# 48 Distribution

Distributional preferences are preferences that relate to the relative amount of money or resources each person gets or has.

It is often easy to incorporate distributional preferences into economic analysis as they are a natural extension of how economists think about individuals’ preferences. We can extend to other people the typical assumption that a person cares about their own material outcomes.

We will examine two types of distributional preferences: altruism and inequality aversion.

## 48.1 Altruism

Altruism is concern for the outcomes of others.

To incorporate altruism, we give a positive weight to the utility of others in the utility function. An example utility function might be:

U_i(x_i,x_j)=x_i+\alpha x_j

where U_i is the utility of agent i, x_i the outcome for agent i and x_j the outcome for agent j. \alpha is some number greater than zero.

Altruism might have different drivers.

For example, the agent might exhibit pure altruism, with genuine concern for others’ wellbeing.

Alternatively, the agent might exhibit impure altruism. They experience a “warm glow” about doing good without actually caring about the other’s wellbeing.

Altruism, however, is insufficient to explain some experimental results, such as those in the ultimatum game. While it could predict non-zero offers by the proposer, it does not predict the rejection of any offers by the responder. Rejection harms both the responder and the proposer.

The proposer could only reject if a negative weight were applied to either their own or the proposer’s outcome.

## 48.2 Inequality aversion

An alternative distributional preference model that may explain some of these results is inequality aversion.

The idea behind inequality aversion is that people may dislike having less than others and dislike having more than others.

### 48.2.1 The Fehr-Schmidt model

One basic mode of inquality aversion comes from the utility function in Fehr and Schmidt (1999). It is of the following form:

u_i(x_i,x_j)=x_i-\alpha\text{max}\{x_j-x_i,0\}-\beta\text{max}\{x_i-x_j,0\}

The three terms in this function represent:

The utility of their own outcome x_i

Their dislike of having less than the other agent (where \alpha>0)

Their dislike of having more than the other agent (where \beta>0)

We can also write this utility function as:

u_i(x_i,x_j)=x_i-\left\{\begin{matrix} \beta(x_i-x_j) \quad &\textrm{if} \quad x_i \geq x_j \\[6pt] \alpha(x_j-x_i) \quad &\textrm{if} \quad x_i < x_j \end{matrix}\right.

Typically \alpha>\beta as people dislike having less than others than they dislike having more than others. We could also set \beta<0 for an agent that likes to be better off than others.

This utility function has a kink at x_j where agent i moves from having less to more than agent j. If 0<\beta<1 as in this diagram, the utility of agent i, U(x_i) continues to increase in x_i above x_j, but at a decreasing rate as inequality degrades the benefits of having more.

### 48.2.2 The ultimatum game

We can examine the Fehr-Schmidt model in the context of the ultimatum game.

Suppose two players of the ultimatum game have Fehr-Schmidt preferences, with \beta=0.25 and \alpha=0.5.

What offers x would the responder reject where the proposer has $10 to split between them?

If the responder rejects, the payoff to the proposer and responder is zero. That is:

x_P=x_R=0

If the responder accepts, the responder receives x, and the proposer keeps the remainder. That is:

\begin{align*} x_P&=10−x \\[6pt] x_R&=x \end{align*}

The responder will accept if the utility of accepting is greater than the utility of rejecting. That is:

\begin{align*} U_R(\text{accept})&>U_R(\text{reject}) \\[6pt] \underbrace{x_R−\alpha\text{max}\{x_P−x_R,0\}−\beta\text{max}\{x_R−x_P,0\}}_{\text{Substituting in the Fehr-Schmidt utility function}}&>0 \\[24pt] \underbrace{x−\alpha\text{max}\{10−x−x,0\}−\beta\text{max}\{x−(10−x),0\}}_{\text{Substituting in the payoffs when accepted}}&>0 \\[24pt] x−\alpha\text{max}\{10−2x,0\}−\beta\text{max}\{2x−10, 0\}&>0 \\ \end{align*}If the offer is more than $5, the \alpha term is multiplied by zero and the inequality becomes:

\begin{align*} x−\beta\text{max}\{2𝑥−10, 0\}>0 \\[6pt] x−\beta(2x−10)>0 \end{align*}This will always hold for any \beta<1 as x>5 and 5\geq 2x-10>0. Therefore, the condition will hold for the agent with \beta=0.5. Recall that if \beta<1 the responder has higher utility from a higher payoff but at a decreasing rate when they have more than the proposer. In this case, if \beta<1 the responder will always accept offers greater than $5.

