How is productivity calculated?

Question

I mean in macroeconomics, and I am looking for actual formulas.

I am not asking how an individual enterprise calculates its productivity or producivity of its employees but how to calculate it top down from national economic data.

Answer 1

Quality and quantity of product output.

Answer 2

Quantity and time less rejects.

Difficult to do for services.

Answer 3

Productivity isn’t everything but in the long run it is almost everything. Productivity improvement is the growth in a nation's output over and above that explained by growth in the inputs to production. Measures of productivity growth are important to understanding long-term improvements in any nations’ living standards and changes in the nation's international competitiveness. For some years, experts have published a variety of productivity estimates for the 'market' sector of the economy. Indeed, it is impossible to divulge the calculation of labor productivity from the national income:
• labour productivity, estimated by dividing an index of real output by an index of labour input;
• capital productivity, estimated by dividing an output index by an index of capital input; and
• multifactor productivity (MFP), estimated by dividing an output index by an index of labour and capital inputs combined
In actuality, economists have not developed any standard macroeconomic way of calculating productivity. This is simply because of the peculiarities involved in macroeconomic calculations. Consequently, when it comes to calculating such vital, but sensitive things like productivity, it is best to leave the calculation at the micro level of individual labour, enterprise, on employer. For clarity however, when GNP of two years are taken, assuming that factor inputs remains unchanged, productivity is measured by subtracting the current GNP from the previous GNP, that is;
Yc-Yp =C+ I+ G+ (NX) – C +I +G + (NX)
Where;
Yc is the national income in the current period while Yp is the national income in the previous period.
In general, productivity is a measure of the efficiency with which one or more inputs are used in the output of goods and services. In Krugman’s point, productivity growth solves a multitude of problems. For example, standards of living rise because profit-maximizing firms can afford to pay higher real wages to workers who are more productive. And budget deficits fall and even turn into budget surpluses as productivity-enhanced growth of output raises tax revenues.
So far we have defined productivity in a general way. Now we turn to two measures of productivity that economists uses and that government agencies track. In Nigeria we have the ministry of labour and productivity and the Federal Office Statistics with and established data bank on Productivity while in the United States, the Bureau of Labor Statistics is the agency that measures and reports on labor productivity and multifactor productivity, also called total factor productivity.
As the name indicates, labor productivity relates output to a single input, labor. That is, labor productivity is output per unit of labor. Multifactor productivity, on the other hand, relates output to a combined set of two or more inputs. In other words, multifactor productivity is output per unit of some combined inputs. To explain these two different measures of productivity, we introduce the most basic and commonly used macroeconomic model of the production process.
In any period of time, in say a year, the firms in an economy produce thousands, or even millions of different goods and services. These agents utilize a variety of factors of factors of production, that is, several kinds of labor, plant and equipment, land, and raw materials. The most common production model for the entire economy assumes that there is a single good called real GDP (Y) and that two factors of production, labor (N) and capital (K), are needed to produce this single good. The production process can be summarized by the production function, which relates output to inputs:

Y = F(N,K, A) (1)
+ + +
The production function tells us that output, Y, depends on the input of labor, the input of capital, and a collective variable, denoted by A. The plus sign under each variable indicates that output variables relates positively with that input.
First, other things being equal, when the input of labor increases, output increases. And when the input of labor decreases, output decreases. In symbols:

Given K and A, N↑ → Y↑
Given K and A, N↓ → Y↓.

Second, with labor and the collective variable held constant, output increases when the input of capital increases and decreases when capital decreases. In symbols:

Given N and A, K↑ → Y↑
Given N and A, K↓ → Y↓.

The third variable, the collective variable, represents all the other factors that affect output, such as the state of technology and management practices. The plus sign under A means that we use the following convention: Factors other than labor and capital that increase output, such as technological progress, we represent by an increase in A. And factors other than labor and capital that decrease output, such as technological deterioration, we represent by a decrease in A. In symbols:

Given N and K, A↑ → Y↑
Given N and K, A↓ → Y↓.


In the production function, A captures the efficiency with which labor and capital are used, which is called multifactor productivity, or total factor, productivity. In other words, multifactor productivity is output per combined set of inputs, in this case, labor and capital. Labor productivity, on the other hand, is the ratio of the output of goods and services to the labor hours devoted to the production of that output.
Labor Productivity

Labor productivity is easier to measure than multifactor productivity because it is observable. Equation 2 represents in symbols labor productivity, which is output per unit of labor:

Labor productivity = Y/N (2)


It follows that the growth of labor productivity equals the growth of GDP minus the growth of the labor force; that is,

Growth of labor productivity = ∆Y/Y – ∆N/N. (3)

Output per hour of all persons--labor productivity--is the most commonly used productivity measure. Labor is an easily-identified input to virtually every production process. In terms of cost, it represents about two-thirds of the value produced
As we have seen, an alternative measure of productivity is multifactor productivity, which is output per unit of some combined inputs, such as labor and capital.

Although labor productivity is the ratio of the output of goods and services to the labor hours devoted to the production of that output, Multifactor productivity relates output to a combination of inputs used in the production of that output, such as labor and capital or labor, capital, energy and materials. Capital includes equipment, structures, inventories, and land.
The most commonly used multifactor productivity measure is for the private business sector of the economy. This sector essentially measures the for-profit sector of the economy and it is the broadest sector for which multifactor productivity measures are available.
Because satisfactory capital measures are unavailable for government enterprises, government enterprises are excluded when multifactor productivity is calculated. The labor productivity measures include government enterprises.
Because multifactor productivity is not directly observable, it is measured indirectly. In 1957, Robert Solow of MIT introduced a method to calculate the rate of growth of multifactor productivity as a residual. Strictly speaking, changes in multifactor productivity embody the effects of changes in all factors other than the combined inputs. The state of technology, of course, is one of the key other factors. Hence, changes in multifactor productivity are often used as a measure of the rate of technological progress. In 1987, Solow received the Nobel Prize in economics for his work, which is still used to calculate the rate of growth of multifactor productivity.
An assumption that underlies Solow’s calculation is that each factor is paid its marginal product. Given this assumption, the Solow residual, which is the rate of growth of multifactor productivity, is given by the following expression:

Rate of growth of multifactor productivity=growth of GDP–
Labor’s share in national income x growth of labor force
–capital’s share in national income x growth of capital stock (4)

We note that labor’s share in national income equals total wages in nominal terms divided by nominal national income. Since there are only two factors of production, labor and capital, in this model, it follows that capital’s share in national income equals (1-labor’s share in national income). Historically, capital’s share in national income has been slightly less than one-third of total factor payments.