101A_problemset6 - Econ 101A Problem Set 6 Due on Th. May 7...

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Econ 101A — Problem Set 6 Due on Th. May 7 in class. No late Problem Sets accepted, sorry! This Problem set tests the knowledge that you accumulated mainly in lectures 20 to 24. The problem set is focused on dynamic games and general equilibrium. General rules for problem sets: show your work, write down the steps that you use to get a solution (no credit for right solutions without explanation), write legibly. If you cannot solve a problem fully, write down a partial solution. We give partial credit for partial solutions that are correct. Do not forget to write your name on the problem set! Problem 1. Dynamic Games (48 points) Two companies produce the same good. In the f rst period, f rm1 seels its product as a monopolist on the West Coast. In the second period, f rm 1 competes with f rm 2 on the East Coast as a Cournot duopolist. There is no discounting between the two periods. Firm 1 produces quantity x W c/α at the West at cost cx W . On the East Coast, and that’s what makes this problem interesting, f rm 1 produces quantity x E at cost ( c αx W ) x E , where 0 <α<c< 1 / 2 . The parameter α captures a form of learning by doing. The more f rm 1 produces on the West Coast, the lower the marginal costs are going to be on the East Coast. As for f rm 2, it produces in the East market with cost cx 2 . The inverse demand functions are p W ( x W )=1 x W and p E ( x E ,x 2 )=1 x E x 2 . Each f rm maximizes pro f t. In particular, f rm 1 maximizes the total pro f ts from its West and East coast operations. 1. Consider f rst the case of simultaneous choice . Assume that f rm 2 does not observe x W before making its production decision. This means that, although formally f rm 1 chooses output x W f rst , that you should analyze the game as a symultaneous game between f rm 1 and f rm 2. Use Nash Equilibrium. Write down the pro f tfun c t iontha t f rm 1 maximizes (careful here) and the pro f t function that f rm 2 maximizes (5 points) 2. Write down the f rst order conditions of f rm 1 with respect to x W and x E , and the f rst order condition of f rm 2 with respect to x 2 . Solve for x W , x E , and x 2 . (4 points) 3. Check the second order conditions for f rm 1 and for f rm 2. (3 points) 4. What is the comparative statics of x W and x E with respect to α ? Does it make sense? How about the comparative statics of x 2 with respect to α ? (4 points) 5. Compute the pro f ts of f rm 2 in equilibrium. How do they vary as α varies? (compute the comparative statics) Why are f
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101A_problemset6 - Econ 101A Problem Set 6 Due on Th. May 7...

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