Single Agent
Dynamic
Models
Paul Schrimpf
Application:
prescription
drug demand
Introduction
Descriptive evidence
Single Agent Dynamic Models
Model
Results
Summary
Identification
Paul Schrimpf
Identification
Fang and
Wang (2015)
Machine
replacement
models
Other
applications
Holmes (2011)
Overview
Model
Results: demand
estimation
Dynamic estimation
Dynamic results
Dynamic demand
References
UBC
Economics 565
March 9, 2017
Single Agent
Dynamic
Models
References
Paul Schrimpf
Application:
prescription
drug demand
Introduction
Descriptive evidence
Model
Results
Summary
Identification
Identification
Fang and
Wang (2015)
Machine
replacement
models
Other
applications
Holmes (2011)
Overview
Model
Results: demand
estimation
Dynamic estimation
Dynamic results
Dynamic demand
References
• Reviews:
• Aguirregabiria (2017) chapters 6-7
• Rust (2008)
• Aguirregabiria and Mira (2010)
• My notes from 628
• Key papers:
• Rust (1987), Hotz and Miller (1993)
Single Agent
Dynamic
Models
Paul Schrimpf
Application:
prescription
drug demand
Introduction
Descriptive evidence
Model
Results
Summary
Identification
1 Application: prescription drug demand
Introduction
Descriptive evidence
Model
Results
2 Summary
3 Identification
4 Identification
Identification
5 Fang and Wang (2015)
Fang and
Wang (2015)
6 Machine replacement models
Machine
replacement
models
7 Other applications
Other
applications
Holmes (2011)
Overview
Model
Results: demand
estimation
Dynamic estimation
Dynamic results
Dynamic demand
References
Holmes (2011)
Overview
Model
Results: demand estimation
Dynamic estimation
Dynamic results
Dynamic demand
Single Agent
Dynamic
Models
Paul Schrimpf
Application:
prescription
drug demand
Introduction
Descriptive evidence
Model
Results
Section 1
Summary
Identification
Identification
Fang and
Wang (2015)
Machine
replacement
models
Other
applications
Holmes (2011)
Overview
Model
Results: demand
estimation
Dynamic estimation
Dynamic results
Dynamic demand
References
Application: prescription drug demand
Single Agent
Dynamic
Models
Medicare part D
Paul Schrimpf
Application:
prescription
drug demand
Introduction
Descriptive evidence
Model
Results
Summary
Identification
Identification
Fang and
Wang (2015)
Machine
replacement
models
Other
applications
Holmes (2011)
Overview
Model
Results: demand
estimation
Dynamic estimation
Dynamic results
Dynamic demand
References
• Paper is focused on Medicare Part D
• The largest expansion of Medicare since inception
• 32 million beneficiaries, 11% of Medicare spending
• Typical coverage highly non-linear
• Many planned and potential changes
• E.g., under ACA, “donut hole” will be “filled” by 2020
• Main objectives:
• Develop a model that would allow us to assess the
contract design impact on drug spending (“moral
hazard”)
• Provide estimates of the spending effects of the
proposed changes in Part D contracts
• More conceptually engage in the analysis of healthcare
utilization under non-linear contracts
Single Agent
Dynamic
Models
Paul Schrimpf
Application:
prescription
drug demand
Introduction
Descriptive evidence
Model
Results
Summary
Identification
Identification
Fang and
Wang (2015)
Machine
replacement
models
Other
applications
Holmes (2011)
Overview
Model
Results: demand
estimation
Dynamic estimation
Dynamic results
Dynamic demand
References
Standard Coverage in Medicare
part D (in 2008)
Single Agent
Dynamic
Models
Approach
Paul Schrimpf
Application:
prescription
drug demand
• Throughout, we follow the health economics jargon
and use the term “moral hazard” to mean the effect of
out-of-pocket price on health care spending.
Introduction
Descriptive evidence
Model
Results
Summary
Identification
1
Identification
• Prescription demand is responsive the out-of-pocket
Fang and
Wang (2015)
price
• Individuals are forward looking when choosing
Machine
replacement
models
Other
applications
Holmes (2011)
Overview
Model
Results: demand
estimation
Dynamic estimation
Dynamic results
Dynamic demand
References
Descriptive analysis of moral hazard, which will provide
evidence that
prescriptions
2
Structural dynamic model of prescription demand to
quantify descriptive results and for counterfactual
analysis of alternative contract designs
Single Agent
Dynamic
Models
Related literatures
Paul Schrimpf
Application:
prescription
drug demand
Introduction
Descriptive evidence
Model
Results
Summary
Identification
Identification
Fang and
Wang (2015)
Machine
replacement
models
Other
applications
Holmes (2011)
Overview
Model
Results: demand
estimation
Dynamic estimation
Dynamic results
Dynamic demand
References
• Large recent literature on Medicare Part D, mostly
focusing on the quality of plan choice (Heiss et al. 2010,
2012; Abaluck and Gruber 2011; Ketcham et al. 2012)
• Large venerable literature on response of healthcare
spending to insurance contracts (“moral hazard”)
• Only recently has attention focused on non-linear
nature of contract (Bajari et al. 2011; Kowalski 2011;
Marsh 2011; Aron-Dine et al. 2012)
• “Bunching” response to progressive income tax (Saez
2010; Chetty et al. 2011; Chetty 2012)
• In our context, need to account for dynamics
Single Agent
Dynamic
Models
Setting
Paul Schrimpf
Application:
prescription
drug demand
Introduction
Descriptive evidence
Model
Results
Summary
Identification
Identification
Fang and
Wang (2015)
Machine
replacement
models
Other
applications
Holmes (2011)
Overview
Model
Results: demand
estimation
Dynamic estimation
Dynamic results
Dynamic demand
References
• Part D introduced in 2006, covering approximately 30M
eligible individuals
• Government sets standard plan, but actual plans often
provide different coverage
• Individuals eligible the month they turn 65, and then
make plan choices prior to every calendar year
Data and sample
20% random sample of all Part D-covered individuals (2007 - 2009)
Baseline sample is about one-quarter of full sample
Restrict attention to those 65+, not dual eligibles, not entitled to low
income subsidies, in stand-alone PDPs
Sample
Full Sample
Baseline Sample
Obs. (beneficiary years)
Unique beneficiaries
Age
Female
Risk score
16,036,236
6,208,076
70.9 (13.3)
0.60
n/a
3,898,247
1,689,308
75.6 (7.7)
0.65
0.88 (0.34)
Einav, Finkelstein, and Schrimpf
Contract Design in Medicare Part D ()
Minnesota, Sept. 2014
7 / 44
Spending patterns
Sample
Annual Total Spending
Mean
Std. Deviation
Pct with no spending
25th pctile
Median
75th pctile
90th pctile
Annual Out of Pocket Spending
Mean
Std. Deviation
Einav, Finkelstein, and Schrimpf
Full Sample
Baseline Sample
2,433
4,065
7.35
378
1,360
2,942
5,571
1,888
2,675
5.65
487
1,373
2,566
3,901
418
744
778
968
Contract Design in Medicare Part D ()
Minnesota, Sept. 2014
8 / 44
Single Agent
Dynamic
Models
Actual contracts
Paul Schrimpf
Application:
prescription
drug demand
Introduction
Descriptive evidence
Samplea
Model
Results
Summary
Identification
Identification
Fang and
Wang (2015)
Machine
replacement
models
Other
applications
Holmes (2011)
Overview
Model
Results: demand
estimation
Dynamic estimation
Dynamic results
Dynamic demand
References
ObsL9Xbeneficiary9yearsDeductible9Amount
Fraction9w49standard9DedL
Deductible9CoinsL9Rate
Has9standard9ICL
ICL9Amount
PreIICL9CoinsL9Rate
Some9Gap9Coverage
Gap9CoinsL9Rate
Gap9CoinsL9Rate9if9some9coverage
Catastrophic9Amount
Catastrophic9ConisL9Rate
Bunching9Sample
Deductible9plans
No9DedL9plans
6B9yLoL9Sample
Deductible9plans
No9DedL9plans
GHPg8Hvv8
vH9vvHB7P
v8H9B8
GGGHBG6
v6BL9
XX
PL88
P
II
II
vB7LG
XX
PL8B
P
II
II
GLPP
vHBvvL6
PLv6
PL98
vHBgBLG
PLg7
GLPP
vHBG6Lh
PLv7
PL97
vHBv6L7
PLg7
PLPG
PL88
XX
PLG7
PL9B
XX
PLPP
PL88
XX
PLGv
PL96
XX
hHPB9L7
PLP7
hHP9PL6
PLP7
hHPh8Lg
PLP7
hHP79LG
PLP7
Single Agent
Dynamic
Models
Paul Schrimpf
Static price response: bunching
at the kink
Application:
prescription
drug demand
Introduction
Descriptive evidence
Model
Results
Summary
Identification
Identification
Fang and
Wang (2015)
Machine
replacement
models
Other
applications
Holmes (2011)
Overview
Model
Results: demand
estimation
Dynamic estimation
Dynamic results
Dynamic demand
References
• Sharp increase in price when go into donut hole
• On average price rises from 34 to 93 cents for every
dollar
• Standard economic theory: with convex preferences
smoothly distributed in population, should see
bunching at the convex kink
Bunching at kink I: 2008 spending distribution
Einav, Finkelstein, and Schrimpf
Contract Design in Medicare Part D ()
Minnesota, Sept. 2014
13 / 44
Bunching II: year-to-year movement in kink
Einav, Finkelstein, and Schrimpf
Contract Design in Medicare Part D ()
Minnesota, Sept. 2014
14 / 44
Heterogeneity across individuals
Population
Excess Mass
All
0.291
Year
2006
2007
2008
2009
0.088
0.150
0.213
0.293
Gender
Male
Female
0.348
0.262
Age group
66
67
68-69
70-74
75-79
80-84
85+
0.519
0.426
0.383
0.334
0.255
0.194
0.136
Einav, Finkelstein, and Schrimpf
Population
Excess Mass
Risk Score Quartile
healthiest
less healthy
sicker
least healthy
0.448
0.155
0.250
0.346
Number of HCCs
0
1
2
3
4
5+
0.837
0.494
0.191
0.197
0.236
0.316
Contract Design in Medicare Part D ()
Minnesota, Sept. 2014
16 / 44
Timing of purchases (December)
Einav, Finkelstein, and Schrimpf
Contract Design in Medicare Part D ()
Minnesota, Sept. 2014
17 / 44
Timing of purchases (Sept - Dec)
Einav, Finkelstein, and Schrimpf
Contract Design in Medicare Part D ()
Minnesota, Sept. 2014
18 / 44
Single Agent
