Signature
Delta_q = D * (p - c * p); WL = 0.5 * D * (p - c * p)^2
| Inputs | Definition | Unit |
|---|---|---|
D | Increase in use for each one-unit fall in the price paid, greater than zero | units of care per person per year per unit of currency |
p | Cost of providing one unit of care, the full price | currency per unit of care |
c | Share of p paid by the patient, at least zero and less than one | proportion |
Delta_q | Increase in units of care used when the price paid falls from p to c × p | units of care per person per year |
|---|---|---|
WL | Welfare loss triangle computed from the demand slope | currency per insured person per year |
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
Extra use and welfare loss from the demand slope
With the demand slope in DemandSlope, the unit cost in UnitCost and the coinsurance rate in CoinsRate, the formulas return extra use and the welfare loss.
=DemandSlope*(UnitCost-CoinsRate*UnitCost); =0.5*DemandSlope*(UnitCost-CoinsRate*UnitCost)^2
Assumptions
Constant demand slope
D is the same at every price between c × p and p. Estimates of D from one range of cost sharing do not carry over to a very different range.
Price reduction as the only change
Insurance changes only the price paid at the point of use. Income effects of the premium and changes in supplier behaviour are ignored.
Worked examples
Demand slope consistent with the visits example
A slope of 0.02 visits per pound reproduces the visits example: 25% coinsurance lowers the price paid by 75 pounds, adding 1.5 visits and a welfare loss of 56.25 pounds. The figures are illustrative.
D = 0.02; p = 100; c = 0.25; Delta_q = 1.5; WL = 56.25
Fifty per cent coinsurance on the same demand curve
At 50% coinsurance the price paid falls by 50 pounds, adding 1 visit and a welfare loss of 25 pounds. Halving the coinsurance rate to 25% more than doubles the loss.
D = 0.02; p = 100; c = 0.5; Delta_q = 1; WL = 25
Common errors
Assuming the loss is proportional to the price reduction
Doubling the price reduction doubles both the extra use and the height of the triangle, so the loss rises fourfold. A linear extrapolation from one coinsurance rate understates the loss at lower rates.
Sources
Welfare loss of insurance-induced demand with a linear demand curve
Zweifel P, Manning WG. Moral hazard and consumer incentives in health care. In: Culyer AJ, Newhouse JP, editors. Handbook of Health Economics. Vol 1A. Amsterdam: Elsevier; 2000. p. 409-459. Welfare loss from moral hazard as the area between the resource cost and the demand curve over the insurance-induced increase in use.
Canonical Identity
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