Capital goods are goods such as factories, machines, and trucks. They are used for the production of other goods and are not completely used up in the production process. Economies build up their capital stocks by devoting some of their gross domestic product (GDP) to new capital goods—that is, investment. As we saw in our discussion of Solovenia in Section 6.2 "Four Reasons Why GDP Varies across Countries", if a country does not interact much with other countries (that is, it is a closed economy) the amount of investment reflects savings within a country. In open economies, the amount of investment reflects the perceived benefits to investment in that country compared to other countries.
Capital goods wear out over time and have to be scrapped and replaced. A simple way to think about this depreciation is to imagine that a fraction of the capital stock wears out every year. A reasonable average depreciation rate for the US economy is 4 or 5 percent. To understand what this means, think about an economy where the capital stock consists of a large number of identical machines. A depreciation rate of 5 percent means that for every 100 machines in the economy, 5 machines must be replaced every year. [***The depreciation rate can be understood in terms of the average lifetime of a typical machine. For example, a depreciation rate of 5 percent is the same as saying that, on average, machines last for 20 years. To see this, imagine that capital stock is kept constant at 100 machines, and each machine lasts for 20 years. Imagine also that 5 machines are 1 year old, 5 machines are 2 years old, and so forth, with the oldest 5 machines being 20 years old. Each year, these 5 oldest machines would wear out (5 percent depreciation) and have to be replaced by 5 new machines. After a year has passed, the situation will be exactly the same as the previous year: there will be 5 machines that are 1 year old, 5 machines that are 2 years, and so forth. Mathematically, we are saying that the lifetime of a machine = 1/depreciation rate: 20 = 1/0.05. ***]
The depreciation of capital goods reduces the capital stock. The total amount of capital lost to depreciation each year is calculated by multiplying the depreciation rate and the capital stock together. If the capital stock is $30 trillion, for example, and the depreciation rate is 5 percent, then $1.5 trillion (= $30 trillion × 0.05) worth of capital is lost each year.
The capital stock increases as long as there is enough new investment to replace the worn out capital and still contribute some extra. The overall change in the capital stock is equal to new investment minus depreciation:
change in capital stock = new investment − depreciation rate × capital stock.
For example, suppose that the current capital stock (measured in trillions of dollars) is 40, and the depreciation rate is 10 percent per year. Then the capital stock after depreciation is 40 − (.1 × 40) = 40 − 4 = 36. Suppose that new investment is $4.8 trillion. Then the new capital stock is 36 + 4.8 = 40.8. In this case, capital stock has increased by $0.8 trillion, or 2 percent.
The equation for the change in the capital stock is one of the fundamental ingredients of economic growth. It tells us that economies build up their capital stock—and therefore their real GDP—by devoting enough output to new investment to both replace worn out capital and then add some more. If we divide both sides of the previous equation by the capital stock, we can obtain the growth rate of the capital stock. (Remember that the growth rate of a variable is the change in the variable divided by its initial level.)
The growth rate of the capital stock depends on three things:
- The amount of investment.The more investment the economy carries out, the more quickly the capital stock grows.
- The current capital stock.The larger the capital stock, other things being equal, the lower its growth rate.
- The depreciation rate.If existing capital wears out faster, the capital stock grows more slowly.
It is intuitive that a higher investment rate increases the growth rate of the capital stock, and a higher depreciation rate decreases the growth rate of the capital stock. It is less obvious why the growth rate of the capital stock is lower when the capital stock is higher. The growth rate measures the change in the capital stock as a percentage of the existing capital stock. A given change in the capital stock results in a smaller growth rate if the existing capital stock is larger. For example, suppose that the current capital stock is 100, and the change in the capital stock is 10. Then the growth rate is 10 percent. But if the current capital stock is 1,000, then the same change of 10 in the capital stock represents only a 1 percent growth rate.
Toolkit: Section 16.11 "Growth Rates"
The toolkit contains more information on how growth rates are calculated.
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