So far this week we've talked about what happens when we scale up just one input, to total output. The concept of returns to scale looks at what happens when we scale up all inputs proportionally. And we can have three different cases. We can have increasing returns to scale. This would be a case where you double all inputs, and you get more than double the initial output. You can have decreasing returns to scale. You double all inputs, and you get less than double output. Or you can have constant returns to scale. You double all input, you get exactly double the initial output. There are certain factors that lead to increasing returns to scale. One is that there can be advantages to teamwork, to specializing, to dividing a task between different members on the same team. Adam Smith was one of the first to point this out, and it's one of his most memorable quotes from The Wealth of Nations. And when he was touring around Europe, he would observe that if somebody just tried to produce pins on their own, metal pins, at the most they could probably produce 20. But then he observed other cases where the same task of pin production was divided into a number of different branches. Each one being a particular trade. He wrote in The Wealth of Nations, one man draws out the wire, another straightens it, a third cuts it, a fourth points it, a fifth grinds it. At the top for receiving the head. To whiten the pins is another. It is even a trade by itself to put them into the paper. And the operations, which in some manufactories are all performed by distinct hands, the same man will sometimes perform two or three of them along the way. And what he saw in these cases is the team of ten individual workers working together could produce, 4,800 total pins, or 480 per person. And before when they worked individually, at the most they could produce 20. So that specialization, and the division of labor allowed dramatic increases in productivity. Scaling up of output beyond the scaling up of the individual inputs. There are also arithmetical relationships that lead to increasing returns to scale. take the case of a pipeline. The circumference of the pipeline is the constant pi, roughly 3.14, multiplied by the radius of the pipeline. The volume though, of, or the area of a particular circle is pi times the radius squared. So let's say a case where we increase the radius of a pipeline from one foot to ten feet, we scale it up by factor of ten. The circumference will go up by pi times the radius, by factor of ten. The volume of that circle will go up, though, by factor of 100. So scaling up the circumference by ten will increase the volume, the throughput that the pipeline can accomplish by ten times this much, by a factor of 100. So, the material applied to the pipeline there's certain factors because of this arithmetical relationship will dramatically increase the throughput. Another example, large scale technologies, using an MRI system, using a large scale irrigation technology, an internet back bone. When we get certain scale levels, we can get dramatic increases in total output. Balanced against that. There are factors giving rise to decreasing returns to scale. When we keep scaling up things, it becomes harder to manage an enterprise. You look at the case of the financial services industry. Prior to 2008, studies had been done indicating that up to about 140 billion in assets under management. There were increasing returns to scale. Beyond that constant, or if anything decreasing returns to scale. And prior to the major financial downturn, there were firms like JP Morgan, Bank of America, Citigroup, that were 1 trillion and above in assets. So, well beyond the point of increasing returns to scale. And one of the factors, arguably, that made those firms harder to manage, and harder to evaluate the risks that they were taking on by trading mortgage backed securities. Let's look what we mean by returns to scale on a graph depicting isoquants units, how units of labor and capital and put translate to the final output. Let's say we go from a point like A, where we're using two units of both labor and capital, to a point like D. Where we're using, we've doubled the amount of both inputs. Over this range we've moved from Isoquant 10 to Isoquant 40. So we've doubled all inputs, we've quadrupled output. Over this range in the Isoquant map, we're experiencing increasing returns to scale. We go from point D to point F. We increase both inputs by 50%. So both capital and labor go from four to six. And over this range, output, we move from Isoquant 40 to Isoquant 60. We increase output by 50%. So over this range of the Isoquant map, we're experiencing constant returns to scale. And now let's go further, from point F to point H. We've doubled usage of both inputs from six to 12, and in doubling usage of both inputs over this range, we've only increased output by 50%, from Isoquant 60 to Isoquant 80. A general way to think about this, when isoquants get scrunched closer and closer together, as we increase proportionally in, input usage, that's a range where increasing returns to scale apply. If isoquants are, are spaced equal distance apart as we over range that we scale up all inputs. That's the situation where we have constant returns to scale. And if isoquants get further and further apart from each other, as we proportionally scale up input usage. That's the case where we have decreasing returns to scale.
