Assignment – due 11 October 2024
1. Take a utility function with a1, a2 > 0 and -1 < β < 0. There are two goods and two agents. Consider endowments wA = (w1, 0), wB = (0, w2), w1, w2 > 0.
(a) [4 points] Normalisep2 = 1 and show that the equilibrium price
( Hint: Using this general form to find the equilibrium price. ∀i, x2 = Y x1 ⇔ w2 = Y w1 )
(b) [6 points] Let a2/a1 = 1, w1 = w2 = 1. What is the competitive equilibrium?
(c) [6 points] Let a2/a1 = 1, w1 = 2, w2 = 1. What is the competitive equilibrium? Show that agent A is better off.
2. Consider an investor with utility of wealth u(w) = 2√w . Her initial wealth w0 = 12. There are two states of the world, and two financial assets. Asset A pays 3 in every states and asset B pays 0 in state 1 and 8 in state 2. The prices of the assets are q1 = 2 and q2 = 4.
(a) [3 points] Let y1 andy2 denote the investor’s wealth levels in states 1 and 2. What is her budget constraint in terms of y1 andy2 ?
(b) [3 points] Suppose the investor maximizes her expected utility, with the probability of state 1 π1 = 0.4. Write down and solve the investor’s optimization problem.
(c) [3 points] What is the optimal portfolio (z1, z2)?