Final answer:
To decide if an individual with a VNM utility function will accept the gamble, we calculate the expected utility. With an expected utility of approximately 6.33 from the gamble and 6 from the current wealth of 36 units, the individual would accept the gamble as it presents higher utility.
Step-by-step explanation:
To determine if an individual will accept the gamble using a Von-Neumann-Morgenstern (VNM) utility function, u = √w; where w is wealth, we first calculate the expected utility of the gamble. With initial wealth of 36 units, winning 13 units with probability 2/3 and losing 11 units with probability 1/3, we find the expected utilities for each outcome:
- If the individual wins: u(36+13) = u(49) = √49 = 7.
- If the individual loses: u(36-11) = u(25) = √25 = 5.
The expected utility (EU) of the gamble is then:
EU = (2/3 × 7) + (1/3 × 5) = (14/3) + (5/3) = 19/3 ≈ 6.33
The utility without gambling (the status quo) is u(36) = √36 = 6. Since the expected utility of the gamble, which is 6.33, is higher than the utility of not gambling, which is 6, the individual adhering to the VNM utility theory would accept the gamble, as it yields greater expected utility.