Question

1. Consider the issue of determining the optimal amount of investment in national defense in a country composed of two regions. Assume that national defense is a pure public good; it is non-rival and non-excludable. Denote by q the amount of national defense. The total cost of producing q units of defense is C(q) = 107. In region 1, the total benefit of national defense is B1(q) = 109 - 22 In region 2, the total benefit of national defense is B2(q) = 200 – 22. (a) Derive the marginal benefit of defense for each region and the marginal cost of pro- viding defense as a function of q. (b) What is the socially optimal amount of investment in national defense? (c) Suppose region 1 has to finance all of the investment in national defense. What is the optimal amount of investment for region 1? (d) Suppose region 2 has to finance all of the investment in national defense. What is the optimal amount of investment for region 2? (e) Suppose the national government knows the socially optimal amount of defense, and proposes to fund it by requiring each region to pay half of the total cost. i. Calculate the change in welfare for each region under this policy, compared to the scenario where region 1 has to finance all the investment in national defense. ii. If the two regions could vote on whether to adopt this policy instead of having region 1 finance all defense investment, would both regions vote yes?

Answer 1

First four parts

B, (q) = 109-1 q² B2 (q) = 209 - 92 ans (92-109, a marginal Renefit for each region mB, and MB2 MB = dB, (q) = 10-9 Edda -) MB, = 10-9 MB = dB2(q) = 20-24 dait dedo IMB2=20-29 marginal cost Cnc = aboc decall wala MC=10 MARGINAL COST socially optimal amount of investment is the level where eggregate marginal benefit aquals marginal cost Aggregate Margina Benefit (AMB) - MBI MB2 = 30-39 - MB = 10-9 + 20-29 so MC- 10

investment : optimal level of MC-MB → 10 = 30- 3a = 39 = 20 = q = 20 = 6.61 Socially optimal 6.67 1. amount of lowestment is () - C Region I has all the investment. to finomce I optimal investment is the level where marginal benefit of region, is equal to marginal Cost =) MB, = mc s 10-9 = 10 => 9=0 cm of national defense for Optimal region amount I is to d. 2 Optimal forestment for region MB2-mc - 20-291 = 10 ) 29=10 =) 9-5 Optimal amount of national defense for regions