07.12.2012 - Aprajit Mahajan

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Time Inconsistency, Expectations and Technology Adoption

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07.12.2012 - Aprajit Mahajan

  1. 1. Introduction Model Identification Estimation Time Inconsistency, Expectations and Technology Adoption Aprajit Mahajan (UCLA, Stanford) Alessandro Tarozzi (Pompeu Fabra) IFPRI Seminar July 12, 2012Dynamic Choice, Time Inconsistency and ITNs
  2. 2. Introduction Model Identification EstimationMotivation: Time (In)consistencyCan self-control based explanations rationalize behavior hard to reconcilewith standard model? Two strands of empirical Work: Significant body of US-based work, e.g. consumption and saving (Laibson 1997, Laibson et al 2009), welfare uptake (Fang and Silverman 2007), job search (Paserman 2008). More recent but growing interest in development: Commitment contracts (Ashraf et al 2007, Tarozzi et al 2009), Fertilizer (Duflo et al. 2009), Banerjee and Mullainathan (2010) Identification of time preferences not easy. Generically, time discounting parameters in standard dynamic discrete choice (DDC) not identified (Rust 1994, Magnac and Thesmar 2002).Dynamic Choice, Time Inconsistency and ITNs
  3. 3. Introduction Model Identification EstimationMotivation (continued) Empirical Work: Strotz (1955) “hyperbolic discounting” (“β − δ”) T E(u({at+s }T )) = u(at ) + β s=0 δ s E(u(at+s )) s=1 Allowing for both time consistent and inconsistent agents seems important. But identifying just δ difficult even with no population heterogeneity. How to account for (time) preference heterogeneity theoretically and empirically in such models? Finally: information and beliefs (E(·)) may also help explain “suboptimal” behavior. Contrast with Fang and Wang (2010) DetailsDynamic Choice, Time Inconsistency and ITNs
  4. 4. Introduction Model Identification Estimation This paper uses 1. Elicited beliefs 2. Survey responses to time preference questions 3. Actual product offers (Insecticide Treated Nets, ITNs) to estimate preference parameters in a dynamic discrete choice (DDC) model of demand with time inconsistent preferences and unobserved types in the malaria-endemic state of Orissa (India) study area We point identify 1. Time preference parameters: β and δ 2. (Normalized) Utility (non-parametrically) We estimate the model and provide estimates of all time preference parameters and other (risk, cost) parameters in the utility function. 1. Inconsistent agents are a majority (Naive & Sophisticated) ... 2. ... but Naive agents are “almost” consistent. 3. Sophisticated agents are more present-biased than naive agents. 4. Other preference differences appear to be small.Dynamic Choice, Time Inconsistency and ITNs
  5. 5. Introduction Model Identification EstimationTalk: Overview 1. Introduction 2. Model 2.1 State Space 2.2 Action Space 2.3 Preferences 2.4 Transition Probabilities 2.5 Maximization Problem 3. Discussion of Agent Types 4. Identification 4.1 Observed Types 4.2 Unobserved Types 5. Monte Carlos 6. Estimation 7. ConclusionDynamic Choice, Time Inconsistency and ITNs
  6. 6. Introduction Model Identification EstimationModel: Timing Study Design Overview: study design Timeline: Agent takes actions in 3 periods 1. At t = 1, given past malaria history, agent decides whether to purchase an ITN and if so, which of 2 possible contracts to choose. contracts 2. At t = 2, malaria status is realized and subsequently, agent decides whether to retreat the ITN to retain effectiveness. 3. At t = 3, malaria status for period 3 is realized and the agent decides again whether to retreat the ITN.Dynamic Choice, Time Inconsistency and ITNs
  7. 7. Introduction Model Identification EstimationModel: PrimitivesWe begin by defining the decision problem: State Space Action Space Preferences Transition Probabilities Maximization ProblemDynamic Choice, Time Inconsistency and ITNs
  8. 8. Introduction Model Identification Estimation Observables State Space ↓ st ≡ (xt , εt ) ↓ Unobservables This Talk: t=1 x1 ∈ {m, h} malaria status at baseline (6 months before ITN intervention) h =healthy, m =malaria. t=2,3 For t ∈ {2, 3} xt ∈ {nm, nh, bm, bh, cm, ch} n: No net purchased b: Purchased treated bednet only c: Purchased ITN with retreatment cost included. {m, h}: Anyone in household had malaria last 6 months Extend to more general set up with xt ∈ {n, b, c} × [0, 1]2 × Y (include: contract choice, fraction of household members coverable by bednets, fraction with malaria, income). General time-varying observeables. Restriction: Need finite state space. Details Easy to incorporate time-invariant observables.Dynamic Choice, Time Inconsistency and ITNs
  9. 9. Introduction Model Identification EstimationAction Space: At Simple: t=1 Choice of contract: a1 ∈ A1 ≡ (n, b, c) Details b Loan for cost of ITN only. Re-treatment offered for cash only later. Rs. 173(223) for single(double) nets paid in 12 monthly installments of Rs.16(21). Retreatment offered at Rs.15 (18). c Loan for the cost of ITN and loan for two re-treatments to be carried out 6 and 12 months later. Rs. 203(259) for single(double) nets paid in 12 monthly installments of Rs.19(23). t=2,3 Re-treatment choice at ∈ At ≡ {0, 1} Richer Structure: fraction of household chosen to be covered and fraction of nets that are re-treated in t = 2, 3Dynamic Choice, Time Inconsistency and ITNs
  10. 10. Introduction Model Identification EstimationTransition Probabilities: P(st+1 |st , at ) Assume Markovian transition probabilities: P(st+1 |st , st−1 , ..., s1 , at , ..., a1 ) = P(st+1 |st , at ) Unobservable t ∈ st has dimension equal to #At and is independent across time with known distribution. independent of (xt−1 , at−1 ) and P(xt |xt−1 , at−1 , t , t−1 ) = P(xt |xt−1 , at−1 ) so P(xt , t |xt−1 , t−1 , at−1 ) = P(xt |xt−1 , at−1 )P( t ) Strong assumptions: rule out unobserved correlated time varying variables – time invariant unobservables (types) can be accommodated.Dynamic Choice, Time Inconsistency and ITNs
