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macromath_v2

macromath_v2 - Macromath 122 These are some important...

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- 1 - Macromath 122 These are some important mathematical concepts used in macroeconomics (and in other parts of economics). They are not a substitute for the serious study of mathematics; rather, they serve as a reminder on important concepts. Any comments or corrections are welcome. I. Single variable calculus: Farmer estimates an agricultural production function from experiments: 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.5 1.0 1.5 2.0 2.5 N CORN C = corn production = 2N – N 2 , where N is nitrogen fertilizer Marginal product of nitrogen is: dC dN = 2 – 2N, which is positive for N < 1. d 2 C dN 2 = -2, which shows diminishing returns. Note that dC/dN = 0 at N = 1, this is maximum output.

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- 2 - II. Multivariate calculus With multivariate calculus, we have multiple variables in our function. For example, we now have capital as well as nitrogen: C = 12 K α (2N – N 2 ) The marginal product of nitrogen is: C/ N = 12 K α (2 – 2N) , holding K fixed. A partial derivative uses instead of d. Like economics, it “holds other things constant.” III. Powers and logs Power functions are widely used in sciences, finance, and economics: y = a x As for example in 10,000 = 10 4 A logarithm is a base for representing a power function: x = log a y as for example y = log 10 (10,000) Since 1 = b 0 , log b (1) = 0 for any base.
- 3 - A plot of the log to the base 10 is shown below. Note that it tends to minus infinite as x goes to 0, and is undefined at 0 or negative numbers: -1.2 -0.8 -0.4 0.0 0.4 0.8 1.2 0 2 4 6 8 10 12 X log10(x) IV. Rules for logs for any base b Two important rules are the following: log (xy) = 1og(x) + log (y) log (x y ) = y log (x)

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