A farmer has the utility function U=C^0.5 and faces a choice between planting a
ID: 1151369 • Letter: A
Question
A farmer has the utility function U=C^0.5 and faces a choice between planting a risky high yield variety of maize or planting the traditional variety. The outcomes and probabilities are summarized in the table below:
Different from the example in class, the farmer is now endowed with an initial wealth level of 500. So his/her consumption is the amount that they get from the Maize planted +500. (e.g., if a farmer gets 1444 income from maize planting, their consumption is 1444+500=1944).
Q3. What is the Certainty Equivalent (CE) of the High-Yield Variety? (Hint: Recall the CE is the fixed amount of money, given with certainty, that makes the farmer indifferent between taking the CE and planting High-Yield Maize. Don't forget the farmer has an initial wealth level!)
[Note: No need for decimal places here]
Variety Probability of Good outcome Income in Good outcome Probability of Bad outcome Income in Bad outcome High-Yield Maize 0.5 1444 0.5 100 Traditional Maize 0.5 625 0.5 625Explanation / Answer
It is given that the farmer faces the following probabilities and their respective rewards from alternatives of planting high yield maize and traditional maize:
Variety
Probability of Good outcome
Income in Good outcome
Probability of Bad outcome
Income in Bad outcome
High-Yield Maize
0.5
1444
0.5
100
Traditional Maize
0.5
625
0.5
625
It has also been given that the utility function of the farmer is U= C^0.5
Certainty Equivalent (CE) as has been clarified in the question is the amount of money that the farmer needs to be given in order to make him/her indifferent between accepting the CE and planting the high yield maize. In other words, ‘indifference’ means that the expected utility from getting the CE is equal to the expected utility from planting the high yield maize.
Thus, (CE)^0.5 = 0.5* (1444+500)^0.5 + 0.5* (100+500)^0.5
Or (CE)^0.5 = 0.5* (1944)^0.5 + 0.5* (600)^0.5
Or (CE)^0.5 = 0.5* (44.090) + 0.5* (24.494)
Or (CE)^0.5 = 34.292
Or CE= 34.292^2
Therefore CE = $1175.94. Since it is mentioned that decimals are not required, the CE figure is rounded off to $1176.
Thus, the farmer has to be given a fixed amount of $1176 as Certainty Equivalent in order to make him/her indifferent between taking the money and planting the high yield maize.
Variety
Probability of Good outcome
Income in Good outcome
Probability of Bad outcome
Income in Bad outcome
High-Yield Maize
0.5
1444
0.5
100
Traditional Maize
0.5
625
0.5
625
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