So far when we talk about the behavioral relationship between the variables of interest, we are talking about the Cobb-Douglas production function. This relates output per worker (real GDP per capita, Y/L), capital stock per worker and labor efficiency (or technology). That function looks something like this:
However, whenever we talk about equilibrium, we always talk in terms of the capital-output ratio. For instance, in the no-growth model, the equilibrium condition was shown to be:
And in the constant-growth case, the equilibrium condition was:
So it might help us if we rewrote the production function relating output per worker (or real GDP per capita) with capital intensity or the capital-output ratio. That production function looks like this:
Let’s call this production function the sequel, since like a Hollywood sequel, it has the same principle characters (but in different form), it is basically the same formula, and you get the same thing from both.
Now let’s take a look at what this function looks like, and what happens when we vary alpha, (the diminishing-returns-parameter):
The Cobb Douglas “sequel” for increasing alpha. Note that increasing alpha increases overall output for any given capital intensity. Also, a lower alpha causes a slower increase in output as capital intensity increases, compared to a higher alpha (the line is flatter and “less curvy” for lower alpha).
As you can see, increasing alpha gives higher capital per worker for a given level of capital intensity. Also, a higher alpha will cause the line to be steeper and increase output per worker faster for a given increase in capital intensity. Below is the change in Cobb Douglas: The Sequel, for increases in labor efficiency, E.
Output per worker increases with increases in efficiency (E), and the growth is proportionate. That is (for instance), a five-fold increase in E will cause a corresponding five-fold increase in output per worker.
Thus, increases in labor efficiency cause corresponding increases in output per worker.
All the datasets that created the graphs above have been uploaded to the ‘Handouts’ section to the right. I encourage you to explore the data, change the numbers and see how that affects the shape and level of the curve.
About the relationship of Cobb Douglas: The Sequel with g (the growth rate of efficiency) This relationship is nonexistent. Since the production function is a “snapshot in time” and talks only about a specific time-period, we cannot have a single production function showing changes g. In class, I meant to have the slide show the relationship with the steady-state capital-labor ratio and g, NOT the Cobb Douglas Sequel and g. Apologies for the confusion. I’ve removed that slide and will replace it for the next class.