  11. 11. Introduction Model Identification EstimationKey Difference: Calculating P(xt+1 |xt , at ) Usually, next invoke rational expectations (e.g. Rust (1994), Magnac and Thesmar (2002)) and assert that agent beliefs P(xt+1 |xt , at ) are equal to observed transition probability in the data. KEY DIFFERENCE: We elicit P(xt+1 |xt , at ) for each household in the survey. beliefs Variation in beliefs is key to identification. Intuitively, beliefs lead to variation in period t value function while holding period t utility fixed. Need sufficient exogeneity in beliefs for argument to hold (precise conditions outlined later)Dynamic Choice, Time Inconsistency and ITNs
  12. 12. Introduction Model Identification EstimationPreference: Types of Agent Three types of agent: Time Consistent agents. (τC ) “Naive” Time Inconsistent agents. (τN ) “Sophisticated” Time Inconsistent agents. (τS ) Types differ by 1. Awareness of future present-bias 2. Extent of present-bias βτ - For time consistent agents βτC = 1 - But also allow βτN = βτS 3. Per-Period utility uτ,t (·)Dynamic Choice, Time Inconsistency and ITNs
  13. 13. Introduction Model Identification EstimationPreferences: Per Period Utility Utility is time separable and per period utility is uτ,t (st , a) = uτ,t (xt , a) + t (a) for xt ∈ Xt Additively separable in unobserved state variables Suppress dependence on other hhd. characteristics. Agent of type τ at time t chooses decision rules {dτ,j }3 , dτ,j : Sj → Aj chosen to maximize j=t 4 uτ ,t (st , dτ ,t (st )) + βτ δ j−t E(uτ ,t (st , dτ ,t (sj ))) j=t+1 Per period utility functions and hyperbolic parameters can vary by type. However, the exponential discount rate is constant across types.Dynamic Choice, Time Inconsistency and ITNs
  14. 14. Introduction Model Identification EstimationAwareness of Future Present-BiasFinite Horizon Dynamic Discrete Choice: Backward Induction Agents of all types 1. use backward induction to formulate optimal policy at t 2. discount t + 1 utility by βτ δ at t. (βτC = 1: time consistent) However, inconsistent types differ in how they view the trade off between periods t + 1 and t + 2 (from the viewpoint of period t) 1. “Sophisticated” types recognize that they will be present biased and at t + 1 they will discount t + 2 utility by βτ δ 2. “Naive” types do not recognize present bias of their future selves. Corresponding discount rate δDynamic Choice, Time Inconsistency and ITNs
  15. 15. Introduction Model Identification EstimationTalk Overview 1. Introduction 2. Model 2.1 State Space 2.2 Action Space 2.3 Preferences 2.4 Transition Probabilities 2.5 Maximization Problem 3. Discussion of Agent Types 4. Identification 4.1 Observed Types 4.2 Unobserved Types 5. Estimation 6. ConclusionDynamic Choice, Time Inconsistency and ITNs
  16. 16. Introduction Model Identification EstimationTwo Step Identification Consider identification in two steps: 1. Identification when types are directly observed: Type is assumed deterministic function of observables Response to hypothetical time preference questions (r) details choice of commitment product (a1 ) details 2. Identification when types are not observed (General Case): (r, a1 ) are only roughly informative about type. Identification in the general case builds on identification arguments for the observed type case.Dynamic Choice, Time Inconsistency and ITNs
  17. 17. Introduction Model Identification EstimationDirectly Observed Types Use 2 pieces of information to directly identify agent type. 1. Choice of commitment product (a1 ∈ {n, b, c}) 2. Displaying time preference reversal in baseline (r ∈ {0, 1}) Classifies (some) agents unambiguously τC ⇐⇒ {r = 0} time consistent τN ⇐= {r = 1, a1 = b} “naive” inconsistent τS ⇐= {r = 1, a1 = c} “sophisticated” inconsistent Advantage: Problem much more tractable. Disadvantage: Not clear if (r, a1 ) map directly into types as defined above. Ambiguities with classifications ({r = 1, a1 = n} =⇒?). problemsDynamic Choice, Time Inconsistency and ITNs
  18. 18. Introduction Model Identification EstimationUnobserved Types Now, types no longer directly observed. Observed choice probabilities now mixtures over type choice probabilities. Additional parameter: πτ (v) ≡ P(type = τ |v) unknown type probabilities. Index types by {τC , τN , τS } ≡ T Model is still identified (under additional conditions). Key is to reduce this problem to previous one. Can impose (and test) whether (r, a1 ) map into agent types. e.g. test πτS (1, c) = 1 Advantage: More agnostic about ability to infer type from observables. Disadvantage: Identification/Estimation requires more workDynamic Choice, Time Inconsistency and ITNs
  19. 19. Introduction Model Identification EstimationIdentification Results: Overview Directly observed types: Point Identification of 1. Time Preference Parameters: (βτ , δ) 2. Normalized utility definition payoff Unobserved types: Point Identification of 1. Time preference parameters: (βτ , δ) 2. Normalized utility 3. Type probabilities πτ Key: Reduce problem to previous one payoffDynamic Choice, Time Inconsistency and ITNs
  20. 20. Introduction Model Identification EstimationIdentification Outline: Directly Observed Types 1. Start in last period (Period 3). Invert relationship between - observed type-specific choice probabilities: Pτ (at |xt , zt ) - model predictions: Pτ (at |xt , zt ; θ) where θ ≡ (δ, {βτ }τ ∈T , {ut,τ (·)}4 ) and zt are beliefs about malaria in t=1 period t (observe beliefs at 2 point in time). 2. Use variation in (x3 , z3 ) to identify (some parts of) θ 3. Repeat Steps (1) and (2) for period 2 to recover (further elements of) θ – including δ 4. Repeat for period 1.Dynamic Choice, Time Inconsistency and ITNs
  21. 21. Introduction Model Identification EstimationPeriod 3 Probability type τ retreats: Pτ (a∗ = 1|x3 , z3 ) = G∆ (gτ,3 (x3 , z3 , θ)) 3 (1) LHS directly identified since {a∗ , x3 , z3 , τ } observed. 3 d G∆ = 0 − 1 known, support over R. Invert (1) to identify gτ,3 (x3 , z3 , θ) = u3,τ (x3 , 1) − u3,τ (x3 , 0)+βτ δ u3,τ (x4 )dF∆ (x4 |x3 , z3 ) Util. Differential u3,τ (x3 , 1) − u3,τ (x3 , 0) measures change in period 3 utility from re-treatment. Next, identify this.Dynamic Choice, Time Inconsistency and ITNs
