ECN700-midterm1 - perfect If an equilibrium is not subgame...

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ECG 700 Midterm In order to receive full points, you must carefully and thoroughly justify your answers. Question 1 - 30 points A person’s utility is given by the Cobb-Douglass function u ( x, y ) = x 1 2 y 1 2 . (a) Show that the optimal demand functions for goods x and y are x ( p x , p y , I ) = I 2 p x and y ( p x , p y , I ) = I 2 p y . (b) Find the Indirect utility function V ( p x , p y , I ). (c) Find the Expenditure function E ( p x , p y , ¯ U ). (d) Find the compensated demand functions x c ( p x , p y , ¯ U ) and y c ( p x , p y , ¯ U ). Question 2 - 20 points In the game below, the first number is player 1’s payo±. The second number is player 2, so (1,5) means player 1 gets a payo± of 1 and player 2 gets a payo± of 5. Player 1 Player 2 R (2,2) L Player 1 R (3,3) L (0,0) d (1,5) u (a) What is the backward induction (subgame perfect) equilibrium of this game? (b) Write the game in normal form. (c) What are all the pure strategy equilibria of the game. For each, identify whether or not it is subgame
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Unformatted text preview: perfect. If an equilibrium is not subgame perfect, explain why not. Question 3 - 15 points For the following game: L R U 1 , 1 3 , 4 D 4 , 3 2 , 2 (a) Solve for all pure strategy Nash equilibria (b) Solve for all mixed strategy Nash equilibria Question 4 - 20 points The members of a hierarchical group of hungry lions face a piece of prey. If Lion 1 does not eat the prey, the prey escapes and the game ends. If it eats the prey, it becomes fat and slow, and Lion 2 can eat it. If Lion 2 does not eat Lion 1, the game ends. If Lion 2 eats Lion 1, then it may be eaten by Lion 3, and so on. Each lion prefers to eat than to be hungry, but prefers to be hungry than to be eaten. There are a total of 139 lions. Find the subgame perfect equilibrium of this game....
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