The Problem of the Uninsured
July 2012
Isaac Ehrlich and Yong Yin
State University of New York at Buffalo
© Preliminary. Not to be circulated or cited without authors’ permission.
0. Motivation and Objectives
2
Motivation
• Being uninsured can be a rational choice – not just a
question of unaffordability. Issue is not new –dealt
extensively in literature on natural hazards.
• Our main argument: 0-insurance may be even more likely
in context of “full insurance”, including market insurance
(MI): self-insurance [SI] & self-protection [SP]
– point is missing from policy debate on “mandate” in ACA
and micro-simulation models by CBO, Rand’s COMPARE
• How relevant is the omission?
3
Motivation
• We give preliminary answer by applying the “full insurance”
paradigm to health insurance & offer calibrated simulations.
• Examples of indiv. self-insurance – potential loss reduction
– Investing in medical knowledge to complement remedial
care by “medics”, make early detections of risks, increase
speed of recovery, use medical savings accounts.
– Special case: free-riding on safety-net system: emergency
rooms and charitable institutions; Hippocratic oath.
• Examples of self-protection – potential hazard reduction
– Preventive medical care (annual checkups) exercise, diet,
safety measures & devices, and life-style choices.
4
Objectives
• We use the analysis to address three basic aspects of
“the problem of the uninsured”
• The “economic“ aspect – Rationalizing variations in the no-insurance decision
• Some policy aspects re: the US Affordable Care Act’s
mandate
– “Compliance rate” by the uninsured.
• Some welfare implications of the mandate
– Repercussions for impact on net losses from sickness
5
I. Set up of the problem
6
Theoretical Background
• “Full insurance” problem includes 3 alternatives, but
one may not necessarily use all 3.
• Condition for not insuring when market insurance
[MI] is the only alternative:
{peU’(I0) /(1-pe)U’(I)} < [(1+λ)pe/(1-pe)] ≡ π
pe= endowed hazard probability I0; = I-Le
– Slope of the market insurance “budget line” (t.o.t) cuts the
indifference curve [IC] from above at endowment point, E
– More likely if probab. of endowed loss is rare (π is high)
and loss is not too high. Illustrate via Ehrlich & Becker JPE
72 (go to Fig. 1)
7
Fig.1 – Rationalizing No-Insurance
8
Background
• Adding self-insurance
-1/[L’(c*) +1)={p(0)U’(I0*) /[1-p(0)]U’(I1*)}<[(1+λ)p(0)/1-p(0)] = π(0)
– At endowment pt. E, absolute slope of the market
insurance budget line exceeds that of the SI’s
transformation curve (see Fig. 1)
– More generally, individual self-insurance crowds out MI
• Adding self-protection
- 1/[L’(c*) +1] = {p(r*)U’(I0*)/[1-p(r*)]U’(I1*)} ≡ π(r*) < π(0)
– At endowment E, SP, by making the probability of hazard
lower, makes the price of insurance too high.
– SP crowds out both MI and SI (moving from S1 to S2- Fig. 1)
9
Background
Propositions
– Adding SP to SI is more likely to induce a corner solution
– For uninsured, SP (c*) & SI (r*) are substitutes. If SP* is
used, SI* falls.
– [- dlnP/dr* > -dlnL/dc*]: Last $ of r* effects a bigger % fall
in p rel’ to the % reduction in L caused by last $ of c*.
Same holds for relative % falls in “expected loss” p*L* But
both lower it.
This implies the corollary that:
– preventive care is more potent than remedial care
10
Applying the problem to health
insurance
Setting up the baseline model
11
Baseline Model
• “To insure or not to insure, that is the question.”
How we model the answer?
• Simplifying assumptions in baseline model
– Heterogeneous population stratified just by endowed
probability of sickness, pe, uniformly distributed in [0 1]. All
other parameters taken to be uniform (income, losses,
premiums, PFs, preferences). Will be relaxed in extensions
– Loss from getting sick, Le is purely monetary: we ignore
consumption aspects, or state preferences to focus on role
of SI & SP. Will be relaxed in extensions.
12
Baseline Model linking w/ real-world insurance plans
– Market health insurance [HI]
•
•
•
•
Fixed Indemnity , menu-based - no “partial coverage”
Payout = La ≤ Le - no full coverage; La = max. coverage
Single premium, R. relaxed if we let premium age-dependent.
