Inv Dec Symmetr Inf
Inv Dec Symmetr Inf BA 855
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Date Created: 10/29/15
7 Representative Agent Theorems Most of the economic models that we will apply can be considered representative agent models A representative agent model is a model in which all agents act in such a man ner that their cumulative actions might as well be the actions of one agent maximizing its expected utility function Economists construct representative agents in order to deal with the complicated issue of aggregation It is relatively simple to model the behavior of one person given some preferences and constraints It is more dliCUIt to model the behavior of a group of people or an entire economy Aggregation is what economists call the summing up of individuals39 behavior to derive the behavior of a market or an economy Models that aggregate individuals without using a representa tive agent device are sometimes called heterogeneity models The representative agent only exists under certain circumstances and those cir cumstances are the subject of this section Most of the material in this section comes from Huang and Litzenberger 1987 chapter 5 71 Complete Markets The biggest assumption involved in creating a representative agent is that markets are complete A complete market is a market in which there are at least as many assets with linearly independent payo s as there are states Returning to the notation used in section 3 we will think again of an N E 8 matrix D that represent the payo s of N securities in 8 states A complete market in this notation is characterized by the condition that the rank of D is S This requires that N 5 S It also means that we can assume that D is square If there are more securities than states then at least 8 i N of the securities must have payo s that are linear combinations of other securities As long as the law of one price holds these redundant securities can be ignored Some of the implications of complete markets are discussed in section 2 For exam ple in complete markets worlds we can create ArrowDebreu securities and every agent 50 can use these securities to smooth consumption When markets are not complete they can sometimes be made complete by the use of derivatives Derivatives generally have payo s that are nonlinear functions of their underlying assets so adding derivatives to the set of assets available can increase the rank of D 72 Pareto Optimality In order to see how to construct a representative agent we are going to need a slightly more general asset pricing model than the CAPM We will consider a consumption based one period model Throughout this section we will assume that there is one perishable consumption good that serves as our numeraire All the uncertainty in our model is about which state will be revealed at the end of period one Agents can consume at time zero and at time one but for their consumptions to be feasible it must be true that X Clo C0 and x c 0 8 2 110 1 where there are l individuals indexed by i C0 denotes aggregate time zero consumption available and C means the total amount of consumption possible in state A set of state contingent consumption allocations is pareto optimal if it is feasible and if there do not eXist other feasible allocations that can strictly increase the utility of one individual without decreasing the utilities of others There is a well know result that is associated with the second welfare theorem of economics described in Varian pp 329335 that states that corresponding to every Pareto optimal allocation there eXist a set of nonnegative numbers f g391 such that the same allocation can be achieved by a social planner maximizing a linear combination of individuals39 utility functions using f5g391 as weights The social planner solves the 51 problem IX 51 uc0c 1 2 X max subject to the constraints 109 and 110 listed above where 1A denotes individual i39s subjective probability of state occuring and u c0c is individual i39s utility over c0 and c What is the intuition behind this result We will show that 5 can be interpretted as the reciprocal of the marginal utility of income of agent i Agents that have relatively large incomes therefore have more weight in the maximization than agents with low incomes Thus how the problem turns out depends on initial endowments Since we have assumed that utility is strictly increasing the weights are strictly positive 5 gt0 i 12 Forming the Lagrangian for the social planner we obtain quot quot x x 5 x 010 A C i on 112 1 27 1 X X L 51 ucl0c A0 COi 27 max Claim 1 Since the utility functions used here are strictly concave and the 5 are strictly positive the 39rst order conditions are necessary and suicient for maximization in this problem Those conditions are X 39 5 MHWAm i12l 113 27 CIO 5xwA 12 i12l 114 CI plus the constraints 109 and 110 We can get rid of the weights 5 by examining the ratio of these two conditions 1A U1210C1 Al 1339 39 2 i12 115 2 1111 U 39l39c A0 From this condition for maximization it is clear that a feasible allocation of state contingent consumption is Pareto optimal if and only if for each state marginal rates of substitution between present consumption and future state contingent consumption are equal across individuals In other words a Pareto optimal outcome is one in which all individuals share risk perfectly This does not mean of course that each individual has equal consumption or equal utility in all states Agents with a larger endowment will have higher consumption in all states than agents with a smaller endowment It just means that everyone39s relative unhappiness in a bad state is the same Pareto optimal allocations are always possible in competitive economies with com plete securities markets Suppose that markets are complete and that A is the price of the ArrowDebreu security that provides one unit of consumption in state Then the individual39s problem can be stated as X max 1Allulcl0cl Cumcu 27 X Sit 30 Alc e0 Age 2r 2 where e0 and e represent agent i39s endowment in period zero and state We assume that these endowments are such that wealth is strictly positive at time zero The Lagrangian for each agent is mgle uc0cu e0ic0 Ale iAgc 118 which has 39rst order conditions XUI oiCn W 119 27 I0 mew g 2 120 plus the budget constraint Once again we can get rid of u by forming the ratio 1 UI Clown All Cll Q Ml Uli amcl Ag 2 121 In a market equilibrium the feasiblity constraints 109 and 110 are always satis39ed If we set A0 1 A A and 5 i then we can see that the conditions for the optimality of a single agent are equivalent to the conditions for a Pareto optimal allocation discussed above Conversely to achieve a Pareto optimal allocation from this competitive economy the social planner assigns a weight of p ii to each individual This derivation reinforces what we already know about the