Question

Suppose that Philip has a total utility of income given by \(\mathrm{U}(\mathrm{I})=\mathrm{I}\), where \(I\) denotes income. In a graph with utility on the vertical axis and income on the horizontal axis, plot Philip's total utility of income. What is Philip's attitude towards risk? Explain.

Short answer

Philip's utility is linear, showing he is risk-neutral.

Step-by-step solution

Understand the Utility Function

The utility function given is \( U(I) = I \), which is a linear function. This indicates that Philip's utility is directly proportional to his income, meaning each additional unit of income increases his utility by the same amount as the previous.

Plot the Utility Function

To plot the utility function, draw a graph with income \( I \) on the horizontal axis and utility \( U \) on the vertical axis. Since \( U(I) = I \), draw a 45-degree line from the origin (0,0) that extends into the first quadrant, making sure it passes through points like (1,1), (2,2), and so on.

Analyze Slope and Risk Attitude

A linear utility function with a constant slope, such as \( U(I) = I \), implies that Philip is risk-neutral. This means that he values each additional unit of income equally, regardless of the level of income he currently possesses.

Key concepts

Risk Neutrality

Risk neutrality is one of the three primary attitudes towards risk that individuals can have. It refers to a situation where a person's choices are unaffected by the risk involved. In other words, a risk-neutral person is indifferent between a certain outcome and a gamble with the same expected value.
For example, if Philip is offered a guaranteed income of \(100 or a 50% chance to receive \)200, he would treat these options equally because the expected value is the same.
In economic terms, risk neutrality implies that a dollar amount is valued equally regardless of certainty or risk.
This concept closely relates to Philip's utility function: \( U(I) = I \), since it's a linear relationship indicating consistent valuation of income, regardless of amount or risk involved.

• Risk neutrality means decisions are based solely on expected outcomes.
• It reflects an even trade-off between risk and certainty.

Income

Understanding income in the context of utility functions is key to analyzing economic models. Income, often denoted as \( I \), represents the total earnings available for spending.
In Philip's case, his utility function relation shows that income is directly translated into utility. More income simply means more utility, showing a straightforward and unweighted relationship.
This linear relationship implies that any increase in income brings about the same increase in satisfaction or happiness.

• Income is a critical variable as it directly affects an individual's utility.
• Higher income levels correlate with higher utility due to the direct proportionality.

With the linear function \( U(I) = I \), we see that any factor increasing \( I \) results in increased utility, without any diminishing returns at higher income levels, a unique characteristic of a risk-neutral perspective.

Economic Graphing

Economic graphing is a pivotal tool in visualizing relationships in economic models, such as the utility function \( U(I) = I \).
Through graphing, concepts like utility, risk neutrality, and income become more tangible and understandable.
The graph in this context has utility on the vertical axis and income on the horizontal axis, illustrating the direct and linear relationship between the two.
Philip's utility graph would be a 45-degree line originating at the origin, reflecting equal increments in income leading to equal increments in utility. Graphs like this are essential for showing:

• Linear relationships between variables, reinforcing the concept of risk neutrality.
• Visual interpretations of mathematical models to simplify complex concepts.

Graphing aids in understanding how different levels of income affect utility and provides a clear visual representation of theoretical ideas, making it easier for students to grasp otherwise abstract economic principles.