Question

This business decision looks at the impact an externality has on an Investment Choice. A dynamic externality occurs when (as a group) our animals impose a cost on the future provision of the good produced in the commons, that is we can cause environmental damage through overgrazing. There is one pasture we share in common—there are just the two of us using the pasture. On this pasture, milk production as a function of total herd size is as follows:

# of animals	Liters of milk produced
0	0
5	10
10	20
15	30
20	36
25	40
30	44

As you can see, the amount of milk produced increases with the size of the herd—however, it increases at a decreasing rate.

For each livestock owner, the share of this total milk produced they receive is a function of your share of the total herd. The Price of milk is $1 per liter.

For each animal put on the pasture, it costs $1 in private labor costs. (5 animals costs $5, 10 animals costs $10,…)

So if I have 5 animals and you have 5 animals, my payoff is (5/10)*20-5, or 5. If you had 15 animals and I had 5, then it is (5/20)*36-5, or 4.

              a—given no market intervention, what is the Nash Equilibrium if there are two farmers?

              b—What is the Pareto Optimal Equilibrium?

c— What would each investor need to do to create the Optimal Equilibrium? Show your work on all answers!

d—This is an arithmetic question—it does not require extremely high level math. If the government wished to maximize its tax revenue on Milk, how much would it charge given the information provided in the introduction (note—you will create a new equilibrium)?

Answer 1

Answer a :-

In the above matrix On the left side , the number of animals held by Player 1 is demonstrated while on the above the choices of player 2 are mentioned .

Pia-I PlaX 2:33 5 Jo i0 5 9.33 3 15 15 7 / 19-3 s 66 30 a0 20 3 467 30 0

It is evident from the above that some spaces are blank .This is because this represents the choices of player 1 can be more than the lowest number of animals livestock held by both .

The Nash equilibrium in the above payoff matrix are :-

1) 5 animals each

2) 10 animals each

3)15 animals each

Answer b :-

Pareto optimum equilibrium is 15 animals each .

Payoff of player 1 = 15/30*44-15=7

Payoff of player 2 = 15/30*55-15=7

Answer c :-

Each player has to purchase equal amount of animals to create equilibrium .

Let us consider another example

Each player buys 10 animals .

Now each one's share will be 10/20*36-10=3

Answer d :-

Tax can only be charged on income .Now let consider following examples

1) When both purchases 5 animals each and the revenue generated is 10 . There is no chance of taxation because cost = revenue

5*2=10

2) When both purchase 10 animals each then

Cost is 20 (10*2)

Revenue is 10/20*36-10=3 each

Now revenue generated is 36

Thus 36 -10*2-3-3=10

Thus maximum revenue charged can 10

3) When both purchase 15 each then

Maximum tax charged can be 44-15*2-7-7=0

Thus if government charge taxes new equilibrium will be formed at 10 animals each with government tax revenue of $10