Solow residual
Encyclopedia
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
.
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.
where:
To measure or predict the change in output within this model, the equation above is differentiated
in time (t), giving a formula in partial derivative
s 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.
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.
, 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.
by doing".
This equation is a "Cobb-Douglas function
", which is used more often than any other output relationship because of its analytical tractability, and because in the long-run, the precise relationship of capital to labour in the production function is not important. The same results can be derived with greater hardship using any production function having constant returns to scale
(and satisfying the technical Inada conditions
.)
Number
A number is a mathematical object used to count and measure. In mathematics, the definition of number has been extended over the years to include such numbers as zero, negative numbers, rational numbers, irrational numbers, and complex numbers....
describing empirical productivity
Productivity
Productivity is a measure of the efficiency of production. Productivity is a ratio of what is produced to what is required to produce it. Usually this ratio is in the form of an average, expressing the total output divided by the total input...
growth in an economy
Economy
An economy consists of the economic system of a country or other area; the labor, capital and land resources; and the manufacturing, trade, distribution, and consumption of goods and services of that area...
from year to year and decade to decade. Robert Solow
Robert Solow
Robert Merton Solow is an American economist particularly known for his work on the theory of economic growth that culminated in the exogenous growth model named after him...
defined rising productivity as rising output
Output (economics)
Output in economics is the "quantity of goods or services produced in a given time period, by a firm, industry, or country," whether consumed or used for further production.The concept of national output is absolutely essential in the field of macroeconomics...
with constant capital
Capital (economics)
In economics, capital, capital goods, or real capital refers to already-produced durable goods used in production of goods or services. The capital goods are not significantly consumed, though they may depreciate in the production process...
and labor input. It is a "residual
Errors and residuals in statistics
In statistics and optimization, statistical errors and residuals are two closely related and easily confused measures of the deviation of a sample from its "theoretical value"...
" because it is the part of growth that cannot be explained through capital accumulation
Capital accumulation
The accumulation of capital refers to the gathering or amassing of objects of value; the increase in wealth through concentration; or the creation of wealth. Capital is money or a financial asset invested for the purpose of making more money...
or the accumulation of other traditional factors, such as land or labor. The Solow Residual is procyclical
Procyclical
Procyclical is a term used in economics to describe how an economic quantity is related to economic fluctuations. It is the opposite of countercyclical. However, it has more than one meaning.-Meaning in business cycle theory:...
and is sometimes called the rate of growth of total factor productivity
Total factor productivity
In economics, total-factor productivity is a variable which accounts for effects in total output not caused by inputs. If all inputs are accounted for, then total factor productivity can be taken as a measure of an economy’s long-term technological change or technological dynamism.If all inputs...
.
History
In the 1950s, many economists undertook comparative studies of economic growth following World War IIWorld War II
World War II, or the Second World War , was a global conflict lasting from 1939 to 1945, involving most of the world's nations—including all of the great powers—eventually forming two opposing military alliances: the Allies and the Axis...
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
Depreciation
Depreciation refers to two very different but related concepts:# the decrease in value of assets , and# the allocation of the cost of assets to periods in which the assets are used ....
, it was a widespread view of the correct industrial policy to adopt. Many economists pointed to the Soviet
Soviet Union
The Soviet Union , officially the Union of Soviet Socialist Republics , was a constitutionally socialist state that existed in Eurasia between 1922 and 1991....
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
Exogenous growth model
The neoclassical growth model, also known as the Solow–Swan growth model or exogenous growth model, is a class of economic models of long-run economic growth set within the framework of neoclassical economics...
' 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 capitalCapital (economics)In economics, capital, capital goods, or real capital refers to already-produced durable goods used in production of goods or services. The capital goods are not significantly consumed, though they may depreciate in the production process...
in the productive economy - which might be measured through the combined value of all companies in a capitalistCapitalismCapitalism is an economic system that became dominant in the Western world following the demise of feudalism. There is no consensus on the precise definition nor on how the term should be used as a historical category...
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 unemploymentUnemploymentUnemployment , as defined by the International Labour Organization, occurs when people are without jobs and they have actively sought work within the past four weeks...
effects, assuming instead that the labour force is a constant fraction of an expanding population. - A(t) represents multifactor productivityMultifactor productivityMultifactor productivity measures the changes in output per unit of combined inputs. In the United States, Indices of MFP are produced for the private business, private nonfarm business, and manufacturing sectors of the economy...
