Time Preferences¶

Most non-trivial economic choices involve

Determine tradeoffs between costs and benefits that occur across time points
Determine values (utility) of costs and benefits, by weighing costs & benefits against each other

Present Bias¶

Humans have a tendency to put more weight into the present rather than the future when making decision

Indifference Curves¶

At any point on the curve, the combination of the two will leave the consumer equally well off or equally satisfied—hence indifferent.

	Shows various combinations of __ that consumers can choose
Goods-Indifference Curve	two commodities at the same time point
Fisher’s Time-Indifference Curve	same commodity at different time points

Utility Function¶

\[ \begin{aligned} \max U_t &= \sum_{t = 0}^\infty D(t) \cdot u_{t_0 + t} \\ u_t &= u(c_t, l_t, \dots) \end{aligned} \]

where


$u_t$	Instantaneous utility	Captures how person feels at a specific moment Function of consumption, leisure
$U_t$	Discounted utility	Captures total utility obtained until a specific moment
$D(t)$	Discount Function	Specifies weights on utility derived in $t$ time periods Measures how utility in alter periods is discounted relative to earlier periods Replaces complex psychology of how people think about future Usually $\in (0, 1]$
$\rho(t)$	Discount Rate	Rate of decline in the discount function Specifies rate at which value of $u$ declines with delay	$\dfrac{-D'(t)}{D(t)}$

Standard Utility Models¶

	Samuelson’s Exponential	Quasi-Hyperbolic
	Non-graphical model of Fisher’s Time-Indifference Curve Developed as a simple approximation as a first start, not meant to be accurate
$D(t)$	$\delta^t$ $\delta \in [0, 1]$	$\beta \delta^t$ $\delta \to 1 \ (0.\bar{9})$ $\beta \in [0, 1]$
$\rho(t)$	$- \log \vert \delta \vert \approx 1-\delta$
Advantages	Not affected by awareness issue	- Separate short & long-run discounting - Great patience for tradeoffs in the future than for tradeoff in present - Deals with preference reversals
Limitation	Constant discount rate 1. Short vs Long-Run impatience 2. Preference reversals 3. Commitment devices	Affected by awareness issue

Estimating $\delta$¶

Ask the person the following question

What $X$ makes you indifferent between receiving $15 today and $X at various time point $t$

$t=1$ day

$t=1$ month

$t=1$ year

Assumes that utility is linear in money, ie marginal utility is constant, ie $u(X) = X$

IDK¶

Short vs Long-Run Impatience¶

People tend to be more patient in the long-run than in the short-run

Eg: Credit card loans

Preference Reversals¶

In reality, dynamic consistency is not followed. People don’t always follow through with their plans.

Hence

When thinking ahead to the future, we want to be patient
When the time actually comes, we are impatient
People are over-confident about their self-control

Eg: Dieting, Gym membership

Dynamic/Time Consistency¶

The action a person thinks they should take in the future always coincides with the action that they actually prefer to take once the time comes
A person’s preferences at different points in time are consistent with each other; there are no “intra-personal conflicts”

Commitment Devices¶

Arrangement taken upon by agent to restrict their future choice set, by making certain choices more costly

Demand for commitment requests (at least partial) sophistication

When you know that your future preferences will be different from present preferences, you may engage in commitment devices to penalize (and hence eliminate) few options from the future. We disapprove of the tendency for instant gratification beforehand, but struggle to actually follow through

Requirements for effectiveness of commitment device¶

Person needs to have a self-control problem: $\beta < 1$
Person needs to at least partly be sophisticated: $\hat \beta < 1$
Commitment devices needs to be effective
Person needs to believe that the commitment device is effective

Things that worsen effectiveness of commitment devices¶

Time-inconsistent preferences: Each time period’s self restrict set of choices for their future selves, and hence there may be difference in assessment of best action at each time period
Substitution: Substitution across temptation goods worsens effectiveness of commitment devices. Avoiding one temptation good may lead to increases consumption of another
Naivete: Person underestimates their present bias, and might
Naive: Not demand a helpful commitment device
Partial naive: Demand a unhelpful commitment device

Goods Types¶

	Leisure	Investment
Example	Eating Candy	Going to Gym Finishing assignments Quitting bad habits Finding a job
Costs	Delayed	Immediate
Rewards	Immediate	Delayed
Result Consumption relative to long-run	Over-consumption	Under-consumption

Behavior Types¶

	Perfect Exponential Discounter	Naïveté	Partial Naïveté	Sophistication
$\beta$	$1$	$< 1$	$<1$	$< 1$
$\hat \beta$ What you think $\beta$ is in the future	$\beta$	$1$	$> \beta$ Measures belief about future $\beta$	$\beta$
Optimism	Perfect	Over-optimistic (Assumes future self will through on optimal plan)	Underestimate degree of future present bias	Pessimistic
Person self-aware of preference reversal	N/A (No preference reversal)	❌	✅	✅
Overcommitment/ Self-control problem	❌	❌	✅	❌
Set deadlines optimally	✅	❌ (No perceived need to choose deadlines)	⚠️ (Tries, but fails)	✅
Deadlines help	Deadline not required	✅	⚠️	✅
Take advantage of commitment devices	Not required	❌	⚠️ (Tries, but fails)	✅
No surprises of future present bias	No present bias	❌	❌	✅
Overcomes short-run impatience	No impatience	❌	❌	✅
Utility evaluation	Both ways gives same result	Forwards	Forwards and backwards	Backwards
Investment Goods: Behavior	No procrastination	Naïve Procrastination		Sophisticated Procrastination
Investment Goods: Welfare Cost	0	Large		Low
Leisure Goods: Behavior	No precrastination	Naïve precrastination		Sophisticated precrastination (self-aware about impatience, and hence consumes earlier)
Leisure Goods: Welfare Cost	0	Low		Large (does not wait until max enjoyment)

Utility Evaluation¶

Forward	Backward
1. Start at beginning 2. Solve for optimal plan, assuming future self follows plan 3. Person takes first step in that plan 4. Go to next period 5. Go to step 2	1. Start at end 2. Solve for optimal action 3. Go back to previous period 4. Solve for optimal action, considering what happens in next period 5. Go to step 3

Example¶

Consider the following table showing the utilities associated with watching a movie

	Naïveté	Sophistication
Utility Evaluation	t=0: Plans to go at t=3, so doesn’t go t=1: Plans to go at t=3, so doesn’t go t=2: Goes	t=2: goes if she hasn’t t=1: realizes she won’t wait until t=3, she goes t=0: realizes she won’t wait until t=2 or 3, she goes
Conclusion	Goes at t=2, even though she planned to go at t=3	Just goes at t=0

Indicators of Behavior Types¶

Naivete: Person mis-predicting future behavior
Sophistication: Person’s use of commitment devices

Uncertainties about Future¶

Present bias
Planning Fallacy

Planning Fallacy¶

Under-estimation of effort costs of tasks, leading to people always under-performing compared to original plan, even if they are aware of this.

Last Updated: 2024-05-12 ; Contributors: AhmedThahir


\(u_t\)	Instantaneous utility	Captures how person feels at a specific moment Function of consumption, leisure
\(U_t\)	Discounted utility	Captures total utility obtained until a specific moment
\(D(t)\)	Discount Function	Specifies weights on utility derived in \(t\) time periods Measures how utility in alter periods is discounted relative to earlier periods Replaces complex psychology of how people think about future Usually \(\in (0, 1]\)
\(\rho(t)\)	Discount Rate	Rate of decline in the discount function Specifies rate at which value of \(u\) declines with delay	\(\dfrac{-D'(t)}{D(t)}\)