잔여성장, 요소생산성

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생산성연구

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Solow residual

The Solow residual is a number describing empirical productivity growth in an economy from year to year and decade to decade. Robert Solow defined rising productivity as rising output with constant capital and labor input. It is a "residual" because it is the part of growth that cannot be explained through capital accumulation or the accumulation of other traditional factors, such as land or labor. The Solow Residual is procyclical and is sometimes called the rate of growth of total factor productivity.

In the 1950s, many economists undertook comparative studies of economic growth following World War II reconstruction. Some said that the path to long-term growth was achieved through investment in industry and infrastructure and in moving further and further into capital intensive automated production. Although there was always a concern about diminishing returns to this approach because of equipment depreciation, it was a widespread view of the correct industrial policy to adopt. Many economists pointed to the Soviet command economy as a model of high-growth through tireless re-investment of output in further industrial construction.

However, some economists took a different view: they said that greater capital concentrations would yield diminishing returns once the marginal return to capital had equalized with that of labour—and that the apparently rapid growth of economies with high savings rates would be a short-term phenomenon. This analysis suggested that improved labour productivity or total factor technology was the long-run determinant of national growth, and that only under-capitalized countries could grow per-capita income substantially by investing in infrastructure—some of these undercapitalized countries were still recovering from the war and were expected to rapidly develop in this way on a path of convergence with developed nations.

The Solow residual is defined as per-capita economic growth above the rate of per-capita capital stock growth, so its detection indicates that there must be some contribution to output other than advances in industrializing the economy. The fact that the measured growth in the standard of living, also known as the ratio of output to labour input, could not be explained entirely by the growth in the capital/labour ratio was a significant finding, and pointed to innovation rather than capital accumulation as a potential path to growth.

The 'Solow growth model' is not intended to explain or derive the empirical residual, but rather to demonstrate how it will affect the economy in the long run when imposed on an aggregate model of the macroeconomy exogenously. This model was really a tool for demonstrating the impact of “technology” growth as against “industrial” growth rather than an attempt to understand where either type of growth was coming from. The Solow residual is primarily an observation to explain, rather than the outcome of a theoretical analysis. It is a question rather than an answer, and the following equations should not obscure that fact.

As a residual term in the Solow model

Solow assumed a very basic model of annual aggregate output over a year (t). He said that the output quantity would be governed by the amount of capital (the infrastructure), the amount of labour (the number of people in the workforce), and the productivity of that labour. He thought that the productivity of labour was the factor driving long-run GDP increases. An example economic model of this form is given below1:

where:

• Y(t) represents the total production in an economy (the GDP) in some year, t.

• K(t) is capital in the productive economy - which might be measured through the combined value of all companies in a capitalist economy.

• L(t) is labour; this is simply the number of people in work, and since growth models are long run models they tend to ignore cyclical unemployment effects, assuming instead that the labour force is a constant fraction of an expanding population.

• A(t) represents multifactor productivity (often generalized as "technology"). The change in this figure from A(1960) to A(1980) is the key to estimating the growth in labour 'efficiency' and the Solow residual between 1960 and 1980, for instance.

To measure or predict the change in output within this model, the equation above is differentiated in time (t), giving a formula in partial derivatives of the relationships: labour-to-output, capital-to-output, and productivity-to-output, as shown:

Observe:

Similarly:

Therefore:

The growth factor in the economy is a proportion of the output last year, which is given (assuming small changes year-on-year) by dividing both sides of this equation by the output, Y:

The first two terms on the right hand side of this equation are the proportional changes in labour and capital year-on-year, and the left hand side is the proportional output change. The remaining term on the right, giving the effect of productivity improvements on GDP is defined as the Solow residual:

The residual, SR(t) is that part of growth not explicable by measurable changes in the amount of capital, K, and the number of workers, L. If output, capital, and labour all double every twenty years the residual will be zero, but in general it is higher than this: output goes up faster than growth in the input factors. The residual varies between periods and countries, but is almost always positive in peace-time capitalist countries. Some estimates of the post-war U.S. residual credited the country with a 3% productivity increase per-annum until the early 1970s when productivity growth appeared to stagnate.

Regression analysis and the Solow residual

The above relation gives a very simplified picture of the economy in a single year; what growth theory econometrics does is to look at a sequence of years to find a statistically significant pattern in the changes of the variables, and perhaps identify the existence and value of the "Solow residual". The most basic technique for doing this is to assume constant rates of change in all the variables (obscured by noise), and regress on the data to find the best estimate of these rates in the historical data available (using an Ordinary least squares regression). Economists always do this by first taking the natural log of their equation (to separate out the variables on the right-hand-side of the equation); logging both sides of this production function produces a simple linear regression with an error term, :

A constant growth factor implies exponential growth in the above variables, so differentiating gives a linear relationship between the growth factors which can be deduced in a simple regression.

In regression analysis, the equation one would estimate is

where:

y is (log) output, ln(Y)

k is capital, ln(K)

ℓ is labour, ln(L)

C can be interpreted as the co-efficient on log(A) – the rate of technological change – (1 − α).

