Joe:

y= 100 (High) prob = 2/5
=25 (medium) prob= 2/5
=0 (low) prob = 1/5

Calculate:
1. José’s risk premium, p, associated with farming and
2. José’s certainty equivalent associated with farming.

The village decides to implement an informal insurance (risk sharing) arrangement. For this part assume there is no asymmetric information or enforcement problems. Let tH, tM, and tL denote the transfer made by a farmer into the village insurance fund under the High, Medium, and Low outcomes (a negative t means the farmer receives a payment). Find the values of tH, tM, and tM in an ideal (first best) insurance arrangement (remember an ideal risk sharing arrangement will achieve the most possible consumption smoothing).

ohh and I forgot to add:

José lives in a village with many identical farmers. Specifically, they all have the same utility function and income levels and probabilities under farming. All risk in the village is idiosyncratic risk.
--extra info for part D!

To calculate José's risk premium and certainty equivalent associated with farming, we need to determine the expected value of the income under each outcome.

1. Risk Premium:
The risk premium represents the additional amount of money that a risk-averse person like José would be willing to pay to avoid uncertainty. It is calculated by comparing the expected value of income in the risky situation to the certain situation.

The expected value of income can be calculated by multiplying each income level by its corresponding probability and then summing them up. In this case, we have:

Expected value = (100 * 2/5) + (25 * 2/5) + (0 * 1/5)
= (200/5) + (50/5) + (0/5)
= 40 + 10 + 0
= 50

Therefore, the expected value of income for José in farming is 50.

To calculate the risk premium, we subtract the expected value of income from the highest income level (100) in order to compensate for the risk:

Risk premium = 100 - 50
= 50

So, José's risk premium associated with farming is 50.

2. Certainty Equivalent:
The certainty equivalent is the guaranteed amount of income that a risk-averse individual would accept instead of facing the uncertainty.

To determine the certainty equivalent, we find the amount of money that, when received for sure, would provide the same utility (satisfaction) to José as the uncertain income.

In this scenario, José's income levels are 100, 25, and 0 with their respective probabilities of 2/5, 2/5, and 1/5.

We need to find the income level that would give José the same utility as the expected value of income (which we already calculated as 50). This can be done by trial and error or using utility functions if provided.

Assuming José's utility is linear (i.e., each additional dollar of income provides the same utility), we can set up the following equation:

CE = p1 * u(x1) + p2 * u(x2) + p3 * u(x3)

Where CE is the certainty equivalent, p1, p2, p3 are the probabilities, and x1, x2, x3 are the income levels.

Using the given probabilities and income levels, we have:

CE = (2/5 * 100) + (2/5 * 25) + (1/5 * 0)
= 40 + 10 + 0
= 50

The certainty equivalent is 50, which is the same as the expected value of income.

Therefore, José's certainty equivalent associated with farming is 50.

Now, let's move on to the second part of the question about the values of tH, tM, and tL in an ideal (first best) insurance arrangement.

In an ideal risk-sharing arrangement, the transfers made by a farmer into the village insurance fund should offset the income losses in the low and medium outcomes and provide additional income in the high outcome. This aims to achieve the most possible consumption smoothing.

Since the probabilities are given as 2/5 for high and medium outcomes, and 1/5 for the low outcome, the transfers into the insurance fund should be designed accordingly.

For a first best insurance arrangement, we want to ensure that the expected income for José remains the same under all outcomes, i.e., 50. Therefore, the transfers should be such that:

tH * (2/5) + tM * (2/5) + tL * (1/5) = 50

However, since the actual values of tH, tM, and tL are not provided in the question, it is not possible to determine their exact values without more information.

In an ideal risk-sharing arrangement, tH should compensate for the expected income loss under the low outcome, tM should compensate for the expected income loss under the medium outcome, and tL should provide additional income in the high outcome to achieve consumption smoothing.

Overall, the exact values of tH, tM, and tL would depend on the specific insurance contract and the village's risk-sharing mechanism, which are not defined in the question.