Alice and Bob both produce Caesar salad dressing in the same kitchen

QUESTION

Alice and Bob both produce Caesar salad dressing in the same kitchen.Each of them has an endowment of one unit of labor that can be used aseffort in producing the dressing, or can be used to play video games. BothAlice and Bob have identical preferences given by:U(x, y) = ln(x) + y, (1)where x is the hours of TV watched and y is the output of Caesar salad dressing.If an amount of effort, zi, is used in Caesar salad dressing production by person i, then the amount of dressing produced is given by:yi = fi(zi) = Sizi, (2)for i ? {alice, bob}. Note that Siis the marginal product of productioneffort for person i. An hour of video game playing is produced one-foronefrom the labor endowment. That is (by substituting in the resourceconstraint), xi = 1 ? zi.The thing is that making the dressing is hard when the other person is not in the kitchen very much (because it is time-consuming to look for the spoons etc, and it gets very boring and lonely). In particular, it turns out that:Sa = czb, and Sb = cza, (3)where c > 2.Suppose that Alice guesses that Bob will spend z?bhours in the kitchen.Write out Alice’s optimization problem.2. What is her best response number of hours in the kitchen, za(z?b)? Graphthis best response in (za, zb) space.(a) Do the same exercise for Bob, but place his best response on the same graph that you have just drawn.Hint: The two are identical,so the problem (and graph) should be very similar.i. Graphically identify the Nash equilibrium levels of productioneffort, (z?a, z?b). Explicitly solve for the solution when c = 3.Hint: You will need to use the quadratic formula.

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