  22. 22. Introduction Model Identification EstimationPeriod 3: Identifying Utility KEY: Use variation in beliefs to identify the utility differential. Need household beliefs not perfectly predicted by observables x3 (formally Assumption 6) Intuition: Evaluate gτ,3 (x3 , z3 , θ) = u3,τ (x3 , 1) − u3,τ (x3 , 0) + βτ δ u4,τ (x4 )dF∆ (x4 |x3 , z3 ) at two different values of z and difference. Lemma 1: The researcher observes an i.i.d. sample on ({a∗ , xt }T −1 , w). With sufficient variation in beliefs t t=1 1. u3,τ (x3 , 1) − u3,τ (x3 , 0) are identified for all x3 ∈ X3 . 2. Fourth period expected discounted (normalized) utility is identified (βτ δ u4,τ (x4 )(dF (x4 |x3 , a3 = 1, z3 ) − dF (x4 |x3 , a3 = 0, z3 ))Dynamic Choice, Time Inconsistency and ITNs
  23. 23. Introduction Model Identification EstimationIdentifying Hyperbolic Parameters βτ Data from t = 3 do not identify all time preference parameters. However, If in addition to previous assumptions 1. Some time consistent agents make a purchase decision. 2. Period 4 utility differentials are constant across time consistent and time inconsistent naive types Restrictive. But preferences in periods < 4 differ by type, so can gauge reasonableness Much less restrictive than previous work. Under these additional assumptions, βτN is identified (Lemma 2)Dynamic Choice, Time Inconsistency and ITNs
  24. 24. Introduction Model Identification EstimationIdentification: Period 2 Need this period to identify remaining time parameters. Use same inversion argument as before. Identification argument more delicate since types further differ in perceptions about future present-bias dτ (s3 ) ≡ argmaxa∈A3 u3,τ (x3 , a) + 3 (a) ˜ + βτ δ u4,τ (x4 )dF (x4 |x3 , a, z3 ) “sophisticated” type: recognizes that period 3 self will be subject ˜ to present-bias. βτS = βτ “naive” type: is present biased (in period 2) but does not recognize that his period 3 self will also be present biased. ˜ βτ N = 1 = βτ N ˜ Time consistent agents: βτC = βτC = 1.Dynamic Choice, Time Inconsistency and ITNs
  25. 25. Introduction Model Identification EstimationIdentification: Period 2 parameters Lemma 3 1. Assuming that beliefs (conditional on the state variables) have two points of support we identify normalized utility: uτ,2 (x2 , 1) − uτ,2 (x2 , 0) 2. Next, using results from the previous section (Lemma 2) we separately identify βτS and δ. Intuition Summary: We identify both utility and time preference parameters given sufficient variation in beliefs about re-treatment effectiveness. Key: beliefs provide variation in the value function term while holding utility differentials constant.Dynamic Choice, Time Inconsistency and ITNs
  26. 26. Introduction Model Identification EstimationIdentification: Period 1 Parameters Survey response (r) can distinguish between consistent and inconsistent types and purchase reveals type (for r=1). However, cannot separate “naive” and “sophisticated” for non-purchasers. Cannot observe types =⇒ Can’t use inversion directly. Insight: All we needed for inversion was type-specific choice probabilities (not individual types). Identification argument here in 2 steps: Identify type-specific choice probabilities Pτ (a1 |x1 , z1 ). As before, recover type-specific utility parameters (θ) by studying mapping b/w Pτ (a1 |x1 , z1 ) and model prediction Pτ (a1 |x1 , z1 , θ) Same argument used in general case.Dynamic Choice, Time Inconsistency and ITNs
  27. 27. Introduction Model Identification Estimation Need sufficient variation in type-specific choice probabilities across and within states. Sufficient conditions: - Conditional on state, ≥ 2 types have different choice probs. - ∃ at least two states such that the corresponding vector of type-specific choice probabilities are different. Weaker condition suffices (Assumption 11) Lemma 4 Under assumptions 1-11 1. The first period utility differences u(x1 , b, τ ) − u(x1 , n, τ ) and u(x1 , c, τ ) − u(x1 , n, τ ) are identified for all x1 ∈ X1 and for all types τ . 2. The type probabilities {πτ (·)}τ ∈T are also identified. In addition to identifying preferences for the different types, we also identify the relative size of all three different types of agent in population. This is useful because we obtain unconditional distribution of types whereas previous work could at best be informative about type distribution conditional upon purchase.Dynamic Choice, Time Inconsistency and ITNs
  28. 28. Introduction Model Identification EstimationOverview 1. Introduction 2. Model 2.1 State Space 2.2 Action Space 2.3 Preferences 2.4 Transition Probabilities 2.5 Maximization Problem 3. Discussion of Agent Types 4. Identification 4.1 Observed Types 4.2 Unobserved Types 5. Estimation 6. ConclusionDynamic Choice, Time Inconsistency and ITNs
  29. 29. Introduction Model Identification EstimationUnobserved Types Previous model useful but relied heavily on types being directly observed. Now consider case where types are not observed. Useful if we are unwilling to believe that survey responses and choice of “commitment” product mechanically identify agent type (test the mapping too). Identification problem much harder now since can’t use the standard inversion argument. Two step identification argument (as in last lemma): 1. Identify type-specific choice probabilities Pτ (and type probabilities πτ ). 2. Use identified type-specific choice probabilities to back out the type specific preferences as before.Dynamic Choice, Time Inconsistency and ITNs
  30. 30. Introduction Model Identification EstimationType-Specific Choice Probabilities: Assumptions Need some apriori knowledge about the relationship between ru ≡ (r, a1 ) and types. In particular, for ru = ru , the three π (r ) π (r ) πτ (r ) ratios πτC (ru ) , πτN (ru ) , πτS (ru ) can be ordered ex-ante τ τ C u N u S u Sufficient Conditions: - Among agents with r = 1, inconsistent agents are more likely to purchase the commitment product (and sophisticated agents the most likely): πS (1, c) ≥ πN (1, c) > πC (1, c) - Among agents with r = 0 time consistent agents are most likely to buy product b and naive agents are more likely to purchase b than sophisticated agents.: πS (0, b) < πN (0, b) ≤ πC (0, b) but weaker condition above suffices.Dynamic Choice, Time Inconsistency and ITNs
