Final answer:
To find the certainty equivalent, asking price, and risk premium, we need to calculate the expected utility of each outcome of the lottery. For the utility function u(x) = x^0.75, the certainty equivalent is 8.8312, the asking price is 91.1688, and the risk premium is -87.3734. For the utility function u(x) = x^1.5, the certainty equivalent is 10.5856, the asking price is 89.4144, and the risk premium is -57.2934.
Step-by-step explanation:
To find the certainty equivalent, asking price, and risk premium, we need to calculate the expected utility of each outcome of the lottery. The expected utility for the first outcome is -10^0.75 * 0.25 = -1.6818, and for the second outcome is 10^0.75 * 0.75 = 5.4772. The expected utility of the lottery is the sum of these two expected utilities, which is 5.4772 - 1.6818 = 3.7954.
The certainty equivalent is the amount of certain wealth that would give the same utility as the expected utility of the lottery. Let's say the certainty equivalent is c. Then we have c^0.75 = 3.7954. Solving this equation, we find that c = 3.7954^(1/0.75) = 8.8312.
The asking price is the amount of wealth the consumer would be willing to give up in order to avoid the lottery. In this case, the asking price is 100 - c = 100 - 8.8312 = 91.1688. The risk premium is the difference between the expected value of the lottery and the asking price, which is 3.7954 - 91.1688 = -87.3734.
The graph would show the utility function u(x) = x^0.75 increasing with x, and the budget constraint line representing the lottery. The certainty equivalent, asking price, and risk premium would be shown on the graph as well.
For the second part, when the utility function is u(x) = x^1.5, the calculations would be similar. The expected utility of the first outcome is -10^1.5 * 0.25 = -11.1803, and for the second outcome is 10^1.5 * 0.75 = 43.3013. The expected utility of the lottery is 43.3013 - 11.1803 = 32.1210.
The certainty equivalent is c^1.5 = 32.1210, so c = 32.1210^(1/1.5) = 10.5856. The asking price is 100 - c = 100 - 10.5856 = 89.4144. The risk premium is 32.1210 - 89.4144 = -57.2934.
The graph for this case would show the utility function u(x) = x^1.5 increasing more rapidly with x than the previous utility function, and the budget constraint line representing the lottery. The certainty equivalent, asking price, and risk premium would be shown on the graph as well.