# Steady State

# Steady State

The concept of the “steady state” refers to a long-term, dynamic equilibrium, where diminishing returns have exhausted all the gains from capital deepening (i.e., increasing capital per worker). In the absence of technological progress, growth in per-worker output stops. A related concept of “dynamic equilibrium” was central to classical theory, particularly to David Ricardo’s growth theory. This dynamic equilibrium was called the “stationary state.” In the stationary state, all growth ends; both income and the size of the population are static. The stationary state occurs because diminishing returns in agriculture eventually lead to the end of growth. In this state the rate of profits is driven to zero, and rent captures all the economic output in excess of the output required to support the population at a subsistence wage rate. Policy proposals were often justified in part by whether they would postpone the emergence of the stationary state.

After a century of neglect, interest in growth theory reemerged with the Harrod-Domar model, which was developed independently by Roy Harrod in 1939 and Evsey Domar in 1946. This model did not have a dynamic equilibrium, however, because it did not allow for the possibility of diminishing returns. It was not until the Solow model emerged in 1956 that a dynamic equilibrium was introduced into neoclassical economics. The steady state in the Solow model occurs because the production function has diminishing returns. In this model, growth potentially has three sources: growth in capital per worker, growth in the labor force, and technological change. The latter two sources of growth are exogenous and are constant, as is the saving rate. For a linearly homogeneous production function, growth rates will converge to a steady-state growth rate, where all the positive effects of saving and investment on growth have been exhausted and where only the rate of population growth and the rate of technological progress impact the growth rate. This steady-state growth rate is expressed as *g* = Ẏ/ *Y* = λ + *n*, where *g* is the rate of growth of income (GDP), λ is the rate of technological change, and *n* is the rate of growth of the labor force.

This steady-state growth rate can best be understood by considering a linearly homogeneous production function, *Y* = *F* (*K, L* ), where *K* is capital and *L* is labor. Output per unit of labor is *Y* / *L* = *y* = *f* (*k* ). The labor input can increase either because the number of workers, *N*, rises, or because Harrod-neutral technological change makes each worker more productive. Assume that *N* grows at a constant rate, *n*, and that there is a constant rate of technological change, λ. Then *L* will grow at the rate *n* + λ. With a constant marginal and average propensity to save and invest, *s*, capital per worker, *k*, will change by *k* = *sy* –(*n* + λ + δ) *k*, where δ is the rate of depreciation. Increases in *k* will increase *y* = *f* (*k* ), and *k* will increase if *sy* > (*n* + λ + δ) *k*. In other words, there will be capital deepening, which increases capital per augmented worker, *L*, if investment is sufficient to offset depreciation and provide each augmented worker with the same capital as before.

Policies that affect saving and investment will only have a temporary impact on growth, but since *s* does not affect steady-state growth, policy will have no impact on growth in the long run. Policy will, however, affect the steady state level of income per augmented worker, *y*. Consider an economy that is at a steady state. An increase in the saving rate will temporarily increase the growth rate because it will increase *sy*. Eventually a new steady-state level of *y* is reached and the growth rate will fall to its original level of λ + *n*. The economy is said to conditionally converge to an equilibrium *y*. The level of *y* is conditional on the saving rate, the labor force growth rate, and the technology.

Endogenous growth models have challenged this notion of the steady state by introducing models where policy matters in the long run because certain types of saving and investment affect λ through positive externalities. If these externalities exhibit constant returns, then policies that affect the saving rate will influence steady-state growth, and growth rates will not converge. Convergence is one of the most tested implications of the neoclassical steady state. If all countries share the same technology because knowledge is freely available, then per capita income should depend only on the rate of saving. Furthermore countries or regions that were late to develop should conditionally converge to countries that began to develop earlier. In other words, everything else being equal, poor countries should experience faster growth than rich countries. Robert Barro and others have tested for this convergence in cross-sectional estimates of growth. One of the explanatory variables is real per capita income for some earlier year. A negative coefficient for past income is evidence that convergence is occurring. Many empirical studies have found a negative coefficient, indicating conditional convergence, but they have also found that the rate of convergence is slow.

Conditional convergence does not necessarily imply a narrowing of gaps in per capita income between rich and poor nations. Many countries, both rich and poor, may be at or close to their conditional steady state, where per capita income is determined by structural variables, such as population growth rates, saving rates, and structural and institutional factors that determine λ, the rate of technological change (see Jones 1997; Darity and Davis 2005). Structural, institutional, and historical differences are receiving attention in the literature on economic development (see Grabowski, Self, and Shields 2007).

**SEE ALSO** *Stability in Economics*

## BIBLIOGRAPHY

Barro, Robert J. 1991. Economic Growth in a Cross-Section of Countries. *Quarterly Journal of Economics* 106 (2): 407–433.

Darity, William, Jr., and Lewis S. Davis. 2005. Growth, Trade, and Uneven Development. *Cambridge Journal of Economics* 29 (1): 141–170.

Domar, Evsey. 1946. Capital Expansion, Rate of Growth, and Employment. *Econometrica* 14 (2): 137–147.

Grabowski, Richard, Sharmisha Self, and Michael P. Shields. 2007. *Economic Development: A Regional, Institutional, and Historical Approach*. Armonk, NY: Sharpe.

Harrod, Roy F. 1939. An Essay in Dynamic Theory. *Economic Journal* 49 (193): 14–33.

Jones, Charles I. 1997. On the Evolution of the World Income Distribution. *Journal of Economic Perspectives* 11 (3): 19–36.

Jones, Charles I. 2002. *Introduction to Economic Growth*. 2nd ed. New York: Norton.

Lucas, Robert E., Jr. 1988. On the Mechanics of Economic Development. *Journal of Monetary Economics* 22 (July): 3–42.

Ricardo, David. [1817] 1965. *The Principles of Political Economy and Taxation*. London: Aldine.

Romer, David. 2006. *Advanced Macroeconomics*. 3rd ed. Boston: McGraw-Hill.

Romer, Paul M. 1986. Increasing Returns and Long-Run Growth. *Journal of Political Economy* 94 (5): 1002–1037.

Solow, Robert M. 1956. A Contribution to the Theory of Economic Growth. *Quarterly Journal of Economics* 70 (1): 65–94.

Solow, Robert M. 2000. *Growth Theory: An Exposition*. 2nd ed. New York: Oxford University Press.

*Michael P. Shields*

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