Consider an obstetrician who can perform two types of deliveries: normal deliveries and cesarean deliveries. Each typeof delivery provides different levels of income for the physician, and the physician has some ability to induce patientsto opt for cesarean deliveries. The model is as follows:The physician’s utility is defined as:U = U(Y, I)where:• Y is the income from performing deliveries.• I is the total disutility from inducementThe income Y from deliveries depends on the type of delivery:Y = Yn · N + YC · Cwhere:• Yn is the income per normal delivery, Yn = 1, 000• YC is the income per cesarean delivery, Yc = 1, 500,• Initial number of births Binitial = 100,• Post-shock number of births Bshock = 90,• a(i) = 0.1 + 0.05i is the fraction of total births that are cesareans, which increases with inducement level i,• the physician sets the inducement level to i = 2.• N = B · (1 − a(i)) is the number of normal deliveries,• C = B · a(i) is the number of cesarean deliveriesDue to a fertility shock, the number of births decreases to B = 90.a) Find the inducement level ishock needed to maintain income after the shock. b) Now consider a specific utility function:U(Y, i) = Y − β · iwhere:• β · i represents the disutility from inducement, with β as a positive constant reflecting the penalty per unit ofinducement. Set β = 3, 000.• B = 100: Total number of births (no fertility shock).• Income for normal and cesarean delivers are as before. Yn = 1, 000 and Yc = 1, 500Show in this model that inducement will not occur. c). What is the minimum income gap between Yc − Yn that would make inducement beneficial (i.e., that wouldlead to an inducement level (i > 0).[Hint: Set up the income function Y as a function of i then substitute into the utility function.]
Consider an obstetrician who can perform two types of deliveries: normal deliveries and cesarean deliveries. Each type
of delivery provides different levels of income for the physician, and the physician has some ability to induce patients
to opt for cesarean deliveries. The model is as follows:
The physician’s utility is defined as:
U = U(Y, I)
where:
• Y is the income from performing deliveries.
• I is the total disutility from inducement
The income Y from deliveries depends on the type of delivery:
Y = Yn · N + YC · C
where:
• Yn is the income per normal delivery, Yn = 1, 000
• YC is the income per cesarean delivery, Yc = 1, 500,
• Initial number of births Binitial = 100,
• Post-shock number of births Bshock = 90,
• a(i) = 0.1 + 0.05i is the fraction of total births that are cesareans, which increases with inducement level i,
• the physician sets the inducement level to i = 2.
• N = B · (1 − a(i)) is the number of normal deliveries,
• C = B · a(i) is the number of cesarean deliveries
Due to a fertility shock, the number of births decreases to B = 90.
a) Find the inducement level ishock needed to maintain income after the shock.
b) Now consider a specific utility function:
U(Y, i) = Y − β · i
where:
• β · i represents the disutility from inducement, with β as a positive constant reflecting the penalty per unit of
inducement. Set β = 3, 000.
• B = 100: Total number of births (no fertility shock).
• Income for normal and cesarean delivers are as before. Yn = 1, 000 and Yc = 1, 500
Show in this model that inducement will not occur.
c). What is the minimum income gap between Yc − Yn that would make inducement beneficial (i.e., that would
lead to an inducement level (i > 0).
[Hint: Set up the income function Y as a function of i then substitute into the utility function.]
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