Question 1. For this exercise t=1+tlie fourth digit of your student number. Consider an economy, where there is a single consumption good and two states of nature S = {a, b). In this eeouiotii there are 2 x t individuals of type M, with preferences and (random) endowments: . U, (x4,(a),x,(b)) = 2logx,(a) + 3logx,(b), (e4,(a),eA,(b)) = (2.2) There are aLso 5 indivi(IuaLs of type N, with preferences and (random) eIl(lowmetlts: . Uv (xv(a),XN(b)) = 4logx,y(a) +logxN(b), (eN(a).eN(b)) = (2 X t,4 X t).
In order to hedge against risk individuals can trade two securities wit h payoffs:r1 = ( ), r2 = ( ), wire the top payoff refers to state a auid the bottom to state b and k O is a parameter.
(j) ¡f k = O conuinetit oiu the nature of the asset structure. What is the unique equilibrium portfolio and contingent commodity allocation in this case?
[10 marks]
(ii) If k > 0, CompUte tlw c1tIwt it ive (lu1ilibriuIu1 asset prices and portfolios as a function of the parameter k. Demonstrate that equilibrium asset prices satisfy the no arbitrage property.
[20 marks]
(iii) lithe parameter k> O was under government control, explain and discuss the government influence on the equilibrium.
[10 marks]
(iv) Suppose that the asset r2 is now replaced by an option on asset r1 with strike j)rice c = 2. Compute time new equilibrium security prices, portfolios, and consumption allocation.
Question 2. (Word Limit: 5(H) words) consider an economy with uncertainty ( {(s, uh,eh) : h e where S denotes a (finite) set of slates, L a set of corn nl()data types. and II a set of individuals. This economy is endowed with an S x J asset structure A where rank(A) < S.
(1) Provide a definition of constrained optionality for this economy. How (does an asset-spots market (Radner) equilibrium in this eco non relate to constrained opt I mality?
(ii) Prove that, generically. equilibrium in this economy cannot be (Pareto) op— t ¡mal.
(iii) Suppose that the government intervenes in asset markets and changes the asset structure to a new one B, where rank(B) = S. Establish whether or not t.his will have an effect on tile real economy. In particular, establish whether or not this intervention will improve the optionality of the new equilibrium allocation. Comment on the possibility that some individual is worse off after time government intervention.
(iv) Consider now an economy with a single commodity aid uncertainty captured by two states of nature S = {a, b}. The economy is endowed with markets for two assets with payoffs r = [1 2] and rl = [2 O]in states a and b respectively. There are iii aggregate ten units of the cornnioditv available in each state of nature.
The economy is populated by two individuals h, k. Individual h has preferences U, (r,(a),xn(b)) = login(a) + logxn(b). At equilibrimmni, xn(a) = 6 while xk(b) = 2.k(a). Find equilibrium asset prices and allocation of the conmiodity iii each state.
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