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 Question from nona, a parent: If inputs increase by 30% and outputs increase by 15%, what is the percentage change in productivity?

Hi Nona.

In economics, the productivity is the amount of output per unit of input. For example, a paper company's productivity could be measured as the kilograms of paper it produces per 1000 kilograms of timber it
fells. If its logging practices are changed, then its productivity can change as well.

Let's make up a context for your question. Say this is a fishing zone and the inputs are the number of ocean trawlers. The outputs are the amount of food they can produce. If the number of ocean trawlers is increased by 30%, then the catch is increased by just 15% in your question. (Productivity says nothing about the "why"; perhaps in this case it is a problem of over-competitiveness or depletion of the fish stock).

The impact on the "productivity" measured as catch per ocean trawler would be C/T.
So if we compare the "old" productivity value C/T to the "new" productivity value (C + 15%C)/(T + 30%T), then we can come up with a percentage change in the productivity:

Percent change in productivity
= (new productivity - old productivity) / (old productivity) x 100%
= ( (C + 15%C)/(T + 30%T) - C/T ) / (C/T) x 100%
= ( (1.15C)/(1.30T) - C/T) / (C/T) x 100%
= ( (C/T) ( 1.15/1.30 - 1) ) / (C/T) x 100%
= ( 1.15/1.30 - 1) x 100%
= ( -0.115 ) x 100%
= -11.5%

So the percentage change in productivity in your question would be -11.5%.

Hope this helps,
Stephen La Rocque.

Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.