Question

Suppose two individuals, Jon and David, form a community and would like to construct a communal fort that would protect them from attacks. They both consume good X, a private good, and the protection of the fort, P. One unit of good X costs 1 unit of currency, and one unit of P costs 2 units of currency. Both Jon and David have an income of 100 and a utility function of the form: U = log(Xi) + 2 × log(PJ + PD) The budget constraint for each is given by: Xi + 2 × Pi = 100

(a) Find the amount of protection Jon will provide as a function of how much David provides and explain why the relationship is the way it is.

(b) How much protection P will be privately provided in this case?

(c) Explain the economic intuition behind this amount and compare it to the socially optimal amount without solving for the socially optimal amount.

Answer 1

a. U = Log(X1) + 2Log (PJ + PD) (The utility given)

(-2/(100 – 2PJ)) + (2/(PJ + PD)) = 0

100 – 2PJ + PJ + PD = 0

100 – 3 PJ + PD = 0(Reaction is the same for PD)

100 – PJ = 3 PD

This is the amount of protection Jon will provide as a function of how much David provides. The relationship is this way because Jon and David have same income and utlity function, and they both also consume the same amount of good X.

b. Let PJ = x , PD = y

3x = 100 – y ( substituing values in above equation)

y= -3x + 100

Put y into equation:3(-3x + 100) = 100 – x

-9x + 300 = 100 – x

-8x = -200(-8x)/-8 = (-200)/-8

x = 25; y = 25

x+y= 25+25= 50

P=50

c. The economic intution behind this is that socially optimal amount is when price is equal to marginal cost in this problem. We can derive from the above problem that if the price is higher than the average total cost, then there would be a profit. It also implies that public goods will be undersupplied by the private market.