NET SOCIETAL BENEFIT OPTIMIZATION

The calculation of economic benefits begins with equations 5.14 and 5.15 such that the private willingness-to-pay for direct protection, WTPdi, as a function of price for subgroup i is

(5.18) WTPdi = (pi* - 1/βpi) · exp(Xiβi) + βpi ⋅ pi*) = (pi* - 1/βpi) · αi · exp(βpi ⋅ pi*), where pi* is the optimal price for subgroup i, βpi is the price coefficient for subgroup i, and αi is the demand intercept for subgroup i, where αi is expressed as the number of people vaccinated at price equal to zero (not the fraction of population expressed as a percentage).

Eq. 5.18 is maximized when price is set equal to zero and coverage rates are maximized. The maximum private WTP for direct protection, WTPMAXdi, is

(5.19) WTPMAXdi = (-1/βpi) · exp(Xiβi) = (-αi /βpi).

In the absence of herd protection, only vaccinated persons’ risk of infection is reduced by a fraction representing the direct efficacy of vaccination. Unvaccinated persons’ risk of infection would remain equal to the pre-vaccination baseline. Before defining economic benefits of herd protection, it is necessary to define functional forms for the coverage-incidence relationships effected by herd protection for both the vaccinated and unvaccinated subgroups. The variable COVi is coverage for subgroup i and is based on the demand relationship: COVi = αi · exp(βpi ⋅ pi*). The coverage-incidence relationship for unvaccinated persons in subgroup i (INCUi) is assumed to be the following exponential relationship

25_{A bivariate normal model was also used to jointly model the responses to the first and second prices. The}

coefficients estimates for the two questions were statistically different at the 1% level. I also estimated interval models based on normal, lognormal, and Weibull distributions. The Weibull distribution best fit the data and provided average WTP estimates that were very similar to those from the probit models. (Median WTP estimates from the Weibull interval model were smaller.)

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(5.20) INCUi = INCU0i ⋅ exp(γi,1 · COV1 + γi,2 · COV2 + …)

where INCU0i is the baseline incidence for group i and γi,1 and γi,2 are herd protection coefficients. The choice of an exponential function ensures that the magnitude of incidence reduction will be greatest at low coverage rates and that the rate of change will decrease as coverage increases. If COVi represents the number of people vaccinated in each subgroup, the herd protection coefficients determine the impact increased coverage on indirect protection by subgroup, allowing for differential rates of impact by subgroup. Thus, an increase in coverage for subgroup 1 may have a greater impact on herd protection than an increase in coverage for

subgroup 2.

Next, I assume that the direct efficacy of vaccination, Eff, is constant such that the vaccinated subgroup incidence is a fraction of the unvaccinated incidence rate, specifically 1 - Eff. Thus, the expected incidence rate for vaccinated persons in subgroup i (INCVi) is

(5.21) INCVi = (1-Eff) · INCUi.

These INCUi and INCVi relationships can be used in combination with the willingness-to- pay for direct protection functions to estimate the value of indirect protection. The estimated benefits for vaccinated persons can be calculated based on eq. 5.19 for WTPdi and the difference in incidence reduction with and without consideration for herd protection. This difference can be calculated from (1-Eff) · (INCU0i – INCUi). I assume that vaccinated persons would value the reduced risk of infection due to herd protection; however, I do not have any empirical data on the prevalence elasticity of demand. In the absence of data, I assume that vaccinated persons’

willingness-to-pay for indirect protection, WTPvii, are proportional to the magnitude of (1-Eff) · (INCU0i – INCUi) according to

(5.22) WTPvii = πv · Eff · (INCU0i – INCUi) / (Eff · INCU0i) · WTPdi

where πv is a correction factor for the value of indirect versus direct protection for vaccinated persons. The fraction, Eff · (INCU0i – INCUi) / (Eff · INCU0i), represents the ratio of the magnitude of indirect protection to the magnitude of direct protection without herd protection.

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For example, assume that the baseline incidence, INCU0i, is 1 case per 1,000 persons, vaccination efficacy, Eff, is 65% and that the incidence for unvaccinated persons, INCUi, is 0.50 cases per 1,000. Without herd protection, the incidence of vaccinated persons would be (1 – Eff) · INCU0i = 0.35 cases per 1,000 persons. With herd protection incidence declines to (1 – Eff) · INCU0i = 0.175 cases per 1,000 persons. Thus, the ratio of indirect to direct protection would be 0.175/0.65 = 0.27, indicating that the additional indirect protection is about a quarter of the expected direct vaccine protection without regard for herd protection. This is multiplied by a correction factor,

πv, such that the sum of willingness-to-pay for direct and indirect protection for vaccination persons is (1 + 0.75 · 0.27) · WTPdi or 1.20 · WTPdi

Next, I develop an expression to estimate unvaccinated persons’ WTP for indirect protection. Recall that αi is the fraction of the population that would receive a free vaccination. For now, I assume that the rest of the population, POPi - αi, has no value for indirect protection. Thus, I focus on the remaining portion of the population that would accept a free vaccination, but would not purchase a vaccination at price pi*. Recall from eq. 5.19 that the total WTP for a free vaccination program without consideration of herd protection is WTPMAXdi or – αi / βpi for subgroup i. Next, I define the change in WTP benefits in moving from a program that offers vaccines at price pi* to a free program.