  31. 31. Introduction Model Identification EstimationType-Specific Choice Probabilities: Assumptions Conditional on state and agent-type, ru is uninformative about actions. Reasonable if ru only informative about choices through predictive power for type. Violated if e.g. r = 1 indicates reflects innumeracy or other flaws in cognition. (Assumption 13) Transition probabilities do not vary by type and are independent of ru . Can test this. (Assumption 13) There is sufficient variation in the type specific choice probabilities Pτ (at = 1|xt , z). In particular, require M − 1 points in xt and a rank condition that rules out using multiple states such that all types have the same choice probabilities for them. (Assumption 14) All types exist with positive probability for at least two values of ru . (Assumption 15) Can potentially test for this (Kasahara and Shimotsu (2009)).Dynamic Choice, Time Inconsistency and ITNs
  32. 32. Introduction Model Identification EstimationType Specific Choice Probabilities: Results Lemma 5: Under additional assumptions 13-15 the choice specific probabilities Pτ (at = 1|xt ) are identified for all xt ∈ XB ∪ XC and t > 1. In addition, the type probabilities {πτ (ru )}τ ∈T are also identified. Uses argument from Kasahara and Shimotsu (2009) (requires fewer assumptions on length of panel). Lemma 6: Under assumptions 1-3,5-15 we can identify 1. The type-specific utility differentials ut (xt , 1, τ ) − u(xt , 0, τ ) ∀ τ ∈ T , xt ∈ XB ∪ XC ∀t 2. The exponential discount parameter δ and the hyperbolic parameters βτ ∀τ ∈ TDynamic Choice, Time Inconsistency and ITNs
  33. 33. Introduction Model Identification EstimationMonte Carlo Simulations uτ (st , at , θ) = ut (xt , at , θ) + t (at ) t i.i.d. Generalized Extreme Value -I (convenient) At t agent solves 4−t ut (st , at , θ) + βτ δ j Et (ut (sj , aj , θ)) j=1 Basic Set Up: Agents only differ in the values of the hyperbolic parameters βτ and the level of “sophistication” among time inconsistent agents. Finite Horizon DDC model (Backward Induction)Dynamic Choice, Time Inconsistency and ITNs
  34. 34. Introduction Model Identification EstimationMonte Carlo Simulations: Per-Period Utility 1. Period 4: x4 ∈ {0, 1} u(x4 ) = −θ4 x4 2. Period 3: x3 ∈ {bm, bh, cm, ch, nh, nm} ≡ {b, c, n} × {h, m} ≡ {0, 1, 2, 3, 4, 5} and a ∈ {0, 1} u(x3 , a) = −θ4 {x3 ∈ {1, 3, 5}} − θ5 pr {x3 ∈ {0, 1}, a = 1} 3. Period 2: x2 ∈ {bm, bh, cm, ch, nh, nm} ≡ {b, c, n} × {h, m} ≡ {0, 1, 2, 3, 4, 5} and a ∈ {0, 1} u(x2 , a) = −θ4 {x2 ∈ {1, 3, 5}} − θ5 pr {x2 ∈ {0, 1}, a = 1} 4. Period 1: x1 ∈ {h, m} ≡ {0, 1} and a ∈ {b, c, n} ≡ {0, 1, 2} u(x1 , a) = −θ4 {x1 = 1} − θ5 pb {a1 = 1} − θ5 pc {a1 = 2}Dynamic Choice, Time Inconsistency and ITNs
  35. 35. Introduction Model Identification Estimation Choice probabilities exp(v(xt , j, βτ ; z)) Pτ (at = j|xt ; z) = J (2) s=1 exp(v(xt , s, βτ ; z)) where v(xt , j, β) is the Emax function. E.g. ∗ v(x2 , j, βτ ; z) = u(x2 , j) + βτ δ vτ (x3 )dF (x3 |x2 , j; z) x3 ∗ vτ (x3 ) = (v(x3 , s, 1) + 3 (s))I(s is chosen)dG( 3 ) ˜ ˜ I(s is chosen) ≡ {v(x3 , k, βτ ; z) + k > v(x3 , s, βτ ; z) + s ∀k = s} ˜ βC = βC = 1 ˜ βN = 1 βN = .7 ˜ βS = βS = .8 Here, types differ (in period 2) in predicting own choice in period 3 Use (2) as moment condition for estimation.Dynamic Choice, Time Inconsistency and ITNs
  36. 36. Introduction Model Identification Estimation Table 1: Monte Carlo Results: Directly Observed Types Mean Median Std.Dev IQR N=300 δ 0.88 0.86 0.52 0.65 βN 0.74 0.71 0.30 0.40 βS 0.83 0.79 0.32 0.41 θ4 4.37 3.09 4.92 3.11 θ5 1.03 1.03 0.56 0.74 N=600 δ 0.90 0.86 0.37 0.52 βN 0.71 0.70 0.18 0.25 βS 0.81 0.78 0.23 0.28 θ4 3.71 3.09 2.10 1.93 θ5 1.04 1.04 0.38 0.51 N=2400 δ 0.89 0.89 0.18 0.26 βN 0.69 0.69 0.09 0.13 βS 0.80 0.79 0.11 0.14 θ4 3.17 3.03 0.73 0.94 θ5 1.00 0.99 0.18 0.24Notes: Each model was simulated 250 times. The true values are (δ, βN , βS , θ4 , θ5 ) = (.9, .7, .8, 3, 1)Dynamic Choice, Time Inconsistency and ITNs
  37. 37. Introduction Model Identification Estimation Table 2: Monte Carlo Results: Unobserved Types Mean Median Std.Dev IQR N=300 δ 0.6669 0.6309 0.3303 0.4147 βN 0.4034 0.2795 0.4306 0.6809 βS 0.9608 0.9315 0.4766 0.6875 θ4 5.0283 4.0879 3.3146 3.0744 θ5 1.0576 1.0462 0.5426 0.6934 N=600 δ 0.7377 0.7051 0.3016 0.4182 βN 0.4330 0.4020 0.4000 0.4674 βS 0.9475 0.9263 0.3027 0.4387 θ4 4.0817 3.6559 1.7953 2.2880 θ5 1.0742 1.0695 0.3836 0.5152 N=2400 δ 0.7865 0.7751 0.2083 0.2920 βN 0.4137 0.4096 0.1782 0.2229 βS 0.9701 0.9552 0.2143 0.2611 θ4 3.2838 3.0896 0.9665 1.2084 θ5 1.0159 1.0165 0.2054 0.2545Notes: Each model was simulated 250 times. The true values are (δ, βN , βS , θ4 , θ5 ) = (.7, .4, .95, 3, 1)Dynamic Choice, Time Inconsistency and ITNs
  38. 38. Introduction Model Identification EstimationOverview 1. Introduction 2. Model 2.1 State Space 2.2 Action Space 2.3 Preferences 2.4 Transition Probabilities 2.5 Maximization Problem 3. Discussion of Agent Types 4. Identification 4.1 Observed Types 4.2 Unobserved Types 5. Estimation 6. ConclusionDynamic Choice, Time Inconsistency and ITNs
  39. 39. Introduction Model Identification EstimationEstimation: Overview Assume that errors are GEV-I (standard - convenient) Additional – relative to standard DDC models – complications: Unobserved Types Time Inconsistent agents Recover time preference parameters. State Variables x : income (y), malaria status (h) and a1 for t>1 Additional household characteristics (v) household size (hhsize), baseline assets (assets), measures of risk aversion (risk). Also used education of household head and finer demographics.Dynamic Choice, Time Inconsistency and ITNs
