S2 2024SQL

Java Python 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,   x2i  =  Y x1i      ⇔  w2i  = Y w1i )

(b) [6 points] Let  a2/a1 = 1, w1   = w2  = 1. What is the competitive equilibrium?

(c) [6 points] Assignment S2 2024SQL 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