ENGR.ECONOMIC ANALYSIS
ENGR.ECONOMIC ANALYSIS
14th Edition
ISBN: 9780190931919
Author: NEWNAN
Publisher: Oxford University Press
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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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Transcribed Image Text:### 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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(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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