Choice becomes “take it or leave it”
– Self Insurance Production Function:
• Loss from illness Le can fall to A(c)Le ; % fall s.t. PF:
A(c) = Ah + (1-Ah)exp(-η1c)
– with A(0) = 1 and A(∞) = Ah = limit on effectiveness of SI
– Safety-net care is available at zero cost
• but through “inferior indemnity” L0 << La.
13
Baseline Model
– Self Protection Production Function
• Probab. of illness pe can be similarly lowered to B(r)pe
With % fall an increasing fn. of SP spending r, via the PF:
B(r) = Bh + (1-Bh)exp(-η2r) ;
– with B(0) = 1 and B(∞) = Bh.
– “log form” PF imposes a limit, Bh, on effectiveness of SP
– Utility function assumed to exhibit CRRA:
U(I) = (Iσ-1 – 1)/(σ-1)
– As commonly used in the literature
– Conventionally calibrated as σ =2
14
Maximizing Problem
• Individual chooses to buy MI if maximized expected
utility is comparatively larger:
– When insured, so wealth prospect becomes:
I1 = I – R – c – r with probability of 1 – B(r)pe
I0 = I – R – c – r – A(c)Le + La with prob. of B(r)pe
– Relative to being uninsured with wealth prospect:
I1 = I – c – r with probability of 1 – B(r)pe
I0 = I –c – r – A(c)Le + L0 with prob. of B(r)pe
– Solving the “full insurance problem” using optimal SI (c*)
and SP (r*) in both cases.
15
Calibration
• Optimality conditions -similar to background slides
• Calibration
– We calibrate the baseline model using data from 2009
Medical Expenditure Panel Survey (MEPS)
– Parameters in legend to Table 1
– Focusing on target group: Non-institutional, US legal
citizens who are: nonelderly adults ( to exclude Medicare
& CHIP); in households where income is up to 133% (lower
bound) of poverty line (to exclude Medicaid)
– Calibrating baseline model to yield 20% of this target pop.
as uninsured
16
Baseline-Model Results (see Table 1)
• Our simulations solve for following decomposition:
• Of the (20%) uninsured:
• 50.3% of uninsured are attributable to 3 alt. choices:
− 19% (3.8%/20%) = free-riders; [3.8 = 20-16.2]
− 31.3% (6.25%/20%) = uninsured due to SI & SP; [6.25 = 20-13.75]
• The remaining 49.7% of uninsured can be explained by
highly “unfair” price of market insurance to individuals
• Alternatively, solving for % uninsured when SI, SP & safety net are not
available we find:
− 55.6% (11.11%/20%) will stay uninsured. But this is an
upper limit: some of the 3 alternatives always exist
17
Table 1: Decomposing the Uninsured
Feasible “insurance” and
“protection” options
% uninsured
All uninsured under: (4) options
20.00*
(3): MI, SI and SP (no safety net)
16.20
(2): Market Insurance & Safety-net
13.75
(1): Just Market Insurance alone
11.11
Baseline model parameters (based on population aged 19 to 64, lived in household
whose income is above 133% of poverty line). Calibrated parameters:
σ = 2; I = $36,000; Le = $8,250; La = (.8) Le (to match ‘average medical expenditure’ in
2009 MEPS); Premium (R) = $960 (ave. employee’s contribution from 2009 MEPS);
L0 = 12.73% Le (calibrated to match percentage uninsured in the total population)
A(c) = Ah + (1-Ah)exp(-η1c) with Ah = 0.8 and η1 = 0.05;
B(r) = Bh + (1-Bh)exp(-η2r) with Bh = 0.7 and η2 = 0.05
* Percentage uninsured calibrated on MEPS data
18
Decomposing the uninsured by pe
• Range of pe yielding a zero insurance solution:
– By our simulation, people with endowed sickness
probability less than 20% would choose to be uninsured
– The estimated upper bound is p = B(r*)pe is 14.32%
• Recovering π = (1+λ)p/(1-p)
– Assuming employer’s contribution equals employee’s
contribution, π = (6600 – 960)/960 = 5.875 we can infer
the magnitudes of personal “loading factors”: e.g.,
– For p = 10%, loading = (90%/10%)/5.875 – 1 = 53.2%
19
Model Results- optimal SI, SP spending & impact
• Optimal SI and SP spending
– Optimal c* and r* are 6-9% of premiums if not insured, but 9-15% if
uninsured. Drop is especially large for r, – see Table 2
– Optimal spent on SP is smaller than on SI if one is insured
– Comparing c* and r* for insured and uninsured:
• People with 10% endowed probability would optimally choose to
be uninsured. Thus, If forced to be insured, they would reduce SI
by 9.53% and SP by 56.48%
• Absolute and relative Impact of SI and SP – see Table 3
– Large impacts on both L(c*) and p(r*) but much bigger for uninsured.
