Solow–Swan model


The Solow–Swan model or exogenous growth model is an economic model of long-run economic growth. It attempts to explain long-run economic growth by looking at capital accumulation, labor or population growth, and increases in productivity largely driven by technological progress. At its core, it is an aggregate production function, often specified to be of Cobb–Douglas type, which enables the model "to make contact with microeconomics". The model was developed independently by Robert Solow and Trevor Swan in 1956, and superseded the Keynesian Harrod–Domar model.
Mathematically, the Solow–Swan model is a nonlinear system consisting of a single ordinary differential equation that models the evolution of the per capita stock of capital. Due to its particularly attractive mathematical characteristics, Solow–Swan proved to be a convenient starting point for various extensions. For instance, in 1965, David Cass and Tjalling Koopmans integrated Frank Ramsey's analysis of consumer optimization, thereby endogenizing the saving rate, to create what is now known as the Ramsey–Cass–Koopmans model.

Background

The Solow–Swan model was an extension to the 1946 Harrod–Domar model that dropped the restrictive assumption that only capital contributes to growth. Important contributions to the model came from the work done by Solow and by Swan in 1956, who independently developed relatively simple growth models. Solow's model fitted available data on US economic growth with some success. In 1987 Solow was awarded the Nobel Prize in Economics for his work. Today, economists use Solow's sources-of-growth accounting to estimate the separate effects on economic growth of technological change, capital, and labor.
The Solow model is also one of the most widely used models in economics to explain economic growth. Basically, it asserts that outcomes on the "total factor productivity can lead to limitless increases in the standard of living in a country."

Extension to the Harrod–Domar model

Solow extended the Harrod–Domar model by adding labor as a factor of production and capital-output ratios that are not fixed as they are in the Harrod–Domar model. These refinements allow increasing capital intensity to be distinguished from technological progress. Solow sees the fixed proportions production function as a "crucial assumption" to the instability results in the Harrod-Domar model. His own work expands upon this by exploring the implications of alternative specifications, namely the Cobb–Douglas and the more general constant elasticity of substitution. Although this has become the canonical and celebrated story in the history of economics, featured in many economic textbooks, recent reappraisal of Harrod's work has contested it. One central criticism is that Harrod's original piece was neither mainly concerned with economic growth nor did he explicitly use a fixed proportions production function.

Long-run implications

A standard Solow model predicts that in the long run, economies converge to their balanced growth equilibrium and that permanent growth of per capita income is achievable only through technological progress. Both shifts in saving and in population growth cause only level effects in the long-run.
An interesting implication of Solow's model is that poor countries should grow faster and eventually catch-up to richer countries. This convergence could be explained by:
  • Lags in the diffusion on knowledge. Differences in real income might shrink as poor countries receive better technology and information;
  • Efficient allocation of international capital flows, since the rate of return on capital should be higher in poorer countries. In practice, this is seldom observed and is known as Lucas' paradox;
  • A mathematical implication of the model.
Baumol attempted to verify this empirically and found a very strong correlation between a countries' output growth over a long period of time and its initial wealth. His findings were later contested by DeLong who claimed that both the non-randomness of the sampled countries, and potential for significant measurement errors for estimates of real income per capita in 1870, biased Baumol's findings. DeLong concludes that there is little evidence to support the convergence theory.

Assumptions

The key assumption of the Solow–Swan growth model is that capital is subject to diminishing returns in a closed economy.
  • Given a fixed stock of labor, the impact on output of the last unit of capital accumulated will always be less than the one before.
  • Assuming for simplicity no technological progress or labor force growth, diminishing returns implies that at some point the amount of new capital produced is only just enough to make up for the amount of existing capital lost due to depreciation. At this point, because of the assumptions of no technological progress or labor force growth, we can see the economy ceases to grow.
  • Assuming non-zero rates of labor growth complicate matters somewhat, but the basic logic still applies – in the short-run, the rate of growth slows as diminishing returns take effect and the economy converges to a constant "steady-state" rate of growth.
  • Including non-zero technological progress is very similar to the assumption of non-zero workforce growth, in terms of "effective labor": a new steady state is reached with constant output per worker-hour required for a unit of output. However, in this case, per-capita output grows at the rate of technological progress in the "steady-state".