If the offer is less than $5, the \beta term is multiplied by zero and the inequality becomes:

\begin{align*} x−\alpha\text{max}\{10−2x,0\}>0 \\[6pt] x−\alpha(10−2x)>0 \end{align*}Whether this holds depends on the value of \alpha and the size of the offer x. If \alpha=1/2, then:

\begin{align*} \bigg(1+\frac{1}{2}\bigg)x−\frac{1}{2} (10−x)&>0 \\ 2x−5&>0 \\[6pt] x&>2.5 \end{align*}A responder with \alpha=1/2 will reject any offer under $2.50.

We can plot the utility function for this game as the size of the offer increases. As the offer is not independent of the proposer’s payoff, I will derive the shape of the utility curve as a function of x_R=x.

\begin{align*} U_R(x_P,x_R)&=x_R-\alpha\text{max}\{x_P-x_R,0\}-\beta\text{max}\{x_R-x_P,0\} \\[6pt] &=x-\alpha\text{max}\{10-2x,0\}-\beta\text{max}\{2x-10,0\} \end{align*}We can also write this as:

U_R(x_P,x_R)=\left\{\begin{matrix} (1+2\alpha)x−10\alpha \quad &\textrm{if} \quad x \geq 0 \\[6pt] (1−2\beta)x+10\beta \quad &\textrm{if} \quad x < 0 \end{matrix}\right.

The slope of each of these curves is twice that we saw earlier as any increase in outcome for the responder is matched by a decrease in outcome for the proposer (and vice versa).

This diagram shows the responder’s utility curve as a function of the offer x.

## 48.3 The Charness-Rabin model

Charness and Rabin (2002) developed a utility function that captures the possible forms of distributional preference. An agent’s attitude toward others depends on their relative position. The utility function is:

u_i(x_i,x_j)=\left\{\begin{matrix} \rho x_j+(1-\rho)x_i \quad &\textrm{if} \quad x_i \geq x_j \\[6pt] \sigma x_j+(1-\sigma)x_i \quad &\textrm{if} \quad x_i < x_j \end{matrix}\right.

Where x_i is the payoff to player i and x_j is the payoff to the other player.

\rho and \sigma capture the agent’s attitudes toward others. When the agent is ahead the other player’s welfare enters their utility via \rho. When the agent is behind the other player’s welfare enters agent’s utility via \sigma. For most people \rho>\sigma. They give more weight to others’ utility when they are better off. \sigma can also be less than zero. If they are behind someone, they place negative weight on further gains by that person.

This utility function is equivalent to that of Fehr and Schmidt (1999). You can rearrange the terms to show that \beta=\rho and \alpha=-\sigma. However, expressing the utility function in this way allows us to consider distributional preferences other than inequality aversion in a more intuitive way.

### 48.3.1 The dictator game

Consider the following example of the dictator game. In the dictator game, the dictator makes a unilateral offer to the receiver. The game then ends. The receiver has an empty strategy set.

In this version of the game, the dictator must decide between the allocations (0, 1) and (1, 5), where (x_D,x_R) represent the payoffs for the dictator and receiver, respectively. The dictator’s \sigma=−1/2. As the dictator has less than the other player under each distribution, \sigma is the relevant parameter.

We can calculate the dictator’s utility of each allocation.

\begin{align*} U(0,1)&=\sigma\times 1+(1−\sigma)\times 0 \\[6pt] &=−1/2×1+(1+1/2)\times 0 \\[6pt] &=−1/2 \\[12pt] U(1,5)&=\sigma\times 5+(1−\sigma)\times 1 \\[6pt] &=−1/2×5+(1+1/2)\times 1 \\[6pt] &=−1 \end{align*}The dictator prefers to allocate (0,1), even though it is worse for them because it is also worse for the other player.

\sigma<0 can also account for the rejection of low offers in the ultimatum game.

### 48.3.2 Forms of distributional preferences

We can adjust the values of \rho and \sigma to capture many forms of distributional preferences. Some are as follows.

If \sigma>0 and \rho>0, the agent is altruistic. A higher payoff to the other player increases the agent’s utility.

If 1\geq\rho\geq 0>\sigma, the agent is inequality averse. If the other player has more, the agent’s utility decreases with further gains for the other player. If the other player has less, the agent’s utility increases with further gains for either agent.

