Question

) Suppose that 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(X_i) + 2 × log(P_J+ P_D)
The budget constraint for each is given by:
X_i+ 2 × P_i= 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

We need to maximize

log(X_i)+2log(PJ+PD)

where X_i+2P_i=100

Rearranging the constraint, we get

X_i=100-2P_i

Putting this into the maximization problem, we get

log(100-2P_i)+2log(PJ+PD)

Differentiating with respect to PJ and equating to zero, we get

(-2/(100-2PJ))+2/(PJ+PD)=0

Rearranging, we get 3PJ=100-PD

Doing the same for PD, we will get

3PD=100-PJ

Solving these, we get

PD=PJ=25

X1=X2=50

Both provide equal protection.

This is because their utility and budget function is same and symmetrical. Hence, they both get the same utility and so provide the same protection.

B. The total private protection provided is 25+25=50.

C. The socially optimal amount will be higher than 50. The reason being that public goods (which protection is) are underprovided by the private market as private goods take higher preference from individuals.