Restrictions proving quasiconcavity of utility function I have the utility function given as $u(x)=x_1+x_2^\gamma$. The answer says that for a utility function to exhibit strict quasi-concavity, $\gamma$ needs to be between 0 and 1; but I don't understand why this is true. I know strict quasi-concavity means that if $f(x) ≥ f(x')$, then $f((1−λ)x + λx') &gt; f(x')$ but I don't know how this can be shown in this example...
Since $u$ is twice differentiable, it is quasi-concave if and only if its marginal rate of substitution declines as $x_1$ increases and $x_2$ falls. You can then compute
$$
MRS = \frac{\frac{\partial u}{\partial x_1}}{\frac{\partial u}{\partial x_2}} = \frac{1}{\gamma x_2^{\gamma - 1}} = \frac{x_2^{1 - \gamma}}{\gamma};
$$
which is increasing in $x_2$ (so quasi-concave) if and only if $\gamma &lt; 1$. [We assume throughout that $\gamma &gt; 0$ since $u$ is increasing.]