0
votes
2answers
42 views

Distance between convex set and non-convex set?

So in http://en.m.wikipedia.org/wiki/Shapley%E2%80%93Folkman_lemma there is some talk about distance between a mintowksi sum and a convex set. But I couldn't get how distance is being defined. Can ...
0
votes
0answers
237 views

Monotonicity of expectation of a concave function of a random variable wrt the variance of the random variable

This is a question motivated from utility function. (See here and here.) I have been trying to develop some common sense in Economics by the way. Given a function $f: \mathbb{R} \to \mathbb{R}$ and a ...
3
votes
3answers
801 views

Using the definition of a concave function prove that $f(x)=4-x^2$ is concave (do not use derivative).

Let $D=[-2,2]$ and $f:D\rightarrow \mathbb{R}$ be $f(x)=4-x^2$. Sketch this function.Using the definition of a concave function prove that it is concave (do not use derivative). Attempt: ...
0
votes
1answer
87 views

Find restrictions on $a>0$ and $b>0$ that ensure that $f(x_1,x_2)$ is concave.

Let $f:\mathbb{R}_{+}^2 \rightarrow \mathbb{R}$ be $f(x_1,x_2)=x_1^a x_2^b$ for $a>0$ and $b>0$. Find restrictions on $a>0$ and $b>0$ that ensure that $f(x_1,x_2)$ is concave. I ...
0
votes
2answers
377 views

Prove the set $M=\{x\in\mathbb{R}^2_+ \mid \alpha x_1+\gamma x_2\leq \beta\}$ is convex

Let $\alpha\gt 0$, $\gamma\gt 0$, and $\beta\gt 0$ be real numbers. Let $$M=\{x\in\mathbb{R}^2_+ \mid \alpha x_1+\gamma x_2\leq \beta\}$$ Prove $M$ is a convex set. Prove that $M$ is bounded. ...
2
votes
3answers
128 views

Problem relating convex sets and optimization

I am working on a microeconomics problem, but I have just kind of just boiled down to the following problem involving convex sets. I have a convex set of vectors in $\mathbb{R^n_+}$ of the form ...
1
vote
1answer
847 views

Proof involving a convex set

So, the problem is actually from a microeconomics class. The problem is this: If preferences are represented by a utility function $u(x,y)=xy$, show that these preferences are convex. Now in case ...
4
votes
1answer
601 views

Lower hemicontinuity of the intersection of lower hemicontinuous correspondences

I have been stumped for long by this exercise (3.12(d)) from Stokey and Lucas's Recursive Methods in Economic Dynamics. Would greatly appreciate any hints. Let $\phi: X \to Y$ and $\psi: X \to Y$ be ...