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A Line Greater Utility Constrained Calculus

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A Line Greater Utility Constrained Calculus Donald Wittman APPENDIX A1 Mathematical Background Constrained Calculus: Example: Maximizing utility subject to a budget Greater Utility constraint. The triangle bounded by the y x and y axes and the budget constraint . is the feasible set. Choose the highest iso-utility curve (indifference curve) Budget Constraint - a line that is feasible. Note that the constraint holds with equality. x Linear Programming: The constraints and iso-curves are all linear. The constraints do not have to y Higher Utility hold with equality. That is, some of the Feasible constraints are not-binding (e.g., the set line going north-east). The solution will be at one of the points of intersection. x Higher Utility Non-Linear Programming: The solution may be anywhere, including the interior of the feasible set if the point of highest utility is interior y Feasible Set to the feasible set. This would be the case in the drawing to the left if the feasible set were expanded to the right so that it covered the innermost circle. x 140 Closed Convex Sets Almost all optimization problems make extensive use of convex sets. This statement holds true for maximization via calculus and linear programming as well as non-linear optimization problems. Today I will show that closed convex sets and hyperplanes are the bases behind the optimization techniques that we will cover. These concepts are meant to give insight into the more mechanical methods that we employ and to show that we are fundamentally using the identical method in all that we have covered in previous courses. The proofs regarding convex sets are thus not meant for memorization but to provide basic understanding. We first start off with a number of definitions: Convex Set: Geometric definition--a set is convex if and only if a (straight) line connecting any two points in the set is also in the set. Convex sets: Non Convex sets: Convex Set: Algebraic definition--Let x,y be two vectors in N space in the set S. E.g., x = (x1, x2) y = (y1, y2) . S is convex if and only if Px + [1−P] y ! S for all P such that 0 ≤ P ≤ 1. 141 2 x = x , x 1 2 y = y , y 1 2 Note that a convex set is not the same thing as a convex function. Boundary point: Intuitively boundary points are points, which are on the edge of a set. More formally, a point is a boundary point if every neighborhood (ball) around the point contains points not in the set and in the set. They may be members of the set; e.g., in the set 0 ≤ x ≤ 1, the numbers 0 and 1 are boundary points and members of the set. In the set 0 < x < 1, the numbers 0 and 1 are boundary points and not members of the set. Closed set: a closed set is a set that includes all its boundary points. The following is a closed set: 0 ≤ x ≤ 1. In economics we almost always consider closed convex sets. The reason can be illustrated by considering the opposite. Maximize the amount of gold (G) you get if G < 1. There is no maximum! Preference set: The set of all points, which are indifferent or preferred to a given point. Economics majors are acquainted with indifference curves. A preference set is the set of all points on the indifference curve plus the all of the points strictly preferred to the indifference curve. In the typical drawing of an indifference curve the preference set would include the indifference curve plus all of the points upwards and to the right of the indifference curve. We can view the ordinary two-dimensional graph of indifference curves as a two-dimensional topographical map showing height in terms of contour lines. Here height is utility and the contour lines are indifference curves. Consider the following diagram. There is a mountain 142 (representing utility) sticking out from the paper, the higher the mountain the higher the utility. Assume that it is an ice cream cone cut in half lengthwise, filled with ice cream and put upside down on the paper. The cone part is the indifference curve formed by a horizontal slice parallel to the paper; the cone plus the ice cream represents all points such that utility is either equal to or greater. We assume the set of such points is convex. y Indifference curves x Theorem 1. Sufficient conditions for a local optimum to be a global optimum. If the feasible set F is a closed and convex set and if the objective function is a continuous function on that set and the set of points (such that the function is indifferent or preferred) create a convex set, then the local optimum is a global optimum. This is a very important result. When these conditions hold, we know that that myopic optimization will end up at a global optimum. If either the feasible set or the preference set is not convex, then the local optimum need not be a global optimum. Feasible set is not convex. 