– Relative impact greater for r*: p*(r*)/pe < L*(c*)/Le esp. for uninsured.
Same holds for p*L* which falls a lot more for uninsured
– This has implications for both our policy and welfare analyses.
20
Table 2: Optimal Spending on
Self-Insurance and Self-Protection
If Insured*
If Uninsured*
Endowed
probability
of sickness
c
(in dollars)
r
(in dollars)
c+r
as a % of
premium
c
(in dollars)
r
(in dollars)
c+r
as a % of
premium
0.1
38.26
20.01
6.07
42.29
45.98
9.19
0.2
51.99
19.95
7.49
55.10
58.91
11.88
0.3
60.05
19.94
8.33
62.51
66.36
13.42
0.5
70.23
19.92
9.39
71.57
75.46
15.32
* At p=0.1 and 0.2 consumers are optimally uninsured. At higher values
of p they are optimally insure
21
Table 3. Impact of Optimal Self-Insurance and
Self-Protection on Prospective sickness losses
If Insured
If Uninsured
Endowed
probability
of sickness
L*/Le
in %
p*/pe
in %
p*L*
in $
(L*/Le)
in %
(p*/pe)
in %
p*L*
in $
0.1
82.95
81.03
5,545
82.41
73.01
4,964
0.2
81.49
81.06
5,449
81.27
71.58
4,799
0.3
80.99
81.07
5,417
80.88
71.09
4,743
0.5
80.60
81.08
5,391
80.56
70.69
4,698
L* = A(c*)Le ; p* = B(r*)pe
22
Comparative Statics- see Table 4
• How basic parameters affect “full-insurance” components
• No-insurance decision:
− rises with: income, as fixed endowed Le & La fall rel’ to I (may change if health is in the U fn.);
safety-net coverage; and premium; but falls with: endowed loss and indemnity coverage
• Self-insurance for the insured:
− falls with: income, like reluctance to purchase market insurance; but rises with Le & premium
(SI is subst.). Effect of higher indemnity depends on how close La gets to Le –positive initially
• Self-Protection for the insured:
–
The same as for self-insurance. But effect of higher indemnity is positive initially.
• Self-insurance and Self-Protection for the uninsured:
–
The same as for the insured, except that no effect can come from MI parameters - indemnity
coverage and premium – from those who choose not to insure
23
Table 4. Comparative Statics
Impact of parameter change on “full insurance” options
If Insured*
If Uninsured**
Parameter
% uninsured
c
r
c
r
Income, I
+
-
-
-
-
Loss from
sickness, Le
-
+
+
+
+
Amount of
indemnity, La
-
+/-
-/+
0
0
Premium, R
+
+
+
0
0
Amount of
safety-net, L0
+
0
0
-
-
* Evaluated for pe = 0.24
** Evaluated for pe = 0.17
24
Policy Analysis
• Success of ACA hinges on Mandate compliance
– We use our simple calibrated model to gauge effectiveness
of the mandated penalty used as chief instrument.
• Design
– We impose 3 different penalties on uninsured:
• Penalty Level 1: $695 mandated in ACA; amounts to 72.4% of ave.
employee premium of $960 assumed in our model. Excessive?
• Penalty Level 2: $455 to achieve a 50% compliance (similar to MA
experience by Census data); amounts to 47.4% of $960 premium
• Penalty Level 3: $222.4 set by applying 23% of the $960 premium,
where 23.2% = $695/$3000 and $3000 is the actual premium in
the “silver plan” offered by Central Exchanges. Realistic?
25
Results – see Table 5
• To what extent will the penalty assure compliance?
– By our baseline model, estimated compliance rates with
the mandate by the uninsured range from 24.5% of pop for
the lowest penalty to 75% for the highest penalty, when SI,
SP, and Safety-net are all taken into consideration.
– If SI, SP, and the Safety-net are ignored, this would lead to
much higher compliance rates.
– By Panel B of Table 5, the overstated compliance rates
would then range from 13.6% = [85.2%/75% - 1] to 135.7%
= [57.75%/24.5% - 1]
– This Inference may apply in principle to the CBO & Rand
estimates of compliance rates (see note).
