Primer on Deriving Demand for Inputs in a Fixed Proportion Production Process

The demand for inputs in the production of final goods is ultimately derived from the demand for the final products themselves, which is why input demands are sometimes called "derived demands." This relationship can sometimes be lost in all the math surrounding modern textbook treatments. I think this is why it is best to introduce students to the concept of derived demand using an example where an industry uses a fixed-proportion production process. Here the math is so simple that it doesn't get in the way of the economics of how output markets influence input markets (and visa versa).

However, few modern textbooks discuss this special case (exceptions include Friedman's Price Theory and Becker's Economic Theory). I think that is a shame. So, I thought I'd write a short primer on deriving an industry's demand for inputs into a fixed proportion production process. First, I provide an intuitive explanation for how to derive the inverse demand for an input using Alfred Marshal's famous knife manufacturing example. Second, I provide a formal discussion of how to derive the elasticity of input demand. Third, I show how this simple example illustrates Marshall's four laws of derived demand. Lastly, I provide some links to additional reading.

There is nothing original here. I am basically just summarizing some old notes that I wanted in one place. Hopefully someone besides me finds them useful.

1. Knives, Blades, and Handles
Suppose that knives are produced using a fixed proportions technology. Specifically, one handle and two blades are combined to create one knife. Figure 1 illustrates the demand curve for completed knives and the supply curves of each input (note that the Pb represents the price of two blades). 

Figure 1. Demand Curve for Final Product and Supply Curves of Inputs

Source: Friedman (1972)

So, how do we derive the demand for one input like handles? Well, let's think about what each curve is telling us. The demand curve for knives shows the most that consumers are willing to pay for a given quantity of knives. Similarly, the supply curve for each input shows the least that suppliers would have to be paid to provide a given quantity of that input. Thus, the most that knife producers would be willing to pay for a given quantity of handles is the difference between the demand for knives and the supply of blades (see Note XIV in Marshall's Mathematical Appendix). Put another way, the "demand price" for handles equals the "demand price for knives" minus the "supply price for two blades":

Figure 2 illustrates the derived demand for handles. Note that no handles are purchased when the price for knives equals the supply price for blades. This is because blades are so expensive at that level of output there is no money left over for handles.

Figure 2. Derived Demand for Handles

Source: Friedman (1972)

2. Deriving the Elasticity of Demand for Handles

We can use the inverse demand function for handles above to derive the elasticity of demand. I provide the details here, but the ultimate result is:

This formula can be useful in applied settings. For example, the EPA used this formula to calculate the elasticity of demand for small, stationary combustion engines (a key input in the product of irrigation equipment among other things) when considering adding regulations on that industry (see page 4-2).

3. Marshall's Laws of Derived Demand

This expression for the elasticity of demand illustrates several of Marshall's 4 laws of derived demand.

1. "The demand for anything is likely to be more elastic, the more elastic is the demand for any further thing which it contributes to produce." (Note that as the elasticity of demand for knives increases, the elasticity of demand for handles increases).
2. "The demand for anything is likely to be more elastic, the more readily substitutes for the thing can be obtained." (Not illustrated here because there is no substitutes for handles)
3. "The demand for anything is likely to be less elastic, the less important is the part played by the cost of that thing in the total cost of some other thing, in the production of which it is employed." (Note that as Ph/Pk decreases, the elasticity of demand also decreases)
   
4. "The demand for anything is likely to be more elastic, the more elastic is the supply of co-operant agents of production." (Note that as the elasticity of supply for blades increases, the elasticity of demand for handles also increases)

Side Note: Hicks later showed that Marshall's third law only holds if the elasticity of final demand is greater than the elasticity of substitution. An intuitive explanation by Saul Hoffman for why this is the case can be found here. However, we don't need to worry about that special case with fixed proportion technologies because the elasticity of substitution across inputs is zero. So the condition will always be satisfied.

4. Additional Reading

For more info on this topic, I'd recommend checking out these resources:

• Becker, Gary. Economic theory. Transaction Publishers, 2007.
• Diewert, W.E. "A Note on the Elasticity of Derived Demand in the N-Factor Case," Economica (May 1971): 192-198.
• Friedman, Milton. Price theory. 1972.
• Muth, R., "The Derived Demand Curve for a Productive Factor and the Industry Supply Curve," Oxford Economic Papers 16 (1964): 221-234
• Hoffman, Saul D. "Revisiting Marshall's Third Law: Why Does Labor's Share Interact with the Elasticity of Substitution to Decrease the Elasticity of Labor Demand?." Journal of Economic Education 40, no. 4 (2009): 437-445.
• Thurman, Walter N. "Applied general equilibrium welfare analysis." American Journal of Agricultural Economics 73, no. 5 (1991): 1508-1516.