2nd peak is a local optimal but not a global optimum (See following page) 143 144 Preference set is not convex. Right-most tangent is not a global optimal. 145 We next turn our attention toward hyperplanes. Hyperplane: a line in N-space dividing the space in two. In two-space a hyperplane is a line. E.g., y = a + bx or a = y - bx or in more general notation, A = c1x1 + c2x2. In three-space we have a plane: y = A + BX + CZ, or in more general notation, …a = c1x1 + c2x2 + c3x3. In four or more space we have a hyperplane. Note that y ≥ a + Bx or a ≥ c1x1 + c2x2 creates a convex halfspace. HYPERPANES CREATE CONVEX HALFSPACES y x x 2 x 1 Supporting hyperplane has one or more points in common with a (closed) convex set but no interior points. The set lies to one side of the hyperplane. SUPPORTING HYPERPLANES 146 Theorem 2. Given z a boundary point of a closed convex set, there is at least one supporting hyperplane at z. Theorem 3: If 2 convex sets intersect but without interior points, there is a supporting hyperplane that separates them. The following diagrams illustrate other cases of supporting hyperplanes. 147 In the following figure, the shaded area is the feasible set and the curved line is the indifference curve, which characterizes the preference set. The straight line is the hyperplane. We usually give it a less technical term – the budget set or price line. Given these prices, the point of intersection maximizes utility. Linear program Y supporting hyperplane of objective function F X Neoclassical optimization: Neoclassical optimization (usually only consider edge) good Y Max U (x, y) + ¬ (g(x) - 5) indifference curve Price line tangent Feasible hyperplane separating the two convex sets good X 148 There are other results that are of use. The following two theorems state that the intersection of closed convex sets is closed and convex. Theorem 4. The intersection of two convex sets is convex. Proof: If x ! (S ∩ T) and y ! (S ∩ T), then Z = Px + [1 - P] y ! S because S convex Z = Px + [1 - P] y ! T because T convex Therefore Z ! (S ∩ T) Theorem 5. The intersection of two closed sets is closed. 149 Donald Wittman APPENDIX A2 Concave and Quasiconcave Functions A. CONCAVE FUNCTIONS f(x) Concave functions x Three definitions of concavity: (1) Concave Function: The line connecting any two points of the function lies on the function or below. (2) Concave Function: Algebraic definition: Pf (x) +[1 ! P] f (y) " f (Px + [1 ! P]y). Note that x and y may be vectors in n-space (3) Concave Function: Tangency definition: The tangent is always outside or on the function. The sum of two concave functions is also concave. 150 In order to determine whether a function is concave, we generally would prefer a method that did not rely on drawing a picture of the function. Hence, we have the second derivative test. d 2 f If f !! (x) = " 0 , for all x, then the function is concave. This is a necessary and sufficient dx 2 condition for concavity. Example A: f(x) = -x4 is concave since f''(x) = -12x2 ! 0 for all x. Example B: f(x) = - x3 is not concave since f''(x) = - 6x and - 6x > 0 for x < 0. Example C: x ≥ 0 and f(x) = − x3 is concave since f''(x) = − 6x ≤ 0 for all x ≥ 0. Notice that the test for concavity has a great similarity to the test for a maximum. To test for a maximum, you find out if the second derivative is less than zero when the first derivative equals zero. To speak imprecisely, the test for a maximum discovers whether the function is locally concave. In contrast, the test for concavity requires the second derivative to be non-positive everywhere. Maximization Problem from Calculus: Max f (x) = 12x ! 3x2 f ! (x) =12 " 6x = 0 x = 2 f !! (x) = "6 < 0 Therefore at x = 2, f(x) is a maximum and not a minimum. tangent tangent above function (locally) where f '(x) = 0. Going down hill. The analogous test for concavity: f !! (x) = "6 < 0. Therefore f is concave. 151 An Important Property of Concavity: If f(x) is concave, then the set of x such that f(x) ≥ k is convex for all k. f(x) k convex set convex set 152 B. MULTI-DIMENSIONAL CONCAVE FUNCTIONS Next, we would like to expand our intuition by visualizing concave functions and convex sets in two dimensions.
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