Saturday, September 14, 2019

Convexity and Nonsatiation

Checking the convexity and nonsatiation assumptions EC201 LSE Margaret Bray October 25, 2009 1 Nonsatiation 1. 1 1. 1. 1 The simple story De? nition and conditions for nonsatiation Informally nonsatiation means that â€Å"more is better†. This is not a precise statement, and it is possible to work with a number of di? erent de? nitions. For EC201 †¢ Nonsatiation means that utility can be increased by increasing consumption of one or both goods. If the utility function is di? erentiable you should test for nonsatiation by ? nding the partial derivatives of the utility function. 1. 1. 2Example: testing for convexity with a Cobb-Douglas utility function A Cobb-Douglas utility function has the form u(x1 , x2 ) = xa xb where a > 0 and b > 0. Here u(x1 , x2 ) = 12 2/5 3/5 x1 x2 . Assuming that x1 > 0 and x2 > 0 the partial derivatives are ? u ?x1 ?u ?x2 = = 2 ? 3/5 3/5 x2 > 0 x 51 3 2/5 ? 2/5 > 0. xx 51 2 (1) (2) You should note that because the partial deriva tives are both strictly1 positive utility is a strictly2 increasing function of both x1 and x2 when x1 > 0 and x2 > 0 so nonsatiation is satis? ed. 1. 1. 3 Implications of nonsatiation 1.If utility is strictly increasing in both goods then the indi? erence curve is downward sloping because if x1 is increased holding x2 constant then utility is increased, so it is necessary to reduce x2 to get back to the original indi? erence curve. 2. If utility is strictly increasing in both goods then a consumer that maximizes utility subject to the budget constraint and nonnegativity constraints will choose a bundle of goods which satis? es the budget constraint as an equality so p1 x1 + p2 x2 = m, because if p1 x1 + p2 x2 < m it is possible to increase utility by increasing x1 and x2 whilst still satisfying the budget constraint. A number is strictly positive if it is greater than 0. function is strictly increasing in x1 if when x0 > x1 and x2 is held constant at x2 then u x0 , x2 & gt; u (x1 , x2 ). 1 1 The important point here is that the inequality > is strict. 2A 1 1. 1. 4 Nonsatiation with perfect complements utility A utility function of the form u (x1 , x2 ) = min (a1 x1 , a2 x2 ) is called a perfect complements utility function, but the partial derivative argument does not work because the partial derivatives do not exist at a point where a1 x1 = a2 x2 which is where the solution to the consumer’s utility maximizing problem always lie.This is discussed in consumer theory worked example 6 1. 2 1. 2. 1 Nonsatiation: beyond EC201 Complications with the Cobb-Douglas utility function A really detailed discussion of nonsatiation with Cobb-Douglas utility would note that the partial derivative argument does not work at points where the partial derivatives do not exist. The partial ? u derivative does not exist if x1 = 0 because the formula requires dividing by 0. Similarly the ? x1 ?u formula for requires dividing by 0 if x2 = 0 so the function does not have a partial derivative with ? x2 respect to x2 when x2 = 0.However observe that if x1 = 0 or x2 = 0 then u(x1 , x2 ) = 0, whereas if x1 > 0 and x2 > 0 then u(x1 , x2 ) > 0 so if one or both x1 and x2 is zero then increasing both x1 and x2 always increases utility. Thus nonsatiation holds for all values of x1 and x2 with x1 ? 0 and x2 ? 0. 1. 2. 2 More general formulations ?u ?u > 0 and > 0 implies nonsatiation. However these conditions can be ?x1 ?x2 weakened considerably without losing the implication that the consumer maximizes utility by choosing a point on the budget line which is what really matters.For example if utility is increasing in good 1 but decreasing in good 2 so good 2 is in fact a â€Å"bad† the consumer maximizes utility by spending all income on good 1 and nothing on good 2. The condition that 2 2. 1 2. 1. 1 Convexity and concavity Concepts Convex sets A set is convex if the straight line joining any two points in the set lies entirely within the set. Figure 1 illustrates convex and non-convex sets. 2. 1. 2 Convex functions A function is convex if the straight line joining any two points on the graph of the function lies entirely on or above the graph as illustrated in ? gure 2.Another way of looking at convex functions is that they are functions for which the set of points lying above the graph is convex. Figure 2 suggests that if the ? rst derivative of a function does not decrease anywhere then the function is convex. This suggestion is correct. If the function has a second derivative that is positive or zero everywhere then the ? rst derivative cannot decrease so the function is convex. This gives a way of testing whether a function is convex. Find the second derivative; if the second derivative is positive or zero everywhere then the function is convex. 