  40. 40. Introduction Model Identification EstimationPreferences Period 4: uτ (x4 ; v) = c(x4 )ατ (v) − cτ (x4 , v) Period 2,3: uτ (xt , at ; v) = (c(xt ) − pr at I{a1 = b})ατ (v) − cτ (x4 , v) Period 1: uτ (x1 , a1 ; v) = (c(x1 ) − pb I{a1 = b} − pc I{a1 = c})ατ (v) − cτ (x4 , v)where ατ (v) = Logit (ατ + α1 hhs + α2 assets + α3 risk) restricted for simplicity. cτ (xt , v) ≡ ht cτ (v) = I{ht = m} exp(κτ + κ1 hhs + κ2 assets) pr =price of retreatment, (pc , pb )=(price of b and c) and c(xt ) is consumption level in state xtDynamic Choice, Time Inconsistency and ITNs
  41. 41. Introduction Model Identification EstimationMapping Model to Type-Specific Choice Probabilities exp(vτ (xt ,a,w,βτ ) ˜ Pτ (at+1 = a|xt ; w) = ˜ J j=0exp(vτ (xt+1 ,aj ,w,βτ ) ˜ vτ (xt , a, w, βτ ) ≡ ˜ ∗ (x uτ (xt , a) + βτ δ vτ t+1 )dF(xt+1 |xt , a) vτ (xt+1 ) = J ∗ s=1 (vτ (xt+1 , s, 1) + s,t+1 )IAs,t+1 dF( τ t+1 ) Aτ ˜ ˜ ≡ {vτ (xt+1 , k, βτ ) + k,t+1 > vτ (xt+1 , s, βτ ) + s,t+1 ∀s = k} k,t+1 Hypothesized optimal action in t + 1 chosen assuming present ˜ bias in t + 1 is βτ Modified value function J ∗ ˜ vτ (x3 ) = P(Aτ ) vτ (x3 , s, 1) − vτ (x3 , s, βτ ) s s=1 J + γeuler + log( ˜ exp(vτ (x3 , j, βτ ))) j=1Dynamic Choice, Time Inconsistency and ITNs
  42. 42. Introduction Model Identification EstimationEstimation: First Step Identify Pτ (at |xt , z, v, ru ) using Lemma 5. Requires flexible estimate of P(at , at+1 , xt , xt+1 |z, v, r) as inputs into Kashara-Shimotsu procedure. Use flexible logit specifications. Implement the proof of Lemma 5 at each value of (z, v, ru ) for all relevant values of (at , at+1 , xt , xt+1 ) (for t > 1). Discretize (z, v, ru ) for tractability. Eigenvalue decomposition yields type probabilities πτ (ru ) and type-specific choice probabilities Pτ (at |xt , z, v)Dynamic Choice, Time Inconsistency and ITNs
  43. 43. Introduction Model Identification EstimationEstimation: Step Two For a given parameter vector θ = (δ, βτN , βτS , α, κ) compute model choice probabilities starting from the last period and working backwards to construct the value functions needed to calculate model choice probabilities for each type. Estimate θ by minimizing the distance between between model probabilities and the type-specific choice probabilities recovered in the first step.Dynamic Choice, Time Inconsistency and ITNs
  44. 44. Introduction Model Identification EstimationResults: Population Distribution of Types Table 3: Type Probabilities πτ (r) Estimate 2.5 97.5 πC (0) 0.3870 0.2894 0.4837 πN (0) 0.5019 0.4172 0.6059 πS (0) 0.1111 0.0593 0.1691 πC (1) 0.4143 0.3092 0.5126 πN (1) 0.4699 0.3851 0.5790 πS (1) 0.1158 0.0639 0.1756Notes: πτ (r) is the probability that an agent is of type τ given response r to the time-inconsistency question. Time consistent agents are about 40% of population Bulk of time-inconsistent agents are naive. The relative sizes of the population are ≈ same irrespective of r. Note that we did not need to assume πC (0) > πC (1) for identification or estimation. Suggests that conventional mapping of time-consistency from survey responses may not be straightforward.Dynamic Choice, Time Inconsistency and ITNs
  45. 45. Introduction Model Identification EstimationResults: Time Preference Parameters Table 4: Unobserved Types: Time Preferences Estimate 2.5 97.5 δ 0.7880 0.0000 0.9351 βN 0.9757 0.9313 0.9798 βS 0.5727 0.0007 0.7311 Notes: δ is the exponential discount parameter. βN is the hyperbolic parameter for naivetime-inconsistent agents, βS is the corresponding parameter for sophisticated time-inconsistent agents. “Naive” and “Sophisticated” agents have different rates of time preference. “Sophisticated” agents appear to me much more present-biased than “naive” agents. Speculation: consistent with idea that highly impatient agents learn how to cope over time (by becoming “sophisticated”).Dynamic Choice, Time Inconsistency and ITNs
  46. 46. Introduction Model Identification EstimationResults: Cost and Risk Aversion Parameters Table 5: Unobserved Types: Cost and Risk Aversion Estimate 2.5 97.5 αC 0.7230 0.6047 1.7890 αN 0.4348 0.2935 1.9277 α4 0.5513 0.3725 1.9736 α5 0.8389 0.6911 2.0000 α6 0.9205 0.7935 1.9445 κC 0.0070 -1.9950 1.0754 κN -0.1998 -0.6869 0.8373 κS -0.5314 -1.9951 1.3667 κS -0.9613 -1.2298 0.3725 κ5 -0.3721 -2.0000 1.6852Notes: The α vector parameterizes the risk-aversion parameter and the κ vector parameterizes the malaria cost function. Some variation in risk and cost parameters across types. However, differences are imprecisely estimated and appear to be substantively small (for counterfactuals considered in paper)Dynamic Choice, Time Inconsistency and ITNs
  47. 47. Introduction Model Identification EstimationCounterfactuals: Summary Ran a set of exercises where we varied the utility and time-preference parameters across types and compared take-up and retreatment results. e.g. compare take up for a model where all types have the same cost and risk preferences but different hyperbolic parameters. The results suggest that the differences in take-up and retreatment across types are driven primarily by the time-preference parameters rather than by the cost and risk parameters. Since the hyperbolic parameter for the naive agents are quite close to 1, their take-up and retreatment behaviour is quite close to that of the time-consistent agents. The behavior of the sophisticated agents is quite different but they are small fraction of the population.Dynamic Choice, Time Inconsistency and ITNs
  48. 48. Introduction Model Identification EstimationConclusions and To Do List Time Inconsistency is often proposed as an explanation for observed choice behaviour but identifying time preferences is usually difficult. Combine information on beliefs along with a field intervention to identify a dynamic discrete choice model with time inconsistency and unobserved types. Results suggest that about 40% of sample was time-consistent and that the bulk of inconsistent agents were “naive” Results suggest that “sophisticated” agents much more hyperbolic than naive ones. Examined other differences (in risk, cost preferences) across types and found these differences to be relatively small. Model Validation needed.Dynamic Choice, Time Inconsistency and ITNs
