Final answer:
The prospect theory utility function for x<0 is not explicitly provided in the options. The theory suggests a value function that is typically concave for gains and convex for losses, reflecting loss aversion and diminishing marginal utility.
Step-by-step explanation:
The prospect theory utility function for x<0 is not clearly defined by the options provided (1 to 4). Prospect theory, introduced by Daniel Kahneman and Amos Tversky, describes how people choose between probabilistic alternatives that involve risk, where the probabilities of outcomes are known. The theory differs from the expected utility theory in that it uses a value function that is defined for gains and losses rather than final wealth, and this value function is typically concave for gains, indicating diminishing marginal utility, and commonly convex for losses, showing that losses are weighed more heavily than gains.
It incorporates a concept called loss aversion, which suggests that losses have a greater emotional impact on an individual than an equivalent amount of gains. Thus, none of the presented options (U(x) = x, U(x) = x², U(x) = x³, U(x) = x´) strictly represents the utility function under prospect theory for negative values of x.
To maximize utility, one would need to make decisions that generate the highest perceived value, taking into account the diminishing marginal utility of gains and the greater impact of losses. The concept of marginal utility is vital as it represents the additional satisfaction a person receives from consuming an additional unit of a good or service. The principle of diminishing marginal utility is observed when each additional unit of consumption provides less additional utility than the previous one.