Signature
eta_arc = ((q_2 - q_1) / (q_2 + q_1)) / ((p_2 - p_1) / (p_2 + p_1))
| Inputs | Definition | Unit |
|---|---|---|
q_2 | Average annual medical spending, or use, under the second arrangement | same unit as q_1 |
q_1 | Average annual medical spending, or use, under the first arrangement | currency per person per year, or units of care |
p_2 | Price measure under the second arrangement, in the same unit as p_1 and different from it | proportion or currency per unit |
p_1 | Price measure under the first arrangement, such as the coinsurance rate, zero for free care | proportion or currency per unit |
eta_arc | Proportional change in spending divided by proportional change in price, each relative to the average of its two values | dimensionless |
|---|
Function
Ex post moral hazard welfare function
Maps the resource cost of care, the share of that cost the insured patient pays and the quantities of care used at the full and insured prices to the welfare loss attributed to insurance-induced use, and maps observed spending under two cost-sharing arrangements to a price elasticity of demand. The welfare measures assume that the demand curve measures the patient's marginal benefit, an assumption that the access motive and behavioural hazard challenge.
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Implementations
Excel
Arc elasticity from two plans
With spending in Spend1 and Spend2 and prices in Price1 and Price2, Excel returns the arc elasticity.
=((Spend2-Spend1)/(Spend2+Spend1))/((Price2-Price1)/(Price2+Price1))
Assumptions
Two arrangements that differ in price only
The two groups differ only in the price of care, as under random assignment to plans. A comparison of self-selected insured and uninsured people mixes the price effect with adverse selection.
A single price summarises each plan
Each plan is represented by one price, such as its coinsurance rate. Real plans have deductibles and out-of-pocket maximums, so the price faced changes during the year and the elasticity depends on which price is used.
Worked examples
RAND free care plan against the 25 per cent coinsurance plan
Aron-Dine and colleagues report mean total annual spending of 2,170 dollars (2011 prices) in the free care plan and 648 dollars less, 1,522 dollars, in the 25% coinsurance plan. The arc elasticity with respect to the coinsurance rate is about minus 0.18, the figure they report for this pair.
q_1 = 2170; q_2 = 1522; p_1 = 0; p_2 = 0.25; eta_arc = -0.18
RAND free care plan against the 95 per cent coinsurance plan
Spending in the 95% coinsurance plan was 845 dollars below free care, 1,325 dollars. The simple arc elasticity from these rounded means is about minus 0.24; Aron-Dine and colleagues report minus 0.234 for this pair from their own estimates.
q_1 = 2170; q_2 = 1325; p_1 = 0; p_2 = 0.95; eta_arc = -0.24
Common errors
Comparing plans with free care alone
When p_1 is zero the price term equals 1 whatever the coinsurance rate of the other plan, so an arc elasticity against free care reflects only the change in spending. The 25% and 95% plans are then treated as the same change in price.
Treating minus 0.2 as a constant of nature
The RAND figure rests on an episode-based model and on assumptions about how people respond to non-linear contracts. Aron-Dine and colleagues obtain estimates from about minus 0.04 to minus 0.6 from simple alternative summaries of the same experiment.
Sources
RAND experiment effects and arc elasticities re-examined
Aron-Dine A, Einav L, Finkelstein A. The RAND Health Insurance Experiment, three decades later. Journal of Economic Perspectives. 2013;27(1):197-222. Table 2 (mean annual spending in the free care plan and differences by plan, 2011 dollars), Table 4 and footnote 11 (arc elasticity defined relative to the average of the two values; used by RAND because the free care plan had a price of zero), and the section on the origin of the minus 0.2 estimate in Keeler and Rolph 1988.
Canonical Identity
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