The Encyclopedia of Economic Principals

Chapter 18

Time Inconsistency, Self-Control, and Projection

When the self who plans and the self who acts disagree.

Four of the chapter's worked examples, made interactive: a present-biased reversal between $50 and $60, a lunch not packed in a cold state, the welfare cost of a tempting menu, and a dessert order placed while hungry. 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

The preference reversal on Monday

Why does a plan to wait for $60 turn into taking $50 once the $50 is immediate?

Present bias multiplies every future reward by beta but leaves the present untouched. A week ahead both rewards carry beta and the larger one wins. When the $50 becomes immediate it loses its beta discount and can overtake the $60.

Equation, written in LaTeX: V_{50}=0.70(0.98)(50)=34.30

Equation, written in LaTeX: V_{60}=0.70(0.98)^2(60)=40.3368.

Equation, written in LaTeX: 0.70(0.98)(60)=41.16.

Equation, written in LaTeX: 0.98(60)=58.80,

Scroll sideways for the whole equation

beta is present bias, applied to every future period, and delta = 0.98 is the weekly discount factor. The $50 arrives one week after the first Monday and the $60 two weeks after. Utility is linear in dollars.

Predict first. With beta = 0.85, does the person still switch to the $50 on the day?

Your prediction

Choose an example

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Figure: The preference reversal on Monday. Two bars for beta = 0.70 valued one week before: the $50 reward at 34.3000 and the $60 reward at 40.3368. The person chooses: Wait for $60.
Present bias beta: 0.7, Evaluation date: One week before
Constructed example: the chapter's hypothetical $50 or $60 choice with beta 0.70 and 1 and delta 0.98; beta values 0.5 and 0.85 are added for comparison.

Calculated values

Value of $50
34.3000
Value of $60
40.3368
Choice
Wait for $60
Gap, $60 minus $50
6.0368
Value of commitment
6.0368

V50 = 0.70 x 0.98 x 50 = 34.3000 and V60 = 0.70 x 0.9604 x 60 = 40.3368. A week ahead the person plans to wait for the $60, since 40.3368 > 34.3000. A commitment device is worth up to 6.0368 utility units. On the day the $50 becomes immediate and is worth 50 against 41.16, so the plan reverses.

Worked steps

  1. V50 = 0.70 x 0.98 x 50 = 34.3000
  2. V60 = 0.70 x 0.98 x 0.98 x 60 = 0.70 x 0.9604 x 60 = 40.3368
  3. 40.3368 > 34.3000, so the plan is to wait for the $60
  4. Value of commitment = 40.3368 - 34.3000 = 6.0368

Use the idea

If you plan to wait but often switch on the day, price a commitment device at no more than the advance gap between the two discounted values.

Where the conclusion applies

Linear utility, a known beta and delta, and no new information between the two dates. The value of commitment treats the earlier evaluation as the welfare benchmark.

Check your understanding: What is the largest beta at which the day-of choice still takes the $50?
The person switches when 50 > beta x 0.98 x 60 = 58.8 beta, so beta < 50 / 58.8 = 0.8503. At beta = 0.85 the $60 is worth 49.98, so the person still switches, barely.

Chapter 18 source: section "Present bias".

Demonstration 2 of 4

Packing lunch in a cold state

What does underestimating future hunger cost when a late meal is still for sale?

The advance decision uses the forecast; the surplus uses the true hot-state value. The cost of the error is the best surplus available under an accurate forecast minus the best action left after the cold-state plan.

Equation, written in LaTeX: 20-15=5.

Equation, written in LaTeX: 20-6=14.

Equation, written in LaTeX: 14-5=9.

Scroll sideways for the whole equation

Packing a meal after lunch costs $6; the late-night vendor charges $15. The worker packs when the cold-state forecast of the meal's evening value exceeds $6. The hot-state value is what the meal is actually worth at hour seven.

Predict first. With a forecast of $8, does the worker pack?

Your prediction

Choose an example

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Figure: Packing lunch in a cold state. Bars for a cold forecast of $5 and an actual hot value of $20, with lines at the packing cost $6 and vendor price $15. Forecast error cost $9.
Predicted evening value ($): 5, Actual hot-state value ($): 20
Constructed example: the chapter's hypothetical worker (packing $6, vendor $15, hot value $20, forecasts $5 and $12); a forecast of $8 and hot values of $12 and $25 are added for comparison.

Calculated values

Plan
Pack nothing, buy late
Realized surplus ($)
5
Best-plan surplus ($)
14
Cost of forecast error ($)
9

Forecast 5 < packing cost 6, so the worker packs nothing. Late purchase: 20 - 15 = 5. Best plan with an accurate forecast: pack, 20 - 6 = 14. Cost of the forecast error = 14 - 5 = 9.

Worked steps

  1. Forecast 5 < packing cost 6, so the worker packs nothing
  2. Late purchase: 20 - 15 = 5
  3. Best plan with an accurate forecast: pack, 20 - 6 = 14
  4. Cost of the forecast error = 14 - 5 = 9

Use the idea

Before committing in a calm state, recall how you valued the item last time you were in the hot state, and compare that number with the advance cost.