  49. 49. Additional MaterialLoan Products Our MF partner offered two loan contract types (20% annual interest rate, equal installments): Calculations C1 Loan for the cost of ITN and loan for two re-treatments to be carried out 6 and 12 months later. Rs. 203(259) for single(double) nets paid in 12 monthly installments of Rs.19(23). C2 Loan for cost of ITN only. Re-treatment offered for cash only later. Rs. 173(223) for single(double) nets paid in 12 monthly installments of Rs.16(21). Retreatment offered at Rs.15 (18). Context: Daily agricultural wages are about Rs.50, the price of 1 kg. of rice is about Rs. 10 and the official poverty line for Orissa (2004-5) was Rs. 326 per capita per month. Intro Intro: Overview Model Model: Action Space Types Study DesignDynamic Choice, Time Inconsistency and ITNs
  50. 50. Additional MaterialLoan Product Calculations Cost of the product is p Monthly interest rate r Number of months to repay: t The identical monthly installment x pr x(p, r, t) = 1 − (1 + r)−t is obtained by solving t 1 p = x j=1 (1 + r)j Return to Loan ProductDynamic Choice, Time Inconsistency and ITNs
  51. 51. Additional MaterialTransition Probabilities For t ∈ {2, 3}, partition the space Xt into the sets B = (bm, bh), C = (cm, ch) and A = (nm, nh). The transition probabilities from states t to t+1 for are given by P(xt+1 = x|xt = y, a, z) = I{y ∈ B ∪ C}(π − δ − γa) for x ∈ {bm, cm} P(xt+1 = x|xt = y, a, z) = I{y ∈ B ∪ C}(1 − π + δ + γa) for x ∈ {bh, ch} P(xt+1 = nm|xt = y, a, z) = I{y ∈ A}π P(xt+1 = nh|xt = y, a, z) = I{y ∈ A}(1 − π) Note that stationarity rules out learning. In fact, don’t need stationarity in transitions. We also elicit beliefs at the end of project (i.e. after period 3) which we can use to directly study belief evolution. Return to Model OutlineDynamic Choice, Time Inconsistency and ITNs
  52. 52. Additional MaterialStudy Design Part of a larger study covering 162 villages in rural Orissa evaluating alternative methods of ITN provision. Here, focus on treatment arm where 627 households were offered loan contracts to purchase ITNs. Details 1. March-April 2007: Baseline Survey 2. September-November 2007: Information Campaign and ITN Offers 3. March-April 2008: First Retreatment 4. September-November 2008: Second Retreatment 5. December 2008-April 2009: Follow Up Survey Baseline and Follow Up surveys: Detailed Information Retreatment and Offer periods: Minimal Information Return to Model OverviewDynamic Choice, Time Inconsistency and ITNs
  53. 53. Additional MaterialLocation Malaria “number one public health problem in Orissa” (Orissa HDR, 2004). Sample: 627 MF client households from 47 villages. ≈ 12% malaria prevalence, almost all P. falciparum. Back to IntroDynamic Choice, Time Inconsistency and ITNs
  54. 54. Additional MaterialElicited Beliefs and P(xt |xt−1 , at−1 ) Elicit P(Malaria|No Net) ≡ π P(Malaria|Untreated Net) ≡ π − δ P(Malaria|ITN) ≡ π − δ − γ. Use this along with a stationarity assumption to construct a transition probability matrix. Can build up transition probabilities for more complicated state spaces. e.g. P(k members sick|No Net) = H π k (1 − π)H−k k Stationarity rules out learning. However, don’t need stationarity for identification. We also elicit beliefs at the end of project (i.e. after period 3) which we can use to directly study belief evolution. Back to Intro Model: Transition ProbabilitiesDynamic Choice, Time Inconsistency and ITNs
  55. 55. 11.04 Rs. _________________________________Fraction Fraction FractionAdditional Materialnexthow likely is year the total income that your household will beincome that your household will be able toand not more larger than So, you think that during the In your opinion, on a scale 0-10, agricultural it that during the next agricultural year the total able to earn will be no less than (11.03), earn will be than (11.02). .5 .5 .5 (11.04)? 11.05 P(y>11.04)= ]e«ê @ûi«û Pûh ahðùe @û_Yu _eòaûee ùcûU @ûd (11.03) Vûeê Kcþ ùja^ûjó Kò´û (11.02) Vûeê ùagò ùja ^ûjó û @û_Yu cZùe 0-10 _~ðý« GK ùiÑfþùe @ûi«û Pûh ahðùe @û_Yue _eòaûee ùcûU @ûd (11.04) Vûeê ùagò ùjaûe i¸ûa^û ùKùZ @Qò ? 0 In your opinion, on a scale 0-10, how likely is it that in the next agricultural year the total income that your household will be able to earn will be smaller than (11.04)? 0 0 11.06 P(y<11.04)= Perceived Protective Power of ITNs @û_Yu cZùe 0-10 _~ðý« 4 ùiÑfþùe6@ûi«û Pûh 8 ùe @û_Yue _eòaûee ùcûU @ûd (11.04) Vûeê Kcþ ùjaûe i¸ûa^û ùKùZ @Qò? 8 0 2 GK No net use ahð 10 0 2 4 6 Regular use of untreated net 10 0 2 4 6 Regular use of ITN 8 10 EXPECTATIONS ABOUT MALARIA ùcùfeò@û aòhdùe i¸ûa^û 1 1 11.07 - Imagine first that your household [or a household like yours] does not own or use a bed net. 1 ]e«ê @û_Yu _eòaûeùe (Kò´û @û_Yu _eò @^ý _eòaûe) cgûeú ^ûjó aû aýajûe Ke«ò ^ûjó ùZùa @û_Yu cZùe (.........) K[û C_ùe @û_Y Kò_eò GKcZ Zûjû 0-10 c¤ùe GK ^´e ùA Kjòùa û i¸ûa^û @]ôK ùjùf @]ôK ^´e ùùa Gaõ Kcþ ùjùf Kcþ ^´e ùùa û In your opinion, and a scale of 0-10, how likely do you think it is that @û_Yu cZùe @û_Y ....... C_ùe 0-10 bòZùe ùKùZ ^´e ùùaFraction Fraction Fraction A child under 6 that does not sleep under a bed net will contract malaria in the next 1 year? A 6 ahðeê Kcþ adie _òfûUòKê cgûeò Zùk ^ gê@ûAùf @ûi«û GK ahð c¤ùe ZûKê ùcùfeò@.5 .5 û ùjaûe i¸ûa^û ùKùZ ? .5 An adult that does not sleep under a bed net will contract malaria in the next 1 year? B RùY adiÑ aýqò cgûeú Zùk ^ ùgûAùf @ûi«û 1 ahð c¤ùe ZûKê ùcùfeò@û ùjaûe i¸ûa^û ùKùZ ? A pregnant woman that does not sleep under a bed net will contract malaria in the next 1 year? C RùY MbðaZú cjòkû cgeú Zùk ^ ùgûAùf @ûi«û 1 ahð c¤ùe ZûKê ùcùfeò@û ùjaûe i¸ûa^û ùKùZ ? 0 0 0 11.08 – Now imagine that your household [or a household like yours] owns and uses a bed net that is not treated with insecticide 0 2 4 6 8 10 0 2 4 6 8 10 0 2 4 6 8 10 No net use Regular use of untreated net Regular use of ITN 26 1 1 1Fraction Fraction Fraction .5 .5 .5 0 0 0 0 2 4 6 8 10 0 2 4 6 8 10 0 2 4 6 8 10 No net use Regular use of untreated net Regular use of ITN π ≡ P (malaria within one year | no net) γ ≡ P (malaria | untreated net) − P (malaria | ITN) δ ≡ P (malaria | no net) − P (malaria | untreated net) Back to Intro Back to ModelDynamic Choice, Time Inconsistency and ITNs
