What is a Suitable Concave Function in 2 Variables for Modelling?

In summary, a concave function is a mathematical function that curves downward and resembles a bowl shape. It is defined as one where the line segment connecting any two points on the graph lies above or on the graph. The main difference between a concave and convex function is the direction of curvature, with concave functions curving downward and convex functions curving upward. To determine if a function is concave, one can look at the second derivative or the slope at different points. Some real-life examples of concave functions include utility and cost functions. They are important in fields like economics and engineering, as well as in mathematical analysis and optimization theory.
  • #1
johnd17
1
0
I need a concave function in 2 variables X and Y for some modelling I'm doing.

For 0<x<1 and 0<y<1, the function needs to satisfy the following:

f(x,y) < 0 for x<0.5 and/or y<0.5
f(x,y) = 0 for x=0.5 and y=0.5
f(x,y) > 0 for x>0.5 and y>0.5

MUST be concave.. i.e. second order partials wrt X and Y must both be < 0.

My maths was never that good and I can't seem to make any progress here.. can anyone help out? Thanks
 
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  • #2
Hi johnd17! Welcome to PF! :smile:

What about:
[tex]f(x,y) = \ln(x + \frac 1 2) + \ln(y + \frac 1 2)[/tex]
 

1. What is a concave function?

A concave function is a mathematical function that is defined as one where the line segment connecting any two points on the graph of the function lies above or on the graph. In simpler terms, it is a function that curves downward, resembling a bowl shape.

2. What is the difference between a concave and convex function?

A convex function is the opposite of a concave function. It curves upward, resembling a hill shape. The main difference between the two is the direction of curvature, with concave functions curving downward and convex functions curving upward.

3. How do you determine if a function is concave?

There are several methods to determine if a function is concave. One way is to look at the second derivative of the function. If the second derivative is negative for all points in the domain, then the function is concave. Another method is to look at the slope of the function at different points. If the slope is decreasing as you move from left to right, then the function is concave.

4. What are some real-life examples of concave functions?

Concave functions are commonly seen in economics, specifically in utility theory where they represent the diminishing marginal utility of a good. Another example is in cost functions, where the cost decreases as the quantity produced increases, but at a decreasing rate.

5. Why are concave functions important?

Concave functions have many practical applications in fields such as economics, finance, and engineering. They are used to model relationships between variables, optimize functions, and make predictions. They also have important theoretical implications in mathematical analysis and optimization theory.

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