Where the conclusion applies

One meal, known prices, and a hot-state value that does not depend on the plan. The late vendor is bought from only when the meal is worth more than its price.

Check your understanding: If the hot value were $12 and the forecast $5, what is the cost of the forecast error?
The vendor surplus is 12 - 15 < 0, so the worker does not buy and gets 0. Packing would give 12 - 6 = 6. The cost is 6 - 0 = 6.

Chapter 18 source: section "Hot-cold empathy gap".

Demonstration 3 of 4

Paying for a smaller menu

Why would a diner who eats fruit either way prefer a menu without cake?

Removing a tempting option can raise welfare even when it would not have been chosen, because resisting it costs the gap between its temptation and the temptation of the chosen item.

Equation, written in LaTeX: W(A)=\max_{x\in A}\{u(x)+v(x)\}-\max_{y\in A}v(y).

Equation, written in LaTeX: W(A)=10-0=10.

Equation, written in LaTeX: W(B)=\max\{10,9\}-\max\{0,5\}=10-5=5.

Equation, written in LaTeX: u(c)+v(c)=4+8=12,

Equation, written in LaTeX: W(B)=12-8=4.

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u is normative utility (fruit 10, cake 4) and v is temptation (fruit 0, cake v(c)). The diner eats the item with the highest u + v. Menu welfare W subtracts the strongest temptation on the menu.

Predict first. At v(c) = 8, which item is eaten from the full menu?

Your prediction

Choose an example

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Figure: Paying for a smaller menu. Bars of combined utility u + v for the items on the full menu with cake temptation 5; the diner eats fruit and menu welfare is 5.
Cake temptation v(c): 5, Menu: Full, fruit and cake
Constructed example: the chapter's hypothetical diner (u = 10 and 4, v(c) = 5 and 8); a temptation value of 2 is added for comparison.

Calculated values

Choice
Fruit
Menu welfare W
5
Self-control cost
5
Loss versus commitment
5

Cake's combined utility 4 + 5 = 9 is below fruit's 10, so the diner resists and eats fruit. Menu welfare is 10 - 5 = 5: the same fruit as under commitment, but 5 units of self-control cost.

Worked steps

  1. Fruit: u + v = 10 + 0 = 10
  2. Cake: u + v = 4 + 5 = 9
  3. Choice: fruit, since 10 > 9
  4. W = max(10, 9) - max(0, 5) = 10 - 5 = 5

Use the idea

Count the self-control cost of options you never take; paying a little to remove them can be worth it.

Where the conclusion applies

Fixed u and v for each item and a single choice from the menu. Removing an item for other reasons (an allergy, missing information) would not identify this mechanism.

Check your understanding: At what v(c) does the diner switch from resisting to yielding?
The diner yields when 4 + v(c) > 10, so for v(c) above 6; at exactly 6 the two items tie.

Chapter 18 source: section "Temptation and self-control preferences".

Demonstration 4 of 4

Ordering dessert while hungry

How does hunger on Friday change a dessert order for Sunday?

Projection bias pulls the forecast of a future taste toward the current one. A hungry shopper overestimates later value and can buy what she will not want; the same shopper after lunch forecasts correctly.

Equation, written in LaTeX: \widehat u=(1-0.50)(2)+0.50(8)=5.

Equation, written in LaTeX: 2(2)-2(4)=-4,

Equation, written in LaTeX: \widehat u=(1-0.50)(2)+0.50(2)=2.

Scroll sideways for the whole equation

u hat is the predicted Sunday value of one dessert, alpha is projection intensity, 2 is the true after-dinner value and the current value depends on the state at the time of ordering. Each dessert costs $4 and the smallest package holds two.

Predict first. With alpha = 0.25 while hungry, does she order?

Your prediction

Choose an example

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Figure: Ordering dessert while hungry. Bars of dessert values: actual $2, current $8, predicted $5.00, price $4. She orders.
Projection alpha: 0.5, Current state at order: Hungry, $8
Constructed example: the chapter's hypothetical subscriber (actual value $2, current $8 or $2, alpha 0.50, price $4); alpha of 0, 0.25 and 0.75 and a slightly hungry current value of $5 are added for comparison.

Calculated values

Predicted value u hat ($)
5.00
Order
Two desserts
Realized gain ($)
-4.00

Ordering while hungry before lunch, her current value is $8. Predicted value is (1 - 0.50) x 2 + 0.50 x 8 = 5.00. 5.00 > 4.00, so she orders the two-dessert package. On Sunday the pair is worth 4 and costs 8, a loss of 4.00.

Worked steps

  1. u hat = (1 - 0.50) x 2 + 0.50 x 8 = 5.00
  2. 5.00 > 4.00, so she orders the two-dessert package
  3. Realized gain = 2 x 2 - 2 x 4 = -4.00

Use the idea

Order perishable or tempting goods in a neutral state, or choose sellers that allow free cancellation after a calmer review.

Where the conclusion applies

Linear utility, a fixed package of two and a single projection weight alpha. Values are in dollar units.

Check your understanding: With current value 8, what alpha makes the predicted value exactly equal the $4 price?
2 + 6 alpha = 4 gives alpha = 1/3.

Chapter 18 source: section "Projection bias".