Dynamic
Models
Forward looking moral hazard
Paul Schrimpf
Application:
prescription
drug demand
Introduction
Descriptive evidence
Model
Results
Summary
Identification
Identification
Fang and
Wang (2015)
Machine
replacement
models
Other
applications
Holmes (2011)
Overview
Model
Results: demand
estimation
Dynamic estimation
Dynamic results
Dynamic demand
References
• Individuals become eligible when they turn 65; contract
resets on January 1st
• Compare spending in first month of eligibility for
people who turn 65 in February versus October — same
spot price, but very different expected future prices
Single Agent
Dynamic
Models
Paul Schrimpf
Application:
prescription
drug demand
Introduction
Descriptive evidence
Model
Results
Summary
Identification
Identification
Fang and
Wang (2015)
Machine
replacement
models
Other
applications
Holmes (2011)
Overview
Model
Results: demand
estimation
Dynamic estimation
Dynamic results
Dynamic demand
References
Forward looking moral hazard
Single Agent
Dynamic
Models
Model I
Paul Schrimpf
Application:
prescription
drug demand
Introduction
Descriptive evidence
Model
Results
Summary
Identification
Identification
Fang and
Wang (2015)
• Model highly stylized, designed to capture key
economic objects that determine expenditure response
to contract
• Risk-neutral fwd-looking individual faces uncertain
health shocks
• Prescriptions are defined by (θ, ω), where θ > 0 is the
prescription’s (total) cost and ω > 0 is the monetized
cost of not taking the drug
Machine
replacement
models
• Arrive at weekly rate λ, drawn from
Other
applications
• Serial corr captured by λ following a Markov process
Holmes (2011)
G(θ, ω) = G2 (ω|θ)G1 (θ)
H(λ|λ′ )
Overview
Model
Results: demand
estimation
Dynamic estimation
Dynamic results
Dynamic demand
References
• Insurance specifies (discrete) covg length T and defines
c(θ, x) – the out-of-pocket cost associated with a
prescription that costs θ when total spending so far is x
Single Agent
Dynamic
Models
Paul Schrimpf
Application:
prescription
drug demand
Model II
• When a shock arrives, individuals make binary choice
• Flow utility given by
{
Introduction
Descriptive evidence
u(θ, ω; x) =
Model
Results
−c(θ, x) if filled
−ω
if not filled
Summary
Identification
• Optimal behavior characterized by finite horizon
dynamic problem
Identification
Fang and
Wang (2015)
Machine
replacement
models
Other
applications
Holmes (2011)
Overview
Model
• Bellman equation given by
v(x, t, λt+1 ) = Eλ|λt+1


λ
∫
{
max

(1 − λ)δv(x, t − 1, λ)+
}

−c(θ, x) + δv(x + θ, t − 1, λ),
dG(θ, ω)
−ω + δv(x, t − 1, λ)
Results: demand
estimation
Dynamic estimation
Dynamic results
Dynamic demand
References
with terminal condition v(x, 0) = 0 for all x
Single Agent
Dynamic
Models
Three key economic objects
Paul Schrimpf
Application:
prescription
drug demand
Introduction
Descriptive evidence
Model
Results
Summary
Identification
Identification
Fang and
Wang (2015)
• Statistical description of distribution of health shocks:
λ and G1 (θ)
• “Primitive” price elasticity capturing substitution
between health and income: G2 (ω|θ)
• If ω ≥ θ, fill even if have to pay full cost
• If ω < θ, fill only if some portion of cost (effectively)
paid by insurer
• Convenient to think about the ratio ω/θ
• Loosely, identified off the bunching
Machine
replacement
models
• Extent to which individuals understand and respond to
Other
applications
• “Full” myopia (δ = 0): don’t fill if ω < c(θ, x)
• Dynamic response (δ > 0): utilization depends on both
Holmes (2011)
Overview
Model
Results: demand
estimation
Dynamic estimation
Dynamic results
Dynamic demand
References
dynamic incentives in non-linear contract: δ ∈ [0, 1]
spot and future price
• δ is context specific! ... Captures salience, discounting,
and perhaps liquidity constraints
• Loosely, identified off the timing patterns
Single Agent
Dynamic
Models
Parameterization I
Paul Schrimpf
Application:
prescription
drug demand
Introduction
Descriptive evidence
• Assume G1 (θ) is lognormal: log θ ∼ N(µ, σ 2 ).
• Assume ω|θ is stochastic, and is drawn from a mixture
distribution:
Model
Results
Summary
Identification
Identification
Fang and
Wang (2015)
Machine
replacement
models
Other
applications
• ω ≥ θ with probability 1 − p (prescription is filled for
sure)
• ω ∼ U[0, θ] with probability p (decision responds to
price)
• Assume λ could obtain two values, and follows a 2-by-2
transition matrix
• Heterogeneity modeled using a finite (five types)
mixture:
Holmes (2011)
Overview
Model
Results: demand
estimation
Dynamic estimation
Dynamic results
Dynamic demand
References
• Individual(is of )type m with(probability
)
∑
′
πm = exp z′i βm / M
k=1 exp zi βk
2
• Almost all parameters vary with type: λm,low , µm , σm
, pm
(exception: δ, λhigh /λlow , λ−transition)
Single Agent
Dynamic
Models
Parameterization II
Paul Schrimpf
Application:
prescription
drug demand
Introduction
Descriptive evidence
Model
Results
Summary
Identification
Identification
Fang and
Wang (2015)
Machine
replacement
models
Other
applications
Holmes (2011)
Overview
Model
Results: demand
estimation
Dynamic estimation
Dynamic results
Dynamic demand
References
• Baseline zi : constant, risk score and 65-year-old
indicator
• Allowing for heterogeneity in both individual health
(λ, µ, σ ) and in responsiveness of individual spending to
cost-sharing (p)
• Do not use panel nature across years (although risk
scores introduce some serial correlation within
individuals across years)
Single Agent
Dynamic
Models
Intuition for identification
Paul Schrimpf
Application:
prescription
drug demand
Introduction
Descriptive evidence
Model
Results
Summary
Identification
Identification
Fang and
Wang (2015)
Machine
replacement
models
Other
applications
Holmes (2011)
Overview
Model
Results: demand
estimation
Dynamic estimation
Dynamic results
Dynamic demand
References
Three key objects:
• Claim sizes θ distributions identified from observed
claims (panel data allows identification despite
unobserved types and selection, similar to Kasahara
(2009), Hu and Shum (2012), and Sasaki (2012) )
• Moral hazard (p’s and λ’s) identified from bunching
described earlier.
Single Agent
Dynamic
Models
Estimation I
Paul Schrimpf
Application:
prescription
drug demand
Introduction
Descriptive evidence
Model
Results
Summary
Identification
Identification
Fang and
Wang (2015)
Machine
replacement
models
Other
applications
Holmes (2011)
Overview
• Moments chosen based on identification intuition
• Summary statistics (mean, standard deviation, P = 0) of
total spending
• Bunching: histogram of total spending near ICL
• Timing patterns for individuals around and below the
kink
• Calculate objective function using simulation
1 Given parameters solve for value function using
backward induction
• Choose grid of values of xg , set v(x, 0; m) = 0∀x
• Given v(x, t − 1; m), calculate
[
{
−oop(θ, xj ) + δv(xg + θ, t
vg,m = (1−λm )δv(xg , t−1)+λE max
−ω + δv(xj , t − 1; m
Model
Results: demand
estimation
Dynamic estimation
Dynamic results
Dynamic demand
References
• Set v(x, t; m) = shape preserving cubic spline
interpolation of {xg , vg,m }, repeat until maximum t
Single Agent
Dynamic
Models
Estimation II
Paul Schrimpf
Application:
prescription
drug demand
• Backward induction can amplify approximation errors so
important to calculate E[max] accurately (our
distributional assumptions give an analytic expression
conditional on θ and we use quadrature to integrate
over θ) and a good interpolation method of v
(polynomials, Fourier series, and non shape preserving
splines fail spectacularly here; linear interpolation is
okay)
Introduction
Descriptive evidence
Model
Results
Summary
Identification
Identification
Fang and
Wang (2015)
Machine
replacement
models
Other
applications
Holmes (2011)
Overview
Model
Results: demand
estimation
Dynamic estimation
Dynamic results
Dynamic demand
References
2
3
Draw m, sequences of θ, ω and simulate the model
Compute moments from observed data and simulated
data
Single Agent
Dynamic
Models
Poor approximation with
polynomials
Paul Schrimpf
Application:
prescription
drug demand
vg
t=1
P(treat)
Introduction
Descriptive evidence
Model
0.8
Results
0.7
Model
Results: demand
estimation
Dynamic estimation
Dynamic results
Dynamic demand
References
Value
Value
-10
-15
-10
-15
-20
-20
0.4
-25
0
2000 4000 6000 8000 1000012000
Cumulative spending (x)
0
2000 4000 6000 8000 1000012000
Cumulative spending (x)
0
2000 4000 6000 8000 10000
Cumulative spending (x)
0
2000 4000 6000 8000 10000
Cumulative spending (x)
t=2
P(treat)
0.8
0
0.75
-5
0.7
-10
0.65
-15
0.6
0.55
0.5
Value
Machine
replacement
models
Overview
0.6
0.55
0.45
Fang and
Wang (2015)
Holmes (2011)
-5
0.65
0.5
Identification
Other
applications
0
-5
Value
Identification
0
0.75
P(treat)
Summary
v(x, t)
-20
-25
-30
0.45
-35
0.4
0
-5
-10
-15
-20
-25
-30
-35
-40
-40
0
2000 4000 6000 8000 1000012000
Cumulative spending (x)
0
2000 4000 6000 8000 1000012000
Cumulative spending (x)
Single Agent
Dynamic
Models
Value function with shape
preserving cubic splines
Paul Schrimpf
Application:
prescription
drug demand
P(treat)
vg
t=1
Introduction
Descriptive evidence
Identification
Identification
0.8
0
0
0.75
-1
-1
0.7
-2
-2
0.65
-3
-3
0.6
0.55
Value
Summary
P(treat)
Results
Value
Model
v(x, t)
-4
-5
0.5
-6
0.45
-7
0.4
-8
-4
-5
-6
-7
-8
Fang and
Wang (2015)
0
Overview
Model
Results: demand