If 0>\rho\geq\sigma, the agent is status-seeking. They gain more utility by having more than the other player. Their utility goes up when either they get more or the other player gets less.

If \rho=\sigma=0 we are left with the classical self-interested utility function. The agent only cares about their own payoff.

If \rho=1 and \sigma=0, then u_i(x_i,x_j)=\text{min}\{x_i,x_j\}. The agent has Rawlsian preferences whereby the agent seeks the greatest benefit for the least advantaged.

If \rho=\sigma=1/2, then u_i(x_i,x_j)=x_i+x_j. The agent has utilitarian preferences whereby the agent seeks to maximise total utility.

We could develop similar forms of preferences by adjusting the values of \alpha and \beta in the Fehr-Schmidt model.

### 48.3.3 Example: the trust game

In the exercises in Section 43.3, I considered whether Linda should invest in Marco’s startup:

Linda is looking for investment opportunities. She identifies a promising crypto-based start-up created by Marco. Marco is looking for seed funding.

Linda can invest $10.

If Linda invests, her investment will triple in value. Marco can then decide to either shut down the start-up and keep the $30 or maintain the start-up in the market and pay a $15 dividend to each of Linda and himself.

If Linda does not invest, Linda keeps the $10. The start-up gets $0.

Macro, who is effectively playing a dictator game, would shut down and keep the $30. As a result, Linda would not invest.

Suppose now that Linda and Marco have preferences as follows:

U_L(x_L,x_M)=\left\{\begin{matrix} \frac{1}{3}x_L+\frac{2}{3}x_M \quad &\textrm{if} \quad x_L \geq x_M\\[6pt] \frac{2}{3}x_L+\frac{1}{3}x_M \quad &\textrm{if} \quad x_L < x_M \end{matrix}\right.

U_M(x_L,x_M)=\left\{\begin{matrix} \frac{3}{4}x_L+\frac{1}{4 }x_M \quad &\textrm{if} \quad x_M \geq x_L\\[6pt] x_M \quad &\textrm{if} \quad x_M < x_L \end{matrix}\right.

Where U_L and U_M are Linda and Marco’s utility functions. x_L and x_M are the outcomes for Linda and Marco.

Both Marco and Linda give positive weight to the payoff of the other in most circumstances, except for Marco, who, when he is behind Linda, only cares about himself.

Marco and Linda know each other’s utility functions.

What is the equilibrium with these distributional preferences?

If Linda chooses trust, Marco has a choice between $15 each and $30 for himself. Marco calculates the utility of each option.

U_M(x_L,x_M)=\left\{\begin{matrix} \frac{3}{4}x_L+\frac{1}{4}x_M \quad &\textrm{if} \quad x_M \geq x_L\\[6pt] x_M \quad &\textrm{if} \quad x_M < x_L \end{matrix}\right.

\begin{align*} U_M(15,15)&=\frac{3}{4}(15)+\frac{1}{4}(15)=15 \\[12pt] U_M(0,30)&=\frac{3}{4}(0)+\frac{1}{4}(30)=7.5 \end{align*}

Marco receives higher utility by paying the dividend to Linda.

Linda also has utility from each distribution.

U_L(x_L,x_M)=\left\{\begin{matrix} \frac{1}{3}x_L+\frac{2}{3}x_M \quad &\textrm{if} \quad x_L \geq x_M\\[6pt] \frac{2}{3}x_L+\frac{1}{3}x_M \quad &\textrm{if} \quad x_L < x_M \end{matrix}\right.

\begin{align*} U_L(15,15)&=\frac{1}{3}(15)+\frac{2}{3}(15)=15 \\[12pt] U_L(0,30)&=\frac{2}{3}(0)+\frac{1}{3}(30)=10 \end{align*}

Linda would prefer that Marco pay a dividend.

For the other node, if Linda does not invest, she will keep $10. Marco will have nothing.

\begin{align*} U_M(10,0)=0 \\ \\ U_L(10,0)=\frac{1}{3}(10)+\frac{2}{3}(0)=3.33 \end{align*}Putting those payoffs into the extensive form of the game, we get the following:

In this game, Marco can return a dividend for utility 15 or shut down for utility 7.5. He chooses to return the dividend. As a result, Linda will invest for utility 15, rather than not invest for utility 3.33. Linda invests.