# Help with Monopolistic Competition Proof, Prove Love for Variety

I need some help with a proof.

Assume η = 2 and there are just two goods. Verify that the following utility function exhibits Love for Variety tastes, show that:

u(λa + [1 − λ]b, λb + [1 − λ]a) > u(a, b) for 1 > λ > 0

Additionally, I am given the utility function: $$\\u(q_1,q_2, ...q_n) = \sum_1^N \sqrt[η]{q_i}\,$$

In the lead up to the question, I am given a different problem to prove that should help me solve the above one.

It uses the same utility function, and I have to prove that the consumer wants a variety of two firms, rather than just one firm, I think I figured it out but I don't know how to apply what I did in the above proof, I have to prove that sqrt(x) + sqrt(y) > sqrt(x + y):

$$\sqrt{x} + \sqrt{y} > \sqrt{x+y},$$ $$(\sqrt{x} + \sqrt{y})^2 > (\sqrt{x + y})^2$$ $$2\sqrt{xy} + x + y > x + y$$ $$2\sqrt{xy} > 0$$

Any help is sincerely appreciated.

• What are $a$, $b$, $q_i$, $x$ and $y$? What do you mean by "a variety of two firms"? Feb 10 '18 at 21:39
• @colorado I disagree with your method of proof for $\sqrt x + \sqrt y > \sqrt{x + y}$. You can't fiddle with a relation - you should rather modify each side separately. Btw, questions like these would be received better on math.stackexchange.com. May 26 '19 at 17:09

## 1 Answer

You have

\begin{align} u(\lambda a + (1 - \lambda) b, \lambda b + (1 - \lambda)a) &= \sqrt{\lambda a + (1 - \lambda)b} + \sqrt{\lambda b + (1 - \lambda)a} \\ &\geq \lambda\sqrt{a} + (1 - \lambda)\sqrt{b} + \lambda\sqrt{b} + (1 - \lambda)\sqrt{a} \\ &= \sqrt{a} + \sqrt{b} \\ &= u(a, b) \end{align}

The inequality in the second line follows from Jensen's inequality for concave functions. I don't think you need to prove Jensen but you can even do that moderately easy with the AM-GM inequality if you want to.