estimation
Dynamic estimation
Dynamic results
Dynamic demand
References
0.8
1000 2000 3000 4000 5000 6000
Cumulative spending (x)
0
0
1000 2000 3000 4000 5000
Cumulative spending (x)
0
-50
-50
0.7
-100
-100
0.65
-150
-150
0.6
0.55
Value
0.75
Value
P(treat)
Holmes (2011)
0
t = 52
Machine
replacement
models
Other
applications
1000 2000 3000 4000 5000 6000
Cumulative spending (x)
-200
-250
0.5
-300
0.45
-350
-200
-250
-300
-350
0.4
-400
-400
0
1000 2000 3000 4000 5000 6000
Cumulative spending (x)
0
1000 2000 3000 4000 5000 6000
Cumulative spending (x)
0
1000 2000 3000 4000 5000
Cumulative spending (x)
Single Agent
Dynamic
Models
Estimation I
Paul Schrimpf
Application:
prescription
drug demand
Introduction
Descriptive evidence
Model
Results
Summary
Identification
Identification
Fang and
Wang (2015)
Machine
replacement
models
Other
applications
Holmes (2011)
Overview
Model
Results: demand
estimation
Dynamic estimation
Dynamic results
Dynamic demand
References
• Objective function minimized using CMA-ES
• Uses quadratic approximation of objective to guide
search, so converges more quickly than most random or
global minimization algorithms (e.g. simulated
annealing, genetic algorithms, pattern search)
• Introduces randomness so able to escape local minima
unlike deterministic algorithms (e.g. Nelder-Mead,
(Quasi)-Newton, conjugate gradient)
• Good performance, especially on non-convex problems
compared to other algorithms
Single Agent
Dynamic
Models
Parameter estimates
Paul Schrimpf
δ
Application:
prescription
drug demand
β0
Introduction
Descriptive evidence
βRS
Model
Results
Summary
Identification
β65
µ
σ
Identification
Fang and
Wang (2015)
Machine
replacement
models
Other
applications
Holmes (2011)
Overview
Model
Results: demand
estimation
Dynamic estimation
Dynamic results
Dynamic demand
References
pω
λ
λ
0.961
( 0.00186 )
j=1
0.00
(0.000000)
0.00
(0.000000)
0.000
(0.00e+00)
-0.00292
(1.44e-05)
2.373
(0.000114)
0.859
(8.02e-05)
0.0114
(8.65e-06)
0.010
(0.000051)
j=2
3.59
(0.001427)
-2.46
(0.001378)
-0.101
(1.17e-02)
3.99789
(1.32e-02)
1.180
(0.010186)
0.902
(9.24e-03)
0.1432
(3.17e-04)
0.127
(0.000609)
λ transition probabilities
0.5518941
(0.00180)
(0.00195)
0.4481059
(0.00180)
(0.00195)
λ marginal probabilities
0.4928264
0.5071736
j=3
3.98
(0.001682)
-2.85
(0.002090)
1.336
(8.85e-04)
2.94797
(8.75e-05)
1.582
(0.004952)
0.495
(3.26e-03)
0.6300
(2.35e-03)
0.557
(0.001905)
0.4354298
0.5645702
(0.00155)
(0.00155)
j=4
-4.37
(0.000696)
4.10
(0.000774)
0.926
(1.53e-14)
4.31604
(1.04e-02)
0.419
(0.003278)
0.505
(4.26e-03)
0.8817
(5.83e-03)
0.779
(0.004545)
j=5
-4.35
(0.000594)
6.18
(0.000296)
-1.596
(4.13e-15)
4.29602
(1.04e-02)
1.431
(0.005838)
0.374
(1.45e-03)
0.4490
(1.25e-03)
0.397
(0.001467)
Single Agent
Dynamic
Models
Parameter estimates: implied
quantities
Paul Schrimpf
Application:
prescription
drug demand
Introduction
Descriptive evidence
Model
Results
Summary
Identification
Identification
Fang and
Wang (2015)
Machine
replacement
models
Other
applications
Holmes (2011)
Overview
Model
Results: demand
estimation
Dynamic estimation
Dynamic results
Dynamic demand
References
P(j)
P(j|age = 65)
P(j|age
[ > 65)
]
E
dP(j|rs)
drs
E[θ|j]
s.d.(θ|j)
E[spend full ins.|j]
E[spend 0.25 coins.|j]
E[spend no ins.|j]
0.05
0.00
0.05
0.29
0.14
0.29
0.35
0.86
0.33
0.03
0.00
0.03
0.29
0.00
0.30
0.01
16.65
278
9.84
7.73
1.39
-0.38
109.36
190
814.53
630.86
79.85
-0.51
66.65
223
2183.38
1913.16
1102.50
0.06
81.76
36
3748.90
3275.54
1855.44
0.83
204.42
531
4772.61
4326.28
2987.30
Single Agent
Dynamic
Models
Fit: Distribution of total costs
Paul Schrimpf
Application:
prescription
drug demand
Introduction
0.09
Descriptive evidence
Model
Results
Identification
Identification
Fang and
Wang (2015)
portion
Summary
observed: P(0)=0.056 median=1315.50, mean=1788.45, and sd=2425.99
0.06
estimated: P(0)=0.057 median=1311.90, mean=1746.42, and sd=1906.83
0.03
Machine
replacement
models
Model
Dynamic estimation
Dynamic results
Dynamic demand
References
total spending
observed
estimated
00
60
40
00
0
Overview
Results: demand
estimation
00
0.00
Holmes (2011)
20
Other
applications
Single Agent
Dynamic
Models
Fit: Total costs near ICL
Paul Schrimpf
8e-04
Application:
prescription
drug demand
Introduction
Descriptive evidence
6e-04
Model
Results
Identification
Identification
Fang and
Wang (2015)
density
Summary
4e-04
2e-04
Machine
replacement
models
Model
Dynamic estimation
Dynamic results
Dynamic demand
References
total spending - ICL
observed
estimated
00
10
50
0
0
0
-50
Overview
Results: demand
estimation
0e+00
00
Holmes (2011)
-10
Other
applications
Single Agent
Dynamic
Models
Fit: Risk score and total
spending
Paul Schrimpf
First Quartile
Application:
prescription
drug demand
Second Quartile
0.3
Introduction
Descriptive evidence
portion
Summary
portion
Results
0.10
0.2
Model
observed: P(0)=0.028 median=1142.16, mean=1420.46, and sd=1466.96
estimated: P(0)=0.073 median=936.12, mean=1239.58, and sd=1394.76
0.05
0.1
Identification
observed: P(0)=0.160 median=459.30, mean=855.79, and sd=1296.82
estimated: P(0)=0.060 median=885.49, mean=1079.75, and sd=1119.21
Identification
0.075
observed: P(0)=0.009 median=2458.09, mean=3072.92, and sd=3703.80
portion
portion
observed: P(0)=0.015 median=1665.11, mean=1953.46, and sd=1918.68
0.050
estimated: P(0)=0.070 median=1387.63, mean=1802.03, and sd=1930.88
0.04
0.025
Dynamic results
Dynamic demand
0.00
0.000
estimated: P(0)=0.026 median=2606.97, mean=3023.66, and sd=2364.60
0
60
0
0
Fourth Quartile
0.08
Model
References
estimated
0.12
Overview
Dynamic estimation
40
0
total spending
observed
estimated
Holmes (2011)
Results: demand
estimation
00
total spending
observed
Third Quartile
Other
applications
20
0
60
0
20
0
0
Machine
replacement
models
00
0.00
40
00
0.0
Fang and
Wang (2015)
Single Agent
Dynamic
Models
Fit: Timing moments
Paul Schrimpf
Application:
prescription
drug demand
Descriptive evidence
Model
Results
Summary
Identification
Identification
Fang and
Wang (2015)
Machine
replacement
models
Other
applications
P(claim in month|total spending)
Introduction
0.86
0.84
0.82
Model
Dynamic estimation
Dynamic results
Dynamic demand
References
5
Overview
Results: demand
estimation
10
Holmes (2011)
month
kink
pre
Single Agent
Dynamic
Models
Average weekly spending
Paul Schrimpf
42.5
Application:
prescription
drug demand
Introduction
Descriptive evidence
40.0
Model
Results
Identification
Identification
Fang and
Wang (2015)
Machine
replacement
models
Other
applications
Average spending
Summary
37.5
35.0
32.5
Holmes (2011)
Overview
Model
Dynamic results
Dynamic demand
References
week
50
40
30
20
10
Dynamic estimation
0
Results: demand
estimation
Single Agent
Dynamic
Models
Counterfactual: “filling the gap”
Paul Schrimpf
Application:
prescription
drug demand
Introduction
Descriptive evidence
Model
Results
Summary
Identification
Identification
Fang and
Wang (2015)
• Main counterfactual exercise considers “filling the gap”
as specified by ACA by 2020:
• Coinsurance rate in standard contract will remain at its
pre-gap level (of 25%) until out of pocket spending puts
individual at CCL
• First consider spending effect of “filling the gap” in the
2008 standard benefit design
• On average, increases total spending by $204 (11.5%)
Machine
replacement
models
Other
applications
Holmes (2011)
Overview
Model
Results: demand
estimation
Dynamic estimation
Dynamic results
Dynamic demand
References
baseline
filled gap
mean
1760
1964
sd
1924
2127
Q25
402
407
Q50
1413
1455
Q75
2513
2862
Q90
3632
4450
Single Agent
Dynamic
Models
Counterfactuals: change in
spending from filling gap
Paul Schrimpf
Application:
prescription
drug demand
1500
Introduction
Descriptive evidence
Model
0
Holmes (2011)
Dynamic demand
References
mean
Q10
Q25
Q50
Q75
Q90
0
10
00
75
0
00
initial spending
Dynamic estimation
Dynamic results
00
-500
Model
50
Overview
Results: demand
estimation
CCL
Dedu
Other
applications
ICL
Machine
replacement
models
500
00
Fang and
Wang (2015)
25
Identification
1000
ctible
Identification
change in spending
Results
Summary
Single Agent
Dynamic
Models
Counterfactuals: change in
weekly spending from filling gap
Paul Schrimpf
Application:
prescription
drug demand
Introduction
Descriptive evidence
37.5
Model
Summary
Identification
Identification
Fang and
Wang (2015)
Machine
replacement
models
Other
applications
average spending
Results
35.0
32.5
30.0
Holmes (2011)
week
Dynamic estimation
Dynamic results
Dynamic demand
References
spending with standard benefit
spending without gap
50
40
30
20
0
Model
Results: demand
estimation
10
Overview
Single Agent
Dynamic
Models
Some subtle implications of
non-linear contracts
Paul Schrimpf
Application:
prescription
drug demand
Introduction
Descriptive evidence
Model
Results
Summary
Identification
Identification
Fang and
Wang (2015)
Machine
replacement
models
Other
applications
Holmes (2011)
Overview
Model
Results: demand
estimation
Dynamic estimation
Dynamic results
Dynamic demand
References
• Change in spending by people far from gap /
endogeneity of people at risk of bunching
• Arises due to dynamic considerations
• Estimate that about 25% of average $204/person
increase in annual spending comes from ”anticipatory”
response by people more than $200 below kink location
• “Filling” donut hole causes some people to decrease
spending
• CCL held constant with respect to out of pocket (vs.
