Fixed versus random volatility with the same mean

Compare two prices for the same European call. In one, volatility is a constant $30\%$. In the other, volatility is random (independent of the stock) with an average of $30\%$. Which call is more valuable?

It depends on moneyness: at the money they are almost equal, away from the money the random-volatility call is worth more. With independent random volatility the price is $\mathbb{E}[C(\sigma)]$. Near the money $C\approx0.4\,\sigma S\sqrt T$ is almost linear in $\sigma$, so averaging changes little; for in- or out-of-the-money calls $C$ is convex in $\sigma$ (positive volga), so by Jensen $\mathbb{E}[C(\sigma)]>C(30\%)$.