Suppose that the economy is composed of two individuals, A and B. Their respective demand curves for a public good, y, are given by: y_A= 50 - p/2 and y_B= 25 - p. Suppose the MC of providing the public good is given by MC = 10 + 2y. A social optimum is achieved if: A)y = 21.7. B)y = 30. C)y = 23. D)y = 15.

Question

Quizplus · Accepted Answer

To find the social optimum, we need to find the quantity of the public good that maximizes the total benefit to society (the sum of individual demands) minus the total cost of providing it (MC multiplied by the quantity provided). In other words, we want to maximize:

T = (50 - p/2) + (25 - p) - (10 + 2y)y

To simplify this expression, we can first use the demand curves to solve for p in terms of y:

50 - p/2 = 25 - p
25 = p/2
p = 50

Substituting this back into T, we get:

T = 75 - 50y - (10 + 2y)y
T = 75 - 50y - 10y - 2y^2
T = -2y^2 - 60y + 75

To maximize T, we can take the derivative with respect to y and set it equal to zero:

dT/dy = -4y - 60 = 0
y = -15 (discarded) or y = 15

Since y can't be negative, the only feasible answer is y = 15. We can check that this is a maximum by taking the second derivative of T with respect to y:

d^2T/dy^2 = -4

Since this is negative, we know that T is maximized at y = 15. Therefore, the social optimum is to provide 15 units of the public good.

Suppose That the Economy Is Composed of Two Individuals, a and B