total) spending, so for some people marginal price
actually rises
• General point: with non-linear contracts, a given
contract change can provide more coverage on margin
to some individuals but less coverage to others
Single Agent
Dynamic
Models
Paul Schrimpf
Application:
prescription
drug demand
Introduction
Descriptive evidence
Model
Results
Summary
Identification
Identification
Fang and
Wang (2015)
Machine
replacement
models
Other
applications
Holmes (2011)
Overview
Model
Results: demand
estimation
Dynamic estimation
Dynamic results
Dynamic demand
References
Policy-relevant counterfactuals
• Most people have more coverage than standard benefit
• So effect will be lower (people have some gap coverage
already)
• Analyze filling gap on existing contracts (ignore
potential firm responses in contract design or
beneficiary contract choice)
• Filling gap increases spending by about $148 (8.5%)
• But insurer (≈Medicare) spending rise by $253 (26%);
absent behavioral response, insurer spending would
increase by only $150
Single Agent
Dynamic
Models
Robustness
Paul Schrimpf
Application:
prescription
drug demand
Introduction
Descriptive evidence
Model
Results
Summary
Identification
Identification
Fang and
Wang (2015)
Machine
replacement
models
Other
applications
Holmes (2011)
Overview
Model
Results: demand
estimation
Dynamic estimation
Dynamic results
Dynamic demand
References
• Results appear quite stable across a (limited) set of
alternative specifications
• Explore sensitivity to:
• Alternative number of discrete types (heterogeneity)
• Set of covariates (e.g. gap coverage indicator as
”reduced form” way to capture potential plan selection)
• Adding risk aversion via recursive utility
• Letting ω vary more flexibly with θ
Single Agent
Dynamic
Models
Paul Schrimpf
Application:
prescription
drug demand
Introduction
Descriptive evidence
Model
Results
Summary
Identification
Identification
Fang and
Wang (2015)
Machine
replacement
models
Other
applications
Holmes (2011)
Overview
Model
Results: demand
estimation
Dynamic estimation
Dynamic results
Dynamic demand
References
Robustness
Single Agent
Dynamic
Models
Cross-year substitution
Paul Schrimpf
Application:
prescription
drug demand
Introduction
Descriptive evidence
Model
Results
Summary
• So far we treated each year of coverage in isolation
• Typical in the literature
Identification
Identification
Fang and
Wang (2015)
Machine
replacement
models
Other
applications
Holmes (2011)
Overview
Model
Results: demand
estimation
Dynamic estimation
Dynamic results
Dynamic demand
References
• But some of the effect could simply reflect substitution
to next calendar year (as in Cabral, 2013)
• Consequences (for health and spending) may be less
important if cross-year substitution is the main story
Single Agent
Dynamic
Models
Paul Schrimpf
Application:
prescription
drug demand
Introduction
Descriptive evidence
Model
Results
Summary
Identification
Identification
Fang and
Wang (2015)
Machine
replacement
models
Other
applications
Holmes (2011)
Overview
Model
Results: demand
estimation
Dynamic estimation
Dynamic results
Dynamic demand
References
Evidence of cross-year
substitution
Single Agent
Dynamic
Models
Paul Schrimpf
Application:
prescription
drug demand
Introduction
Descriptive evidence
Model
Results
Summary
Identification
Identification
Fang and
Wang (2015)
Machine
replacement
models
Other
applications
Holmes (2011)
Overview
Model
Results: demand
estimation
Dynamic estimation
Dynamic results
Dynamic demand
References
But it does not explain all ...
Single Agent
Dynamic
Models
Paul Schrimpf
Application:
prescription
drug demand
Introduction
Descriptive evidence
Model
Results
Summary
Identification
Identification
Fang and
Wang (2015)
Machine
replacement
models
Other
applications
Holmes (2011)
Overview
Model
Results: demand
estimation
Dynamic estimation
Dynamic results
Dynamic demand
References
Quantifying cross-year
substitution
• Back of the envelope: January effect is ≈ $40, so if
applied to everyone would be about 25% of estimated
average response
• Extend (or “tweak”!) the model by making terminal
values depend on unfilled prescriptions, and allowing
individual to fill in the beginning of next year
• Extension suggests overall effect may be reduced by
70%, from $153 to $44
Single Agent
Dynamic
Models
Paul Schrimpf
Application:
prescription
drug demand
Introduction
Descriptive evidence
Model
Results
Section 2
Summary
Identification
Identification
Fang and
Wang (2015)
Machine
replacement
models
Other
applications
Holmes (2011)
Overview
Model
Results: demand
estimation
Dynamic estimation
Dynamic results
Dynamic demand
References
Summary
Single Agent
Dynamic
Models
Summary
Paul Schrimpf
Application:
prescription
drug demand
Introduction
Descriptive evidence
Model
Results
Summary
Identification
Identification
Fang and
Wang (2015)
Machine
replacement
models
Other
applications
Holmes (2011)
Overview
Model
Results: demand
estimation
Dynamic estimation
Dynamic results
Dynamic demand
References
• Spending response to non-linear health insurance
contracts (vs ”an elasticity” with respect to ”the” price)
• Context: Medicare Part D (lots of current policy interest)
• Results:
• “Filling the gap” (ACA) will increase Part D spending by
about $150 per beneficiary (8.5%), and government
spending by $250 (26%)
• Importance of non-linearity of insurance contracts
• Much of spending increase comes from ”anticipatory”
behavior of individuals whose predicted spending is
below the gap (would not be captured in static model)
• A big part of the effect (but not all) could be explained
by cross-year substitution
Single Agent
Dynamic
Models
Normative analysis
Paul Schrimpf
Application:
prescription
drug demand
Introduction
Descriptive evidence
Model
Results
Summary
Identification
Identification
Fang and
Wang (2015)
Machine
replacement
models
Other
applications
Holmes (2011)
Overview
Model
Results: demand
estimation
Dynamic estimation
Dynamic results
Dynamic demand
References
• Focus of paper has been entirely positive
• Normative implications are more tricky
• Some of our findings regarding nature of response to
kink may be useful for informally beginning to assess
normative implications. e.g.,
• Larger response by healthier individuals
• Larger response of chronic (vs. acute) drugs
• (Evidence on spillover effects to non-drug spending and
health would also be useful)
• Conceptual question: optimality of drug consumption
in absence of insurance?
• Prices marked up above social MC (patents)
• Failures of rationality may produce under-consumption
w/o insurance?
Single Agent
Dynamic
Models
Paul Schrimpf
Application:
prescription
drug demand
Introduction
Descriptive evidence
Model
Results
Section 3
Summary
Identification
Identification
Fang and
Wang (2015)
Machine
replacement
models
Other
applications
Holmes (2011)
Overview
Model
Results: demand
estimation
Dynamic estimation
Dynamic results
Dynamic demand
References
Identification
Single Agent
Dynamic
Models
Paul Schrimpf
Application:
prescription
drug demand
General Setup
• Discrete time t, maximum τ ≤ ∞
• State sit ∈ S, follows a controlled Markov process
F(sit+1 |It ) = F(sit+1 |sit , ait )
Introduction
Descriptive evidence
Model
Results
Summary
Identification
Identification
Fang and
Wang (2015)
Machine
replacement
models
Other
applications
Holmes (2011)
• Action ait ∈ A
∑∞ j
• Preferences:
j=0 β U(ai,t+j , si,t+j )


τ
∑
ait ∈ arg max E 
β j U(ai,t+j , si,t+j )|ait = a, sit  .
a∈A
• Bellman equation
V(sit ) = max U(a, sit ) + βE[V(si,t+1 )|a, sit ]
a∈A
Overview
Model
Results: demand
estimation
Dynamic estimation
Dynamic results
Dynamic demand
References
j=0
• Policy function
α(s) = arg max U(a, sit ) + βE[V(si,t+1 )|a, sit ]
a∈A
Single Agent
Dynamic
Models
Example: Retirement
Paul Schrimpf
Application:
prescription
drug demand
Introduction
Descriptive evidence
Model
Results
Summary
Identification
Identification
Fang and
Wang (2015)
Machine
replacement
models
Other
applications
Holmes (2011)
Overview
Model
Results: demand
estimation
Example (Retirement)
1
Consider the choice of when to retire. Let ait = 1 if an
agent is working and ait = 0 if retired. Suppose τ is the age
at death. The payoff function could be
(
)
t
θ1
U(ait , xit , εit ) = E[cit |ait , xit ] exp θ2 + θ3 hit + θ4
−θ5 ait +ε(a
1+t
where cit is consumption, θ1 is the coefficient of relative risk
aversion, hit is health, and the expression in the exp
captures the idea that the marginal utility of consumption
could vary with health and age. −θ5 ait captures the
disutility of working.
Dynamic estimation
Dynamic results
Dynamic demand
References
1
From Aguirregabiria and Mira (2010).
Single Agent
Dynamic
Models
Example: Entry / Exit
Paul Schrimpf
Application:
prescription
drug demand
Introduction
Descriptive evidence
Model
Results
Summary
Identification
Identification
Fang and
Wang (2015)
Machine
replacement
models
Other
applications
Example (Entry/Exit)
2
A firm is deciding whether to operate in a market. Its
per-period profits are
(
)
U(ait ) = ait θR log(St ) − θN log (1 + nt ) − θF − θE (1 − ai,t−1 ) + εit
where ait is whether the firm operates at time t. St is the size
of the market, nt is the number of other firms operating. θF
is a fixed operating cost, and θE is an entry cost.
Holmes (2011)
Overview
Model
Results: demand
estimation
Dynamic estimation
Dynamic results
Dynamic demand
References
2
From Aguirregabiria and Mira (2010).
