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In a traditional Solow-Swan growth model, you can decompose growth in labor productivity (output/hour) into a component stemming from total factor productivity growth and another stemming from capital deepening (increases in the capital/worker ratio).

Let's assume that I create a new machine which doubles the output of soybean production and try to sell it to soybean farmers. Prior to adopting my machine, farmers used horses and primitive plows to grow their soybeans. From the perspective of the farmer, after buying my machine and using it, they work the same amount, but they produce 2x as much, so their labor productivity has doubled. However, to what would you attribute the growth in output? Would you attribute it to capital deepening (because the farmer is now using more capital than before), or total factor productivity?

The reason I ask is because total factor productivity growth is generally considered by economists to be a measure of "the pace of technological progress" (See: The Rise and Fall of American Growth and other books of the same nature). But the act of creating my fancy soybean picking machine was in of itself a technological act. I had to combine some other capital inputs (steel, etc.) and some labor (my own and perhaps my employees) with insight to create the machine in the first place. I'm a bit confused about the growth accounting here.

A follow up question makes the situation a bit harder. Let's say that again I have invented a new machine tool. This machine tool let's car makers make smooth surfaces that are more aerodynamic and look cooler. This improves the look and miles/gallon figure of cars made using my machine tool. Car companies realize that consumers' are going to love the new look and higher MPG of cars made with the machine, so they purchase it in droves. In this situation, would labor productivity (output/hour) or even raw output of cars increase? After all, nothing has changed about the fundamental efficiency of producing the car - if anything the cost of producing the car has gone up because the fixed costs the car manufacturer pays to buy my machine must now be amortized over every car they sell. If the former did rise, would you attribute it to capital deepening - the car OEM is using more capital than before - or total factor productivity?

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The increase in output in Solow-Swan model can come both from total factor productivity and capital deepening. Let’s step back from the real life example you have given because there many factors come to place and it’s hard to satisfy the ceteris paribus.

Consider simple Cobb-Douglas production function - that is actually almost always used in standard Solow-Swan model:

$$f=AK^{\alpha}L^{1-\alpha}$$

Now here the total factor productivity is represented by the $A$ parameter.

If you look at this this way you should quickly see that growth can come both from capital deepening (if K increases and L stays constant) or from increase in total factor or even both at the same time.

In your farmer example, the new technology increases $A$ so output should increase even if the investment in the new capital would be exactly equal to depreciation of old capital (and hence no net increase in K).

However, this of course abstracts from markets for factors of production where now demand for new technology increases because the new capital is perceived as more valuable as old one.

So both things can happen at the same time in Solow-Swan model, you can have both change in output due to change in A or K. Unfortunately due to this just from the raw change in output $\Delta f$ you can’t directly deduce which part of change is due to capital deepening and which part due to increase in $A$. You either have to keep one of them constant or you need to get data on the changes in either total factor productivity or capital accumulation to be able to deduce the other (assuming labor is always fixed constant).

In the second example if there is no change in total factor productivity (not every new technology has to change it) then you could attribute changes in output solely to capital deepening.

However, the problem there is that here you have another confounding variable and that is that the quality of car probably changed in the eyes of consumer so the output will have higher value. Here you would have to modify the Solow-Swan model to account for that the new output is not same as old and you basically switch from production $f$ to some $f^*$ - the problem is here that you are also trying to fit the macro model that can easily abstract from quality differentials to microeconomic problem where those things matter, but if you would hold the quality constant you could still use it. Or you could include the quality change explicitly with some pice wise production function or assuming multiple plants with different production functions etc.

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