Economists' Mathematical Manual, 4th Edition by Knut Syds?ter, Arne Strom, Peter Berck

By Knut Syds?ter, Arne Strom, Peter Berck

This quantity provides mathematical formulation and theorems prevalent in economics. It contains either formulation like Roy`s identification which are odd to economics and formulation like Leibniz's rule which are universal to many parts of utilized arithmetic. the amount is intended to be a reference paintings, for use by means of scholars along side a textbook and by way of researchers wanting unique statements of mathematical effects. the amount is the 1st grouping of this fabric for a in particular economist viewers.

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The constant γ is called Euler’s constant. g. Edwards and Penney (1998). g. Graham, Knuth, and Patashnik (1989). 1 f (x) dx = F (x) + C ⇐⇒ F (x) = f (x) Definition of the indefinite integral. 2 (af (x) + bg(x)) dx = a f (x) dx + b g(x) dx Linearity of the integral. a and b are constants. 3 f (x)g (x) dx = f (x)g(x) − f (x)g(x) dx Integration by parts. 10 xn ax n e − a a n = −1 n = −1 a > 0, a = 1 xn−1 eax dx, a = 0 loga x dx = x loga x−x loga e+C, a > 0, a = 1 Change of variable. ) Special integration results.

Xn , and m endogenous variables, y1 , . . , ym . ∂f (x, y) ∂x The Jacobian matrix of f1 , . . , fm with respect to y1 , . . , ym . The general implicit function theorem. 1) to define the endogenous variables y1 , . . , ym as differentiable functions of the exogenous variables x1 , . . , xn . ) A general system of m equations and n variables. 4) has k degrees of freedom if there is a set of k of the variables that can be freely chosen such that the remaining n − k variables are uniquely determined when the k variables have been assigned specific values.

2 P (x, f (x)) y = f (x) Illustration of Marshall’s rule. t. x at the point P in the figure, first draw the tangent to the curve at P . Measure the distance Ay from P to the point where the tangent intersects the y-axis, and the distance Ax from P to where the tangent intersects the x-axis. Then Elx f (x) = ±Ay /Ax . Marshall’s rule. The distances are measured positive. Choose the plus sign if the curve is increasing at P , the minus sign in the opposite case. 4 • If | Elx f (x)| > 1, then f is elastic at x.

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