Single Agent
Dynamic
Models
Identification: Setup
Paul Schrimpf
Application:
prescription
drug demand
Introduction
Descriptive evidence
Model
Results
Summary
Identification
Identification
Fang and
Wang (2015)
Machine
replacement
models
Other
applications
Holmes (2011)
Overview
Model
Results: demand
estimation
Dynamic estimation
Dynamic results
Dynamic demand
References
• Panel data on N individuals for T periods
• Observe
• Actions ait
• Some state variables xit , sit = (xit , εit )
• i.e. observe joint distribution of xi· and ai·
• Goal: recover U, F(sit+1 |sit , ait ), β
Single Agent
Dynamic
Models
Paul Schrimpf
Application:
prescription
drug demand
Introduction
Non-identification without
more restrictions I
• Value function
Descriptive evidence
Model
Results
V(s) = max U(a, s) + βE[V(s′ |a, s)]
a∈A
Summary
Identification
• Change U(a, s) to
Identification
Fang and
Wang (2015)
Machine
replacement
models
Other
applications
Holmes (2011)
Overview
Model
Results: demand
estimation
Ũf (a, s) = U(a, s) + f(s) − βE[f(s′ )|a, s]
, new value function
Ṽ(s) = max U(a, s) + f(s) − βE[f(s′ )|a, s] + βE[Ṽ(s′ )|a, s]
a∈A
Ṽ(s) − f(s) = max U(a, s) + βE[Ṽ(s′ ) − f(s′ )|a, s]
a∈A
Dynamic estimation
Dynamic results
Dynamic demand
References
• So, V(s) = Ṽ(s′ ) − f(s′ )
Single Agent
Dynamic
Models
Non-identification without
more restrictions II
Paul Schrimpf
Application:
prescription
drug demand
Introduction
Descriptive evidence
Model
• Policy functions,
Results
Summary
Identification
Identification
Fang and
Wang (2015)
Machine
replacement
models
Other
applications
Holmes (2011)
Overview
Model
Results: demand
estimation
Dynamic estimation
Dynamic results
Dynamic demand
References
α(s) = arg max U(a, s) + βE[V(s′ |a, s)]
a∈A
α̃(s) = arg max U(a, s) + βE[Ṽ(s′ ) − f(s′ )|a, s]
a∈A
so α(s) = α̃(s)
• U leads to same policy as Ũf ; they are observationally
equivalent
Single Agent
Dynamic
Models
Identification: discrete A
Paul Schrimpf
Application:
prescription
drug demand
Introduction
Descriptive evidence
Model
Results
Summary
Identification
Identification
Fang and
Wang (2015)
Machine
replacement
models
Other
applications
Holmes (2011)
Overview
Model
Results: demand
estimation
Dynamic estimation
Dynamic results
Dynamic demand
References
• Assume:
• A is discrete and finite
• U(a, x, ε) = u(a, x) + ε(a),
• ε has known CDF G, ε ⊥
⊥ x and εit ⊥
⊥ εis for t ̸= s
• Partial identification if G unknown, see Norets and Tang
(2013)
• β is known
• u(0, x) = 0
• Then u(a, x) is identified
• Proof: see my notes from 628 and references therein
Single Agent
Dynamic
Models
Paul Schrimpf
Application:
prescription
drug demand
Introduction
Descriptive evidence
Model
Results
Section 4
Summary
Identification
Identification
Fang and
Wang (2015)
Machine
replacement
models
Other
applications
Holmes (2011)
Overview
Model
Results: demand
estimation
Dynamic estimation
Dynamic results
Dynamic demand
References
Identification
Single Agent
Dynamic
Models
Paul Schrimpf
Application:
prescription
drug demand
Introduction
Descriptive evidence
Model
Results
Summary
Identification
Identification
Fang and
Wang (2015)
Machine
replacement
models
Other
applications
Holmes (2011)
Overview
Model
Results: demand
estimation
Dynamic estimation
Dynamic results
Dynamic demand
References
Identification – discrete controls
• Given {xit , ait } want to recover U
• Compare with discrete control identification from
Magnac and Thesmar (2002) or Bajari, Chernozhukov,
Hong, and Nekipelov (2009)
• Assume:
1 Transition distribution is identified
2 Payoff additively separable in ε, U(x, a, ε) = u(x, a) + ε(i)
3 Distribution of ε known and εit independent across f and
t
4 u(x, a0 ) is known for all x ∈ X and some a0 ∈ A
5 Discount factor, δ, is known
• Proof:
• Additive separability and knowing distribution of ε
allows Hotz-Miller inversion to recover differences of
choice specific value functions
• Given u(x, a0 ) and differences in choice specific value
functions, can recover choice specific value functions
from Bellman equation
• Given choice specific value functions, can get u(x, a)
from Bellman equation
Single Agent
Dynamic
Models
Identification – continuous
controls
Paul Schrimpf
Application:
prescription
drug demand
Introduction
Descriptive evidence
Model
Results
Summary
• Key assumptions:
1 Transition density, fxit+1 |xit ,ait , is identified
2 Distribution of ε, Fε , is normalized
• Not a restriction because ε enters U(x, a, ε) without
restriction
Identification
Identification
Fang and
Wang (2015)
Machine
replacement
models
3
4
• ∂x∂U(k) (x, α(x, ε), ε) is known
• There exists
χk (a0 , x(−k) , ε) such
(
) that
a0 = α χk (a0 , x(−k) , ε), x(−k) , ε
Other
applications
Holmes (2011)
Overview
Model
Results: demand
estimation
Dynamic estimation
Dynamic results
Dynamic demand
References
εit is independent across f and t.
Discount factor, δ, is known
For some k,
5
Initial condition: for some a0 , U(x, a0 , ε) is known
Single Agent
Dynamic
Models
Paul Schrimpf
Application:
prescription
drug demand
Introduction
Descriptive evidence
Model
Results
Summary
Identification
Identification
Fang and
Wang (2015)
Machine
replacement
models
Other
applications
Identification – continuous
controls
• Key assumptions (continued):
6 Completeness: let ∂U
∂a ∈ G, define
[∞
]
∑
∂
τ
D (g)(x, ε) =
E
δ g(xt+τ , εt+τ )|xt = x, at = α(x, ε)
∂at
τ=0
∫ x(k)
∂α
L(g)(x, ε) =
g(x̃(k) , x(−k) , ε) (k) (x̃(k) , x(−k) , ε)dx̃(k)
∂x
(−k)
χk (a0 ,x
,ε)
K(g)(x, ε) =D (L(g))(x, ε)
The only solution in G to
0 = g(x, ε) + K(g)(x, ε)
Holmes (2011)
Overview
Model
Results: demand
estimation
Dynamic estimation
Dynamic results
Dynamic demand
References
is g(x, ε) = 0
• Result: U identified
Single Agent
Dynamic
Models
Proof sketch
Paul Schrimpf
Application:
prescription
drug demand
Introduction
Descriptive evidence
• Policy function: Fε (ε) = Fa|x (α(x, ε)|x)
• First order condition for at :
Model
Results
Summary
Identification
Identification
Fang and
Wang (2015)
Machine
replacement
models
Other
applications
Holmes (2011)
Overview
Model
Results: demand
estimation
Dynamic estimation
Dynamic results
Dynamic demand
References
0=
∂U
(xt , α(xt , εt ), εt )+
∂a
∞
∂ ∑ τ
+
δ E [U (xt+τ , α(xt+τ , εt+τ ), εt+τ ) |xt , α(xt , εt )]
∂a τ=1
• Write payoff function in terms of its derivatives:
∫
x(k)
U(x, α(x, ε), ε) =
χk (a0 ,x(−k) ,ε)
( ∂U
∂a
(x, α(x, ε), ε)
)
∂U
(k)
(x,
α(x,
ε),
ε)dx̃
∂x(k)
+ U(χk (a0 , x−k , ε), x−k , a0 , ε)
+
∂α
(x, ε)+
∂x(k)
Single Agent
Dynamic
Models
Proof sketch
Paul Schrimpf
Application:
prescription
drug demand
Introduction
Descriptive evidence
• Let
Model
Results
Summary
Identification
Identification
Fang and
Wang (2015)
Machine
replacement
models
Other
applications
Holmes (2011)
Overview
Model
Results: demand
estimation
Dynamic estimation
Dynamic results
Dynamic demand
References
ϕ(x, ε) =U(χk (a0 , x−k , ε), x−k , a0 , ε)+
∫ x(k)
∂U
+
(x, α(x, ε), ε)dx̃(k)
(k)
∂x
(−k)
χk (a0 ,x
,ε)
• Substitute into first order condition:
0 =(1 + K)(
∂U
) + D (ϕ)
∂a
• Integrate to recover U(x, α(x, ε), ε)
Single Agent
Dynamic
Models
Paul Schrimpf
Application:
prescription
drug demand
Introduction
Descriptive evidence
Model
Results
Section 5
Summary
Identification
Identification
Fang and
Wang (2015)
Machine
replacement
models
Other
applications
Holmes (2011)
Overview
Model
Results: demand
estimation
Dynamic estimation
Dynamic results
Dynamic demand
References
Fang and Wang (2015)
Single Agent
Dynamic
Models
Fang and Wang (2015)
Paul Schrimpf
Application:
prescription
drug demand
Introduction
Descriptive evidence
Model
Results
Summary
“Estimating dynamic discrete choice models with
hyperbolic discounting, with an application to
mammography decisions”
Identification
Identification
Machine
replacement
models
• Last week’s presentations — choice of discount factor?
• This paper: identification of discount factor (and more)
• Key restriction: variables that shift expectations, but do
not enter current payoff
Other
applications
• Find substantial present bias and naivety in
Fang and
Wang (2015)
Holmes (2011)
Overview
Model
Results: demand
estimation
Dynamic estimation
Dynamic results
Dynamic demand
References
mammography screening
Single Agent
Dynamic
Models
Model
Paul Schrimpf
Application:
prescription
drug demand
Introduction
Descriptive evidence
• Dynamic discrete choice, i ∈ {0, ..., I}
ui (x, ε) = ui (x) + εi
Model
Results
Summary
• Infinite time horizon, hyperbolic discounting
Identification
Identification
Ut (ut , ut+1 , ...) = ut + β
Fang and
Wang (2015)
Machine
replacement
models
Other
applications
Holmes (2011)
Overview
Model
Results: demand
estimation
Dynamic estimation
Dynamic results
Dynamic demand
References
∞
∑
δ k−1 uk
k=t+1
• Present bias ≡ β
• Discount factor ≡ δ
• Time t self believes that future selves will make choices
with present bias β̃ ∈ [β, 1] and discounting δ
• Completely naive if β̃ = 1
• Sophisticated if β̃ = β
Single Agent
Dynamic
Models
Model
Paul Schrimpf
Application:
prescription
drug demand
Introduction
Descriptive evidence
• Continuation strategy profile σt+ = {σk }k=t∞ ,
• Continuation utility:
Model
Results
Summary
[
+
Vt (xt , εt ; σt+ ) = uσt (xt ,εt ) (xt , εσt (xt ,εt )t )+δE Vt+1 (xt+1 , εt+1 ; σt+1
)|
Identification
Identification
Fang and
Wang (2015)
Machine
replacement
models
Other
applications
Holmes (2011)
Overview
Model
Results: demand
estimation
Dynamic estimation
Dynamic results
Dynamic demand
References
• Perceived continuation strategy
[
]
+
σ̃t (xt , εt ) = arg max ui (xt , εit )+β̃δE Vt+1 (xt+1 , εt+1 ; σ̃t+1
)|xt , i
i
• Perception-perfect strategy with partial naivety
[
]
+
σt∗ (xt , εt ) = arg max ui (xt , εit )+βδE Vt+1 (xt+1 , εt+1 ; σ̃t+1
)|xt , i
i
Single Agent
Dynamic
Models
Identification I
Paul Schrimpf
Application:
prescription
drug demand
Introduction
Descriptive evidence
Model
Results
Summary
Identification
Identification
Fang and
Wang (2015)
Machine
replacement
models
Other
applications
• Assumptions:
• Stationarity
• Conditional independance:
π(xt+1 , εt+1 |xt , εt , dt ) = q(εt )π(xt+1 |xt , dt )
• εt iid extreme value
• Perceived long-run choice-specific value function:
Vi (x) = ui (x) + δ
∑
x′
• Current choice-specific value function
Holmes (2011)
Overview
Model
Results: demand
estimation
Dynamic estimation
Dynamic results
Dynamic demand
References
V(x′ )π(x′ |x, i)
Wi (x) = ui (x) + βδ
∑
x′
V(x′ )π(x′ |x, i)
Single Agent
Dynamic
Models
Identification II
Paul Schrimpf
Application:
prescription
drug demand
Introduction
• Naively perceived next-period choice-specific value
function
Descriptive evidence
Model
Results
Zi (x) = ui (x) + β̃δ
∑
V(x′ )π(x′ |x, i)
x′
Summary
Identification
Identification
Fang and
Wang (2015)
Machine
replacement
models
Other
applications
• Last three bullets, V(x) = expected value from following
perceived continuation strategy
V(x) = E[Vσ̃ (x,ε) (x) + εsigma(x,ε)
]
˜
• Definition of W, Z
Holmes (2011)
Overview
Model
Results: demand
estimation
Dynamic estimation
Dynamic results
Dynamic demand
References
Zi (x) − ui (x) =
β̃
[Wi (x) − ui (x)]
β
Single Agent
Dynamic
Models
Identification III
Paul Schrimpf
Application:
prescription
drug demand
• Distribution of ε implies
eWi (x)
Pi (x) = ∑ W (x)
j
je
Introduction
Descriptive evidence
Model
Results
Summary
Identification
and
Identification
Fang and
Wang (2015)
Machine
replacement
models
Other
applications
Holmes (2011)
Overview
Model
Results: demand
estimation
Dynamic estimation
Dynamic results
Dynamic demand
References
V(x) = log
(
∑
i
)
eZi (x) +(1−β̃)δ
∑
j
eZj (x) ∑
∑ Z (x)
V(x′ )π(x′ |x, j
k
e
k
′
x
• Given δ, β, β̃, variant of usual inversion of choice
probabilities identifies ui (x) − u0 (x)
• Assumption: ∃x1 ̸= x2 such that (i) ui (x1 ) = ui (x2 )∀i and
π(x′ |x1 , i) ̸= π(x′ |x2 , i) for some i and (I + 1) × |X| ≥ 4
Single Agent
Dynamic
Models
Identification of δ, β, β̃
Paul Schrimpf
• Collect equations relating u, β, β̃, δ to observed pi, P
Application:
prescription
drug demand
G(
Introduction
u, β, β̃, δ
| {z }
;
π, P
|{z}
x, dim n=(I+1)|X|+3 s=|X|(|X|−1)(I+1)+(I+1)|X|
Descriptive evidence
0
|{z}
)=
b, dim m=(I+1)|X|+|Xe ||Xr |(I+1)
Model
Results
Summary
• Transversality theorem: let G : Rn × Rs →Rm by
Identification
Identification
Fang and
Wang (2015)
Machine
replacement
models
•
•
Other
applications
Holmes (2011)
Overview
•
Model
Results: demand
estimation
•
Dynamic estimation
Dynamic results
Dynamic demand
References
•
max{n − m, 0} times continuously differentiable.