Given the form of the regression equation, we can interpret the coefficients as elasticities.

Why the productivity growth is attached to labour

The Solow residual measures total factor productivity, but is normally attached to the labour variable in the macroeconomy because return on investment doesn't seem to change very much in time or between developing nations, and developed nations—not nearly as much as human productivity seems to change, anyway.

Critique of the measurement in rapidly developing economies

Rapidly expanding countries (catching up after a crisis or trade liberalization) tend to have a rapid turn-over in technologies as they accumulate capital. It has been suggested that this will tend to make it harder to gain experience with the available technologies and that a zero Solow residual in these cases actually indicates rising labour productivity. In this theory, the fact that A (labour output productivity) is not falling as new skills become essential indicates that the labour force is capable of adapting, and is likely to have its productivity growth underestimated by the residual—This idea is linked to "learning by doing".

Multifactor productivity

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Multifactor productivity (MFP) measures the changes in output per unit of combined inputs. Indexes of MFP are produced for the private business, private nonfarm business, and manufacturing sectors of the economy. MFP is also developed for 2-and 3-digit Standard Industrial Classification (SIC) thru 1987, and NAICS (North Atlantic Industrial Classification System) thru 2005 for manufacturing industries, the railroad transportation industry, the air transportation industry, and the utility and gas industry.

Multifactor productivity measures reflect output per unit of some combined set of inputs. A change in multifactor productivity reflects the change in output that cannot be accounted for by the change in combined inputs. As a result, multifactor productivity measures reflect the joint effects of many factors including new technologies, economies of scale, managerial skill, and changes in the organization of production.

Whereas labor productivity measures the output per unit of labor input, multifactor productivity looks at a combination of production inputs (or factors): labor, materials, and capital. In theory, it’s a more comprehensive measure than labor productivity, but it’s also more difficult to calculate.

(1)

(2)

Multi-factor productivity is the same as total factor productivity, a certain type of Solow residual.

where:

• f is the global production function;

• Y is output;

• t is time;

• sL is the share of input costs attributable to labor expenses;

• sK is the share of input costs attributable to capital expenses;

• L is a dollar quantity of labor;

• K is a dollar quantity of capital.

Total factor productivity

In economics, total-factor productivity (TFP) is a variable which accounts for effects in total output not caused by inputs. If all inputs are accounted for, then total factor productivity (TFP) can be taken as a measure of an economy’s long-term technological change or technological dynamism.

If all inputs are not accounted for, then TFP may also reflect omitted inputs. For example, a year with unusually good weather will tend to have higher output, because bad weather hinders agricultural output. If a variable like the weather is not considered as an input, then weather will be included in the measure of total-factor productivity.

TFP can not be measured directly. Instead it is a residual, often called the Solow residual, which accounts for effects in total output not caused by inputs.

The equation below (in Cobb–Douglas form) represents total output (Y) as a function of total-factor productivity (A), capital input (K), labor input (L), and the two inputs' respective shares of output (α and β are the capital input share of contribution for K and L respectively). An increase in either A, K or L will lead to an increase in output. While capital and labor input are tangible, total-factor productivity appears to be more intangible as it can range from technology to knowledge of worker (human capital).

Technology Growth and Efficiency are regarded as two of the biggest sub-sections of Total Factor Productivity, the former possessing "special" inherent features such as positive externalities and non-rivalness which enhance its position as a driver of economic growth.

Total Factor Productivity is often seen as the real driver of growth within an economy and studies reveal that whilst labour and investment are important contributors, Total Factor Productivity may account for up to 60% of growth within economies.

•

Criticism

Growth accounting exercises and Total Factor Productivity are open to the Cambridge Critique. Therefore, some economists believe that the method and its results are invalid.

On the basis of dimensional analysis, TFP is criticized as not having meaningful units of measurement.[1] The units of the quantities in the Cobb–Douglas equation are:

• Y: widgets/year (wid/yr)

• L: man-hours/year (manhr/yr)

• K: capital-hours/year (caphr/yr; this raises issues of heterogeneous capital)

• α, β: pure numbers (non-dimensional), due to being exponents

• A: (widgets * yearα + β – 1)/(caphrα * manhrβ), a balancing quantity, which is TFP.

The units of A do not admit a simple economic interpretation, and the concept of TFP is accordingly criticized as a modeling artifact.

Estimation

As a residual, TFP is also dependent on estimates of the other components. A 2005 study[2] on human capital attempted to correct for weaknesses in estimations of the labour component of the equation, by refining estimates of the quality of labour. Specifically, years of schooling is often taken as a proxy for the quality of labour (and stock of human capital), which does not account for differences in schooling between countries. Using these re-estimations, the contribution of TFP was substantially lower.

Robert Ayres and Benjamin Warr have found that the model can be improved by using the efficiency of electrical generation, which roughly tracks technological progress.[3][4]