Click the link above to submit the assignment.Week 3 Homework Problems:Chapter 5: 5.1(a-b), 5.2(a-c), 5.3(a-c), 5.5(a-c), 5.10(a-c)Chapter 6: 6.2(a-e), 6.4(a-e), 6.6(a-d), 6.7(a-e), 6.9 (a-d)Note: Homework problems are worth 2 points for each part of the problem.5.3
Return to the game given by the payoff matrix in Problem.
Two classmates A and B are assigned an extra credit group
project. Each student can choose to Shirk or Work. If one or more players
choose Work, the project is completed and provides each with extra credit
valued at 4 payoff units each. The cost of completing the project is that 6
total units of effort (measured in payoff units) is divided equally among all
players who choose to Work and this is subtracted from their payoff. If both
shirk, they do not have to expend any effort but the project is not completed,
giving each a payoff of 0. The teacher can only tell whether the project is
completed and not which students contributed to it.
a. Write down the extensive form for the simultaneous- move
game.
b. Suppose the game is now sequential move, with A moving
first and then B. Write down the extensive form for this sequential-move game.
c. Write down the normal form for the sequential-move game.
Find all the Nash equilibria. Which Nash equilibrium is sub game-perfect?
5.5
Two classmates A
and B are assigned an extra credit group project. Each student can choose to
Shirk or Work. If one or more players choose Work, the project is completed and
provides each with extra credit valued at 4 payoff units each. The cost of
completing the project is that 6 total units of effort (measured in payoff
units) is divided equally among all players who choose to Work and this is
subtracted from their payoff. If both shirk, they do not have to expend any
effort but the project is not completed, giving each a payoff of 0. The teacher
can only tell whether the project is completed and not which students
contributed to it.
a. Write down the normal form for this game, assuming students choose to Shirk
or Work simultaneously.
b. Find the Nash equilibrium or equilibria.
c. Does either player have a dominant strategy? What game from the chapter does
this resemble?
5.10
Consider the Tragedy of the Commons game from
the chapter with two shepherds, A and B, where sA and sB denote the number of
sheep each grazes on the common pasture. Assume that the benefit per sheep (in
terms of mutton and wool) equals
300 â sA â sB
Implying that the total benefit from a flock of sA sheep is
sA (300 â sA â sB)
And that the marginal benefit of an additional sheep (as one can use calculus
to show or can take for granted) is
300 â 2sA â sB:
Assume the (total and marginal) cost of grazing sheep is zero since the common
can be freely used.
a. Compute the flock sizes and shepherdsâ total benefits in the Nash
equilibrium.
b. Draw the best-response-function diagram corresponding to your solution.
c. Suppose Aâs benefit per sheep rises to 330 â sA â sB. Compute the new Nash
equilibrium flock sizes. Show the change from the original to the new Nash
equilibrium in your best-response function diagram.
Chapter 6
Problem 6.2
Frisbees are produced according to the production
function q = 2K+L where q =output of frisbees per hour, K =capital input per
hour, L =labor input per hour.
a) If K = 10, how much L is needed to
produce 100 frisbees per hour?
b) If K = 25, how much L is needed to
produce 100 frisbees per hour?
c) Graph the q = 100 isoquant. Indicate the
points on that isoquant de?ned in part a and part b. What is the RTS along this
isoquant? Explain why the RTS is the same at every point on the isoquant.
d) Graph the q = 50 and q = 200 isoquants
for this production function also. Describe the shape of the entire isoquant
map.
e) Suppose technical progress resulted in
the production function for frisbees becoming q = 3K + 1.5L. Answer part a through
part d for this new production function and discuss how it compares to the
previous case.
Problem 6.6
Power
Goat Lawn Company uses two sizes of mowers to cut lawns. The smaller mowers
have a 24-inch blade and are used on lawns with many trees and obstacles. The
larger mowers are exactly twice as big as the smaller mowers and are used on
open lawns where maneuverability is not so difficult. The two production
functions available to Power Goat are:
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a. Graph the q = 40,000 square
feet isoquant for the first production function. How much k and l would be used
if these factors were combined without waste?
b. Answer part (a) for the
second function.
c) how
much k and l would be used without waste if half of the 40,000-square-foot lawn
were cut by the method of the first production function and half by the method
of the second? How much k and l would be used if three fourths of the lawn were
cut by the first method and one fourth by the second? What does it mean to
speak of fractions of k and l?
d. On the basis of your
observations in part (c), draw a q = 40,000 isoquant for the combined
production functions.
