Question

Suppose that two individuals, Ramzi and Yi-Fan, form a community along a river. They would like to construct a dam that would protect them from floods. They both consume X, a private good, and flood protection, F. One unit of good X costs $1, and one unit of F costs $1. Both Ramzi and Yi-Fan each have an income of $200 and a utility function of the form:

U = 2 × ln(X_i) + ln(F_R + F_Y)
The budget constraint for each is given by:
X_i + F_i = 200

How much total flood protection F will be provided in the social optimum (when Ramzi and Yi-Fan agree to pool their resources and build a dam that maximizes their total benefits)? Answer to the nearest whole unit.

Answer 1

Suppose that two individual Ramzi and Yi-Fan form a commodity along a river. They would like to construct a dam that would protect them from floods. So thay both consume X, a private good and Y, flood protection. One unit of good X costs $1 and one unit of F costs $1 , both Ramzi and Yi - Fan each have an income of $200 and a utility function of form :

U = 2 * ln(X_i) + ln(F_R+F_Y) the budget constraint for each is given by : X_i + F_i = 200

Here, i = consumer 1 and consumer 2. Both are consume private good X and flood protection F

F_R = for Ramzi consumes flood protection F_Y = for Yi-Fan consumes flood protection.

We can assume that F_i= F_R + F_y for the total flood protection.

We can put this on utility function and we can get : U = 2 * ln (X_i) + ln (F_i) [where F_i= F_R + F_Y] the budget constraint : X_i + F_i= 200

By lagrangian method to calculate their maximization total benefit and we get

L = 2 * ln (X_i) + ln (F_i) + [200 - X_i - F_i]

The first order conditions are - dL/ dX_i = 2/X_i - = 0 ---- (1) dL/dF_i = 1/F_{​​​​​​i} - = 0 ------ (2) dL/d = 200 - X_i - F_i = 0 ------- (3)

Equation (1) and (2) are equal so we can write

2/X_i = 1/F_i    or, X_i = 2F_i

We can put this value on equation (3) and we can get the total flood protection for both consumers.

200 - 2F_i - F_i = 0 or, 200 - 3F_i = 0 or, 3F_i = 200 or, F_i = 200/3 or, F_i = 66.666

So the total flood protection F will be 66.666. [ F_i = 66.666 , or, F_R + F_Y = 66.666]

And also we can get the private goods which the consumes by putting the value of F_iI in equation 3

200 - X_i - 66.666 = 0 or , 133.334 - X_i = 0 or , X_i = 133.334