(5.23) WTPMAXdi - WTPdi = (-αi / βpi) - (pi* - 1/βpi) ·αi · exp (βpi · pi*) This would represent the potential value of protection for the unvaccinated if their incidence were reduced to exactly INCUi = (1-Eff) · INCU0i, i.e. the expected direct protection from vaccination without herd protection. Of course, it is unlikely that the unvaccinated incidence would be exactly (1-Eff) · INCU0i. I set up a second ratio that relates the unvaccinated incidence reduction due to herd protection relative to the direct-protection-induced incidence reduction effected by vaccination in the absence of herd protection: (INCU0i - INCUi) / (Eff · INCU0i). Using this ratio, the estimated willingness-to-pay for indirect protection for the unvaccinated, WTPuii is

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where πu is a correction factor for the value of indirect versus direct protection for the

unvaccinated subgroup. In the sensitivity analysis, I can also assume that persons unwilling to purchase vaccines would have some non-zero WTP for protection. This would occur if people were unwilling to spend time procuring vaccinations or if they were frightened of vaccination side effects. I again assume that WTP is proportional to the magnitude of protection, similar to eqs. 5.22 and 5.24. I have already accounted for the monetary benefits for those with non-zero willingness-to-pay, as represented by the demand function intercept, αi. The remainder of the population is POPi – αi. The willingness-to-pay for indirect protection for those with zero WTP for vaccinations is

(5.25) πu · (POPi – αi) · (INCU0i – INCUi) / (Eff · INCU0i) · TIME · MED_WAGE · CORR.

where TIME is the average time required to receive two doses of cholera vaccine, MED_WAGE is median hourly wage rate, and CORR is a correction factor that relates the value time required for vaccination and the median wage rate. This assumes that the maximum willingness-to-pay for indirect protection by those unwilling to receive a free vaccination is equal to the opportunity cost of time required to pursue a free vaccination.

Next, I examine the public cost of illness avoided. This is equal to the number of cases avoided multiplied by the average public COI per case. I discount these savings over three years at 8% interest using a present worth function, PWF. The following expression summarizes the public COI savings, PUBSAVi, for group i.

(5.26) PUBSAVi = PUBCOIi · PWF · (POPi · (INCU0i – INCUi) + COVi · (INCUi – INCVi)

The cost function is simply assumed to be a fixed cost plus a constant variable cost multiplied by the coverage for each group i.

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Now, I am prepared to calculate the set of optimal prices that maximize societal net benefits via unconstrained optimization. I exclude the unvaccinated benefits for those with zero WTP for vaccinations, eq. 5.25. I revisit these benefits in the sensitivity analysis. Without these benefits, net societal benefits are calculated from the sum of eqs. 5.18, 5.22, 5.24 and 5.26 less eq. 5.27. If I assume that there are four subgroups, the complete expression would be

(5.28)

∑

i – INCUi) / (Eff · INCU0i)) · (pi* - 1/βpi) · αi ·

exp(βpi ⋅ pi*) + πu · (INCU0i - INCUi) / (Eff · INCU0i) · ((-αi / βpi) - (pi* - 1/βpi) ·

αi · exp (βpi · pi*)) + PUBCOIi · PWF · (POPi · (INCU0i – INCUi) + COVi · (INCUi – INCVi)] - F - C · (COV1 + COV2 + COV3 + COV4)

=

∑

1

i [ WTPdi + WTPvii + WTPuii + PUBSAVi] - F - C · (COV1 + COV2 + COV3 + COV4)

An optimal set of prices, p1*, p2*, p3*, p4*, can be solved for via Lagrangian analysis or similar numerical methods. If public COI benefits are small relative to WTP benefits and herd protection benefits are minimal, the optimal solution would set price equal to the marginal cost of vaccination. Eq. 5.28 assumes that either the public health ministry or an external donor would be willing to pay the difference between socially optimal prices and revenue neutral prices. However, the program may be constrained to be revenue neutral. The net revenue constraint is program revenues plus public COI savings less program costs as shown in eq. 5.29.

(5.29)

∑

= 4

1

i [COVi · pi* + PUBSAVi] - F - C · (COV1 + COV2 + COV3 + COV4) ≥ Z where COVi · pi* represents the revenue generate through vaccination sales to subgroup i at price pi* and Z is a fixed external contribution which may be zero. The maximum societal net benefits subject to a revenue constraint can be solved via a Lagrange multiplier approach using the following equation

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∑

4 1

i [ WTPdi + WTPvii + WTPuii + PUBSAVi] - F - C · (COV1 + COV2 + COV3 + COV4) – λ1· (

∑

4 1

i [COVi · pi* + PUBSAVi] - F - C · (COV1 + COV2 + COV3 + COV4) – Z)

where λ1 is the undetermined Lagrangian multiplier, which denotes the marginal change in the net

societal benefits per unit change in present value net revenue. The optimal prices pi* can be found numerically or by using calculus to solve for i = 1 to 4, ∂L1/∂(pi) = ∂L1/∂(π)=0.

In summary, the net societal benefit maximization equations contain many variables despite a number of simplifying assumptions, including the following:

♦ I assume that demand is independent of herd protection effects. While this is reasonable for the first vaccination period, it seems likely that prospective purchasers would change their behavior in subsequent periods as they become aware of herd protection impacts.

♦ I model demand based on an average uptake of vaccinations for each subgroup. However, there may be spatial differences in uptake rates within subgroups. ♦ I assume that indirect protection benefits can be calculated based on simple

fractional incidence reduction relationships because I do not have any data regarding valuation of indirect protection.

♦ I assume that the population can be modeled based on relatively simplistic subdivisions of the population. I assume four subdivisions based on the subdivisions I use in the empirical models in Chapter 7.

♦ I assume a linear marginal cost function.