Demand for Factor Inputs (35 minute read)


Learning Outcome: 

After completing this lesson, you will be able to define factors, know how producers choose the optimal mix of factor inputs in the value added nest, and understand the significance of factor intensities and employment shares in interpreting factor market results of a CGE model.  

What Are Factors?

Factors of production in CGE models include labor, capital, and land.  These may be further divided into subsets like rural/urban workers. Factors of production are a country's "primary" resources.  This means that countries are endowed with a factor resource base. Factors cannot be manufactured, although in some CGE models they can accumulate through inflows of migrant labor or foreign capital investment (or decumulate through outflows).  

Choosing the Best Mix of Factors

Charlie Chaplin's classic movie, "Modern Times," dramatized how labor, more or less expertly, worked with machinery to produce steel widgets (Figure 1). In a CGE model, producers have a similar production process. They combine labor with machinery, and any other factors, to assemble their products. The problem facing producers is how to determine the optimal mix of workers and machines to use in their production process. 

Figure 1.  Workers and Machines 

Assembly line scene from Charlie Chaplin's movie, "Modern Times." It is in the public domain.


The producer's decision on the quantity mix of factors is described in the value-added nest of the technology tree (Figure 2).  Factor inputs are termed "value-added" because labor and capital (and other factors) add value to the intermediate inputs that they transform into final products. After the ratio of factor inputs is determined in the value-added nest, this "bundle" of factors is combined with a "bundle" of intermediate inputs, whose quantity mix has been determined in the intermediates nest. The two bundles are then combined in the assembly of the final good.

CES Value-Added Production Function

The UNI-CGE model, like many CGE models, depicts decision-making in the value-added nest with a Constant Elasticity of Substitution (CES) production function. This function describes the producers' technological flexibility to substitute between capital, labor and any other factors in the production of a given quantity of output when their relative prices change. This flexibility remains the same - it is constant -  at any level of output and with any quantity mix of factors. 

Figure 2.  Nested production function
A graph of the nested production function, with a box drawn around the value added nest for emphasis.

Larger version available HERE.

In a nutshell, an important thing to know about CES factor demand is that the technological flexibility is defined by a substitution elasticity, ESUBVA.  Its name signals that it is an Elasticity of SUBstitution in Value Added. When this parameter has a low value, flexibility is limited and producers will make minimal changes to their factor ratios when wages change relative to capital rents. When the parameter has a high value, producers can respond more flexibly and make larger changes in their factor input mix as relative factor prices change.

Factor Substitution Parameter and Supply Response
The CES elasticity parameter describes how easily factor ratios can be changed to produce a given level of output.  But the CES elasticity also impacts the size of producers' supply responses to a model shock.  

An auto dealership with a sign "Lets make a deal"Let's first consider the example of an increase in a factor input price - a wage hike for auto workers.  Some share of the increase in their wages is likely to be passed on to consumers through higher auto prices, leading to lower demand and output. With a high factor substitution elasticity value, the automaker can more easily replace workers with machinery, limiting the effect of higher wages on production costs, auto prices and demand. A low substitution parameter means higher labor costs are less avoidable, causing a greater increase in auto prices, and a greater reduction in demand and output.  

On the other hand, consider the effects of a positive demand shock in which the demand for autos has increased.  A producer who wants to expand output must draw in additional workers and capital from other industries. A producer who can be more flexible about the ratio of capital and labor inputs in their production process can expand output by using more of the lower-cost factors. But a producer who requires a relatively fixed ratio of factor inputs may need to increase their wages or rents to attract the necessary factor mix. Their input costs are likely to be higher than the more flexible producer, leading to a higher sales price, softer demand and a smaller supply response.     

A Graphical View of the CES Value-Added Demand Function

Let's explore the CES value-added production function graphically (Figure 3), starting with the case of a high parameter value for the elasticity of factor substitution.  In this graph, the two factors are labor, L, and capital, K.  Curve Q is an isoquant that shows all combinations of quantities of labor and capital that can produce the same quantity, Q, a bundle of value-added. The further Q lies from the origin, the higher the quantity of value-added bundles. The blue line, C, is an isocost lie, with a slope of -PK/PL, the rent/wage ratio. The isocost line shows all combinations of labor and capital that cost the same amount. The further it lies from the origin, the higher the cost of inputs. In the initial equilibrium, the producer uses QL of labor and QK of capital, at the tangency of the isocost and isoquant curves. It is the least cost combination of labor and capital to produce quantity Q.

Figure 3.  Factor Input Ratios with High Substitution Elasticity

Factor input ratio with high elasticity

Larger version available HERE.