    Variations in the effects of productivity

In the Solow–Swan model the unexplained change in the growth of output after accounting for the effect of capital accumulation is called the Solow residual. This residual measures the exogenous increase in total factor productivity during a particular time period. The increase in TFP is often attributed entirely to technological progress, but it also includes any permanent improvement in the efficiency with which factors of production are combined over time. Implicitly TFP growth includes any permanent productivity improvements that result from improved management practices in the private or public sectors of the economy. Paradoxically, even though TFP growth is exogenous in the model, it cannot be observed, so it can only be estimated in conjunction with the simultaneous estimate of the effect of capital accumulation on growth during a particular time period.
The model can be reformulated in slightly different ways using different productivity assumptions, or different measurement metrics:
  • Average Labor Productivity is economic output per labor hour.
  • Multifactor productivity is output divided by a weighted average of capital and labor inputs. The weights used are usually based on the aggregate input shares either factor earns. This ratio is often quoted as: 33% return to capital and 67% return to labor.
In a growing economy, capital is accumulated faster than people are born, so the denominator in the growth function under the MFP calculation is growing faster than in the ALP calculation. Hence, MFP growth is almost always lower than ALP growth. MFP is measured by the "Solow residual", not ALP.

Mathematics of the model

The textbook Solow–Swan model is set in continuous-time world with no government or international trade. A single good is produced using two factors of production, labor and capital in an aggregate production function that satisfies the Inada conditions, which imply that the elasticity of substitution must be asymptotically equal to one.
where denotes time, is the elasticity of output with respect to capital, and represents total production. refers to labor-augmenting technology or “knowledge”, thus represents effective labor. All factors of production are fully employed, and initial values,, and are given. The number of workers, i.e. labor, as well as the level of technology grow exogenously at rates and, respectively:
The number of effective units of labor,, therefore grows at rate. Meanwhile, the stock of capital depreciates over time at a constant rate. However, only a fraction of the output is consumed, leaving a saved share for investment. This dynamic is expressed through the following differential equation:
is the derivative of the capital stock with respect to time. It is positive when the absolute amount of savings, exceeds the absolute decay of the capital stock,.
Since the production function has constant returns to scale, it can be written as output per effective unit of labour, which is a measure for wealth creation:
The main interest of the model is the dynamics of capital intensity, the capital stock per unit of effective labour. Its behaviour over time is given by the key equation of the Solow–Swan model:
The first term,, is the actual investment per unit of effective labour: the fraction of the output per unit of effective labour that is saved and invested. The second term,, is the “break-even investment”: the amount of investment that must be invested to prevent from falling. The equation implies that converges to a steady-state value of, defined by, at which there is neither an increase nor a decrease of capital intensity:
at which the stock of capital and effective labour are growing at rate. Likewise, it is possible to calculate the steady-state of created wealth that corresponds with :
By assumption of constant returns, output is also growing at that rate. In essence, the Solow–Swan model predicts that an economy will converge to a balanced-growth equilibrium, regardless of its starting point. In this situation, the growth of output per worker is determined solely by the rate of technological progress.
Since, by definition,, at the equilibrium we have
Therefore, at the equilibrium, the capital/output ratio depends only on the saving, growth, and depreciation rates. This is the Solow–Swan model's version of the golden rule saving rate.
Since, at any time the marginal product of capital in the Solow–Swan model is inversely related to the capital/labor ratio.
If productivity is the same across countries, then countries with less capital per worker have a higher marginal product, which would provide a higher return on capital investment. As a consequence, the model predicts that in a world of open market economies and global financial capital, investment will flow from rich countries to poor countries, until capital/worker and income/worker equalize across countries.
Since the marginal product of physical capital is not higher in poor countries than in rich countries, the implication is that productivity is lower in poor countries. The basic Solow model cannot explain why productivity is lower in these countries. Lucas suggested that lower levels of human capital in poor countries could explain the lower productivity.
If the rate of return equals the marginal product of capital then
so that is the fraction of income appropriated by capital. Thus, the Solow–Swan model assumes from the beginning that the labor-capital split of income is constant.