Economists have measured large differences in GDP per capita over time and across countries. Our first impulse is to interpret this data using the production function, which relates per capita output to the capital/labor ratio. If the exponent is 0.25, then this function is

(Y/L) = (K/L)^0.25

Suppose that we are interested in the percentage difference in per capita GDP between two points in time or between two countries. Mathematically, percentage differences behave rather like logarithms. If we were to take the logs of both sides of the production function, we would have

log(Y/L) = .25 log(K/L)

Thinking of this as an equation in percentage changes, it says that for every one percentage point difference in the capital-labor ratio, we should get a .25 percentage point difference in output per worker. Conversely, if we observe that one country has 10 percent higher output per worker than another country, then we would expect the more productive country to have 40 percent more capital per worker.

In theory, differences in the capital-labor ratio should explain all of the differences in output per worker. There is nothing else in the equation.

The capital-labor ratio certainly is important. Countries increase this ratio through capital accumulation. This means that a large share of output goes to investment, which helps to increase the stock of capital. DeLong has a chart in his text which demonstrates that most of the countries with high rates of labor productivity have investment shares of more than twenty percent of output. Conversely, the majority of low-productivity countries have investment rates below twenty percent.

However, differences in the capital-labor ratio can explain no more than half of differences in output per worker. This is true whether you are trying to explain output per worker over time in one country or you are trying to explain differences in output per worker across different countries.

Another way of putting this is that the differences in output per worker are larger than what you would predict on the basis of the capital-labor ratio. In the United States, growth in output per worker has been faster than what one have predicted based on the increase in the capital-labor ratio. Moreover, the difference between per capita output in the U.S. and that in other countries is larger than what one would predict on the basis of differences in the capital-labor ratio.

This phenomenon of unexplained differences in output per worker was first discovered in the 1950's, and dubbed "the residual." The residual is so important that we need to find a place for it in the production function. DeLong's Macroeconomics textbook calls it E, the efficiency of labor. Using this formulation, the production function is

(Y/L) = (K/L)^0.25E^0.75

Efficiency of Labor and Growth Accounting

This new construct, the efficiency of labor, gives us another element in the equation. Growth in output per worker is explained as a weighted average of the growth in capital per worker and growth in the efficiency of labor. Taking logs of both sides of the new production function gives

log(Y/L) = .25 log(K/L) + .75 log(E)

Now, we have an equation that says that economic growth is a weighted average of growth in the capital-labor ratio and growth in the efficiency of labor. Keep in mind that the efficiency of labor is not a number you can look up in the Economic Report of the President or other compendium of government statistics. It is whatever is needed to enable a production function to fit the observed data on output per worker and capital per worker.

Having coined the term "efficiency of labor," economists are obligated to produce some analysis of what determines it. Some plausible factors include:

• education per worker
• knowledge
• economic, political, and social systems

Of these factors, the only one that has a ready scale of measurement is education. In fact, some of the differences in the efficiency of labor across time and across countries can be explained by differences in the average years of schooling per worker. However, education does not explain enough to make us comfortable that it is the overwhelming factor that determines E.

Knowledge is an important factor in explaining differences in E over time. We simply know things today that we did not know years ago. For example, even if we lost all of our medical equipment and our doctors, we would still know much more about sanitation and health than people did hundreds of years ago.

Some of our knowledge is scientific and technical. Other knowledge is more prosaic. When you start a new job, you typically are given a formal orientation, company manuals, and help from senior employees who through trial and error have learned better ways of doing the work. All of this knowledge, from abstract science to everyday experience, contributes to E.

Some knowledge is in the public domain, and some knowledge is proprietary. Most scientific knowledge is available to anyone who can understand it. However, other knowledge, from the formula for Coke to the source code for Microsoft software, is considered a secret by its corporate owners.

Because most knowledge is in the public domain, knowledge does not provide a promising explanation for variations in E across countries. Even proprietary knowledge is not limited to a single country. For example, Coke has manufacturing plants throughout the world, so that its secret formula is used by workers everywhere.

When we attempt to explain differences in the efficiency of labor in different countries, economists almost inevitably are forced to focus on differences in economic, political, and social systems. The contrast that DeLong draws between output per worker in neighboring pairs of Communist and non-Communist countries certainly underlines this issue.

The production function provides a framework for accounting for growth. It leads to an approach that subdivides growth into two components--the capital-labor ratio and the efficiency of labor.

The efficiency of labor is constructed indirectly, based on the residual that results from trying explain differences in output per worker on the basis of differences in the capital-labor ratio. Economists believe that the efficiency of labor is affected by education, knowledge, and the social system.