The curvature of the isoquant describes the flexibility in the production process to shift between the two inputs. The higher the ESUBVA parameter value, the flatter the isoquant. In Figure 3, an increase in the relative price of capital to labor rotates the the isocost curve to ratio PK'/PL' (shown by the red isocost line). The producer shifts to a more labor-intensive production process, with a change in the factor input mix to QL' and QK' in the production of quantity Q of value-added bundles.

Contrast the outcome in the high-elasticity case with that of a low value of the ESUBVA parameter, shown in Figure 4. The producer's more limited flexibility to change the factor input mix is depicted by the more sharply curved isoquant curve, Q. In this case, the same increase in capital rents relative to wages leads to a smaller change in the quantity ratio of labor to capital in producing the same level of output, Q.

    Figure 4.  Factor Input Ratios with Low Substitution Elasticity

Factor input ratios with low substitution elasticity

Larger version available HERE.

As you gain experience in carrying out CGE analyses, one point to think about is the sensitivity of your model results to the assumed values for ESUBVA, the elasticity of factor substitution within the value added nest. A good practice to follow is to check the sensitivity of your model's factor market results by re-running the same model experiment with a range of ESUBVA parameter values. If results are robust across values, you can have more confidence in your findings. 

CES Value Added Function in the UNI-CGE Model

Figure 5 presents the equations in the UNI-CGE model for the CES value added function and its first order condition, which is activity demand for factors. In the equation, variable QAa is the quantity of output from activity a. Notice that this equation describes the factor content of the activity output quantity, rather the quantity of value added bundles. This is a shortcut that is possible because the UNI-CGE model assumes a Leontief relationship between bundles of value-added and total output. 

Parameter alphaa
a is the level of total factor productivity (TFP) in activity a. It is a shift parameter - you can change its value as a model experiment to explore changes in factor productivity. The delta parameters are the shares of each factor f  in output of activity a; they sum to one. Variable QFf,a is the quantity of factor f used in activity a. The exponent term for each activity is a transformation of the elasticity of factor substitution, ESUBVAa. The equation shows that the quantity of output is a function of the quantities of factor inputs and their productivity level.

Figure 5. CES Value-Added Production Function in UNI-CGE Model

CES valaue-added equations in the UNI-CGE model

A larger version is available HERE.

The first order condition of the CES production function defines the demand by activities for each factor. It requires the wage or rent paid by an activity to be equal to that factor's marginal revenue. WFf is the economy-wide average wage and WFDISTf,a is the wage or rent premium that a factor may earn in an activity.  Surgeons, for example, earn more than the average wage in most countries, so their wage premium has a value greater than one. PVAa is the weighted average price of all factors used by an activity, minus the tax on factor usage in activity a (tfaf,a). 

Factor Intensity, Factor Shares

Structural features of factor markets also influence producer behavior and the general equilibrium results in factor markets.  Factor intensity describes the share of a factor in an activity's total value added payments. Table 1 reports the factor intensity of each activity in the US333 SAM.  For example, labor accounts for 70 percent of factor costs in the production of services.  

Table 1. Factor intensity in US333 SAM
Land 0.27 0.00 0.00
Labor 0.35 0.68 0.70
Capital 0.39 0.32 0.30
Total 1.00 1.00


Activities' shares in the total national employment of a factor are called factor shares. Table 2 reports factor shares in the US333 SAM.  For example, the AGR activity accounts for 100 percent of activity payments to US land.  

Table 2. Activity shares in national employment
Land 1.00 0.00 0.00 1.00
Labor .01 0.17 0.83 1.00
Capital .01 0.18 0.08 1.00

These are some observations about the structural features of US factor markets based on factor intensities and factor shares:

  • Agriculture is the most capital-intensive production activity in the US.
  • Services is the most labor-intensive production activity in the US.
  • Most US labor is employed in the services production activity.

When the price of a factor changes, its greatest impact will be on the activity that is most intensive in its use. For example, labor accounts for 70 percent of factor inputs used in services in the US, compared to a 35-percent labor share in agriculture.  An increase in wages will have a greater effect on the cost of production in services than it does in agriculture. 

Given the large share (83 percent) of workers employed in services, even small changes in the size of labor employment in services will have significant impacts on the other sectors of the economy, who must compete for, or absorb, these same workers. The ability of AGR and MFG to substitute labor for capital as SER employment changes will affect the labor market as a whole. The higher the value of parameter ESBUVA the more flexible are activities' production processes and the more labor market adjustment can occur through changes in every activities' factor input ratios. 

Copyright: Cornerstone CGE  CC BY-NC-SA 4.0

Last modified: Saturday, 27 April 2024, 9:33 PM