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
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
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