Behavioral Economics & Household Finance
Laibson Quasi-Hyperbolic (beta, delta) Discount Formula
Model time-inconsistent preference reversals using Phelps-Pollak and David Laibson’s (1997) quasi-hyperbolic discount sequence {1, beta*delta, beta*delta^2, ...}.
Show how a single present-bias parameter beta < 1 causes tomorrow’s plans to reverse today!
Standard exponential discounting D_exp(t) = delta^t applies the exact same per-period patience factor delta between t = 0 and 1 as between t = 10 and 11 — guaranteeing dynamic consistency so a plan chosen on Sunday is always executed on Monday! Building on Phelps and Pollak (1968), David Laibson (1997) introduced Quasi-Hyperbolic (beta, delta) Discounting: D(0) = 1 and D(t) = beta * (delta^t) for all future periods t >= 1, with Present-Bias Factor 0 < beta < 1! Between Today (t = 0) and Period 1, the effective discount factor is beta * delta, whereas viewed from Today between Period 1 and Period 2 it is (beta * delta^2) / (beta * delta) = delta — creating Dynamic Preference Reversals whenever an upfront investment cost C incurs immediate pain today for delayed benefit B in k periods (beta * delta^k * B < C < delta^k * B)!
Verify on Upfront_Task_Cost C = 100.00 (B2), Delayed_Reward B = 150.00 (B3) arriving k = 1 period after action (B4), Long-Run_Patience delta = 0.90 (B5), and Present_Bias beta = 0.70 (B6): Planned in Advance (comparing t = 1 vs t = 2 from t = 0), Net Value is beta*delta*(delta^1 * B - C) -> Future-Self Decision Surplus delta^k * B - C = 0.90 * 150 - 100 = +35.00 (> 0, "Plan to Act!"); but when Today arrives (t = 0 vs t = 1), Today’s Present-Biased NPV drops to beta * (delta^1) * 150 - 100 = 0.70 * 135 - 100 = 94.50 - 100 = -5.50 (< 0 -> "Preference Reversal: Procrastinate Today!")!
| Quasi-Hyperbolic Metric (C=100, B=150, k=1, delta=0.90, beta=0.70) | Formula | Verified Output |
|---|---|---|
| Quasi-Hyperbolic Factor D(k) | IF(k = 0, 1, beta * delta^k) | 0.6300 (vs Exponential 0.9000) |
| Advance-Commitment Net Surplus | (delta^k) * B - C | +35.00 (Plans to Invest!) |
| Today’s Present-Biased NPV | beta * (delta^k) * B - C | -5.50 (Balks Today!) |
| Behavioral Time-Consistency Audit | IF(AND(Advance > 0, Today < 0), ...) | Preference Reversal (Procrastinate) |
Laibson Quasi-Hyperbolic Present-Biased NPV Formula
Validate C >= 0, B >= 0, integer delay k >= 1, and discount factors 0 < beta <= 1, 0 < delta <= 1.
=LET(c_up, B2, b_rw, B3, k_dl, B4, dl_lr, B5, bt_pb, B6, IF(OR(c_up < 0, b_rw < 0, k_dl < 1, INT(k_dl) <> k_dl, dl_lr <= 0, dl_lr > 1, bt_pb <= 0, bt_pb > 1), "Invalid Discount Input", ROUND(bt_pb * (dl_lr ^ k_dl) * b_rw - c_up, 2)))Dynamic Preference-Reversal & Commitment-Device Classifier Formula
Whenever Advance_Surplus > 0 while Today_NPV < 0, a sophisticated agent strictly values illiquid commitment devices (automatic 401(k) payroll deduction, cancellation penalties)!
=LET(c_up, B2, b_rw, B3, k_dl, B4, dl_lr, B5, bt_pb, B6, IF(OR(c_up < 0, b_rw < 0, k_dl < 1, INT(k_dl) <> k_dl, dl_lr <= 0, dl_lr > 1, bt_pb <= 0, bt_pb > 1), "Invalid Discount Input", LET(npv_ex, (dl_lr ^ k_dl) * b_rw - c_up, npv_pb, bt_pb * (dl_lr ^ k_dl) * b_rw - c_up, IFS(npv_pb >= 0, "Time-Consistent: Acts Immediately", npv_ex > 0, "Preference Reversal (Procrastinate)", TRUE, "Time-Consistent: Rejects Always"))))Try the example
Use the original values or change them to test your understanding.