Suppose 0 is a regular value of G, i.e. G(x, b) = 0 implies
rank DGx,b = m, then generically in b, G(·, b) : Rn →Rm
has 0 as a regular value, i.e. rank Dx Gx,b = m
G satisfies these conditions, but n < m, so 0 is never a
regular value, so generically in b, G(x, b) ̸= 0
Paper says, so generically G(x, b) = 0 has no solution,
except at true x∗ , but ...
Theorem says except at true b∗ , given true b∗ , theorem
says nothing about how many x satisfy equation ...
Introduce obviously non-identified reparameterization,
e.g. β = β1 + β2 , where does argument breakdown?
Theorem does imply that model is falsifiable
Single Agent
Dynamic
Models
Identification of δ, β, β̃
Paul Schrimpf
Application:
prescription
drug demand
Introduction
Descriptive evidence
Model
Results
Summary
Identification
Identification
Fang and
Wang (2015)
Machine
replacement
models
Other
applications
Holmes (2011)
Overview
Model
Results: demand
estimation
Dynamic estimation
Dynamic results
Dynamic demand
References
• I believe discount factors are identified here, but proof
appears incorrect
• Possible approaches to prove: either construct them
from observables, or locally show Dx G has rank n
• Related identification results:
• Magnac and Thesmar (2002)
• Bajari et al. (2013)
• An, Hu, and Ni (2014)
Single Agent
Dynamic
Models
Estimation
Paul Schrimpf
Application:
prescription
drug demand
Introduction
Descriptive evidence
Model
Results
Summary
Identification
Identification
Fang and
Wang (2015)
Machine
replacement
models
Other
applications
Holmes (2011)
Overview
Model
Results: demand
estimation
Dynamic estimation
Dynamic results
Dynamic demand
References
• Maximum pseudo-likelihood
Estimate ûi (x; β, β̃, δ) using choice probabilities and
Bellman equations
2 Maximize pseudo likelihood P̂i (x; ûi (x; β, β̃, δ), β, β̃, δ)
1
3
Single Agent
Dynamic
Models
Paul Schrimpf
Empirical application:
mammography
Application:
prescription
drug demand
Introduction
Descriptive evidence
Model
Results
Summary
Identification
• Data from HRS, women 51-64
• Instantaneous utility from getting mammogram:
Identification
Fang and
Wang (2015)
Machine
replacement
models
Other
applications
Holmes (2011)
Overview
Model
Results: demand
estimation
Dynamic estimation
Dynamic results
Dynamic demand
References
u1 (x) − u0 (x) = α0 + α1 BadHealth + αLogIncome
• Demographics excluded from payoffs, but enter
transition probabilities
Single Agent
Dynamic
Models
Paul Schrimpf
Application:
prescription
drug demand
Introduction
Descriptive evidence
Model
Results
Section 6
Summary
Identification
Identification
Fang and
Wang (2015)
Machine
replacement
models
Other
applications
Holmes (2011)
Overview
Model
Results: demand
estimation
Dynamic estimation
Dynamic results
Dynamic demand
References
Machine replacement models
Single Agent
Dynamic
Models
Machine replacement models
Paul Schrimpf
Application:
prescription
drug demand
Introduction
Descriptive evidence
Model
Results
Summary
Identification
Identification
Fang and
Wang (2015)
Machine
replacement
models
Other
applications
Holmes (2011)
Overview
Model
Results: demand
estimation
Dynamic estimation
Dynamic results
Dynamic demand
References
• Firm operates many machines independently, machines
fail with some probability that increases with age, firm
chooses when to replace machines to minimize costs of
failure and replacement
• Classic Rust (1987) about bus-engine replacement
• Many follow-ups and extensions
• Das (1992): cement kilns
• Kennet (1994): aircraft engines
• Rust and Rothwell (1995): nuclear power plants
• Adda and Cooper (2000): cars
• Kasahara (2009): response of investment to tariffs
Single Agent
Dynamic
Models
Model
Paul Schrimpf
Application:
prescription
drug demand
Introduction
Descriptive evidence
Model
Results
Summary
Identification
Identification
Fang and
Wang (2015)
Machine
replacement
models
Other
applications
Holmes (2011)
Overview
Model
Results: demand
estimation
Dynamic estimation
Dynamic results
Dynamic demand
References
• Choose: ait = 1 (replace) or 0 (don’t replace)
• Machine age: xit+1 = (1 − ait )xit + ξi,t+1
• Profits: Y(x) − aRC(x) + ε(a)
• Firm’s problem:


∞
∑
max Et 
β j (Y ((1 − ait )xit ) − ait RC(xit ) + εit (ait ))
a
j=0
s.t. xi,t+1 = (1 − ait )xit + ξi,t+1
• ε and ξ i.i.d.
• Often ξ non-stochastic, e.g. x = age, ξ = 1.
Single Agent
Dynamic
Models
Value functions
Paul Schrimpf
Application:
prescription
drug demand
• Value function
Introduction
Descriptive evidence
Model
Results
Summary
Identification
Identification
Fang and
Wang (2015)
Machine
replacement
models
Other
applications
Holmes (2011)
Overview
Model
Results: demand
estimation
Dynamic estimation
Dynamic results
Dynamic demand
References
V(x, ε) = max Y((1−a)x)−aRC(x)+ε(a)+βE[V(x′ , ε ′ )|x(1−a)]
a
• Expected (or integrated) value function
[
]
V̄(x) = E V(x′ , ε ′ )|x
• Choice specific value function
[
]
′
′
′
v(x, a) =Y((1 − a)x) − aRC(x) + βE max
v(x , a ) + ε(a )|x, a
′
a
=Y((1 − a)x) − aRC(x) + β V̄(x(1 − a))
Single Agent
Dynamic
Models
Paul Schrimpf
Application:
prescription
drug demand
Introduction
Descriptive evidence
Model
Identification I
• Observe: P(a|x)
P(a = 1|x) =P (v(x, 1) + ε(1) ≥ v(x, 0) + ε(0)|x)
=P (ε(0) − ε(1) ≤ v(x, 1) − v(x, 0))
(
(
))
=P ε(0) − ε(1) ≤ Y(0) − Y(x) − RC(x) + β V̄(0) − V̄(x)
Results
Summary
Identification
Identification
Fang and
Wang (2015)
Machine
replacement
models
Other
applications
Holmes (2011)
Overview
Model
Results: demand
estimation
• Choice probabilities identify
v(x, 1) − v(x, 0) = log P(1|x) − log P(0|x)
• Choice probabilities not enough to separately identify
RC(x) and Y(x), only identify the sum RC(x) + Y(x)
• Normalize Y(x), solve for v(x, 0) from
[
]
v(x, 0) =Y(x) + βE max v(x′ , a) − v(x′ , 0) + ε(a)|x + βE[v(x′ , 0)|x]
a
])
(
[
v(x, 0) =(I − βE )−1 Y(·) + βE max v(x′ , a) − v(x′ , 0) + ε(a)|· (x)
a
(
[
(
) ])
∑ ′
−1
v(x ,a)−v(x′ ,0)
=(I − βE )
Y(·) + βE log
e
|·
(x)
Dynamic estimation
a
Dynamic results
Dynamic demand
References
where E (f)(x) =
E[f(x′ )|x]
Single Agent
Dynamic
Models
Identification II
Paul Schrimpf
Application:
prescription
drug demand
• Then
v(x, 1) = v(x, 0) + [v(x, 1) − v(x, 0)]
Introduction
Descriptive evidence
Model
and
Results
Summary
Identification
Identification
Fang and
Wang (2015)
Machine
replacement
models
[
Overview
Model
Results: demand
estimation
Dynamic estimation
Dynamic results
Dynamic demand
References
′
′
V̄(x) =E max
v(x , a ) + ε(a )|x
a′
[
]
′
′ ′
′
′
=E v(x , 0) + max
v(x , a ) − v(x , 0) + ε(a )|x
a′
[
(
) ]
∑
′
′
′
=E v(x′ , 0) + log
ev(x ,a )−v(x ,0) |x
a
Other
applications
Holmes (2011)
]
′
and
RC(x) = −v(x, 1) + β V̄(0)
Single Agent
Dynamic
Models
Paul Schrimpf
(Non)-identification of discount
factor I
Application:
prescription
drug demand
Introduction
Descriptive evidence
Model
Results
Summary
Identification
Identification
Fang and
Wang (2015)
Machine
replacement
models
Other
applications
Holmes (2011)
Overview
Model
Results: demand
estimation
Dynamic estimation
Dynamic results
Dynamic demand
References
• Given β and Y(x), above steps identify RC(x), change β
and get a new observationally equivalent RC(x)
• Does it matter? Consider counterfactuals that change Y
or distribution of ξ. Check whether
∂P ∂v
∂Y , ∂Y
depend on β.
