3. Farmer Jones and Farmer Smith graze their cattle on the same field. If there are 20 cows grazing in the field, each cow produces $4000 of milk over its lifetime. If there are more cows in the field, then each cow eats less grass, and its milk production falls. With 30 cows on the field, each produces $3000 of milk; with 40 cows, each produces $2000 of milk. Cows cost $1000 apiece. (a) Assume Farmer Jones and Farmer Smith can each purchase either 10 or 20 cows but that neither knows how many the other is buying when he makes his purchase. Calculate the payoffs of each outcome; (b) What is the likely outcome of this game? What would be the best outcome? Explain.

What is the answer to the above question

To determine the payoffs of each outcome, we can create a pay-off matrix.

(a) Pay-off Matrix:

Jones buys 10 cows Jones buys 20 cows
Smith buys 10 cows $60,000, $60,000 $80,000, $40,000
Smith buys 20 cows $40,000, $80,000 $60,000, $60,000

In the pay-off matrix:
- The first number represents the pay-off of Farmer Jones.
- The second number represents the pay-off of Farmer Smith.

Let's calculate the payoffs:

- If both farmers buy 10 cows each:
- If Farmer Jones buys 10 cows and Farmer Smith buys 10 cows, each will have 10 cows grazing in the field. The total milk production will be $4000 x 10 = $40,000 per cow, resulting in $40,000 x 10 = $400,000 total milk production. The total cost of purchasing cows will be $1000 x 10 = $10,000 per cow, resulting in $10,000 x 10 = $100,000 total cost. Therefore, the payoffs will be ($400,000 - $100,000) = $300,000, for both farmers.

- If Farmer Jones buys 10 cows and Farmer Smith buys 20 cows:
- In this case, Farmer Smith will have 20 cows grazing in the field, and each cow's milk production will be reduced to $3000. The total milk production will be $3000 x 20 = $60,000 per cow, resulting in $60,000 x 20 = $1,200,000 total milk production. The total cost of purchasing cows will be $1000 x 10 + $1000 x 20 = $30,000 cost for Jones and $1000 x 20 = $20,000 cost for Smith, resulting in a combined cost of $50,000. Therefore, the payoffs will be ($1,200,000 - $50,000) = $1,150,000 for Smith and ($60,000 x 10 - $30,000) = $570,000 for Jones.

- If Farmer Jones buys 20 cows and Farmer Smith buys 10 cows:
- In this case, Farmer Jones will have 20 cows grazing in the field, and each cow's milk production will be reduced to $3000. The total milk production will be $3000 x 20 = $60,000 per cow, resulting in $60,000 x 20 = $1,200,000 total milk production. The total cost of purchasing cows will be $1000 x 20 = $20,000 cost for Jones and $1000 x 10 + $1000 x 20 = $30,000 cost for Smith, resulting in a combined cost of $50,000. Therefore, the payoffs will be ($1,200,000 - $20,000) = $1,180,000 for Jones and ($60,000 x 10 - $30,000) = $570,000 for Smith.

- If both farmers buy 20 cows each:
- If Farmer Jones buys 20 cows and Farmer Smith buys 20 cows, each will have 20 cows grazing in the field. The total milk production will be $2000 x 20 = $40,000 per cow, resulting in $40,000 x 20 = $800,000 total milk production. The total cost of purchasing cows will be $1000 x 20 = $20,000 per cow, resulting in $20,000 x 20 = $400,000 total cost. Therefore, the payoffs will be ($800,000 - $400,000) = $400,000, for both farmers.

(b) The likely outcome of this game depends on the rational decision-making of the farmers. However, considering the payoffs from the pay-off matrix, the Nash equilibrium (likely outcome) is for both farmers to buy 20 cows each, resulting in equal payoffs of $400,000 for each farmer. This is because, by buying 20 cows each, they maximize their milk production while minimizing their costs, leading to a relatively balanced outcome.

The best outcome in this game would be if both farmers coordinate and agree to buy only 10 cows each. This would result in the highest combined milk production ($400,000) and lower costs ($100,000). However, achieving this outcome would require cooperation and communication between the farmers, which cannot be assumed in this scenario.

To calculate the payoffs of each outcome in this game, we need to consider the different strategies and potential outcomes for Farmer Jones and Farmer Smith. Let's break it down step by step:

(a) The possible number of cows each farmer can purchase is either 10 or 20. Since neither farmer knows how many cows the other is buying when they make their purchase, we can analyze the payoffs for each combination of choices.

Possible outcomes for Farmer Jones:
1. If Farmer Jones buys 10 cows and Farmer Smith buys 10 cows:
- Each cow produces $4000 of milk, so the total milk production for both farmers is $4000 * 10 = $40,000.
- The cost of the cows is $1000 * 10 = $10,000.
- Jones' payoff in this case would be $40,000 - $10,000 = $30,000.

2. If Farmer Jones buys 10 cows and Farmer Smith buys 20 cows:
- Each cow produces $3000 of milk, so the total milk production for both farmers is $3000 * 30 = $90,000.
- The cost of the cows is $1000 * 10 = $10,000.
- Jones' payoff in this case would be $90,000 - $10,000 = $80,000.

3. If Farmer Jones buys 20 cows and Farmer Smith buys 10 cows:
- Each cow produces $3000 of milk, so the total milk production for both farmers is $3000 * 30 = $90,000.
- The cost of the cows is $1000 * 20 = $20,000.
- Jones' payoff in this case would be $90,000 - $20,000 = $70,000.

4. If Farmer Jones buys 20 cows and Farmer Smith buys 20 cows:
- Each cow produces $2000 of milk, so the total milk production for both farmers is $2000 * 40 = $80,000.
- The cost of the cows is $1000 * 20 = $20,000.
- Jones' payoff in this case would be $80,000 - $20,000 = $60,000.

Possible outcomes for Farmer Smith can be derived in a similar manner.

(b) To determine the likely outcome of this game, we need to consider the payoffs for each combination of choices and the rational behavior of the farmers. We can observe the following:

- If Farmer Jones buys 10 cows, Farmer Smith's best choice is to buy 20 cows, as it gives him a higher payoff ($90,000) than buying 10 cows ($40,000).
- If Farmer Jones buys 20 cows, Farmer Smith's best choice is again to buy 20 cows, as both options provide the same payoff ($80,000).

Based on this analysis, it is likely that both farmers will choose to buy 20 cows. This is known as the Nash equilibrium, where neither player has an incentive to change their strategy unilaterally.

The best outcome in terms of total payoff would be if both farmers chose to buy 10 cows, resulting in a total milk production of $40,000 and a higher joint payoff ($60,000) compared to the other possibilities.

However, since the farmers are acting independently and do not have information about each other's choices, it is unlikely that they will reach this best outcome. Instead, they are likely to settle on the Nash equilibrium of both buying 20 cows, resulting in a joint payoff of $160,000 ($80,000 per farmer).

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