Concavity

EC487_Advanced_Microeconomics

To determine whether a function is concave or not, we could use two aspects:

1. Concavity: In Economic context, it usually refers to diminishing Marginal Utility.

If there is only one good in the economy, simply calculate $u^{\prime \prime }(x)$. If $u^{\prime \prime }(x)<0$, then $u(\cdot )$ is concave.

2. Quasi-Concavity: (拟凹性): It refers to the “Love for variety”. For any $x$ and $y$, if the utility function satisfies:

$$u(\alpha x+(1-\alpha )y)\ge \min \{u(x),u(y)\}\forall \alpha \in [0,1]$$

Then we say $u(\cdot )$ is quasi-concave.