Suppose Liz and Scott are writing a report together. They have 24 hours to do ba

Question

Suppose Liz and Scott are writing a report together. They have 24 hours to do background research. The quality of their research will affect their prospects for raises and promotions. The more time they spend on the research, the better, but each wants the other to do most of the work. Let’s use the symbol X to indicate the number of hours Scott spends on research and the symbol Y to indicate the number of hours Liz spends. Both these numbers must be positive and neither can exceed 24. Suppose we can measure Scott and Liz’s costs and benefits on a utility scale (recall the discussion of utility in Section 4.4). When Scott works for X hours and Liz works for Y hours, each receives a total benefit of 60(X + 2Y ) (X + 2Y )2. The marginal benefit of Scott’s extra time is 60 2(X + 2Y ), while the marginal benefit of Liz’s extra time is 120 4(X + 2Y ). The cost of their effort is X2 for Scott and Y 2 for Liz, so the marginal cost is 2X for Scott and 2Y for Liz.

(a) Find the Nash equilibrium of this game.
(b) Is it a good outcome for Liz and Scott? Can they do better?

Explanation / Answer

a) Equilibrium for Scott: MC = Marginal benefit

2X = 60 - 2(X+2Y)

X+Y = 15

Equilibrium for Liz: MC = marginal benefit

2Y = 120 - 4X - 8Y

2X + 5Y = 60

Solving these two equations for X and Y,

Y* = 10

X* = 5

So nash equilibrium is (5,10)

b) No they cannot do better.

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