What is a concave utility function?
A function of a single variable is concave if every line segment joining two points on its graph does not lie above the graph at any point. Symmetrically, a function of a single variable is convex if every line segment joining two points on its graph does not lie below the graph at any point.
What is an increasing concave function?
If a function is increasing and concave down, then its rate of increase is slowing; it is “leveling off.” If the function is decreasing and concave down, then the rate of decrease is decreasing. The function is decreasing at a faster and faster rate.
Are utility functions concave or convex?
A utility function is strictly quasi–concave if and only if the preferences represented by that utility function are strictly convex.
What are concave and convex functions?
What are Convex and Concave Functions? A function that has an increasing first derivative bends upwards and is known as a convex function. On the other hand, a function, that has a decreasing first derivative is known as a concave function and bends downwards.
Is concave up the same as convex?
A function is concave up (or convex) if it bends upwards. A function is concave down (or just concave) if it bends downwards. I personally would always mix these two up.
How do you maximize utility in economics?
A Rule for maximizing Utility If a consumer wants to maximize total utility, for every dollar that they spend, they should spend it on the item which yields the greatest marginal utility per dollar of expenditure.
Are concave functions continuous?
This alternative proof that a concave function is continuous on the relative interior of its domain first shows that it is bounded on small open sets, then from boundedness and concavity, derives continuity. If f : C → R is concave, C ⊂ Rl convex with non-empty interior, then f is continuous on int(C).
What is the difference between concave and convex functions?
The major difference between concave and convex lenses lies in the fact that concave lenses are thicker at the edges and convex lenses are thicker in the middle. These distinctions in shape result in the differences in which light rays bend when striking the lenses.
How do you prove a function is convex?
There are many ways of proving that a function is convex: By definition. Construct it from known convex functions using composition rules that preserve convexity. Show that the Hessian is positive semi-definite (everywhere that you care about) Show that values of the function always lie above the tangent planes of the function.
Which of the functions is convex?
Functions of n variables LogSumExp function, also called softmax function, is a convex function. The function − log det ( X ) {\\displaystyle -\\log \\det (X)} on the domain of positive-definite matrices is convex. Every real-valued linear transformation is convex but not strictly convex, since if f is linear, then f ( a + b ) = f ( a ) + f (
When is a function concave or convex?
In mathematics, a real-valued function defined on an n-dimensional interval is called convex (or convex downward or concave upward) if the line segment between any two points on the graph of the function lies above or on the graph, in a Euclidean space (or more generally a vector space) of at least two dimensions.