The Encyclopedia of Economic Principals

Chapter 20

Social Preferences, Fairness, Identity, and Reciprocity

Put envy, trust, gifts and matching into the payoffs and see the choices change.

Four of the chapter's worked examples, made interactive: an envious responder in an ultimatum game, the trust game, a gift wage, and conditional cooperation in a public-goods game. Change one value at a time and watch the figure, the numbers and the hand calculation respond.

Every example here is a constructed teaching example: it uses the hypothetical numbers of the chapter's worked examples, plus a few values added for comparison and labelled as such in each panel. Nothing here measures a real market, firm or household.

Demonstration 1 of 4

How much envy forces a fair offer

How large must an ultimatum offer be before an envious responder accepts it?

The line is the responder's utility from accepting each offer. Where it lies below zero, rejecting is better, so the shaded offers fail. A larger envy weight steepens the penalty and pushes the threshold s* to the right.

Equation, written in LaTeX: U_R=4-0.5(16-4)=-2.

Equation, written in LaTeX: s-0.5[(20-s)-s]\geq0.

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A proposer splits $20 and offers s to the responder, keeping 20 - s. Alpha is the responder's weight on being behind. U_R is the responder's utility of accepting; rejecting gives both zero.

Predict first. Will the responder accept $4 out of $20 with alpha = 0.5?

Your prediction

Choose an example

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Figure: How much envy forces a fair offer. Responder utility of accepting rises with the offer. With envy weight 0.50 it crosses zero at s* = 5.00; the offer of 4 gives utility -2.00.
Envy weight alpha: 0.5, Offer to responder ($): $4
Constructed example: the chapter's hypothetical $20 ultimatum game (offer 4, alpha 0.5, threshold 5); envy weights 0, 0.25 and 1 are added for comparison.

Calculated values

Proposer keeps ($)
16
U of accepting
-2.00
U of rejecting
0.00
Minimum acceptable offer s*
5.00
Smallest whole-dollar offer
$5
Decision
Reject

U = 4 - 0.50 x (16 - 4) = 4 - 6.00 = -2.00. Accepting is worse than the zero from rejection, so the responder rejects. The threshold solves s - 0.50(20 - 2s) = 0, so s* = 0.50 x 20 / (1 + 2 x 0.50) = 10.00 / 2.00 = 5.00.

Worked steps

  1. Proposer keeps 20 - 4 = 16
  2. Disadvantage = 16 - 4 = 12
  3. U = 4 - 0.50 x 12 = -2.00
  4. s* = 10.00 / 2.00 = 5.00, so offers of $5 or more are accepted
  5. Decision: reject

Use the idea

When a counterpart may care about relative shares, estimate the smallest split they will accept rather than assuming any positive amount will do.

Where the conclusion applies

One shot, known envy weight, no weight on being ahead, and rejection leaves both with zero. Envy weights differ across people and are not observed directly.

Check your understanding: With alpha = 1, what is the smallest acceptable offer?
s - 1(20 - 2s) >= 0 gives 3s >= 20, s >= 6.67, so $7 in whole dollars.

Chapter 20 source: section "Inequity aversion".

Demonstration 2 of 4

Trust game surplus

When does sending money to a stranger pay the sender, and how much surplus does it create?

Each dollar sent becomes three, so total payoff grows by 2 for every dollar sent. Who gets the gain depends on the return: the sender profits only when r exceeds s.

Equation, written in LaTeX: \pi_1=12-6+8=\$14,

Equation, written in LaTeX: \pi_2=18-8=\$10.

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The sender holds E = 12 and sends s; the amount is tripled (m = 3). The receiver returns r. pi1 = E - s + r is the sender's payoff and pi2 = 3s - r the receiver's. The return is set as a share of what the receiver gets; 4/9 of 18 is the book's 8.

Predict first. Does sending $6 leave the sender better off than sending nothing?

Your prediction

Choose an example

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Figure: Trust game surplus. Stacked payoff bars. Without trust the sender has 12 and the receiver 0. Sending 6 with 4/9 returned gives the sender 14 and the receiver 10, total 24.
Amount sent ($): $6, Share of tripled amount returned: 4/9 (book: 8 of 18)
Constructed example: the chapter's hypothetical game (E 12, m 3, send 6, return 8, and the selfish benchmark of sending 0); sends of 3 and 12 and a one-third return share are added.

Calculated values

Receiver gets ($)
18
Returned r ($)
8
Sender payoff pi1 ($)
14
Receiver payoff pi2 ($)
10
Total payoff ($)
24
Sender vs no trust
gains

The receiver gets 3 x 6 = 18 and returns 4/9 of it, r = 8. pi1 = 12 - 6 + 8 = 14 and pi2 = 18 - 8 = 10, a total of 24. Trusting pays the sender 2 more than keeping the 12.

Worked steps

  1. Receiver gets 3 x 6 = 18
  2. r = 4/9 x 18 = 8
  3. pi1 = 12 - 6 + 8 = 14
  4. pi2 = 18 - 8 = 10
  5. Total = 12 + 2 x 6 = 24

Use the idea

Before extending trust, compare the amount at risk with the return you can reasonably expect; the surplus exists only if trust is extended at all.

