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Suppose a consumer has log-utility over wealth, defined by $u(W) = \ln(W)$. Suppose this consumer has $100$, and is considering taking a gamble in which the consumer flips a coin, and gets $20$ she flips heads and $0$ if she flips tails. What is the most amount of money the consumer would be willing to pay to play take this gamble? What if the consumer has $1, 000$ dollars of wealth?

I feel like this question is incomplete since we do not know how much the person need to give in order to play the game. Do we know that information in order to solve this problem?

If no, could anyone tell me the direction and piece of information needed to solve for this question?

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The question states: "What is the most amount of money the consumer would be willing to pay to play take this gamble?". What you need to find is the number the person willing to give. –  Bob Jansen Feb 15 at 9:28
    
You haven't provided enough info. The 0 or 20 payoff is after what initial investment? Do they lose their initial investment?? –  user2763361 Feb 15 at 11:23
1  
This looks like homework question –  user2763361 Feb 15 at 11:23

2 Answers 2

This is a basic utility exercise. I would guess the additional assumption you are missing to solve the exercise is that the player would be willing to accept the fair game but nothing worse.

To make this a fair game, the maximal amount which could be paid to enter the game, would result in zero expected loss of utility (nobody would accept anything worse). In other words, the expected utility must match the utility of the starting wealth $W_0$: $$ \mathrm{E}[\ln(W)]=\ln(W_0)$$ Since this game has only two outcomes you can easily compute the expectation from the tree of possible outcomes

    / Head: 100 + 20 - entry fee
   / (50%)
  /
100
  \
   \ (50%)
    \ Tail: 100 + 0 - entry fee

From this we can compute the expectation directly: $$\mathrm{E}[\ln(W)] = 0.5\ln(120-x) + 0.5\ln(100-x)=\ln(100)$$

The result is approximately 9.5 (you can check here), and if you repeat with a starting wealth of 1000, the result is approx. 9.9, since the richer person doesn't care as much about loosing a few bucks as the poorer guy does.

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If there is not a entry fee, I cannot lose money for at worst, I win 0USD.

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The question is about the entry fee. –  user1157 Feb 15 at 11:38

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