Please see the attached file.

A farmer's payoff matrix per plot is:
Good Rain Year Bad Rain Year
Coffee 140 27
Corn 64 50
Probability 1/4 3/4

Assume that the farmer has no access to insurance and has to make the plantint decision before she knows the outcome of the rain. The farmers utility for income is given by U(I)=I^(1/3).

1. Which crop does the farmer plant? I know it has something do with expected value being higher.

2. A device is invented that gives the farmer perfect information on the weather, how much would the farmer be willing to pay for the device? Is this similar to what value the farmer would be willing to pay for the information (which would be the difference)? Does it have to do with expected payoff again?

3. Assume, the farmer recieces less than a payoff of 50, but her son will send her enough money to ensure that her payoff is equal to 50, this should make her chose the more risky coffee shouldn't it?

Question 1
As you mention, the farmer should chose the crop that provides the highest expected utility. If he plants coffee, there's a ¼ probability of getting an income of 140, and ¾ of getting 27. Therefore, the expected utility from planting coffee is:

On the other hand, if he chooses to plant corn:

Therefore, since his expected utility is higher when he plants corn, then the farmer should plant that.

Question 2
This question has to do with the "expected value of perfect information". Let's see how to apply it in order to answer this. If the farmer had this device, he would know the weather in advance. Therefore, if the device told him that there would be Good rain, then he would plant Coffee (as his payoff is 140 in this case, versus 64 if he planted corn). On the other hand, if the device told him that there would be Bad rain, then the farmer would plant Corn (as his payoff would be 50 in this case, versus 27 if he planted coffee). So we've found that with this device:
• If there is Good Rain, payoff is 140
• If there is Bad Rain, payoff is 50

Since there is a ¼ chance that there is good rain, and ¾ that there is bad rain, the expected utility of the farmer, if he had this device, would be:

So, how much should he be willing to pay for the device. We've already established that, when he doesn't have the device, his expected utility is 3.7630. Therefore, the maximum he would be willing to pay for it would be an amount that takes his expected utility down to a minimum of 3.7630. If the device were more expensive than that, then the farmer would be better off without it. Calling X to the maximum amount he should pay for the device, we get the equation:

As you can see, these are the payoffs the farmer gets with the device, minus the amount he paid in order to buy it. We equate it to 3.7630, which is the expected utility he gets when he doesn't use the device.

Now, the above equation cannot be solved analytically, so I used Excel's Goal Seeker in order to do it. I found that he should a maximum of approximately $12.72 for the device. Let's check that this is correct:

We've found that when the device costs $12.72, he's indifferent between buying it or not. If it's more expensive, then he would be better off not buying it.

Question 3
In this case, there's no reason to plant Corn. If there is Good Rain, it's clearly better to plant Coffee. If there is Bad Rain, and the farmer knows that he will get at least 50, then it's the same to plant Coffee or Corn. Therefore, no matter the weather outcome, the payoff for the farmer if he plants Coffee will be greater than or equal to his payoff if he plants Corn.