Suppose that there is a common resource of size y in a two period society. Each of two citizens, one and two, can withdraw a nonnegative amount c1 or c2 for consumption in period one, provided that c1+c2 <=y . In the event that they attempt to consume in excess of what is available, the total amount will be divided equally between them, i.e., each citizen gets to consume y/2. When total consumption in period one is less than y , the leftover amount,y-c1-c2 , forms the resource base for consumption in period two. In period two, each citizen gets to consume (y-c1-c2)/2 .
Suppose that citizen has the utility function,
Ui (c1,c2)= ln(ci) + ln [(y-c1-c2)/2]
, for i=1,2.
Find the Nash equilibrium pattern of resource use in this game of individual competition and compare it to the pattern of resource use that would be socially optimal.
P.S: i know how to solve the Nash quilibrium for individual, but i don't know how to come up with the social optimization.
Thank you
To find the Nash equilibrium pattern of resource use in this game of individual competition, we need to determine how each citizen would choose their consumption levels.
First, let's analyze citizen 1's decision-making process. Citizen 1 wants to maximize their utility function, which is U1(c1, c2) = ln(c1) + ln[(y - c1 - c2)/2]. To find the best response of citizen 1, we can maximize this utility function by taking the partial derivative with respect to c1 and setting it equal to zero:
dU1/dc1 = 1/c1 - 1/(y - c1 - c2) = 0.
Solving this equation will give us citizen 1's best response consumption level for c1, denoted as c1*.
Similarly, we can analyze citizen 2's decision-making process. Citizen 2 wants to maximize their utility function, which is U2(c1, c2) = ln(c2) + ln[(y - c1 - c2)/2]. To find the best response of citizen 2, we can maximize this utility function by taking the partial derivative with respect to c2 and setting it equal to zero:
dU2/dc2 = 1/c2 - 1/(y - c1 - c2) = 0.
Solving this equation will give us citizen 2's best response consumption level for c2, denoted as c2*.
The Nash equilibrium occurs when both citizens are choosing their best responses simultaneously. Therefore, we need to find the values of c1* and c2* that satisfy both equations simultaneously.
Once we find the Nash equilibrium pattern of resource use, we can compare it to the pattern of resource use that would be socially optimal. In this case, social optimization would mean maximizing the sum of the utilities of both citizens, U1 + U2, subject to the constraint of c1 + c2 <= y. This can be solved using mathematical optimization techniques, such as Lagrange multipliers or the Kuhn-Tucker conditions.
To find the solution for social optimization, we need to set up the Lagrangian function:
L(c1, c2, λ) = U1(c1, c2) + U2(c1, c2) + λ(y - c1 - c2).
Then, we take the partial derivatives with respect to c1, c2, and λ and set them equal to zero. This will give us the necessary conditions for the social optimum consumption levels of c1 and c2.
By comparing the Nash equilibrium pattern of resource use to the socially optimal pattern, we can determine if there is any difference between the individual competition outcome and the socially optimal outcome.