Capital Deepening vs. Technological Progress

In the short run, businesses often have the capital that they have. That is, buildings and machines are fixed assets today, this week, and maybe even for the month or year. It depends. There's more flexibility when it comes to hiring and firing laborers. A company can add some more hours very quickly, or send people home early. With labor being so flexible, perhaps it makes sense to talk about output as a function of labor. In other words, you might start thinking of everything in a "per unit of labor" sense.

Start with a production function. If $$\displaystyle Y = AF(K,L) $$, then dividing everything by _L_ gives you $$\displaystyle Y/L = AF(K/L,L/L) $$. Two things at this point: First, just let the "per unit of labor" values be lowercase, and second, recognize that the number 1 isn't a variable of a function. This leads to $$\displaystyle y = Af(k) $$. If you assume a Cobb–Douglas production function, then $$\displaystyle y = \frac{AK^{\alpha} L^{1 - \alpha}}{L} = \frac{AK^{\alpha} L^{1 - \alpha}}{L^{\alpha} L^{1 - \alpha}} = Ak^{\alpha} $$. What do you think is this production function's returns to scale?

No. That was the case with the production function assuming equal increases in capital and labor. But now consider what the logical conclusion would be on a per-worker basis.

Not exactly. This would require exponents to the production function that sum to more than 1. That's not the case here.

Yes! The alpha here is still the same as before; probably 0.3 or so. Whatever it is, it's less than 1, and so the function exhibits decreasing returns to scale, as labor is scaled (since this is all on a per-worker basis now). So the production function is concave when graphed, like this: 

Output per worker is also called __labor productivity__. Obviously laborers are more productive when they have machines and tools. But if you keep adding more, it will help less and less. Each laborer can only do so much with the capital available, so eventually the slope of the production function in per-worker terms tends toward 0. Suppose new capital is added to an economy. What is the result, using this production function and its graphical representation?

No, that's not right. This would suggest that the same level of capital per worker is used to produce more output per worker, and that's not the case here.

Not so. There's no change of shape in the production function with the addition of more capital.

Exactly! More capital means more capital per worker. The function tells you what that will do to output, and that it will provide decreasing returns to scale. The graph tells you the same story, and the decreasing returns are seen in the lower slope. This movement along the production function is called __capital deepening__, and is simply an increase in the capital per capita. That's one way to produce more.

Another way is technology. The total factor productivity, represented as _A_, could simply increase. A new technology may make workers more efficient, faster, or allow workers to simply use capital more effectively. Wherever the innovation exists, more can be produced than before with a given amount of capital and labor. What do you suppose this change would look like on a graph?

No. A shift upward would imply that the increase of output per worker is the same at every level of capital per worker. This includes zero capital, and wouldn't make sense.

That's right! Every point on the production function would be augmented by this higher _A_ value. This __technological progress__ would allow for a higher level of output per worker no matter what the level of capital per worker might be. 

Actually, it wouldn't. This is a change of shape, which would suggest increasing returns to scale. There would still be decreasing returns to scale of the capital-to-worker ratio.

If you got to choose an increase of 1% in either _k_ or _A_, which do you think would offer the greatest impact on a nation's standard of living?

No. Consider the returns to scale of both variables.

Incorrect. It would matter. Consider the returns to scale of each.

You got it! A 1% increase in total factor productivity means a 1% increase in output per worker. But a 1% increase in capital per worker is subject to decreasing returns to scale, leading to a lower impact.

To summarize: [[summary]]

Constant returns to scale

Increasing returns to scale

Decreasing returns to scale

The function shifts

The function changes shape

The economy moves up and to the right on the existing function

The function would shift upward

The function would stretch upward

The function would become convex upward

Capital per worker

Total factor productivity

It wouldn't matter which one increased

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