Feldstein and Gruber reduction in the deadweight loss of insurance when coinsurance rises

Signature

dDWL = (E_0 - E_1) * (1 - P_1) + 0.5 * (P_1 - P_0) * (E_0 - E_1)

Function

Deadweight loss of a wedge between marginal value and marginal cost

Computational function

• Computational function: Feldstein and Gruber deadweight loss reduction across insurance units

  Takes each insurance unit's spending and coinsurance rate under the existing plan, the new coinsurance rate and a constant price elasticity, and returns each unit's spending under the new rate, its reduction in deadweight loss (HE-FM-DWL-003) and the average across units, the calculation Feldstein and Gruber ran for each individual in their survey data. The inputs differ from the formula's: E_1 is derived from the elasticity instead of being entered.

  Inputs and outputs: E_0: Spending at full cost under the existing plan for each unit; required, vector, zero or above. Unit: currency.; P_0: Existing coinsurance rate for each unit; required, vector, above zero. Unit: proportion.; P_1: New coinsurance rate; required, scalar, above every P_0 and at most 1. Unit: proportion.; eta: Absolute price elasticity; required, zero or above. Unit: none.; E_1: Spending at full cost under the new rate. Unit: currency.; dDWL: Reduction in deadweight loss for each unit, and mean_dDWL its average. Unit: currency.

  Assumption: Constant cost, constant elasticity over the range of prices, out-of-pocket spending below any cap under both plans, and the change in risk bearing valued separately.

  Worked example (Three insurance units at 20 per cent coinsurance moved to 50 per cent): With spending of 500, 2,000 and 8,000 and an elasticity of 0.33, the reductions are about 84.81, 339.22 and 1,356.89, averaging 593.64, against an average fall in spending of about 913.29 (computed here for illustration). E_0 = [500, 2000, 8000]; P_0 = [0.2, 0.2, 0.2]; P_1 = 0.5; eta = 0.33; E_1 = [369.5301, 1478.1203, 5912.4814]; dDWL = [84.8054, 339.2218, 1356.8871]; mean_dDWL = 593.6381

  Worked example (Zero elasticity): With no price response spending is unchanged and every reduction is zero, as Feldstein and Gruber found. E_0 = [500, 2000, 8000]; P_0 = [0.2, 0.2, 0.2]; P_1 = 0.5; eta = 0; dDWL = [0, 0, 0]; mean_dDWL = 0

  Excel: With spending in column B and coinsurance in column C from row 2, and NewCoins and Elasticity named, =B2*(C2/NewCoins)^Elasticity in D2 gives E_1 and =(B2-D2)*(1-NewCoins)+0.5*(NewCoins-C2)*(B2-D2) in E2 the reduction; =AVERAGE(E2:E1000) averages it.

  R: fg_dwl <- function(E_0, P_0, P_1, eta) { E_1 <- E_0*(P_0/P_1)^eta; d <- (E_0-E_1)*(1-P_1)+0.5*(P_1-P_0)*(E_0-E_1); list(E_1 = E_1, dDWL = d, mean_dDWL = mean(d)) } Returns reductions of 84.8054, 339.2218 and 1356.8871 and a mean of 593.6381 for the first example.

  Python: def fg_dwl(E_0, P_0, P_1, eta): E_1 = [e*(p/P_1)**eta for e, p in zip(E_0, P_0)]; d = [(e-f)*(1-P_1)+0.5*(P_1-p)*(e-f) for e, f, p in zip(E_0, E_1, P_0)]; return {"E_1": E_1, "dDWL": d, "mean_dDWL": sum(d)/len(d)} Returns the same values as the R function.

  Test (Reduction proportional to spending at a common coinsurance rate): When every unit starts at the same rate, each reduction is the same share of initial spending, about 0.1696 here. Expected result: TRUE. Excel check: =ABS(MAX(E2:E4/B2:B4)-MIN(E2:E4/B2:B4))<1E-9

  Test (Zero elasticity gives zero reduction): With Elasticity set to 0 every reduction is zero. Expected result: TRUE. Excel check: =IF(Elasticity=0,SUMPRODUCT(ABS(E2:E1000))=0,TRUE)

  Common error (Coinsurance entered as a percentage): Entering 20 and 50 instead of 0.2 and 0.5 leaves E_1 unchanged, because only their ratio enters it, but makes 1 minus P_1 negative, so the rectangle and the reduction turn negative.

  Source: Feldstein M, Gruber J. A major risk approach to health insurance reform. In: Poterba JM, editor. Tax Policy and the Economy. Vol 9. Cambridge, MA: MIT Press; 1995. p. 103-130. Section 4.1 and Figure 1.