(often generalized as "technologyTechnologyTechnology is the making, usage, and knowledge of tools, machines, techniques, crafts, systems or methods of organization in order to solve a problem or perform a specific function. It can also refer to the collection of such tools, machinery, and procedures. The word technology comes ;...
"). 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
Derivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
in time (t), giving a formula in partial derivative
Partial derivative
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant...
s 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 yearsTime series
In statistics, signal processing, econometrics and mathematical finance, a time series is a sequence of data points, measured typically at successive times spaced at uniform time intervals. Examples of time series are the daily closing value of the Dow Jones index or the annual flow volume of the...
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
Regression analysis
In statistics, regression analysis includes many techniques for modeling and analyzing several variables, when the focus is on the relationship between a dependent variable and one or more independent variables...
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
Simple linear regression
In statistics, simple linear regression is the least squares estimator of a linear regression model with a single explanatory variable. In other words, simple linear regression fits a straight line through the set of n points in such a way that makes the sum of squared residuals of the model as...
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 productivityTotal factor productivity
In economics, total-factor productivity is a variable which accounts for effects in total output not caused by inputs. If all inputs are accounted for, then total factor productivity can be taken as a measure of an economy’s long-term technological change or technological dynamism.If all inputs...
, but is normally attached to the labour variable in the macroeconomy because return on investment
Return on investment
Return on investment is one way of considering profits in relation to capital invested. Return on assets , return on net assets , return on capital and return on invested capital are similar measures with variations on how “investment” is defined.Marketing not only influences net profits but also...
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 "learningLearning
Learning is acquiring new or modifying existing knowledge, behaviors, skills, values, or preferences and may involve synthesizing different types of information. The ability to learn is possessed by humans, animals and some machines. Progress over time tends to follow learning curves.Human learning...
by doing".
See also
- Solow computer paradox is based on finding a zero residual in many countries even as information technologyInformation technologyInformation technology is the acquisition, processing, storage and dissemination of vocal, pictorial, textual and numerical information by a microelectronics-based combination of computing and telecommunications...
was becoming more widely available. - Capital controversy over whether the level of capital in an economy can be measured even in theory; if not, neither can the Solow residual.
- The Solow growth model is a model of economic development into which the Solow residual can be added exogenousExogenousExogenous refers to an action or object coming from outside a system. It is the opposite of endogenous, something generated from within the system....
ly to allow predictions of GDP growth at differing levels of productivity growth. - The Balassa–Samuelson effect describes the effect of variable Solow residuals: it assumes that mass-produced traded goods have a higher residual than does the service sector. This assumption has been used to explain the PPP-deviationsPenn effectThe Penn effect is the economic finding associated with what became the Penn World Table that real income ratios between high and low income countries are systematically exaggerated by gross domestic product conversion at market exchange rates...
, and may create a 'drag' on the overall-residual as more effort is moved into services industries precisely because they have low productivity growth (being harder to automate.) - Multifactor productivityMultifactor productivityMultifactor productivity measures the changes in output per unit of combined inputs. In the United States, Indices of MFP are produced for the private business, private nonfarm business, and manufacturing sectors of the economy...
External links
- Does the Solow Residual for Korea Reflect Pure Technology Shocks? - a paper showing how modern econometric techniques such as cointegrationCointegrationCointegration is a statistical property of time series variables. Two or more time series are cointegrated if they share a common stochastic drift.-Introduction:...
are being used to draw a more reliable inference on the Solow residual, because the real world is not like the smoothly evolving model described in the simple regression here.
Footnotes
1This equation is a "Cobb-Douglas function
Cobb-Douglas
In economics, the Cobb–Douglas functional form of production functions is widely used to represent the relationship of an output to inputs. Similar functions were originally used by Knut Wicksell , while the Cobb-Douglas form was developed and tested against statistical evidence by Charles Cobb and...
", which is used more often than any other output relationship because of its analytical tractability, and because in the long-run, the precise relationship of capital to labour in the production function is not important. The same results can be derived with greater hardship using any production function having constant returns to scale
Returns to scale
In economics, returns to scale and economies of scale are related terms that describe what happens as the scale of production increases in the long run, when all input levels including physical capital usage are variable...
(and satisfying the technical Inada conditions
Inada conditions
In macroeconomics, the Inada conditions are assumptions about the shape of a production function that guarantee the stability of an economic growth path in a neoclassical growth model....
.)