A guide to input output model multipliers (part 1)

Photo by Thomas Galvez, togalearning.com

Photo by Thomas Galvez, togalearning.com

This series of posts is intended for individuals with a basic understanding of input output models but with no practical knowledge on how to derive output, income and employment multipliers.

From my own personal experience, I have found that the online literature regarding multipliers is neither easily accessible, clear nor concise.  Furthermore, this topic was not covered during both my undergraduate and post graduate economics degrees and I feel this may also apply to other university courses.  I therefore hope this post will be useful for any individual looking for a quick and practical guide to derive multipliers using input output models.  Please feel free to post any questions or comments below.  This first post will deal with the calculation of the simple (or Type I) output multiplier.

To get the ball rolling, it is necessary to calculate the direct requirements matrix (A).  The direct requirements matrix describes the amount of inputs needed from other industries to produce one unit (or $1 in this case) of output in a given industry. This is relatively straightforward to calculate.  Using the intermediate outputs and primary inputs table (see Table 1), simply divide each column by the column total to derive the direct requirements matrix (see Table 2).

Table 1: Combined intermediate and primary table

Input output 1

Table 2 shows that to produce an extra dollar’s worth of output in the agricultural industry, the manufacturing industry must produce an extra $0.13 worth of output, the transportation industry an extra $0.07 worth of output and so on.  This is known as the first round effect i.e. the amount of output each industry must produce to meet an one unit increase in demand/output in a given industry.  The total first round effect can be calculated from summing the column values for each industry e.g the first round effect of a $1 increase in output is $0.37.

Table 2. Direct requirements matrix (A)

Input output 2

Similarly, if the manufacturing industry (as an example) increases output by $0.13 it will induce additional outputs from other industries across the economy and these in turn induce extra output and so on.  It is therefore necessary to generate the Leontief inverse matrix to assess these effects.  The Leontief inverse matrix is formally defined as:


This is the inverse of the identity matrix minus the direct requirements matrix (derived above).  I will not discuss the derivation of this formula as it would constitute a separate post altogether!  Nevertheless, I have found a useful four part video series on Youtube that I belief provides a good intuitive introduction to the Leontief inverse matrix – the link can be found here.  Using simple matrix functions in Excel gives the Leontief inverse matrix in Table 3 below.

Table 3. Derivation of the Leontief inverse (I-A)-1 in excel

Input output 3

Note: when using Excel for matrix operations use the array command (i.e. shift + ctrl +enter), I have used the MINVERSE function for the (I-A)-1 matrix  

The output multipliers are simply the column totals for each industry.  For example, the (I-A)-1  matrix shows that a $1 increase in agricultural sector output will induce an additional $1.63 of output in the overall economy.  This is the derivation of the simple or Type I multiplier as it is now more commonly know.  In the next post I will move on to the income and Type II multipliers.

Additional guidance:



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