# Convexity and Nonsatiation

EC201 LSE

Margaret Bray

October 25, 2009

1

Nonsatiation

1.1

1.1.1

The simple story

Deﬁnition and conditions for nonsatiation

Informally nonsatiation means that "more is better". This is not a precise statement, and it is possible to work with a number of diﬀerent deﬁnitions. For EC201

• Nonsatiation means that utility can be increased by increasing consumption of one or both goods. If the utility function is diﬀerentiable you should test for nonsatiation by ﬁnding the partial derivatives of the utility function. 1.1.2

Example: 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 derivatives are both strictly1 positive utility is a strictly2 increasing function of both x1 and x2 when x1 > 0 and x2 > 0 so nonsatiation is satisﬁed. 1.1.3

Implications of nonsatiation

1. If utility is strictly increasing in both goods then the indiﬀerence 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 indiﬀerence 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 satisﬁes 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. 1A

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 > 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 "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...

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