Economic problem 2?

Question

Five stores have a problem with teft, and security is a public good. Use S to stand for the number of person-hourse of security patrols per week. The marginal benefit of security patrols to each of the stories is given by the formula

MB=300/ (1+S)

Patrols cost $ 20 per hour. What is the socially efficient level of security? If security is left to the independent decisions od the stores, what will they choose?

Answer 1

MC-Marginal cost

MC=20

TB-total benefit

TC-total cost

Equilibrium condition (individual choice):

MB=MC

300/(1+S)=20

300/20=1+S

15=1+S

S=15-1=14

Equilibrium condition (social):

TB=TC

TC=P*S=20S

TB=∫MB  (we will assume that there is no any fixed residuals)

∫MB=20S

∫MB/20=S

S=11'187

Answer:

If security is left to the independent decisions od the stores, they will choose S=14

Socially efficient level of security is at S=11'187 (though I may solved integral wrong)