### Problem 2: Analysis of the St. Petersburg Paradox**(a) St. Petersburg Paradox Overview**The St. Petersburg Paradox, first discussed by Daniel Bernoulli in 1738, involves a theoretical game where a coin is tossed repeatedly. The player earns a payoff of $2^n$, where $n$ is the number of tosses required to get the first head. For example, if the sequence of tosses is TTTT followed by H, the payoff is $2^4$. This outcome occurs with a probability of $\frac{1}{2^4}$. Calculate the expected value of playing this game.**(b) Expected Payout with Utility Function**Assume the utility $U$ is a function of wealth $X$ given by $U = X^5$ and $X = \$1,000,000$. Assume further that the game concludes if a head does not appear after 40 tosses. The payoff then is $2^{40}$. Determine the expected payout of this modified game.**(c) Maximum Payment for Utility Equivalence**Determine the maximum amount one would be willing to pay to play the game, given that the expected utility after playing must be equal to the pre-game utility. Utilize Excel’s Goal Seek function (located in Data, What-If Analysis) to find this value.

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Answered: 2 (a) Consider the St. Petersburg…

ENGR.ECONOMIC ANALYSIS

14th Edition

ISBN:9780190931919

Author:NEWNAN

Publisher:NEWNAN

Chapter1: Making Economics Decisions

Section: Chapter Questions

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### Problem 2: Analysis of the St. Petersburg Paradox

**(a) St. Petersburg Paradox Overview**

The St. Petersburg Paradox, first discussed by Daniel Bernoulli in 1738, involves a theoretical game where a coin is tossed repeatedly. The player earns a payoff of $2^n$, where $n$ is the number of tosses required to get the first head. For example, if the sequence of tosses is TTTT followed by H, the payoff is $2^4$. This outcome occurs with a probability of $\frac{1}{2^4}$. Calculate the expected value of playing this game.

**(b) Expected Payout with Utility Function**

Assume the utility $U$ is a function of wealth $X$ given by $U = X^5$ and $X = \$1,000,000$. Assume further that the game concludes if a head does not appear after 40 tosses. The payoff then is $2^{40}$. Determine the expected payout of this modified game.

**(c) Maximum Payment for Utility Equivalence**

Determine the maximum amount one would be willing to pay to play the game, given that the expected utility after playing must be equal to the pre-game utility. Utilize Excel’s Goal Seek function (located in Data, What-If Analysis) to find this value.

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(b) Assume that utility U is a function of wealth X given by U = X^.5 and that X = $1,000,000. In this part of the question, assume that the game ends if the first head has not occurred after 40 tosses of the coin. In that case, the payoff is 2⁴⁰ and the game is over. What is the expected payout of this game?

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