Suppose that X and Y are partners in business. Each puts in a non-negative amount of effort (x and y, respectively), and the resulting profits are…

Suppose that X and Y are partners in business. Each puts in a non-negative amount of effort (x and y, respectively), and the resulting profits are given by equation 1, below.

The partners split the profits equally. They make their effort decisions simultaneously. Each bears the cost of his or her own effort, so X’s and Y’s utilities are given by equations 2 and 3, respectively.

F(x,y)=2xy-0.5x^2-0.5y^2     (1)

uX(x,y)=0.5F(x,y)-x                 (2)

uY(x,y)=0.5F(x,y)-y                (3)

This game has a unique pure strategy Nash equilibrium. In this equilibrium, how much effort does X provide?

 
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