We introduce some mathematical notation to describe Josh's lawnmowing business. In particular, let

K be the number of lawnmowers
L be the number of workers
Y be the number lawns per day that can be mowed using those inputs

(To keep things simple, we will omit the need for the pickup truck and other inputs.) As economists, we are particularly interested in the ratio of output per worker, or Y/L. This ratio, called labor productivity, is likely to depend on the ratio of capital (lawnmowers) per worker, or K/L. We say that there is a production function

Y/L = f(K/L)

This can be read as "the ratio of output to labor is a function of the ratio of capital to labor." If we raise the number of lawnmowers per worker, we increase output per worker. That is because the more lawnmowers we have, the more backup we have in case a lawn mower breaks down. More lawnmowers per worker means less time that workers have to spend idle while waiting for a working lawnmower.

The properties of returns to scale and substitution depend on the characteristics of the function, f(). Mathematically, the simplest function is a constant multiplied by the capital/labor ratio. That is, the simplest function would be something like Y/L = 8K/L. This would say that the number of lawns each worker mows is equal to 8 times the ratio of lawnmowers to worker.

Unfortunately, this constant function cannot be realistic. It says that if you start with one lawnmower per worker, then each worker can mow 8 lawns, which might be accurate. But then if you have two lawnmowers per worker, the constant function would say that each worker can then mow 16 lawns. If you have ten lawnmowers per worker, then each worker can mow 80 lawns! This is absurd. The problem is that a constant function violates the law of diminishing returns.

To obtain diminishing returns, we can use a function with an exponent that is less than one. For example, we could have

Y/L = 8(K/L)^0.25

If this were the production function, then with one lawnmower per worker each worker can again mow eight lawns. However, now if we were to have two lawnmowers per worker, instead of doubling productivity we only increase it to 9.5 lawns per worker. This smaller increase is more realistic. That is, it is more realistic to estimate that doubling the ratio of lawnmowers to workers results in some increase in productivity, but the increase is quite a bit less than double.

An extreme form of diminishing returns is to set the exponent equal to zero, so that we have something like Y/L = 8, regardless of the number of lawnmowers, as long as there is at least one lawnmower per worker. This type of production function, called a fixed-coefficient production function, gives us no opportunities to substitute capital for labor. This is very unrealistic for a complex economy. Nonetheless, environmentalists have on many occasions made predictions that we will run out of resources (such as oil, or fresh water), and these predictions are based implicitly on a model of zero substitution. Because that model is not realistic, the predictions have been badly off base.

We want to keep the concept of a production function as we change perspective from looking at Josh's lawn mowing busines to looking at the economy as a whole. For the economy as a whole, however, there are many types of output besides lawn mowing services. Also, there are many types of capital goods besides lawnmowers and pickup trucks--including office buildings, factory equipment, airplanes, and other durable goods. Finally, there are many types of labor, from unskilled workers to brain surgeons.

Economists use a process called aggregation to come up with a single measure of output that summarizes all of the different goods and services produced in the economy. Think of aggregation as taking a weighted average of lawns mowed, apples bought, movies rented. The weights are closely related to the relative prices of the goods. That is, an expensive surgery will have a higher weight in the aggregation process than an inexpensive pen. The aggregate measure of output is called real gross domestic product, or real GDP.

Similarly, economists take a weighted average of the number of lawnmowers, office buildings, and other capital goods to arrive at a measure of the aggregate stock of capital. One complicating factor in measuring the capital stock is computing the rate of depreciation. A drill press purchased in 1995 will have lost some of its value by now. A computer purchased in 1995 will have lost nearly all of its value by now.

In theory, economists could contruct a weighted average measure of labor input, in which a brain surgeon gets higher weight than someone with only a high school education. However, we leave the measure of labor unweighted. We do so because we are interested in average output per worker (unweighted), in order to compare across countries and to measure improvement over time.

Economists also draw a distinction between stocks and flows. In this case, a stock does not refer to the securities traded on the stock market. It means any quantity that is measured at a snapshot at a point in time. In contrast, a flow measures a quantity used or produced within a period of time. Josh could mow the Millers' lawn five times over the course of the season. We would say that the Millers' lawn represents a stock of one lawn. However, Josh's mowing represents a flow of five units of lawn mowing services.

Output and labor input are flows. The aggregate measure of real GDP is output per year. When we talk about labor input, we talk about number of workers per year. We can measure labor productivity by dividing output per year by the number of workers employed in that year. Alternatively, we can measure labor input as total hours worked, and divide this into output to obtain output per hour.

The aggregate measure of capital is a stock. We measure the capital stock as of a point in time, such as the end of 2001.

The stock of capital is related to the flow of investment. The change in the aggregate stock of capital goods between the end of 2000 and the end of 2001 is equal to the amount of capital goods produced in 2001 minus the depreciation of the capital stock that was in place at the end of 2000. Both production and depreciation of capital goods are flows. In algebraic terms, we can write

K₂₀₀₁ - K₂₀₀₀ = I₂₀₀₁ - dK₂₀₀₀

Using the Production Function to Choose K/L

Suppose that we have a business where workers are paid $100 a day and each unit of capital equipment costs $10 a day to lease. Suppose that the production function is

Y/L = 6(K/L)^0.2

where Y is the number of units of output that we have to produce each day. If we typically sell 25 units of output per day, how many workers and how many units of capital should we use?

Suppose we try using one worker. Then Y/L has to at least equal 25. If we use 100 units of capital, then Y/L is only 15. It turns out that with one worker we need 1255 units of capital to bring Y/L up to 25. Using one worker and 1255 units of capital costs $100 + (1255)($10) = $12,650.

Next, we try using two workers. Now, we need to bring Y/L up to 25/2 = 12.5 It turns out that we need 79 units of capital to do this. Using 2 workers and 79 units of capital costs (2)($100) + (79)($10) = $990.

Next, we try using three workers. With 3 workers, we would need 16 units of capital. This combination costs 3($100) + 16($10) = $460.

Next, we try using four workers. This requires 5 units of capital, for a cost of 4($100) + 5($10) = $450.

If we use five workers, we know that the cost will be at least $500. (Why?)