BasicMathematicalEconomics

BasicMathematicalEconomics - Basicsfor Mathematical...

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1 Basics for Mathematical Economics Lagrange multipliers __________________________________________________2 Karush–Kuhn–Tucker conditions ________________________________________12 Envelope theorem ___________________________________________________16 Fixed Point Theorems_________________________________________________20 Complementarity theory ______________________________________________24 Mixed complementarity problem _______________________________________26 Nash equilibrium ____________________________________________________30

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2 Lagrange multipliers Figure 1: Find x and y to maximize f ( x , y )subject to a constraint (shown in red) g ( x , y ) = c . Figure 2: Contour map of Figure 1. The red line shows the constraint g ( x , y ) = c . The blue lines are contours of f ( x , y ). The intersection of red and blue lines is our solution. In mathematical optimization , the method of Lagrange multipliers (named after Joseph Louis Lagrange ) is a method for finding the maximum/minimum of a function subject to constraints . For example (see Figure 1 on the right) if we want to solve: maximize subject to We introduce a new variable ( λ ) called a Lagrange multiplier to rewrite the problem as: maximize Solving this new equation for x, y, and λ will give us the solution (x, y) for our original equation.
3 Introduction Consider a two-dimensional case. Suppose we have a function f ( x , y ) we wish to maximize or minimize subject to the constraint where c is a constant. We can visualize contours of f given by for various values of d n , and the contour of g given by g ( x , y ) = c . Suppose we walk along the contour line with g = c . In general the contour lines of f and g may be distinct, so traversing the contour line for g = c could intersect with or cross the contour lines of f . This is equivalent to saying that while moving along the contour line for g = c the value of f can vary. Only when the contour line for g = c touches contour lines of f tangentially , we do not increase or decrease the value of f - that is, when the contour lines touch but do not cross. This occurs exactly when the tangential component of the total derivative vanishes: , which is at the constrained stationary points of f (which include the constrained local extrema, assuming f is differentiable). Computationally, this is when the gradient of f is normal to the constraint(s): when for some scalar λ (where is the gradient). Note that the constant λ is required because, even though the directions of both gradient vectors are equal, the magnitudes of the gradient vectors are generally not equal. A familiar example can be obtained from weather maps, with their contour lines for temperature and pressure: the constrained extrema will occur where the superposed maps show touching lines ( isopleths ). Geometrically we translate the tangency condition to saying that

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BasicMathematicalEconomics - Basicsfor Mathematical...

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