2. 1. 3Concave functions Concave functions are important in the theory of the ? rm. A function is concave if the straight line joining any two points on the graph of the function lies entirely on or below the graph as illustrated in ? gure 3. Another way of looking at concave functions is that they are functions for which the set of points lying below the graph is convex. Figure 3 suggests that if the ? rst derivative of a function does not increase anywhere then the function is concave. This suggestion is correct. If the function 2 Convexity Mathematically a set is convex if any straight line joining wo points in the set lies in the set. Which of these sets are convex? B A non-convex convex C D convex non-convex Figure 1: Convex sets A function is convex if a straight line joining two points on its graph lies entirely on or above the function. If the second derivative of the function is positive or zero at every point then x2 the function is convex. 0 x1 Figure 2: A convex function 3 A f unc tio n is c on ca ve if a s tra ight lin e joining tw o po ints on its g ra ph lies en tirely o n or be low the fun ction . If the s ec on d de riv a tiv e o f the fun ction is ne ga tive or zero a t e very p oint the n 2 the fun ction is c on ca ve . ca ve 0 x1 Figure 3: A concave function has a second derivative that is negative or zero everywhere then the ? rst derivative cannot increase so the function is concave. This gives a way of testing whether a function is convex. Find the second derivative; if the second derivative is negative or zero everywhere then the function is concave. You may ? nd it easier to remember the di? erence between convex and concave functions if you think that a function is concave if it has a cave underneath it. 2. 2 2. 2. 1 Convexity in consumer theory De? nitionThe convexity assumption in consumer theory is that for any (x10 , x20 ) the set of points for which u(x1 , x2 ) ? u (x10 , x20 ) is convex. If utility is strictly increasing in both x1 and x2 so the indi? erence curve slopes downwards the convexity assumption is is equivalent to an assumption that thinking of the indi? erence curve as th e graph of a function that gives x2 as a function of x1 the function is convex. ?u ?u > 0 and > 0 so the indi? erence Thus if the test for nonsatiation establishes that both ?x1 ?x2 curves are downward sloping the convexity assumption can be tested by rearranging the equation for an indi? rence curve to get x2 as a function of x1 and u, and then ? nding whether the second derivative ? 2 x2 > 0. ?x2 1 2. 2. 2 Example: testing for convexity with a Cobb-Douglas utility function 2/5 3/5 Here u(x1 , x2 ) = x1 x2 . Write 2/5 3/5 u = x1 x2 . (3) Rearranging to get x2 as a function of x1 and u ?2/3 x2 = u5/3 x1 . Holding u constant so staying on the same indi? erence curve ? x2 2 ?5/3 = ? u5/3 x1 ?x1 3 and 10 5/3 ? 8/3 ? 2 x2 = >0 u x1 ?x2 9 1 4 (4) ?u ?u > 0 and > 0 the indi? erence ?x1 ?x2 curve is downward sloping and the preferred set is above the indi? rence curve so the convexity condition is satis? ed. so on an indi? erence curve x2 is a convex function of x1 . Beca use 2. 2. 3 Algebra problems You should know how to rearrange equation 3 to get equation 4. If this is causing you problems note ? rstly that equation 3 implies that ? ?5/3 2/5 3/5 2/3 u5/3 = x1 x2 = x1 x2 so x2 = 2. 3 u5/3 2/3 x1 ?2/3 = u5/3 x1 . Beyond EC201 Concavity and convexity can be de? ned algebraically and this is essential if you want to prove any results about concavity and convexity rather than appealing to intuition as I have done here.The procedure I have given for checking the convexity condition in consumer theory requires that the ? rst ? u ?u derivatives > 0 and > 0 and does not work with more than two goods. There is a much ? x1 ?x2 more general method; write down the matrix of second derivatives of the function u (x1 , x2 ). If this matrix is positive semide? nite everywhere the function is convex, if the matrix is negative semide? nite everywhere the function is concave. You do not need to know about this for EC201. 5

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