Abstract
In this chapter we deal with difference equations. In Sect. 10.1 we will present first-order linear difference equations. In particular, we will discuss solution by iteration (Sect. 10.1.1) and by general method (Sect. 10.1.2). In Sect. 10.2 we will learn how to solve second-order linear difference equations. Section 10.3 is devoted to systems of linear difference equations while in Sect. 10.4 we will learn how to transform high-order difference equations. Examples of applications in Economics include: a problem with interest rate, the cobweb model, the Harrod-Domar growth model, the law of motion for public debt, and the autoregressive process. Examples of functions coded from scratch include: a function to solve difference equations by iteration, a function to solve numerically systems of first-order linear difference equations, a function to plot the results of the cobweb model, and a function that simulates the law of motion for public debt.
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Notes
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The quadratic formula is in the normalized form, i.e. the coefficient of b 2 needs to be 1.
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Note that we wrote (10.41) to be consistent with the previous example. However, you may find (10.42) with 0 and 1 inverted on the main diagonal, implying that the equations in (10.41) are written with a different order. However, the interpretation of the results does not change. To be noted that as we arranged the equations and consequently the matrix and the column vectors, periods in sys_folde() returns the desired period at index \( \left [1, 1 \right ] \). That is, in the example, 89 corresponds to t = 11, and consequently 144 corresponds to t = 12. For example, if you set periods = 0, the values 0 and 1, i.e. the initial values, are returned at index \( \left [1, 1 \right ] \) and \( \left [2, 1 \right ] \), respectively. Naturally the function also works if you appropriately rewrite (10.41) and consequently rewrite (10.42). However, make sure you correctly interpret the results.
- 6.
This is the simplest assumption about expected price. Other possible specifications include adaptive expectations and Goodwin expectations.
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Porto, M. (2022). Difference Equations. In: Introduction to Mathematics for Economics with R. Springer, Cham. https://doi.org/10.1007/978-3-031-05202-6_10
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