Dynamic Choice, Time Inconsistency and ITNs: Identification and Estimation from Elicited Beliefs

Introduction Model Identiﬁcation Estimation

Time Inconsistency, Expectations and
Technology Adoption

Aprajit Mahajan (UCLA, Stanford)
Alessandro Tarozzi (Pompeu Fabra)
IFPRI Seminar

July 12, 2012

Dynamic Choice, Time Inconsistency and ITNs


Motivation: Time (In)consistency

Can self-control based explanations rationalize behavior hard to reconcile
with 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).



Motivation (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) Details



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.



Talk: 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. Conclusion


Model: 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.



Model: Primitives

We begin by deﬁning the decision problem:
State Space
Action Space
Preferences
Transition Probabilities
Maximization Problem



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.


Action 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, 3



Transition 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.


Key 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)



Preference: 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 (·)



Preferences: 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.



Awareness of Future Present-Bias

Finite 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 δ



Talk Overview
1. Introduction
2. Model
2.1 State Space
2.2 Action Space
2.3 Preferences
4. Identiﬁcation
4.1 Observed Types
5. Estimation
6. Conclusion



Two 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.



Directly 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} =⇒?). problems



Unobserved 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 work



Identification 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
payoff



Identiﬁcation 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.



Period 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.



Period 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 ))



Identifying 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)



Identification: 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.



Identification: 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.



Identification: 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.



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.



Overview
1. Introduction
2. Model
2.1 State Space
2.2 Action Space
2.3 Preferences
4. Identiﬁcation
4.1 Observed Types
5. Estimation
6. Conclusion



Unobserved 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.



Type-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.



Type-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)).



Type 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 ∈ T



Monte 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)



Monte 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}



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.



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 0.24
Notes: Each model was simulated 250 times. The true values are (δ, βN , βS , θ4 , θ5 ) = (.9, .7, .8, 3, 1)



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 0.2545
Notes: Each model was simulated 250 times. The true values are (δ, βN , βS , θ4 , θ5 ) = (.7, .4, .95, 3, 1)



Estimation: 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.



Preferences
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 xt


Mapping 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=1



Estimation: 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)



Estimation: 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.



Results: 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 0.1756
Notes: 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.


Results: 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 naive
time-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”).



Results: 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 1.6852
Notes: 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)



Counterfactuals: 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.


Conclusions 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.

Additional Material

Loan 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 Design


Additional Material

Loan 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 Product


Additional Material

Transition 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 Outline


Additional Material

Study 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 Overview


Additional Material

Location

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 Intro


Additional Material

Elicited 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 Probabilities


11.04 Rs. _________________________________

Fraction

Fraction

Fraction
Additional 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 ^é ùA Kjòùa û i¸ûa^û @]ôK ùjùf @]ôK ^é ùùa Gaõ Kcþ ùjùf Kcþ ^é 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 ^é ùùa
Fraction

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 1
Fraction

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 Model


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 12
Additional 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
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given to you once the time comes. For each of the two possible games below, please tell us which option you would prefer.
Order
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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 É¸) @^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
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
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
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 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
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
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
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
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 Types


Additional Material

Identifying 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 I


Additional Material

Identifying 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 I


Additional Material

State 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.


Additional Material

Preferences

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.


Additional Material

Preferences

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.


Additional Material

Why 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: Overview


Additional Material

Advantages 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.
Identification


Additional Material

Type 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 = 0


Additional Material

Diﬀerences 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: Identification and Estimation from Elicited Beliefs

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