# Why is utility concave?

I have read that the utility function is usually concave. I assume this requirement arises in order to meet the diversification effect:$$f(\lambda_1c_1+\lambda_2c_2)\ge \lambda_2 f(c_1)+\lambda_2f(c_2)$$ for some $\lambda_1,\lambda_2\leq1$ and $c_1,c_2$ two assets with some payoff. An investor is happier to hold $c_1,c_2$ in a portfolio than separately.

Question: Why do we restrict $\lambda_1$,$\lambda_2$ to be less than 1? Is it not useful to require $-f$ to be sub linear so I can consider any arbitrary composition of $c_1$ and $c_2$, i.e. $\lambda_1, \lambda_2 \in \Bbb R^+$

• Usually, $\lambda_1$ and $\lambda_2$ represent the weights of your initial wealth allocated to the available tradable assets ($c_1$ and $c_2$ in your case). Bearing this in mind, it makes sense to have a weight $\lambda \in [0,1]$. – JejeBelfort May 2 '17 at 14:45
• Put differently, by imposing $\lambda \in [0,1]$ we are leaving out the possibility of shorting one asset to leverage up the other. And this seems a reasonable restriction, "long only" restriction. – noob2 May 2 '17 at 14:54
• @JejeBelfort Could you elaborate this? What is the meaning of $\lambda_1\cdot c_1?$ – quallenjäger May 2 '17 at 15:07
• @quallenjäger Actually, my answer was incomplete as indeed if $\lambda \in [0,1]$ this means that the weights are only positive; therefore, short positions are ruled out. Regarding the weights interpretation itself, in a portfolio context you set $\lambda_i$ such that $\sum_{i \in [1,2,...,N]} \lambda_i = 1 = 100\%$, with $\lambda_i$ representing the weight associated to asset $i$. – JejeBelfort May 2 '17 at 15:49
• @JejeBelfort Thank you! I understand it now. – quallenjäger May 2 '17 at 19:28