Consider an investor who has the von Neumann-Morgenstern utility index
u(x) = 3 + 4√x
An investment provides income according to two possible future scenarios ("states of nature") that may unfold as follows:
State s Income xs Probability πs
1 2 1/4 3/4
2 1 2 1/4 1/4
(a) Depict the von Neumann-Morgenstern utility index u in a diagram.
(b) Determine the expected value of the investment graphically in the diagram of part a) and calculate its value. Explain.
(c) What is meant by the certainty equivalent of an investment? Explain.
(d) Determine the certainty equivalent of the investment for the investor graphically in the diagram of part a) and calculate its value. Explain.
(e) What is meant by the risk premium on an investment? Explain.
(f) Determine the risk premium of the investment for the investor graphically in the diagram of part a). Calculate the value of the risk premium. Explain.
Now this consumer has the opportunity to insure the outcome of the investment project by trading state contingent income in state 1, x1, against state contingent income in state 2, x2, (and the other way around) at prices q1 = 3/4 and q2 = 1/4.
(g) Determine the investor's expected utility function U(x1,x2) based on the probabilities of the states of nature.
(h) Depict the indifference curve through his initial endowment with state contingent income (i.e. the income derived from the investment project) in a diagram with state contingent income. Also depict the investor's budget line in this diagram.
(i) Determine the investor's optimal state contingent income bundle graphically in the diagram of part h).
(j) Calculate the investor's optimal state contingent income bundle.
(k) Would the investor purchase full insurance if the prices for state contingent income did not reflect the probabilities of the states of nature? Discuss.
Consider an economy consisting of a consumer C with the utility function
U(xC , yC) = xC + v(yC) with v(yC) = 24yC - 2(yC)2
and a firm F with the cost function
c(yF) = 4 + (yF)2.
Assume that good x represents money and has the px = 1. The consumer owns all the profit shares of the firm, but the firms is run by an independent manager who maximises its profit for given prices py and px = 1. The consumer obtains the total profits of the firm.
The consumer has the initial endowment (wx , wy) = (50,0), i.e. she starts out with nothing of good y but with 50 units of money.
a) Derive the indifference curve of the consumer for the utility level of her initial endowment. Depict this indifference curve of the consumer in an (xC , yC) - diagram. Also depict the indifference curve for the utility level of 94.
Assume that the firm has all the money (good x) in the economy available as its working capital, wF. So wF = 50.
b) Draw the production possibility curve for this economy in the (xC , yC) diagram of part a).
(Notes to b):
1) The production possibility curve consists of all combinations (xF , yF) for which xF = wF - c(yF), where wF denotes the firm's working capital.
2) to depict (xF , yF) combinations, please use yC = yF for the produced good and xC = wC - xF because money is an input in the production process of the firm.
c) Graphically determine the allocation on the production possibility frontier that the consumer considers to be optimal.
d) What is meant by "Pareto Efficiency"? Why is the optimal allocation depicted in part d) is the only Pareto Efficient allocation in this economy?
The price py (remember px = 1 by definition), a consumption bundle (xC , yC) and production bundle (xF , yF) with profits ?F are a Walrasian Equilibrium if
1) (xC , yC) maximises the consumer's utility at price py given for her initial endowment and income from profits ?F
2) (xF , yF) maximises the firm's profits for at price py given the firm's production possibility set
3) all markets clear, i.e.
xC + xF = wx (because x is an input for the firm) and
yC = wy + yF (because y is produced by the firm)
Some suggest that the following constitute a Walrasian Equilibrium
py = 8 ; (xC , yC) = (30,4); (xF , yF) = (20,4) and ?F = 12.
e) Check if the proposed numbers satisfy condition 1) for a Walrasian Equilibrium.
f) Check if the proposed numbers satisfy conditions 2) and 3) for a Walrasian Equilibrium.
g) State the First Theorem of Welfare Economics? How does it apply here? Explain.