weights 5 The weight of agent i is the reciprocal of agent i39s shadow price of the budget constraintquot The shadow price of the budget constraint is set equal to the marginal utility of consumption in period zero here This all makes intuitive economic sense the social planner con siders people with more at stake people with a higher initial wealth more important in the total maximization than people with a smaller market weight Where does this derivation break down if markets are not complete We need state prices to evaluate the agent39s endowment But we have assumed that we have an equilibrium so there must not be any arbitrage opportunities available in our economy This means of course that a state price vector exists regardless of whether or not markets are complete The problem is that the state price vector is not unique if markets are not complete If state prices are not unique then we can sort of think of one representative agent existing for each set of state prices that we assume people use to value their claims This construction however assumes that all agents use the same state prices to value their endowments If di erent people use di erent state price vectors then the whole representative agent idea breaks down Du ie has a nice concise discussion of the representative agent in his 39rst chapter He says that it is not always necessary for markets to be complete for Pareto optimality and thus for a representative agent to exist Complete markets and other assumptions 54 arejust suicient for the representative agent they are not necessary However Duie also points out that it can be shown that with incomplete markets and under natural assumptions on utility for almost every endowment the equilibrium allocation is not Pareto optimalquot Thus we should not expect to have a representative agent with incomplete markets 73 Constructing the Representative Agent Now we are ready to derive the representative agent result We assume that markets are complete and that the economy is competitive We also assume that individuals have homogeneous beliefs and timeadditive stateindependent utility functions that are strictly concave increasing and di erentiable This means that the conditions for maximization by a single agent 119 and 120 can be stated as UI0CI0 I W I 122 and UI1CI 1 l A c illM 2 123 where 11 is the subjective probability of state 2 that all agents agree on Let A represent state prices as de39ned in section 2 of the notes De39ne uo and u1 to be P u0z maxlegL1 L1 UoZu 124 st 12 z and PI u1z maxlegL1 1 U1Zu 125 st 12 z where 5 i is the social planner39s weight for each individual An immediate conse quence of these de39nitions is that 0 X 0 010 X 010 u0C0 l1IUoCI0 C0 l1nlln0 125 X CIO 127 I1 CO and X c X A C quot quot 39 39 U1C I 5u1c C 15 C 128 A X 01 A I Kb lw These results use the constraints 109 and 110 to conclude that X CIO 1 130 I1 CO and X Cll 39 1 131 I1 C Now think of a representative agent that has endowments of C0 and Cl 2 in periods one and two respectively Let the representative agent39s subjective proba bilities be 111 and let its utility for period zero and one consumption be u0C0 and u1C respectively The state prices in this type of economy must be equal to A for the representative agent to exist In this economy for the markets to clear the representative agent must not want to trade away from its endowment Thus using time zero consumption as the numeraire the state price for state must be equal to the representative agent39s marginal rate of substitution MRS between time zero consumption and time one state consumption This MRS is 11 UU1C U0Co I 56 132 After substituting 127 and 129 into 132 we know that the respresentative agent39s MRS is equal to 111U01C WA W M A 133 Thus the representative agent exists in this case This representative agent however never wants to trade Equilibriums in which this agent is satis39ed will never involve trading This has led many to look for models that say something about the quantity of trade going on 74 Other Aggregation Results The representative agent39s utility function in the previous results depended on the initial endowments of individuals through the parameter Therefore the prices in I the economy will vary with the initial endowments of agents For some utility functions the utility function of the representative agent does not depend on the endowments of the agents in the economy Such utility function have what is called the aggregation property Two utility functions that satisfy the aggregation property are the power utility function uz 1iLBm Bz1i 134 and the negative exponential utility function P Z ll UZ M exp iK I 135 l The power function includes log utility uz lnA 2 136 for the case when B 1 In the case of negative exponential utility it can even be shown that if agents have 57 di erent time preference parameters and di erent subjective probabilities then the rep resentative agent39s time preference and probabilities are composites of the individuals39 parameters See Huang and Litzenberger for more discussion of these issues 75 Thoughts on the Representative Agent We have shown that under fairly general conditions a representative agent exists We have not shown however that the representative agent is always an interesting con struction It may be that while a representative agent exists in most circumstances a simpler way to characterize prices is possible However most of the economic models that are prominent in Finance use some sort of representative agent formulation A few models that don39t rely on the representative agent argument have been suggested in recent years Most of these models rely on some sort of simulation in order to determine prices and things because they are fairly complicated Heterogeneity models or models with incomplete markets seem like a fairly fertile ground for future study For examples of heterogeneity models see Heaton and Lucas 19955 Aiyagari 19936 Telmer 19937 and Constantinides and Du e 19968 76 Homework Problem 1 Show that uo and u1 de39ned in 124 and 125 are both strictly concave and increasing Huang and Litzenberger problem 51 2 Provide some intuition for equation 115 In particular without using any math 5Heaton J and D Lucas 1995 The Importance of Investor Heterogeneity and Financial Market Imperfections for the Behavior of Asset Pricesquot CarnegieRochester Conference Series on Public Policy 42132 6Aiyagari S R 1993 Explaining Financial Market Facts The Importance of Incomplete Mar kets and Transaction Costsquot Federal Reserve Bank of Minneapolis Quarterly Review Winter 1731 7Temer C 1993 AssetPricing Puzzles and Incomplete Marketsquot Journal of Finance XLVIII 18031832 8Constantinides 3 and D Dure 1996 Asset Pricing with Heterogeneous Consumersquot Journal of Political Eoonomy 219240
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