I'd like some guidance on the below practice question on uncertainty in consumer theory. I think I am confused on the certainty equivalent & risk premium or I'm not understanding the question. Please explain your process.

A player has initial wealth $w$ and a vNM utility function $u(w)$ that satisfies $u'(w)>0$ for all $w \in R $. For some fixed payment he can play a game with the following payoffs: probability $\frac{1}{2}$ player wins $z>0$, and with probability $\frac{1}{2}$ player wins nothing. Assume the player does not get utility from gambling but only from potential monetary rewards.

1) Let $s$ denote the maximum amount of money the player will be willing to pay to play the game. Explain why $s$ satisfies the equation:

$u(w) = \frac{1}{2}u(w-s) + \frac{1}{2}u(w+z-s)$

2) Provide argument as to why $s$ must be strictly less than $z$.

My first thought for part 1 was that $s$ was a risk premium as it's the amount of money the player is willing to pay to play the game. I'm not sure that this would make sense? The risk premium by definition: $p=E(g)-CE$ where $E(g)$ is the expected value of a gamble and $CE$ is the certainty equivalent. $E(g)=\frac{1}{2}z+\frac{1}{2}0$. Now I need the $CE$ but I can't find it because I don't have the functional form of $u()$ - correct?

For part 2, I'm assuming the player is risk loving (but without sufficient justification) in which case if $s$ is the risk premium and the player is risk adverse then the premium should be less than the value of taking the gamble.


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