Abstract
In this chapter we cover functions of several variables (Sect. 6.1), partial and total derivatives (Sect. 6.2), and unconstrained optimization (Sect. 6.3). The chapter concludes with a simple example of integration with multiple variables (Sect. 6.4). Examples of applications in Economics include: Cobb-Douglas function and CES function, multi-product firm and ordinary least square.
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Notes
- 1.
The rules describing how the data are generated are referred to Data Generating Process (DGP). DGP goes beyond the scope of the example. Here, we just use a naive approach to generate the data to estimate the model. You may think of the steps to build a simulated data set as follows:
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specify the model to simulate;
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determine the coefficients of the model;
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build the data for the independent variables and the error term based on probability distributions;
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compute the dependent variable by using the coefficients, the simulated data for the independent variables and the error.
However, in R there is the simstudy package that allows users to generate simulated data sets to explore modeling techniques or better understand data generating processes. The interested reader may refer to the following link for more details about the simstudy package https://cran.r-project.org/web/packages/simstudy/vignettes/simstudy.html.
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- 2.
Or, in statistical terminology, linear in the parameters, i.e., the unknown parameters of the model to be estimated do not appear, for example, as exponent or multiplied by another parameter.
- 3.
If you do not have LATE X installed on your computer export the results as text. In out = replace tex with txt.
- 4.
The gradient is associated with the storage of partial derivatives of a scalar function, i.e., a function that assigns a scalar (real number) to a set of real variables, whereas the Jacobian is associated with the storage of partial derivatives of a vector function, i.e., a function that assigns a vector value to a set of real variables. For a clear and concise explanation of vector functions the reader may refer to Moore and Siegel (2013).
- 5.
Refer to an advanced textbook for a proof of the Young’s theorem.
- 6.
max and min are abbreviations for maximum and minimum.
- 7.
It may be helpful to think about f(x) = x 4. This function has a minimum at x ∗ = 0. The first order condition, 4x 3 = 0 implies that x ∗ = 0. The second order condition, 12x 2, evaluated at x ∗ is 0. Therefore, despite f ′′(x ∗) = 0 we reached a minimum. Plot f(x) = x 4 to visualize the function.
- 8.
In case of a 2 × 2 H as in the example, |H 2| = |H|. If |H| < 0, then H is indefinite. Refer to an advanced textbook for the proof of this theorem.
References
Moore, W. H., & Siegel, D. A. (2013). A mathematical course for political & social research (1st edn.). Princeton: Princeton University Press.
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Porto, M. (2022). Multivariable Calculus. In: Introduction to Mathematics for Economics with R. Springer, Cham. https://doi.org/10.1007/978-3-031-05202-6_6
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