Determine the nash equilibria in pure strategies

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Reference no: EM133205697

Assignment:

1. Calculate the expected value of each of the following gambles.

a. win $100 with probability 1/6; win $10 with probability 1/3; lose $5 with probability ½.

b. win $1,000,000 with probability .0001; lose $5 with probability .9999.

c. win $700 with probability .5; lose $300 with probability .5

d. win $10 with probability .9; lose $400 with probability .1.

2. Bill has a utility of money or wealth function given by u(x) = x^1/2 and faces a coin flip (equal .5 probabilities) to win or lose $1,000. Calculate his expected utility for this gamble if his initial wealth level is a) $5,000; b) $20,000; c) $100,000. Then take this expected utility from the gamble and determine the level of wealth that would give Bill this level of utility. [Hint: you will be solving a problem of the form x^1/2 = 8.769, so take the square of the numerical value for expected utility.] This value is known as the certainty equivalent. Divide the certainty equivalent by the initial wealth level in each case. What is happening to this ratio as Bill's wealth level goes up?

3. Consider a game of the following form: Two players interact and each has two actions after picnicking at a park: Throw their trash away in garbage cans (we'll call this Throw), and leave their trash out (we'll call this litter). Consider the payoff matrix given below, with all the payoffs specified except one for each player designated X. Each player will have the same payoff value in this cell of the matrix. Give a number value for X for which this game is the traditional prisoner's dilemma, and then give a number value for X for which this game is not the prisoner's dilemma. [There are multiple values which correctly answer each part; you only need pick one to answer the question.]

	Throw	Litter
Throw	5, 5	-1, X
Litter	X, -1	1, 1

4. Using the underlining method, find all the Nash equilibria in pure strategies of the following normal form game.

	W	X	Y	Z
A	0, 2	7, 5	2, 4	4, 0
B	4, 3	5, 4	-1, 6	5, -2
C	2, 5	2, 3	6, 8	1, 2
D	1, 7	4, 4	4, 1	2, 5

5. Show whether the following production functions exhibit increasing, constant, or decreasing returns to scale.

a. Q = 2L + 3K

b. Q = L + 5K + 10

c. Q = min (2*L, K)

d. Q = 10*K*L

e. Q = L² + K² f. Q = K^.5*L^.5/2 + 10

6. Let the following combinations of capital and labor allow Acme Products to produce 10 Road Runner Traps. Find the cost of producing Q = 10 using each method if a) the wage is $10 and the price of capital is $20; b) the wage is $25 and the price of capital is $10.

Labor	Capital
1	10
2	5
3	3
6	2
12	1

7. Let the production function for thingamajigs be Q = L*K^1/2, the price of labor be $10 and the price of capital be $20. In the short run, the amount of capital is fixed at K = 100. Complete the following short run cost table for this firm (on your own paper, not this sheet!).

Quantity	Labor	Total Fixed Cost	Average Fixed Cost	Total Variable Cost	Average Variable Cost	Total Cost	Average Total Cost
10
50
100
200
300
400
500

8. In each of the following cases, determine if Acme Products is minimizing cost.

a. Wage = $10, Price of Capital = $20, MP_Labor = 3, MP_Capital = 8;

b. Wage = $5, Price of Capital = $15, MP_Labor = 2, MP_Capital= 6;

c. Wage = $15, Price of Capital = $10, MP_Labor = 6, MP_Capital= 4;

d. Wage = $20, Price of Capital = $15, MP_Labor = 6, MP_Capital = 8;

9. On an isocost-isoquant diagram, draw the situation of affirm which is initially minimizing the cost of producing Q units of output. Then show how this firm's cost-minimizing combination of capital and labor to produce Q changes when the wage of labor rises.

Reference no: EM133205697

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