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Risky assets are financial instruments or investments that come with a certain level of uncertainty concerning their returns. Unlike risk-free assets, which have guaranteed returns, risky assets can fluctuate significantly. This is why they can offer both the potential for high returns and the risk of substantial losses.

Investors are drawn to risky assets due to the potential reward for taking on more uncertainty, a concept known as the risk-return tradeoff. Common examples include stocks, commodities, and real estate. These assets are appealing because if the market goes well, they can provide significant profits. However, if the market doesn't perform as expected, losses can also be substantial.

When dealing with risky assets, an investor must consider their risk tolerance, or their comfort level with the uncertainty of returns. Understanding how one’s wealth level impacts decision-making regarding investments in risky assets is essential in managing investment portfolios.

A linear utility function is a type of utility equation where the utility received from wealth is a straight-line relationship. This means that each additional unit of wealth provides the same amount of added utility or satisfaction regardless of how much wealth the person already has.

Mathematically, we define a linear utility function as follows:

• Let the utility function be expressed as \( U(W) = aW + b \) where \( a \) and \( b \) are constants.
  
• Here, \( W \) indicates wealth, \( a \) represents the marginal utility, and \( b \) is a constant term that can shift the utility function up or down on the utility axis.
  

The central concept of a linear utility function in expected utility theory is that it implies constant marginal utility of wealth. This means no matter how rich or how poor you are, each dollar added to wealth provides the same increase in utility. This has major implications for decision making, particularly in contexts involving risky assets. With a linear utility function, the investment in a risky asset does not depend on the level of wealth.

Marginal utility refers to the additional satisfaction or benefit gained from consuming one more unit of a good or service. In the context of wealth, it's how much extra utility, or happiness, you receive when your wealth increases by one unit, say a dollar.

In economic decisions, the concept of marginal utility plays a crucial role. It impacts how resources are allocated and how decisions are made. When marginal utility is constant, as seen in linear utility functions, each additional dollar of wealth is valued the same irrespective of the total amount of wealth one has.

Understanding marginal utility helps in analyzing choices under uncertainty, particularly concerning investments. If we assume that the marginal utility of wealth doesn't change, then changes in wealth do not affect an investor's appetite for risky assets. This insight is essential in expected utility theory, as it explains why certain investors make specific investment choices regardless of their wealth levels.