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):

$$
\sqrt{x} + \sqrt{y} &gt; \sqrt{x+y},
$$
$$
(\sqrt{x} + \sqrt{y})^2 &gt; (\sqrt{x + y})^2
$$
$$
2\sqrt{xy} + x + y &gt; x + y
$$
$$
2\sqrt{xy} &gt; 0
$$

Any help is sincerely appreciated.
You have 

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

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.