# Convexity and Nonsatiation

Informally non-satiation means that “more is better”. This is not a precise statement. Nonsatiation means that utility can be increased by increasing the consumption of one or both goods. Reliable you should test. ending the partial derivatives of the utility function.

Example: testing for convexity with a Cobb-Douglas utility function has the form u(x1, x2 ) = xa xb. If the utility is strictly increasing in both goods then the 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 curve.

If the 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 satisfy? Es the budget constraint as equality so p1 x1 + p2 x2 = m, because if p1 x1 + p2 x2 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.

The important point here is that the inequality is strict.

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 lies. This is discussed in consumer theory worked example

Nonsatiation: beyond EC201 Complications with the Cobb-Douglas utility function A really detailed discussion of non-satiation 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 derivative does not exist if x1 = 0 because the formula requires dividing by 0. Similarly the x1 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, so if one or both x1 and x2 is zero then increasing both x1 and x2 always increases utility. Thus non-satiation holds for all values of x1 and x2.

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 the utility is increasing in good 1 but decreasing in good 2 so good 2 is in fact a “bad” the consumer maximizes utility by spending all income on good 1 and nothing on good

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 the convex and non-convex sets.

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.

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 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 zeroes everywhere then 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.

Concave functions are important in the theory. 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.

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 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 two points in the set lies in the set. 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. The convexity assumption in consumer theory is that for any (x10, x20) the set of points for which u(x1, x2,)(x10, x20) is convex. If the utility is strictly increasing in both x1 and x2 so the curve slopes downwards the convexity assumption is equivalent to an assumption that thinking of the curve as the graph of a function that gives x2 as a function of x1 the function is convex. Thus if the test for establishes that both x1 x2 curves are downward sloping the convexity assumption can be tested by rearranging the equation for a curve to get x2 as a function of x1 and u, and then whether the second derivative.

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 and2/3 x2 = u5/3 x1 .

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