quantity of inputs per unit of output is called a Leontief production function, named after Vassily Leontief, the economist who developed this idea.

##### Leontief Intermediate Production Function

Figure 1 describes a Leontief intermediate production function. It shows how quantities of two inputs, steering wheels and tires, are assembled into the bundle of intermediate inputs used to produce an auto. The axes of the figure show the quantities of each of the two inputs. C is an iso-cost line that shows all combinations of steering wheels and tires that cost the same amount. The closer is C to the origin, the lower the cost of the intermediate bundle.

Figure 1.  A Leontief Intermediate Production Function

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Q is an isoquant that shows all possible combinations of steering wheels and tires that produce the same quantity Q of input bundles. Its L-shape plots the fixed input-output relationship between each input and the output of bundles. Moving up the red isoquant, adding more steering wheels without also adding tires will not change the number of intermediate bundles that can be assembled. And moving horizontally along the isoquant shows that adding more tires without adding more steering wheels also does not increase the quantity of intermediate bundles produced.

In the initial equilibrium, cost-minimizing automakers operate at the tangency between the isocost and the isoquant curves. That point represents the lowest possible cost of producing a given quantity of intermediate bundles. In the figure, the producer uses quantity QS of steering wheels and quantity QT of tires to produce quantity Q of bundles.

##### Intermediate Input Ratios Do Not Change When Input Prices Change

The main implication of a Leontief intermediate production function in CGE models is that the quantity ratio of intermediate inputs does not change when relative input prices change. The elasticity of input substitution is therefore zero. Figure 2 illustrates this point. In the figure, an increase in the price of tires relative to steering wheels is shown by the rotation of the isocost curve to price ratio PT'/PS', marked in green. As the price of tires increases relative to wheels, the producer's input ratio for output of bundle Q remains unchanged. Despite the change in their relative prices, the producer cannot substitute more steering wheels in place of tires within the bundle.

Figure 2.  A Leontief Intermediate Production Function with a Change in an Input Price

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##### Intermediate Input Demand in the UNI-CGE Model

In the UNI-CGE model, demand by activities for intermediate inputs is described by a Leontief intermediate production function, expressed in the equation shown in Figure 3.  In the equation, variable QINTCAc,a is the quantity of commodity c demanded by activity a for use as an intermediate input. Parameter icac,a is an input-output ratio. For each commodity c, the parameter is the ratio of the quantity of that input to the quantity of output of activity a, variable QAa. Input-output ratios are calculated from the base data in the SAM. (Notice that the input-output ratio in the model is between the input and the output of the final good, not the intermediate bundle. This is a short-cut in the UNI-CGE model that is possible because the input-output ratio between the intermediate input bundle and the final good is also a Leontief function.)

Input-output ratios are parameters that remain fixed when the model is shocked in an experiment. There is no need to specify a substitution elasticity parameter for the intermediates nest - its zero value is implied by the fixed input-output ratios of the Leontief function.

Figure 3.  Leontief Intermediate Demand in the UNI-CGE Model

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##### An Example

Let's apply this equation to our example of tires and steering wheels. The equation states that the quantity of tires used as an input into the assembly of a bundle of intermediate inputs is determined by the quantity of tires per auto observed in the base data, and the quantity of autos being produced. If the price of tires rises, and auto output remains the same, auto makers will still demand the same quantity ratio of tires and wheels per auto. If all prices remain the same, but the quantity of autos being produced increases, the quantity demanded of both tires and wheels will increase by the same proportion as the change in auto output.

Copyright:  Cornerstone CGE CC 4.0 BY-NC-SA