  56. 56. made available to you in the future but at different times. For instance, we may ask you if you would rather have Rs 10 one month from NATURAL ORD TWELVE QUESTIONS FOLLOWING THE ORDER INDICATED IN THE COLUMN LABELED “ORDER”, AND NOT FOLLOWING THE now, or Rs 12Additional Materialwe will ask you to choose between two alternatives. At the end of the questionnaire, we will select one of the 12 games at random, an times. Each time, QUESTIONNAIRE. have@ûe¸ _ìaðeê ùLkòaûe _¡Zò @^òŸòðÁthe time indicated iûlûZKûe option gÜ @Wðe Kfcþ selected. We~will make Gaõ _âgÜ @^êprizeiû]ûeYbe given to ùa ^ûjó û ^òcÜfòLòZ 2Uò ùLk c¤e ùLk selected in that game, at bûaùe aûQòaûKê ùja û by the 12Uò _â you have (^òŸòðÁ ɸ) @^ê ûdú _Pûeòùa sure the ~ûdú will _¡Zòùe _Pûeò you through BISWA, so given to you once the time comes. For each of the two possible games below, please tell us which option you would prefer. Order @ûùc @û_Yu iûwùe KòQò ùLk ùLkòaê ~ûjû @û_Yuê aò^û cìfýùe ißÌ Rs 10 paid to @ûùc one monthUòfromÜ @[ðe eûgò c¤eêRs 10G aûQòtoûKê Kjòfour months fromýZùe bò[2]? ^Ü icdù Would you prefer a prize of Uuû RòZûAa û you @û_Yuê êA bò^ today [1], or ùMûUò paid a you aê û ~ûjû @û_Y baòh today ^Ü bò 8.01 cûùi _ùe Pûjó_ûe«ò Kò´û 12 Uuû 4 cûi _ùe iÑûe @ûRòVûeêû cûùi ùLk @ûùcaiûlûZKûe c¤ùe 12‘^û’ ùLk ùLkòaê û _âZ10 Uuû _ê@û_Yuê 2Uòûeêaò4 Ì c¤eê ùMûUòGaaûQòaûKê Kjòaê û iûlûZK 10 Uuû _êe Pûjó _ûe«ò Gjò _ùe Pûjóù - [e ò[e @ûùc eiÑûe @ûRòV K cûi _ùe Pûjóù] └─┴─┘ Gaõ @û_Y aûQò[ôaû @[ðeûgò C_~êq icdùe _êeiÑûe ißeì_ a_ûAùa of Rs 10 paid to you ÄéZ @[ðe eûgòfrom cû¤cùe ù~ûMûAaû_ûAñpaidògtoò you four months ½òZ ùjûA_ûeòùa ù~ C_~êq Would you prefer prize û @ûùc @û_Yuê Gjò _êe one month aògß today [1], or Rs 12 _âZ îZ ùjCQò û ~ûjûßûeû ^ò from today [2]? 8.02 TimeSURVEYOR: BEFORE THEyouandûeêprize of RsPûjóùTHE toTHEone month from today [1], or Rs 14SHOULDVBE4ASKED HASTHE BE RANDOMLY Preferences_êGAMES Va cûùi _ùe 10 paid ORDER IN WHICH THE QUESTIONS _êpaide NOTûeêfour months from today [2]? ORD └─┴─┘ 10 Uuû eiÑûe @ûRò ARE PLAYED, a - “Hyperbolic Discounting”: @ûRò cûi _ùe PûjóùaTO NATURAL ‘^û’ 12 Uuû eiÑû TWELVE QUESTIONS FOLLOWING THE ORDER INDICATED IN you COLUMN LABELED “ORDER”, ANDto youFOLLOWING Would prefer 8.03 └─┴─┘ QUESTIONNAIRE. 10 Uuû _êeiÑûe @ûRòVûeê cûùi _ùe Pûjóùa- ‘^û’ 14 Uuû _êeiÑûe @ûRòVûeê 4 cûi _ùe Pûjóùa ùLk @ûe¸ _ìaðeê ùLkòaûe _¡Zò @^òŸòðÁ Would aûQòaûKê ùja ûaiûlûZKûe 12Uò _âgÜ @Wðe Kfcþ (^òŸòðÁ ɸ) @^ê~ûdú _Pûeòùa Gaõ _âRs@^ê~ûdú iû]ûeY _¡Zòùe months^ûjó û ^òtoday [2]?ùLk c¤e bûaùe you prefer prize of Rs 10 paid to you one month from today [1], or gÜ 16 paid to you four _Pûeòùa from cÜfòLòZ 2Uò 8.04 └─┴─┘ 10 Uuû _êeiÑûe @ûRòVûeê cûùi _ùe Pûjóùa - ‘^û’ 16 Uuû _êeiÑûe @ûRòVûeê 4 cûi _ùe Pûjóùa Order Would you prefer a prize of Rs 10 paid to you one month from today [1], or Rs 10 paid to you four months from today [2]? Would you prefer a prize of Rs 10 paid to you one month from today [1], or Rs 10 paid to you seven months from today [2]? 8.01 8.05 10 Uuû _êeiÑûe @ûRòVûeê cûùi _ùe Pûjóùa -- ‘^û’ 10 Uuû _êeiÑûûe @ûRòVûeê 7 cûi _ùe Pûjóùa └─┴─┘ └─┴─┘ 10 Uuû _êeiÑûe @ûRòVûeê cûùi _ùe Pûjóùa ‘^û’ 10 Uuû _êeiÑ e @ûRòVûeê 4 cûi _ùe Pûjóùa Would you prefer a prize of Rs 10 paid to you one month from today [1], or Rs 12 paid to you four months from today [2]? Would you prefer a prize of Rs 10 paid to you one month from today [1], or Rs 15 paid to you seven months from today [2]? 8.02 8.06 10 Uuû _êeiÑûe @ûRòVûeê cûùi _ùe Pûjóùa -- ‘^û’ 12 Uuû _êeiÑûe @ûRòVVûeê47cûi _ùe Pûjóùaa └─┴─┘ └─┴─┘ 10 Uuû _êeiÑûe @ûRòVûeê cûùi _ùe Pûjóùa ‘^û’ 15 Uuû _êeiÑûe @ûRò ûeê cûi _ùe Pûjóù Would you prefer a prize of Rs 10 paid to you one month from today [1], or Rs 14 paid to you four months from today [2]? Would you prefer a prize of Rs 10 paid to you one month from today [1], or Rs 20 paid to you seven months from today [2]? 8.03 8.07 10 Uuû _êeiÑûe @ûRòVûeê cûùi _ùe Pûjóùa- ‘^û’ 14 Uuû _êeiÑûûe @ûRòVûeê 7 cûi _ùe Pûjóùa └─┴─┘ └─┴─┘ 10 Uuû _êeiÑûe @ûRòVûeê cûùi _ùe Pûjóùa- ‘^û’ 20 Uuû _êeiÑ e @ûRòVûeê 4 cûi _ùe Pûjóùa Would you prefer a prize of Rs 10 paid to you one month from today [1], or Rs 16 paid to you four months from today [2]? Would you prefer a prize of Rs 10 paid to you one month from today [1], or Rs 25 paid to you seven months from today [2]? 8.04 8.08 10 Uuû _êeiÑûe @ûRòVûeê cûùi _ùe Pûjóùa - ‘^û’ └─┴─┘ └─┴─┘ 10 Uuû _êeiÑûe @ûRòVûeê cûùi _ùe Pûjóùa- ‘^û’ 25 Uuû _êeiÑûûe @ûRòVûeê 7 cûi _ùe Pûjóùa 16 Uuû _êeiÑ e @ûRòVûeê 4 cûi _ùe Pûjóùa Would you prefer a prize of Rs 10 paid to you one month from today [1], or Rs 10 paid to you seven months from today [2]? Would you prefer a prize of Rs 10 paid to you four month from today [1], or Rs 10 paid to you seven months from today [2]? 8.05 8.09 10 Uuû _êeiÑûe @ûRòVûeê cûùi _ùe Pûjóùa a- └─┴─┘ └─┴─┘ 10 Uuû _êeiÑûe @ûRòVûeê 4 cûi _ùe Pûjóù - ‘^û’ ‘^û’ 10 Uuû _ê_êeiÑee@ûRòVVûeê7 cûi _ùe Pûjóùùa 10 Uuû eiÑû û @ûRò ûeê 7 cûi _ùe Pûjó a Would you prefer a prize of Rs 10 paid to you one month from today [1], or Rs 15 paid to you seven months from today [2]? Would you prefer a prize of Rs 10 paid to you four month from today [1], or Rs 12 paid to you seven months from today [2]? 8.06 8.10 10 Uuû _êeiÑûe @ûRòVûeê cûùi _ùe Pûjóùaù- - ‘^û’ 15 Uuû _êe_êeiÑûe @ûRòVûeê cûi _ùe PûjóPûjóùa └─┴─┘ └─┴─┘ 10 Uuû _êeiÑûe @ûRòVûeê 4 cûi _ùe Pûjó a ‘^û’ 12 Uuû iÑûe @ûRòVûeê 7 7 cûi _ùe ùa Would you prefer a prize of Rs 10 paid to you one month from today [1], or Rs 20 paid to you seven months from today [2]? Would you prefer a prize of Rs 10 paid to you four month from today [1], or Rs 14 paid to you seven months from today [2]? 8.07 8.11 10 Uuû _êeiÑûe @ûRòVûeê cûùi _ùe Pûjóùa- ‘^û’ 20 Uuû _êeiÑûiÑûe @ûRòûeêûeê 7 cûi _ùe Pûjóùa └─┴─┘ └─┴─┘ 10 Uuû _êeiÑûe @ûRòVûeê 4 cûi _ùe Pûjóùa- ‘^û’ 14 Uuû _êee @ûRòV V 7 cûi _ùe Pûjóùa Would you prefer a prize of Rs 10 paid to you one month from today [1], or Rs 25 paid to you seven months from today [2]? Would you prefer a prize of Rs 10 paid to you four month from today [1], or Rs 16 paid to you seven months from today [2]? 