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ECOS3022

S1CIFE 2023

Assignment – due 10 February 2023

1. Consider an investor with utility of wealth u(x) =  √x . Her initial wealth 业0  = 10. There are two states of the world, and two financial assets. The financial market is described by (r, q), with

r = [3    5]  and  q = [4, 8]

(a)  (4 points) Let y1  and y2  denote the investor’s wealth levels in states 1 and 2. What is

her budget constraint in terms of y1  and y2 ?

(b)  (4 points) Suppose the investor maximizes her expected utility, with the probability of

state 1 几1  = 0.6. Write down and solve the investor’s optimization problem.

(c)  (2 points) What is the optimal portfolio (z1, z2)? [ Hint: r ∙ z = y ]

2. Take a utility function  u(x1, x2 ) = (a1xF  + a2xF )  with  a1, a2  > 0, F > −1, F ≠ 0. There are two goods and two agents. Consider endowments 业1  = (业1, 0), 业2  = (0, 业2), 业1, 业2  > 0.

(a) (2 points) Normalise p2  = 1 and show that the equilibrium price p1  = (业2)F+1 a1

[ Hint: Using this general form to find the equilibrium price. ∀i,   x2(i)  =  y x1(i)      ⇔  业 2(i)  = y 业 1(i)  ]

(b) (4 points) Let   = 1, 业1  = 业2  = 1. What is the competitive equilibrium?

(c) (4 points) Let  = 1, 业1  = 2, 业2  = 1. What is the competitive equilibrium? Show that agent 1 is better off.