The point I would like to add and emphasize is that we are not just talking about cause and effect (let alone mere correlation) in the relationship between productivity and employment terms. This is instead a matter of reference from one thing to the other in the construction of the indexes. Productivity is a ratio between the monetary value of output and hours worked. Productivity increases if the same value of output is produced in fewer hours, regardless of whether that change was produced by technological improvements, increased work effort or by layoffs of redundant workers. Productivity growth declines if GDP growth is constrained, again regardless of what specifically is limiting GDP.
Productivity is the quotient. So when Dean Baker says "productivity growth has slowed sharply in the last decade," there are many ways to parse that number. Productivity growth has slowed because the numerator, GDP, hasn't been growing as fast as before. Or, productivity growth has slowed because the denominator, hours of work, is not declining. Or some combination, again regardless of the reasons for the changes in the components.
It's not just about the machines. It's also about the cost of replacing workers with machines compared to the level of wages. It's also about the performance of GDP relative to its potential. High levels of unemployment and underemployment can thus impose a constraint on "productivity growth" such that the resulting slow growth doesn't appear to present a threat to employment. But that is like the parricide throwing himself on the mercy of the court because he is an orphan. Or as Excel's circular reference warning explains:
One or more formulas contain a circular reference and may not calculate correctly. Circular references are any references within a formula that depend on the results of that same formula. For example, a cell that refers to its own value or a cell that refers to another cell which depends on the original cell's value both contain circular references.
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