  E_1_i = E_0_i * (P_0_i / P_1)^eta; dDWL_i = (E_0_i - E_1_i) * (1 - P_1) + 0.5 * (P_1 - P_0_i) * (E_0_i - E_1_i); mean_dDWL = sum_(i=1)^n [dDWL_i] / n

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Implementations

• Excel

  Reduction in deadweight loss from named spending and coinsurance cells

  With spending at full cost in SpendInitial and SpendNew and the coinsurance rates in CoinsInitial and CoinsNew, the first formula returns the reduction, held in DWLReduction. Under a constant elasticity in Elasticity, the second gives SpendNew from SpendInitial.

  =(SpendInitial-SpendNew)*(1-CoinsNew)+0.5*(CoinsNew-CoinsInitial)*(SpendInitial-SpendNew); =SpendInitial*(CoinsInitial/CoinsNew)^Elasticity

Assumptions

• Care supplied at constant cost with uninsured price normalised to 1

  Feldstein and Gruber assume constant cost, so there is no change in producer surplus, and measure care so that its price without insurance is 1. With prices in pounds, E is spending at full cost and P the patient's share.

• Out-of-pocket spending below any cap

  The patient faces P_1 on the margin, so out-of-pocket spending stays below the plan's maximum; above the cap the marginal price is zero and the formula does not apply.

• Risk-bearing change evaluated separately

  The formula values reduced distortion only. Feldstein and Gruber evaluate the change in the risk people bear as a separate effect of ambiguous sign and add the two.

Worked examples

• Raising coinsurance from 20 to 40 per cent in the physiotherapy example

  In money terms, 9,000 and 8,000 sessions at 50 pounds are 450,000 and 400,000 pounds of care at full cost. The rectangle is 50,000 times 0.6, 30,000 pounds, the triangle 0.5 x 0.2 x 50,000, 5,000 pounds, and the reduction 35,000 pounds, as in the article.

  E_0 = 450000; E_1 = 400000; P_0 = 0.2; P_1 = 0.4; dDWL = 35000

• Constant elasticity of 0.33 with coinsurance rising from 20 to 50 per cent

  With 1,000 units of care at 20 per cent coinsurance and a constant elasticity of 0.33, raising the rate to 50 per cent cuts use to about 739.06 units and the deadweight loss by about 169.61 (computed here for illustration).

  E_0 = 1000; E_1 = 739.0602; P_0 = 0.2; P_1 = 0.5; dDWL = 169.61

• No change in use gives no reduction in deadweight loss

  If use does not respond to the higher coinsurance rate, E_1 equals E_0 and there is no gain from reduced distortion, as Feldstein and Gruber found at zero elasticity.

  E_0 = 1000; E_1 = 1000; P_0 = 0.2; P_1 = 0.5; dDWL = 0

Common errors

• Reading the fall in spending as the welfare gain

  In the article's example spending falls by 50,000 pounds but the gain is 35,000. At an elasticity of 0.5, Feldstein and Gruber found that a deductible of 15 per cent of income cut spending more than 50 per cent coinsurance up to the same 15 per cent limit (USD 1,629 against USD 1,426 per insurance unit) but reduced deadweight loss less (USD 735 against USD 1,045).

• Treating the triangle gain as the whole effect of more cost sharing

  Higher coinsurance also exposes people to more financial risk. A case for more cost sharing needs that change alongside the reduction in deadweight loss.

Sources

• Rectangle and triangle reduction in the deadweight loss of excess insurance

  Feldstein M, Gruber J. A major risk approach to health insurance reform. In: Poterba JM, editor. Tax Policy and the Economy. Vol 9. Cambridge, MA: MIT Press; 1995. p. 103-130. Section 4.1 and Figure 1: raising the coinsurance rate from P0 to P1 reduces the deadweight loss by the area BCDE, equal to (E0 minus E1) times (1 minus P1) plus 0.5 times (P1 minus P0) times (E0 minus E1), with E1 = E0 (P0/P1) raised to the absolute price elasticity. Section 4.3 and Table 5: zero elasticity gives no gain; 50 per cent coinsurance with an out-of-pocket limit of 10 per cent of income reduces deadweight loss by USD 534 per insurance unit at an elasticity of 0.33 and USD 902 at 0.5; with a 15 per cent limit the reduction at 0.5 is USD 1,045, against USD 735 for a deductible of 15 per cent of income, which cuts spending more (USD 1,629 against USD 1,426).

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Canonical Identity