8.08 8.12 10 Uuû _êeiÑûe @ûRòVûeê cûùi _ùe Pûjóùa- ‘^û’ 25 Uuû _êe_êeiÑû@ûRòVûeê ûeê cûi _ùe Pûjóùaùa └─┴─┘ └─┴─┘ 10 Uuû _êeiÑûe @ûRòVûeê 4 cûi _ùe Pûjóùa- ‘^û’ 16 Uuû iÑûe e @ûRòV 7 7 cûi _ùe Pûjó Would you prefer a prize of Rs 10 paid to you four month from today [1], or Rs 10 paid to you seven months from today [2]? 8.09 └─┴─┘ 10 Uuû _êeiÑûe @ûRòVûeê 4 cûi _ùe Pûjóùa- ‘^û’ 10 Uuû _êeiÑûe @ûRòVûeê 7 cûi _ùe Pûjóùa 8.10 “Hyperbolic Uuû youiÑpreferVaûeêprize of _ùe10ifa choicemonth earlier (lower)ûe reward isfrom today [2]? └─┴─┘ Would 10 Discounting” Pûjóù - _êe ûe @ûRò 4 cûi Rs paid to you four of‘^û’from today [1], or12 Uuû paid to youVsevencûi _ùe Pûjóùa Rs 12 _êeiÑ @ûRò ûeê 7 months 8.11 followed byWould youiÑpreferof 4laterPûjó(higher)month from todaywhen14theto time monthsùfrom today [2]? └─┴─┘ choice a prize of Rs 10 paid to you four reward [1], or Rs _êpaide @ûRòVûeê seven horizon 10 Uuû _êe ûe @ûRòVûeê cûi _ùe ùa- ‘^û’ 14 Uuû eiÑû you 7 cûi _ùe Pûjó a 8.12 of both rewardsiÑprefershifted10by to you four month from today [1], or Rs 16 _êeiÑûeto@ûRòVûeêseven months ùa today [2]? Would you is a prize of Rs paid same amount 10 Uuû _êe ûe @ûRòVûeê 4 cûi _ùe Pûjóùa- ‘^û’ 16 Uuû paid you 7 cûi _ùe Pûjó from └─┴─┘ Back to Intro Back to Model Back to TypesDynamic Choice, Time Inconsistency and ITNs
  57. 57. Additional MaterialIdentifying Utility Period 3: Intuition Recall g3 (x3 , w) = u(x3 , 1) − u(x3 , 0) + βτ δ u(x4 )dF∆ (x4 |x3 , z) Depends on γ By assumption γ conditional on (x3 , wγ) has at least two points of support. Evaluating the above at the two different points and taking differences we can identify the second term. Identification of the utility differential follows. Back to Identification IDynamic Choice, Time Inconsistency and ITNs
  58. 58. Additional MaterialIdentifying Utility Period 2 First, note g(·) is identified by standard inversion argument. Next, note g2,k (x2 , w) = u(x2 , k) − u(x2 , 0) + βδH(x2 , w) where (βτ , δ, u2 (·)) are unknown objects and H(·) is known. Use variation in γ ∈ w to identify βδ. Next identify utility differential. Finally, use previous lemmas to separately identify β and δ Back to Identification IDynamic Choice, Time Inconsistency and ITNs
  59. 59. Additional MaterialState Space Extensions Most importantly, need to consider evolution of income, consumption and assets over panel period. We have information on income and expenditures at baseline as well as elicited beliefs about income for periods 1,2 and 3. In addition, we observe realized income for period 3 and 4 as well as some consumption. Some information on household assets. Use realized income and income expectations information to develop a transition probability for income (varying at the household level) P(yt+1 |yt , xt , at ). Use elicited information on income losses from malaria to construct income under alternative states.Dynamic Choice, Time Inconsistency and ITNs
  60. 60. Additional MaterialPreferences Preferences are defined (in addition to the state variables) over consumption which is observed at baseline and followup. Consumption in intervening periods is imputed using time-invariant household characteristics and income beliefs. Preferences are allowed to vary by time-invariant household characteristics as well.Dynamic Choice, Time Inconsistency and ITNs
  61. 61. Additional MaterialPreferences By normalized utility differentials we mean that utility in each state and action in each period is normalized with respect to a utility level at a base action (for all states x3 ). For instance, we will only be able to identify u(x3 , a) − u(x3 , 0) Typically, will normalize and assume that u(x3 , 0) is known.Dynamic Choice, Time Inconsistency and ITNs
  62. 62. Additional MaterialWhy Should We Care? Point identification of hyperbolic (and exponential) parameters allow direct assessment of whether agents are time inconsistent and whether they are differentially so. Can do more: Specify model where types only differ by hyperbolic discount rates to get predictions for model “weighted” towards present bias explanations (“upper bound” on the role present-bias explanations can play). Next, specify model where both hyperbolic parameters as well as utility function parameters (e.g. costs) vary by type. Allows the relative importance of present-bias explanations in ITN adoption and retreatment decisions. Identification: OverviewDynamic Choice, Time Inconsistency and ITNs
  63. 63. Additional MaterialAdvantages of Unknown Types Model Can use model to address the same sets of questions (about time preferences and their relative importance as earlier). New results agnostic about the precise mapping between types and ru . Recall that we only required a “MLR-like” condition. Can use second model as specification check on mappings in first model. IdentificationDynamic Choice, Time Inconsistency and ITNs
  64. 64. Additional MaterialType Classification Choosing to classify agents by (r, f ) may be a problem if choice of products driven by other feature. e.g. time varying credit constraints. Also, while not clear whether the complement (of the identified types) are homogenous. e.g. households with (r = 1, f = 0)? One potential “solution” is to posit 6 types based on (r, f ). Allow all types to have different β parameters. Need that at least one known type is time consistent. Strong assumption, but in application there are many potential candidates for this. e.g. with households r = 0Dynamic Choice, Time Inconsistency and ITNs
  65. 65. Additional MaterialDifferences with Fang and Wang (2010) In FW all agents are identical with the same preferences. So no heterogeneity in terms of types. All agents are inconsistent. Preferences are statiionary, so no changes in preferences over time. Results are only proved for the logit case. Do allow for partially naive agents.Dynamic Choice, Time Inconsistency and ITNs

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