Definition
Convexity
MWG - Definition 3.B.4
The preference relation /equation.png)
on X is convex if for every x /equation001.png)
X, the upper contour set {/equation002.png)
: /equation003.png)
} is convex.
That is, if y /equation004.png)
x and z /equation005.png)
x, then /equation006.png)
for any /equation007.png)
/equation008.png)
[0,1]
Strict convexity
MWG - Definition 3.B.5
The preference relation /equation009.png)
on X is strictly convex if for every x, we have that y /equation010.png)
x, z x and y /equation011.png)
z implies /equation012.png)
for any [0,1]
In other word, x is not on the line linking y and z.
Note
- Convexity can be interpreted in terms of diminishing marginal rates of substitution.
- With convex preferences, from any initial consumption situation x, and for any two commodities, it takes increasingly larger amounts of one commodity to compensate for successive unit losses of the other.
- Convexity can also be viewed as the formal expression of a basic inclination of economic agents for diversification.