1. Let f and g be two concave functions from Rn into R. Verify that h de¯ned via
h(x) = f(x) + g(x) is concave as well.
2. Find two quasiconcave functions f and g such that h de¯ned via
h(x) = f(x) + g(x) is not quasiconcave. (Don't forget to describe the domain of f and g.)
3. Given an example of a quasiconcave function that is not concave.
4. Suppose f is concave. Show that it is quasiconcave.
數學題疑問
2007-10-11 11:31 am
回答 (1)
2007-10-11 12:37 pm
✔ 最佳答案
1.Let y denote ax+bx' where a+b=1
h(y)
= f(y) + g(y)
>= af(x)+bf(x')+ag(x)+bg(x')
= a[ f(x)+g(x)] + b[ f(x')+g(x')]
= ah(x) +bh(x') #
2.
http://www.economics.utoronto.ca/osborne/MathTutorial/QCCX1S.HTM
Q4 Solution
3. exp(x)
Because it's increasing, it's quasiconcave; while it is strictly convex.
4.
Let f(x')>f(x) , y=ax+bx' where a+b=1
then min( f(x), f(x')) = f(x)
f(y)>=af(x)+bf(x') >=f(x) = min( f(x), f(x') ) #
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