Background
A second-order approximation is a method used to approximate an arbitrary function using its expansion in a Taylor series by including the linear and quadratic terms while neglecting higher-order terms. This approach is particularly valuable in various economic models where capturing nonlinear behaviors and more intricate dynamical properties is crucial.
Historical Context
The development of the Taylor series and the concept of approximations rooted in it can be traced back to the mathematical advancements in the 18th century, primarily attributed to Brook Taylor. Economists have adapted these mathematical principles to better understand economic phenomena, paving the way for more complex models that account for risk aversion and income distribution effects.
Definitions and Concepts
Second-Order Approximation
An approximation method that expands a function in a Taylor series and retains only the first-order (linear) and second-order (quadratic) terms. Higher-order terms are assumed to be negligible.
Major Analytical Frameworks
Classical Economics
Classical economics primarily focused on linear models, so second-order approximations were less frequently utilized in this tradition.
Neoclassical Economics
In neoclassical economics, the use of second-order approximations becomes prominent in analyzing individual choice under uncertainty. This framework often employs these approximations to model utility functions, capturing attitudes towards risk.
Keynesian Economics
Keynesian models utilize second-order approximations to enhance predictions related to macroeconomic stability and to quantify effects resulting from demand shocks.
Marxian Economics
While Marxian economics traditionally focuses on linear critique of capitalist structures, newer adaptations may use second-order approximations to understand nonlinear relationships within multifaceted social systems.
Institutional Economics
Institutional economics might adopt second-order approximations in policy modeling to account for the nonlinearity in institutional changes and their impacts on economic performance.
Behavioral Economics
In behavioral economics, second-order approximations help to model complex human behaviors, capturing cognitive biases and decision processes more accurately.
Post-Keynesian Economics
Post-Keynesian economics leverages second-order approximations to analyze income distribution impacts on macroeconomic variables in a more nuanced manner.
Austrian Economics
While less commonly used, Austrian economists could apply second-order approximations to study nonlinear dynamic systems within market processes and entrepreneurial activities.
Development Economics
Development economics applies second-order approximations to model the effects of income distribution and policy impacts on economic growth trajectories in developing countries.
Monetarism
In monetarist analysis, second-order approximations can be used to understand the nonlinear impacts of monetary policy changes on inflation and output.
Comparative Analysis
Different economic schools of thought incorporate second-order approximations to varying extents. Neoclassical and Keynesian models make extensive use of these tools to capture the complex systems dynamics, while Classical and Marxian traditions use them more sparingly.
Case Studies
- Risk Aversion in Neoclassical Models: Using second-order Taylor expansions to approximate utility functions enables better modeling of risk preferences.
- Income Distribution in Welfare Economics: Employing second-order approximations to study income distribution helps to understand the magnitudes and phases of welfare changes due to economic policy.
Suggested Books for Further Studies
- “Microeconomic Theory” by Andreu Mas-Colell, Michael D. Whinston, and Jerry R. Green
- “Advanced Macroeconomics” by David Romer
- “Foundations of Economic Analysis” by Paul Samuelson
Related Terms with Definitions
Taylor Series
An infinite series of mathematical terms that when summed together approximate a mathematical function.
Linear Approximation
The estimation of the behavior of a function using its tangent line at a given point.
Utility Function
A depiction of preferences over a set of goods and services, used in economic models to describe choice under uncertainty.
