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Pareto optimal levels of effort

There is a team technology in which two workers must work together in order
to produce. The revenue from their efforts is given by
R = (e1e2)^1/2
where ei is the effort that worker ∈ {1, 2} puts into production. Each worker's
utility is given by
ui = wi − ei^2
where wi is worker i's income.

(a) What are the Pareto optimal levels of effort (this maximizes total surplus
from production minus the costs of effort)?

(b) If the sharing rule is wi =R/2 , for both agents. How much effort will each
worker put in a Nash equilibrium if they cannot observe each other's effort
level?

(c) Try and obtain the sharing rule wi = bR, where b is a scalar, such that in a
Nash equilibrium the workers will have an incentive to put in the optimal
effort obtained in part a). Is this sharing rule reasonable? How could this
sharing rule be implemented?

Solution Preview

(a) What are the Pareto optimal levels of effort (this maximizes total surplus from production minus the costs of effort)?
For worker 1, his utility is
The total output is R = (e1e2)^1/2
And the cost is assume to be linear to effort: C = e1+e2
Then maximize Surplus = R-C = (e1e2)^1/2 - (e1+e2)
First order condition gives:
dS / de1 = 1/2(e2/e1)^1/2 - 1 = 0
dS / de2 = 1/2(e1/e2)^1/2 - 1 = 0
or,
1/2(e2/e1)^1/2 =1 (1)
1/2(e1/e2)^1/2 =1 (2)
Then (2) / (1) gives:
e1 / e2 = 1
i.e.
e1= e2
which maximizes the total surplus.

(b) If the sharing ...

Solution Summary

Pareto optimal levels of effort are determined.

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