• Identify β by having some components of x affect
E[·|x, a], but not Y or RC
• Previous slide gives RC as a function of β, identify β
from restriction on RC
Single Agent
Dynamic
Models
Paul Schrimpf
Application:
prescription
drug demand
Introduction
Descriptive evidence
Model
Results
Section 7
Summary
Identification
Identification
Fang and
Wang (2015)
Machine
replacement
models
Other
applications
Holmes (2011)
Overview
Model
Results: demand
estimation
Dynamic estimation
Dynamic results
Dynamic demand
References
Other applications
Single Agent
Dynamic
Models
Holmes (2011): “The diffusion of
Wal-Mart and economies of
density”
Paul Schrimpf
Application:
prescription
drug demand
Introduction
Descriptive evidence
Model
Results
Summary
Identification
Identification
Fang and
Wang (2015)
Machine
replacement
models
Other
applications
Holmes (2011)
Overview
Model
Results: demand
estimation
Dynamic estimation
Dynamic results
Dynamic demand
References
Movie
Single Agent
Dynamic
Models
Holmes (2011) overview I
Paul Schrimpf
Application:
prescription
drug demand
Introduction
Descriptive evidence
Model
Results
Summary
Identification
Identification
Fang and
Wang (2015)
Machine
replacement
models
Other
applications
Holmes (2011)
Overview
Model
Results: demand
estimation
Dynamic estimation
Dynamic results
Dynamic demand
References
• Observation: Wal-Mart opens its new stores close to
existing ones
• Benefit from high store density: distribution
• Shipping costs
• Rapid response to demand shocks
• Question: how large are the benefits of density for
Wal-Mart?
• Challenge: Wal-Mart logistics data is confidential, even
if detailed cost data available some benefits of density
might not be reflected by it
• Solution: use revealed preference
• Walmart’s choices reveal tradeoff between benefit and
cost of density
• Cost of high store density: cannibalization
Single Agent
Dynamic
Models
Holmes (2011) overview II
Paul Schrimpf
Application:
prescription
drug demand
Introduction
Descriptive evidence
Model
Results
Summary
Identification
Identification
Fang and
Wang (2015)
Machine
replacement
models
Other
applications
Holmes (2011)
Overview
Model
Results: demand
estimation
Dynamic estimation
Dynamic results
Dynamic demand
References
• Two Wal-Marts close together will take sales away from
one another
• Can be inferred from demand estimates
• Sequence of store openings clearly important, so need a
dynamic model
• Wal-Mart’s dynamic decisions:
How many new Wal-Marts and how many new
supercenters should be opened?
2 Where should the new Wal-Marts and supercenters be
put?
3 How many new distribution centers should be opened?
4 Where should the new distribution centers be put?
1
Focus on 2 and take 1, 3, and 4 as given
Single Agent
Dynamic
Models
Model: dynamic choice of
locations I
Paul Schrimpf
Application:
prescription
drug demand
Introduction
Descriptive evidence
Model
Results
Summary
Identification
Identification
• Complete information
• Take as given choice of number of stores, NWal
t , and
Super
supercenters, Nt
• Choose new store locations to maximize discounted
sum of profits
Fang and
Wang (2015)
Machine
replacement
models
Other
applications
Holmes (2011)
Overview
Model
Results: demand
estimation
Dynamic estimation
Dynamic results
Dynamic demand
References
max
a
[∑
∞
∑
(ρt β)
t=1
g
j∈BtWal (πjt
t−1
+
∑
Super
j∈Bt
g
g
− cjt − τdjt )+
f
f
]
f
(πjt − cjt − τdjt )
• g, f superscripts for goods and food
• π is variable profits
πjte = µ Rejt −Wagejt Laborejt − Rentjt Landejt − Otherejt
|{z}
revenue
Single Agent
Dynamic
Models
Paul Schrimpf
Application:
prescription
drug demand
Introduction
Descriptive evidence
Model
Results
Summary
Identification
Model: dynamic choice of
locations II
• Rejt = revenue comes from demand estimates; demand
at store j depends on whether there is a store at nearby
location k through a distance term in consumers’ utility
of shopping at a store
• cjt is a fixed cost
cjt = ω0 + ω1 log(Popdenjt ) + ω2 log(Popdenjt )2
Identification
Fang and
Wang (2015)
• djt is distance to the nearest distribution center, τdjt is a
(fixed) distribution cost
Machine
replacement
models
• BtWal is set of all Wal-Marts open at time t
Super
• Bt
⊂ BtWal is set of Wal-Mart Supercenters
Other
applications
• a = (A1Wal , A1
, ...) is sequence of sets of new stores
• Stores never close
Holmes (2011)
Overview
Model
Results: demand
estimation
Dynamic estimation
Dynamic results
Dynamic demand
References
Super
Wal
BtWal =Bt−1
+ AtWal
Super
Bt
Super
=Bt−1
Super
+ At
Single Agent
Dynamic
Models
Paul Schrimpf
Application:
prescription
drug demand
Introduction
Descriptive evidence
Model
Results
Model: demand
• Consumers make discrete choice among Wal-Marts
within 25 miles and an outside option
ui0 =α0 + α1 log(Popdenl(i) ) + α2 log(Popdenl(i) )2 + εi0
uij =(ξ0 + ξ1 log(Popdenl(i) ))Distancel(i)j + StoreCharj γ + εij
Summary
Identification
Identification
Fang and
Wang (2015)
Machine
replacement
models
Other
applications
Holmes (2011)
Overview
Model
Results: demand
estimation
Dynamic estimation
Dynamic results
Dynamic demand
References
• l(i) = location of consumer i
• εi ∼ logit
• Revenue:
g
Rj =
∑
l
λg
|{z}
spending per i
×
g
pjl
|{z}
P(l shops at j)
nl
|{z}
×
number of consumers
• Revenue data is store-level sales estimate from Trade
Dimensions, so must have measurment error
log(RData
) = log(RTrue
) + ηjSales
j
j
where ηjSales ∼ N
Single Agent
Dynamic
Models
Estimation strategy
Paul Schrimpf
Application:
prescription
drug demand
Introduction
Descriptive evidence
Model
Results
1
Estimate revenue using demand model and data on
store sales
2
Construct variable costs based on local wages and
property values
3
Estimate fixed costs (ω) and densities of scale (τ) using
moment inequalities derived from profit maximization
Summary
Identification
Identification
Fang and
Wang (2015)
Machine
replacement
models
Other
applications
Holmes (2011)
Overview
Model
Results: demand
estimation
Dynamic estimation
Dynamic results
Dynamic demand
References
Single Agent
Dynamic
Models
Data
Paul Schrimpf
Application:
prescription
drug demand
Introduction
Descriptive evidence
Model
Results
Summary
• Store level sales and employment in 2005
Identification
• Store openings and locations
Identification
• Demographic data from census
Fang and
Wang (2015)
• Local wages and land rents
Machine
replacement
models
Other
applications
Holmes (2011)
Overview
Model
Results: demand
estimation
Dynamic estimation
Dynamic results
Dynamic demand
References
• Information from Wal-Mart’s annual reports
Single Agent
Dynamic
Models
Paul Schrimpf
Application:
prescription
drug demand
Introduction
Descriptive evidence
Model
Results
Summary
Identification
Identification
Fang and
Wang (2015)
Machine
replacement
models
Other
applications
Holmes (2011)
Overview
Model
Results: demand
estimation
Dynamic estimation
Dynamic results
Dynamic demand
References
Single Agent
Dynamic
Models
Paul Schrimpf
Application:
prescription
drug demand
Introduction
Descriptive evidence
Model
Results
Summary
Identification
Identification
Fang and
Wang (2015)
Machine
replacement
models
Other
applications
Holmes (2011)
Overview
Model
Results: demand
estimation
Dynamic estimation
Dynamic results
Dynamic demand
References
Single Agent
Dynamic
Models
Paul Schrimpf
Application:
prescription
drug demand
Introduction
Descriptive evidence
Model
Results
Summary
Identification
Identification
Fang and
Wang (2015)
Machine
replacement
models
Other
applications
Holmes (2011)
Overview
Model
Results: demand
estimation
Dynamic estimation
Dynamic results
Dynamic demand
References
Single Agent
Dynamic
Models
Variable costs
Paul Schrimpf
Application:
prescription
drug demand
Introduction
Descriptive evidence
Model
Results
Summary
Identification
Identification
Fang and
Wang (2015)
Machine
replacement
models
Other
applications
Holmes (2011)
Overview
Model
Results: demand
estimation
Dynamic estimation
Dynamic results
Dynamic demand
References
• Labor costs = average employees per million dollars of
sales (3.61) (measure in 2005) × average retail wage in
county in year
• Land value to sales ratio constructed from property
values based on census data (for each year) and
property tax data for Wal-Marts in Minnesota and Iowa
• Scale demand estimates from 2005 by average Wal-Mart
revenue in each year
Single Agent
Dynamic
Models
Dynamic estimation I
Paul Schrimpf
Application:
prescription
drug demand
Introduction
Descriptive evidence
Model
Results
Summary
• Given demand estimates and variable costs only
unknown parameters are fixed costs, ω, and economies
of density, τ
• Total profits from action a = variable profits plus fixed
costs plus economics of density
Identification
Identification
Fang and
Wang (2015)
Machine
replacement
models
ΠT (a) =
∞
∑
(ρt β)t−1
t=1
[∑
g
Wal
+
j∈Bt
∑
Super
j∈Bt
Overview
Model
Results: demand
estimation
Dynamic estimation
Dynamic results
Dynamic demand
References
f
(πjt − (ω0 + ω1 log(Popdenjt ) + ω2 log(Pop
=Π(a) + ω0 + ω1 C1,a + ω2 C2,a + τDa
• Profit maximization implies that
Other
applications
Holmes (2011)
(πjt − (ω0 + ω1 log(Popdenjt ) + ω2 log(Popde
ΠT (a) ≤ΠT (ao )
ω1 (C1,a − C1,ao ) + ω2 (C2,a − C2,ao ) + τ(Da − Dao ) ≤ Π(ao ) − Π(a)
{z
}
|
{z
} |
≡x′a θ
≡ya
where ao is observed choice, a is any other choice
Single Agent
Dynamic
Models
Dynamic estimation II
Paul Schrimpf
Application:
prescription
drug demand
• Estimation of demand and variable costs ⇒ observe ya
