question archive 1) In a small town, there are three individuals
1) In a small town, there are three individuals
Subject:EconomicsPrice:2.86 Bought7
1) In a small town, there are three individuals. Individual A owns a mill. Individual B owns a field that can grow wheat. Individual C is a worker without ownership of anything. If they work on their own, Individual A produces nothing (nothing to mill), Individual B produces 1 unit of output (they can grow wheat and work it by hand), and Individual C produces 0 units of output (nowhere to work). If any two get together: Individuals A and B working together product 6 units of output. Individuals A and C working together produce 0 units of output. Individuals B and C working together produce 3 units of output. If all three work together, they produce 10 units of output. For each individual A, B, and C: What is the most this individual could receive in a core allocation, and what is the least this individual could receive in a core allocation? (Note - not necessarily the same core allocation for all questions. This is "of all the many core allocations here, what is the most A ever gets? What is the least A ever gets?" etc.)
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Start by writing the participation constraints:
Individual:
A ≥ 0 ..............(1)
B ≥ 1 ..............(2)
C ≥ 0 ..............(3)
Any two participants:
A + B ≥ 6 .......... (4)
A + C ≥ 0 ..........(5)
B + C ≥ 3.......... (6)
And the resource constraint:
A + B + C = 10 ........... (7)
Try rewriting the resource constraint and plugging it into the two-player participation constraints. But keep the individual participation constraints in mind...
A = 10 − B − C by the resource constraint. Plug that into (4) above:
10 − B − C + B ≥ 6
10 − C ≥ 6
C ≤ 4
Rewrite the resource constraint as C = 10 − A − B and plug that into (5):
A + 10 − A − B ≥ 0
B ≤ 10 (not very informative...but true.)
Rewrite the resource constraint as B = 10 − A − C and plug into (6):
10 − A − C + C ≥ 3
A ≤ 7
So, we have a core where 0 ≤ A ≤ 7, 1 ≤ B ≤ 10, and 0 ≤ C ≤ 4. A wide range, but anything where these are true while maintaining A + B + C = 10 is in the core. For the answer to this question, you needed to give these upper bounds.
