The Basic Differential Equation and The Solow Growth Model

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shift in demand curve



Historical and Mathematical Background:

Robert Solow is a Nobel prize winning economist who made great contributions to the Theory of Economic Growth. Solow as educated at Harvard and held various positions including Associate Professor of Economics at the Massachusetts Institute of Technology, Senior Economist at the Council for Economic advisers and was a member of the Presidents Commission on Income Maintenance.
Solow developed a model which was published in his seminal paper on economic growth titled, “A Contribution to the Theory of Economic Growth”, which attributes 3/4 of the economic growth in the U.S. economy to technological progress. Solow’s model consist of 3 key assumptions and from these assumptions one Solow derives the “fundamental differential equation” used to describe the equilibrium solution to the system. The system is described in the assumptions and is composed of a production function, capital growth, and growth in the labor force.
Differential equations are prevalent in the physical sciences and are fundamental for the understanding of complex engineering systems. Dynamic systems whose rates of change are observable in the data via laboratory experiments and measurements are commonly described by differential equations. The economy is a dynamic system and Solow’s model is dynamic in this sense and in the sense that it can be changed with different production functions, depending on the underlying assumptions and whether one wants a numerical solution or merely a qualitative diagram of macroeconomic behavior; in this sense the Solow model accurate predicts and models real world data of economic growth. The parallels between the data and the theory will be outlined in the “Example” section.
Elements of Model:
Production Function
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The production function is expressed in terms of a unit of labor and the capital to labor ratio. The assumption of constant returns to scale allows the simplified f(k). Output is a homogeneous good which will also be a composite for the formation of capital stock.
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The growth of the capital stock (K) is equivalent to growth in investment; depreciation of capital stock will be accounted for so that I is really net investment=investment-depreciation. We are assuming that investments are equivalent to the marginal propensity to save (s) multiplied by the production function.

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The assumption in the labor market is that the labor supply is equivalent to the population. There is no unemployment and the growth of labor as function of time follows an exponential growth pattern.
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The differential equations and production functions outlined in these 3 assumptions are the building blocks for Solow’s Basic Differential Equation. The Basic Differential Equation is the rate of change in capital (k) over time.
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The equilibrium solution to the basic differential equation imply that sf(k)=nk. The implications of this equilibrium is that Labor, Output per Capita, and the Capital to labor ratio grow at the same rate n, hence the importance of introducing another factor in the form of technological progress or a factor productivity enhancing variable which is exogenous from the model. This productivity enhancing factor can be in the form of entrepreneurship, greater efficiency in the allocation of resources, and/or technological progress.

Numerical Example with Cobb-Douglas Production Function: (Coming Soon)