The indifference curves representing altruistic preferences have a downward slope, which shows that the individual is willing to trade off some of their own payoff for the benefit of another. An important aspect of these curves is their slope, which quantifies this trade-off at any given point and reflects the individual's specific degree of altruism.

Characteristics of Altruistic Indifference Curves

The left-hand panel of Figure 4.10 depicts Zoë's preferences as altruistic, yet this altruism is limited. While her utility increases with both her own and Yvonne's monetary gain, the indifference curves are shown to be quite steep. This steepness, especially where their shares are similar, illustrates that she does not value Yvonne's gains as much as her own. For example, the middle curve contains a point where each person has £120.

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The decision Zoë faces with her £200 lottery winnings can be graphically modeled using indifference curves and a feasible frontier. In this model, the two 'goods' being considered are the amount of money for Zoë and the amount for Yvonne. Her altruistic preferences for different allocations of these goods are represented by the indifference curves, while the feasible frontier shows all possible sharing arrangements. The graphical analysis helps identify the optimal distribution by finding the point on the feasible frontier that touches the highest possible indifference curve.

Graphical Modeling of Zoë's Altruistic Choice with Indifference Curves

The feasible set for Zoë's lottery dilemma includes all possible ways she can divide the £200 prize. The boundary of this set is the feasible frontier, which in this case is a budget constraint mathematically expressed by the equation $y + z = 200$. In this formula, $z$ represents the money Zoë keeps, and $y$ is the money she gives to Yvonne. On a graph, this frontier is a straight, downward-sloping line that joins the points (200, 0) and (0, 200). The complete feasible set consists of this line and the entire area underneath it, representing all affordable allocation combinations.

Zoë's Feasible Set and Budget Constraint in the Lottery Dilemma

To cite the ebook, use The CORE Econ Team 2023 The Economy 2.0: Microeconomics Open access e-text https://core-econ.org/the-economy/.

To cite a unit, use The CORE Econ Team 2023 The Economy 2.0: Microeconomics Open access e-text https://core-econ.org/the-economy/ Unit [unit number] (url).

CORE Econ - The Economy 2.0: Microeconomics

Figure 4.10 - Zoë's Limited Altruism

Zoë's choice of how to distribute the lottery winnings is caused by her personal preferences. She will select the point from the feasible set of all possible distributions that lands on her highest possible indifference curve, thereby maximizing her utility. This outcome directly depends on whether her preferences are altruistic or self-interested.

Preferences Determine Optimal Choice in Zoë's Dilemma

Zoë's altruistic preferences are mathematically defined by the utility function $U(z, y) = (z-100)^2 + (y-100)^2$. In this equation, $z$ represents the amount of money Zoë keeps for herself, and $y$ represents the amount she gives to Yvonne. This specific function is used to generate the indifference curves depicted in Figure E4.1, which illustrate her choices.

Zoë's Utility Function for Altruistic Choice

This diagram shows Zoë's choice scenario under the assumption of purely self-interested preferences. The graph plots Zoë's share of the money on the horizontal axis and Yvonne's on the vertical axis. The feasible set is the area defined by the axes and the feasible frontier, a line connecting (0, 200) to S (200, 0). With self-interested preferences, Zoë's utility depends only on her own payoff, represented by vertical indifference curves. Her utility increases as she moves to curves further to the right. Therefore, her optimal choice is point S, where she keeps the entire £200 prize and gives nothing to Yvonne.

Figure 4.10 - Optimal Choice with Self-Interested Preferences

This diagram, identified as Figure 4.11 and reproduced as E4.1, illustrates how an altruistic Zoë chooses to distribute her £200 lottery winnings. It graphs 'money for Zoë' (horizontal axis) against 'money for Yvonne' (vertical axis). The feasible frontier, representing all possible splits, is a line from (200, 0) to (0, 200). Zoë's preferences are depicted by downward-sloping, convex indifference curves. Her optimal choice, which maximizes her utility, is at point A (140, 60), where the feasible frontier is tangent to the highest attainable indifference curve. This means she keeps £140 and gives £60 to Yvonne.

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