26
Table 5. Assessing Compliance
Penalty Level
(as % of
premium)
% pop
Remaining
uninsured
Change in
% uninsured*
Compliance
Rate**
Overstated
Compliance
(B/A)
A. (Accounting for all 4 options)
1 (72.4%)
5%
15%
75%
2 (47.4%)
10%
10%
50%
3 (23.2%)
15.1%
4.9%
24.5%
B. (Ignoring SI, SP, and safety-net)
1 (72.4%)
2.96%
17.04%
85.2%
13.6%
2 (47.4%)
5.72%
14.28%
71.4%
42.8%
3 (23.2%)
8.45%
11.55%
57.75%
135.7%
* Change in % uninsured is calculated as the difference between the
initial % uninsured (20%) and remaining % uninsured
** Compliance rate is calculated as ratio of the change in % uninsured
and the initial % uninsured
27
Note on Linking with CBO Compliance Estimates
• Table 4’s estimated compliance rates do not link directly with other studies
• According to Table 4 of the CBO/CJT (2010) report* assessing the effects of
the insurance-coverage provisions of the Reconciliation Proposal,
Combined with H.R. 3590 as passed by the Senate, 52 million nonelderly
people would be uninsured under the current law in 2016. The report also
stated that the post-policy uninsured nonelderly would become 21 million.
The implied compliance rate is then (52-21)/52 = 59.6%
• According to our Table 5, if we ignore all 3 other options as does CBO, we
predict a compliance rate of 57.75% for the lowest level of penalty, which
links well with CBO. But then, by Panel A - when all 4 options are
employed – compliance rate is only 24.5%.
• This suggests that the CBO estimates, although not precisely comparable,
are significantly overstated, perhaps by over 135%, see case 3.
* CBO, letter to the Honorable Nancy Pelosi, an estimate of the direct spending and revenue
effects of the Reconciliation Act of 2010, (March 20, 2010).
28
Welfare Implications – see Table 6
– Those becoming insured
• Would lower SI and SP by Table 4, which results in higher expected
losses from sickness
• Even though they may not rely on the safety-net, they now pay for
it in premiums they have deemed to be too high
– Those staying uninsured
• Would now have to pay the penalty, although by Table 4:
• Their expected health losses get smaller as they increase SI and SP
• All the previously insured:
– Would not benefit from the mandate directly. If they gain income
transfers, however, this may result in lower SI and SP, and thus in
higher expected health losses. But they may also be subsidizing some
previously uninsured via Medicaid expansion and lowered premiums
29
Impact on Prospective losses from incidence of illness
• In assessing the impacts of the mandated penalties to
induce compliance by the uninsured, we treat all
between-groups income transfers as a wash
• This leads to a focus on resulting changes in expected
losses from ill-health (changes in p*L* of the affected
pop. groups). See Table 6.
• These are also dead-weight losses which affect
personal real income [I-(p*L*)] in our baseline model
• The assessment may change if model allowed for
health gains from offering insurance-financed care to
the uninsured, or from income redistribution [mixed evidence]
30
Table 6. Changes in Prospective Losses from incidence of illness
Penalty Level
(as % of premium)
Group
1 (72.4%)
2 (47.4%)
3 (23.2%)
Who stay uninsured
+$0.12
+$0.09
+$0.04
Who become insured
-$76.49
-$94.42
-$112.77
Previously uninsured
Previously insured
Negative change conceptually
Net effect (upper bound)
-$76.37
-$94.33
-$112.73
As % of premium
-7.96%
-9.83%
-11.74%
31
Extensions
• We next pursue 2 extensions of the baseline model
1. Theoretical extension in which we distinguish health as a
separate argument in the utility function and allow for ex-post
moral hazard.
2. Calibrated simulations in which the heterogeneous population
is stratified not just by endowed probabilities of monetary
losses from sickness but also by: age, income, and the type of
insurance purchased: individual vs. employer group
insurance.
3. Extensions could provide more accurate estimates of impact
of mandated health insurance, esp. on the low-income young.
32
Conclusion
• Baseline model is oversimplified
– Yet sufficiently general to indicate why the “problem of the uninsured”
has to be assessed in the context of a ”full insurance” decision. To wit:
– Self-insurance & self-protection can significantly reduce prospective
losses from incidence of illness, as indicated by Tables 3, 6.
• Calibrated simulations indicate why
– a significant % of pop. may avoid insurance b/c of SI and SP. This applies
not just to health insurance, but more generally
– Why initial estimates of compliance rates with mandated provisions
prove to be overstated, as indicated by Table 5.
– More generally, why there is mixed evidence in the literature on the
role of insurance in affecting health status
– Why impact on health may not be favorable; raising dead-weight losses
• But our policy and welfare implications are s.t. many caveats
– We plan to relax some of these in planned extensions
33