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Why energy and capital are complementary factors of production?

In the literature of environmental taxation, it's often argued that energy and capital are complementary factors of production. So for instance if price of energy goes up, so will the price of capital. I am struggling to understand the relationship.

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You say "energy and capital are complementary factors of production. So for instance if price of energy goes up, so will the price of capital.". I think you might have confused yourself because this kind of positive correlation in prices is not necessarily a property of complements (indeed, often one would expect the reverse to be true).

It is more helpful to think of complementarity as "if the price of energy goes up then the demand for capital will go down" (in much the same way that people will buy less butter if bread becomes more expensive).

The reason for this is not far from that proposed by andersan. Capital is often invested in machines (e.g., computers, robots, trucks, farm equipment) that require energy to operate.

Consider a factory producing cars. It can assemble the car in various ways (a) by having a team of workers (labour) do the work, (b) installing robots on an assembly line, or (c) some combination of the two. If the price of electricity doubles then having a bunch of robots on the factory floor suddenly becomes more expensive because the energy needed to run them became more costly. Thus, the guy who was just indifferent between having a robot and a worker will now strictly prefer to hire a worker and the demand for capital decreases.

Even if you have a business (e.g. long-distance transportation) where you cannot easily substitute capital for labour, there will buy some entrepreneur who is indifferent between buying some capital goods (e.g. planes) to start a business and not. An increase in the cost of energy (e.g. fuel) will reduce the prospective profitability of his firm and thus leave him strictly preferring not to invest.

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Energy and capital are complements. Would be good if you give an example of the situation that puzzled you.

There is a growing list of empirical papers that attempt to estimate elasticity of substitution between energy and capital stock.

Energy and capital stock are complements - you need minimum energy to operate capital stock.

Examples:

  • Transition to coal in the British Industrial Revolution lead to the development of coal-powered machinery. As price of coal declined over time, the demand for coal-driven machinery increased. More and more businesses replaced their horse-driven or charcoal powered capital stock with coal-using machines. Econ textbooks explain this situation using Cross Price Elasticity.
  • In the 1970s oil price hikes meant businesses were not able to run their oil-driven machinery (also think of cars in this category). Not sure oil price hikes induced any price hikes in capital stock at the time. Quite the opposite.
  • Electric cars. As the price of energy (i.e. stored in electric batteries) is going down, demand for electric cars is on the rise. Clear case of complentarity.

Hope this helps.

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One explanation I can think of is that it takes energy to create many forms of capital. In order to build a skyscraper (developing real estate capital), it takes energy to mine iron and other minerals, to forge steel and form pieces of the building, to transport those materials, etc.

Not an explanation informed by any studies, just trying to use my intuition.

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If you take "energy" as an alternative for "labor," it can then be a complementary factor of production with capital.

Most factor substitution models deal with labor versus capital. Capital is just a fixed "stock" of machinery, etc., while labor is the source of the energy that makes these machines "run."

So yes, if you are treating "energy" is a special case of "labor," then it can complement capital.

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