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Lecture 45 Features of the Discounted Utility Model

1. Integration of new alternatives with existing plans

  • New prospects will affect the existing consumption profile \((C_{t},\dots ,C_{T})\)
  • If the "accepted" consumption profile provides more utility, then we accept the prospect.
  • Though, people may not have sufficient information and it would be beyond mental capacities to calculate consumption profile

2. Utility independence

  • DUM only focuses on the sum, and not how the utilities are distributed.
  • A flat or increasing one is preferred
  • Dispersed is preferred over a concentrated one.

3. Consumption independence

  • Utility at \(t\) is independent of consumption at \(t'\)
  • Meaning preferences won't be affect by consumption periods in the two profiles where consumption is identical.
    • So choice between Italian and Thai today shouldn't depend on whether I had Thai or Italian yesterday
  • \(\implies\) Anomalies in this model

4. Stationary instantaneous utility

  • Instantaneous utility function is constant over time
  • Same activity yields same utility in the future
  • Unrealistic, and classic case of projection bias

5. Stationary discounting

  • Same discount rate over their lifespan
  • The choice between consuming \(x\) now (at \(t\)) or a larger quantity \(x'\) later (at \(t'\)) depends on the interval of time that passes, \(t'-t\) and NOT the point in time when \(x\) could be consumed.

But discount rates vary according to age… older people have higher willingness to delay gratification. However, this relationship (age and discounting factor) is complex.
- Study: Middle aged < Younger < Older (discounting amount

M: Middle aged, O: Old Aged. I: Impaired

A patient individual will have a smaller discount rate.
- For Gains
- O(!I) < M = O(I)
- For losses
- O(I) > M = O(!I)
- O(I) are more impatient (shorter expected lifetime and lower quality of life)

6. Constant discounting

  • The same discount rate is applied to all future periods.
  • \(D(k)\) is constant.
  • This condition ensures time-consistent preferences
  • Empirical evidence suggests \(\implies\) discount rate declines over time.

Stationary vs constant: NOT stationary means at \(t+1\), we will observe a different discount function, which would be constant for the future period, i.e. \(t+2\) onwards given we are at \(t+1\).

non-Stationary means that the discounting function is time-dependent.

\[ D_{t+1}(k) = \left[\dfrac{1}{1+\rho'}\right]k \quad \text{where } \rho' \neq \rho \]

7. Independence of discounting from consumption

  • All forms of consumption are discounted at the same rate.
  • Frederick, Loewenstein and O'Donoghue
    • Label time preference according to the object being delayed
    • "banana time preference", "vacation time preference"
  • Evidence: perceived effort and perceived price are discounted differently.
    • DIY involve effort and appear more attractive when the purchase is planned in the future.

8. DMU & Positive time preference

  • Predate the DUM
  • ICs require assumption of diminishing marginal utility / concave utility functions.
  • Implication = delay consumption (I can't consume a lot of it now, let me do it later). DMU acts in the opposite direction to that normally of time preference.
  • But there are many confounding factors and this time preference cannot be attributed to DMU alone.