Where the conclusion applies

One anonymous round, a known multiplier and a return written as a fixed share. Real returns vary and may depend on intentions, not only on the amount received.

Check your understanding: If the receiver returns one third of what she gets, does trusting still pay the sender?
She returns 18/3 = 6, so pi1 = 12 - 6 + 6 = 12, exactly break even; the total is still 24.

Chapter 20 source: section "Trust-game reciprocity".

Demonstration 3 of 4

When a gift wage pays for itself

How much extra effort must a generous wage buy before it raises profit?

The premium is a fixed $4. It pays only if the effort it induces adds more than $4 of output value. A lower value per unit raises the effort needed to break even.

Equation, written in LaTeX: 8(\$6)=\$48,

Equation, written in LaTeX: \$60-\$22=\$38.

Equation, written in LaTeX: \$6[e(22)-e(18)]>\$4.

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The customary wage is $18 with effort e(18) = 8 units; the gift wage is $22 with effort e(22). Each unit is worth v dollars before labor cost. Profit is v x effort - wage.

Predict first. Does the $4 premium raise profit if effort rises to 10 units?

Your prediction

Choose an example

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Figure: When a gift wage pays for itself. Two profit bars: $30.00 at the $18 wage and $38.00 at the $22 wage with effort 10 and value 6 per unit.
Effort at the $22 wage (units): 10, Value per unit ($): $6
Constructed example: the chapter's hypothetical warehouse ($18 and $22 wages, $6 per unit, effort 8 and 10, alternative 8.5); effort 9 and values $4 and $8 are added for comparison.

Calculated values

Profit at $18
$30.00
Profit at $22
$38.00
Change in profit
$8.00
Break-even effort at $22
8.67
Gift wage
pays

At $18: 6 x 8 - 18 = 48 - 18 = $30.00. At $22: 6 x 10 - 22 = 60 - 22 = $38.00. The gift wage raises profit by $8.00. The premium pays when 6(e - 8) > 4, that is e > 8 + 4/6 = 8.67.

Worked steps

  1. Profit at $18 = 6 x 8 - 18 = $30.00
  2. Profit at $22 = 6 x 10 - 22 = $38.00
  3. Extra output value = 6 x (10 - 8) = 12, against a $4 premium
  4. Break-even effort = 8 + 4 / 6 = 8.67

Use the idea

Before paying above the going wage to buy goodwill, estimate the effort response and compare its value with the premium.

Where the conclusion applies

One worker, one hour, a known effort response and a constant value per unit. The effort response is the uncertain part and may fade over time.

Check your understanding: At value $6 per unit, what effort at $22 just breaks even?
6(e - 8) = 4 gives e = 8.67 units.

Chapter 20 source: section "Gift-exchange efficiency wage".

Demonstration 4 of 4

Conditional cooperation fixed point

Where do public-good contributions settle when each person matches what they expect from others?

A belief above the fixed point produces a contribution below it, so beliefs drift down as feedback arrives until contributions equal expectations. A steeper slope moves the fixed point up.

Equation, written in LaTeX: 4(0.4)=1.6>1.

Equation, written in LaTeX: g_L=2+0.6E_L(\bar g_{-L}),

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Four tenants hold 20 tokens each. A contributed token costs its owner 1 and returns 0.4 to each of the four. Lina contributes g = 2 + k x belief, capped at 20, where belief is her expectation of the others' average contribution.

Predict first. If everyone shares Lina's schedule, where do contributions settle?

Your prediction

Choose an example

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Figure: Conditional cooperation fixed point. Response line g = 2 + 0.6 x belief against the 45 degree line. At belief 15 the contribution is 11.00; the line meets the 45 degree line at 5.00.
Belief about others' average (tokens): 15, Slope of response: 0.6
Constructed example: the chapter's hypothetical four-tenant game (intercept 2, slope 0.6, beliefs 15 and 5); beliefs 10 and 20 and slopes 0 and 0.9 are added for comparison.

Calculated values

Contribution g
11.00
Private net cost g(1 - 0.4)
6.60
Group benefit 1.6 g
17.60
Symmetric fixed point
5.00

g = 2 + 0.6 x 15 = 11.00. Her private net cost is 11.00 x 0.6 = 6.60 tokens and the group gains 11.00 x 1.6 = 17.60. If all four share the schedule, g = 2 + 0.6g gives g = 2 / 0.4 = 5.00.

Worked steps

  1. g = 2 + 0.6 x 15 = 11.00
  2. Net cost = 11.00 x 0.6 = 6.60
  3. Group benefit = 11.00 x 1.6 = 17.60
  4. Fixed point: g = 2 + 0.6g gives g = 2 / 0.4 = 5.00

Use the idea

When a group relies on voluntary contributions, raise what members expect others to give, not only what they are asked to give.

Where the conclusion applies

A linear response, identical tenants and beliefs that end up correct. Real schedules differ and some members contribute nothing whatever they expect.

Check your understanding: With slope 0.9, what is the symmetric fixed point?
g = 2 + 0.9g gives g = 20, at the cap.

Chapter 20 source: section "Conditional cooperation".