with error
Introduction
Descriptive evidence
Model
Results
• Assume measurement has zero mean given xa , then
conditional moment inequalities,
Summary
Identification
E[ya − x′a θ|xa ] ≥ 0
Identification
Fang and
Wang (2015)
Machine
replacement
models
Other
applications
Holmes (2011)
Overview
Model
Results: demand
estimation
Dynamic estimation
Dynamic results
Dynamic demand
References
can be used to form objective function
• Must choose deviations a and unconditional moment
inequalities for estimation
• Uses pairwise resequencing deviations (i.e. change
order a pair of stores opens)
• Group deviations according to their affect on density to
aggregate conditional moment inequalities
Single Agent
Dynamic
Models
Paul Schrimpf
Application:
prescription
drug demand
Introduction
Descriptive evidence
Model
Results
Summary
Identification
Identification
Fang and
Wang (2015)
Machine
replacement
models
Other
applications
Holmes (2011)
Overview
Model
Results: demand
estimation
Dynamic estimation
Dynamic results
Dynamic demand
References
Single Agent
Dynamic
Models
Paul Schrimpf
Application:
prescription
drug demand
Introduction
Descriptive evidence
Model
Results
Summary
Identification
Identification
Fang and
Wang (2015)
Machine
replacement
models
Other
applications
Holmes (2011)
Overview
Model
Results: demand
estimation
Dynamic estimation
Dynamic results
Dynamic demand
References
Single Agent
Dynamic
Models
Paul Schrimpf
Application:
prescription
drug demand
Introduction
Descriptive evidence
Model
Results
Summary
Identification
Identification
Fang and
Wang (2015)
Machine
replacement
models
Other
applications
Holmes (2011)
Overview
Model
Results: demand
estimation
Dynamic estimation
Dynamic results
Dynamic demand
References
• τ ≈ $3.50 = cost savings per year in thousands of
dollars when a store is 1 mile closer to its distribution
center
• Shipping costs ≈ $0.85
• Results robust to splitting sample, changing revenues,
labor, or rent, including/excluding supercenters
Single Agent
Dynamic
Models
Dynamic demand
Paul Schrimpf
Application:
prescription
drug demand
Introduction
Descriptive evidence
Model
Results
Summary
Identification
• Storable goods
• Hendel and Nevo (2006)
• Erdem, Imai, and Keane (2003)
Fang and
Wang (2015)
• Durable goods
• Gowrisankaran and Rysman (2009)
Machine
replacement
models
• Health care
• Gilleskie (1998)
Identification
Other
applications
Holmes (2011)
Overview
Model
Results: demand
estimation
Dynamic estimation
Dynamic results
Dynamic demand
References
Single Agent
Dynamic
Models
Paul Schrimpf
Application:
prescription
drug demand
Introduction
Descriptive evidence
Model
Results
Summary
Identification
Identification
Fang and
Wang (2015)
Machine
replacement
models
Other
applications
Holmes (2011)
Overview
Model
Results: demand
estimation
Dynamic estimation
Dynamic results
Dynamic demand
References
Adda, Jérôme and Russell Cooper. 2000. “Balladurette and
Juppette: A Discrete Analysis of Scrapping Subsidies.”
Journal of Political Economy 108 (4):778–806. URL
http://www.jstor.org/stable/10.1086/316096.
Aguirregabiria, Victor. 2017. “Empirical Industrial
Organization: Models, Methods, and Applications.” URL
http://www.individual.utoronto.ca/vaguirre/
courses/eco2901/teaching_io_toronto.html.
Aguirregabiria, Victor and Pedro Mira. 2010. “Dynamic
discrete choice structural models: A survey.” Journal of
Econometrics 156 (1):38 – 67. URL
http://www.sciencedirect.com/science/article/
pii/S0304407609001985.
An, Yonghong, Yingyao Hu, and Jian Ni. 2014. “Dynamic
models with unobserved state variables and
heterogeneity: time inconsistency in drug compliance.”
URL https://research.chicagobooth.edu/~/media/
C321ABC7FAC5455886671CF64B84B82C.pdf.
Single Agent
Dynamic
Models
Paul Schrimpf
Application:
prescription
drug demand
Introduction
Descriptive evidence
Bajari, Patrick, Chenghuan Sean Chu, Denis Nekipelov, and
Minjung Park. 2013. “A Dynamic Model of Subprime
Mortgage Default: Estimation and Policy Implications.”
Working Paper 18850, National Bureau of Economic
Research. URL http://www.nber.org/papers/w18850.
Model
Results
Summary
Identification
Identification
Fang and
Wang (2015)
Machine
replacement
models
Other
applications
Holmes (2011)
Overview
Model
Results: demand
estimation
Dynamic estimation
Dynamic results
Dynamic demand
References
Das, Sanghamitra. 1992. “A Micro-Econometric Model of
Capital Utilization and Retirement: The Case of the U.S.
Cement Industry.” The Review of Economic Studies
59 (2):277–297. URL http://restud.oxfordjournals.
org/content/59/2/277.abstract.
Erdem, Tülin, Susumu Imai, and Michael P Keane. 2003.
“Brand and quantity choice dynamics under price
uncertainty.” Quantitative Marketing and Economics
1 (1):5–64. URL http://www.ingentaconnect.com/
content/klu/qmec/2003/00000001/00000001/05121258.
Single Agent
Dynamic
Models
Paul Schrimpf
Application:
prescription
drug demand
Introduction
Descriptive evidence
Model
Results
Summary
Identification
Identification
Fang and
Wang (2015)
Machine
replacement
models
Other
applications
Holmes (2011)
Overview
Model
Results: demand
estimation
Dynamic estimation
Dynamic results
Dynamic demand
References
Fang, Hanming and Yang Wang. 2015. “Estimating dynamic
discrete choice models with hyperbolic discounting, with
an application to mammography decisions.” International
Economic Review 56 (2):565–596. URL
http://dx.doi.org/10.1111/iere.12115.
Gilleskie, Donna B. 1998. “A dynamic stochastic model of
medical care use and work absence.” Econometrica
:1–45URL
http://www.jstor.org/stable/10.2307/2998539.
Gowrisankaran, Gautam and Marc Rysman. 2009.
“Dynamics of consumer demand for new durable goods.”
Tech. rep., National Bureau of Economic Research. URL
http://www.nber.org/papers/w14737.
Hendel, Igal and Aviv Nevo. 2006. “Measuring the
implications of sales and consumer inventory behavior.”
Econometrica 74 (6):1637–1673. URL
http://onlinelibrary.wiley.com/doi/10.1111/j.
1468-0262.2006.00721.x/abstract.
Single Agent
Dynamic
Models
Paul Schrimpf
Application:
prescription
drug demand
Introduction
Descriptive evidence
Model
Results
Summary
Identification
Identification
Fang and
Wang (2015)
Machine
replacement
models
Other
applications
Holmes (2011)
Overview
Model
Results: demand
estimation
Dynamic estimation
Dynamic results
Dynamic demand
References
Holmes, T.J. 2011. “The Diffusion of Wal-Mart and
Economies of Density.” Econometrica 79 (1):253–302. URL
http://onlinelibrary.wiley.com/doi/10.3982/
ECTA7699/abstract.
Hotz, V. Joseph and Robert A. Miller. 1993. “Conditional
Choice Probabilities and the Estimation of Dynamic
Models.” The Review of Economic Studies 60 (3):pp.
497–529. URL http://www.jstor.org/stable/2298122.
Hu, Yingyao and Matthew Shum. 2012. “Nonparametric
identification of dynamic models with unobserved state
variables.” Journal of Econometrics 171 (1):32 – 44. URL
http://www.sciencedirect.com/science/article/
pii/S0304407612001479.
Kasahara, Hiroyuki. 2009. “Temporary Increases in Tariffs
and Investment: The Chilean Experience.” Journal of
Business & Economic Statistics 27 (1):113–127. URL
http://amstat.tandfonline.com/doi/abs/10.1198/
jbes.2009.0009.
Single Agent
Dynamic
Models
Paul Schrimpf
Application:
prescription
drug demand
Introduction
Descriptive evidence
Model
Results
Summary
Identification
Identification
Fang and
Wang (2015)
Machine
replacement
models
Other
applications
Holmes (2011)
Overview
Model
Results: demand
estimation
Dynamic estimation
Dynamic results
Dynamic demand
References
Kennet, D. Mark. 1994. “A structural model of aircraft engine
maintenance.” Journal of Applied Econometrics
9 (4):351–368. URL
http://dx.doi.org/10.1002/jae.3950090405.
Magnac, Thierry and David Thesmar. 2002. “Identifying
Dynamic Discrete Decision Processes.” Econometrica
70 (2):801–816. URL http:
//www.jstor.org.libproxy.mit.edu/stable/2692293.
Norets, A. and X. Tang. 2013. “Semiparametric Inference in
Dynamic Binary Choice Models.” The Review of Economic
Studies URL http://restud.oxfordjournals.org/
content/early/2013/12/15/restud.rdt050.abstract.
Rust, J. 2008. “Dynamic programming.” In The New Palgrave
Dictionary of Economics. London, UK, Palgrave Macmillan,
Ltd, . URL http://gemini.econ.umd.edu/jrust/
research/papers/dp.pdf.
Single Agent
Dynamic
Models
Paul Schrimpf
Application:
prescription
drug demand
Introduction
Descriptive evidence
Model
Results
Summary
Identification
Identification
Fang and
Wang (2015)
Machine
replacement
models
Other
applications
Holmes (2011)
Overview
Model
Results: demand
estimation
Dynamic estimation
Dynamic results
Dynamic demand
References
Rust, John. 1987. “Optimal Replacement of GMC Bus
Engines: An Empirical Model of Harold Zurcher.”
Econometrica 55 (5):pp. 999–1033. URL
http://www.jstor.org/stable/1911259.
Rust, John and Geoffrey Rothwell. 1995. “Optimal response
to a shift in regulatory regime: The case of the US nuclear
power industry.” Journal of Applied Econometrics 10:75.
URL http://search.proquest.com.ezproxy.library.
ubc.ca/docview/218751608?accountid=14656. Name Nuclear Regulatory Commission; Copyright - Copyright
Wiley Periodicals Inc. Dec 1995; Last updated - 2011-10-21;
CODEN - JAECET; SubjectsTermNotLitGenreText - US.
Sasaki, Yuya. 2012. “Heterogeneity and selection in dynamic
panel data.” Working paper. URL
http://www.sfu.ca/content/dam/sfu/economics/
Documents/SeminarPapers/Sasaki_Nov_8_2012_
seminar_paper.pdf.