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 privately (when Ramzi and Yi-Fan each optimize, in reaction to the other's optimization, but without considering external benefits)? Answer to the nearest whole unit.

Answer 1

For Ramzi:

The budget constraint for each is given by:

The utility function U = 2 × ln(X_R) + ln(F_R + F_Y)

Substituting X_R = 200 - F_R, we get

U = 2 × ln(200 - F_R) + ln(F_R + F_Y)

For utility to be maximum for Ramzi, dU / dF_R = 0

d[2 ln(200 - F_R) + ln(F_R + F_Y)] / dF_R = 0

2*(2)/(200 - F_R) + 1/(F_R + F_Y) = 0

1/(F_R + F_Y) = -4/(200 - F_R)

(200 - F_R) = -4F_{R +} 4F_Y

3F_R = 200 ₊ 4F_Y

Similarly, since it is symmetric function, for Yi-Fan

3F_Y = 200 ₊ 4F_R

Their Nash equilibrium would exist at F_R = F_Y = F_E (as both of them have exact same reaction functions)

3F_E = 200 ₊ 4F_E

F_E = -200

Total flood protection (F) = 2F_E

2 * (-200)

Total flood protection (F) = -400

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