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ption relative to the average consumption in the group. For  , agents are conformist which means   if  ( relative risk aversion is larger than unity. And we know under this condition,  (the agent is overconformist) if   is larger than  .
(2) different time preferences
We define too well-known indexes of sensitivity of marginal utility with respect to t and with respect to c.
The instantaneous rate of pure time preference of agent   consuming c at time t equals
 
 
  Which measures the rate at which marginal utility decreases with time while consumption is held constant. This definition can be written in another form:
 = 
 Impatient people have a positive . In the special case of a multiplicatively separable utility function  = , then from the definition of  , we know that  is independent of c. If we add the assumption that the discounting rate is exponential: . We see in this case  is independent of t. In the case of hyperbolic discounting, is independent of t, but is decreasing in t.
  In a similar way, we can define the absolute aversion to consumption fluctuations over time:
 
  measures the rate at which marginal utility decreases with consumption at a given time. It is also the absolute risk aversion in the sense of Arrow-Pratt. Under our assumptions,   is nonnegative. If we assume utility function is exponential:  ,   would be independent of c. Generally it is assume that   is decreasing in c i.e. DARA applies.  As stated by Pratt 1964), in a static framework it satisfies:
 
  and  are two important to understand agents’ saving and investment behavior.

3 Efficient sharing rules
 Characterizing the optimal investment decision of heterogeneous groups of heterogeneous agents requires understanding how the group will share the cash flows generated by any investment. Here we have some restriction on the group’s sharing of the cake is that it is Pareto –efficient. We define that an allocation   is Pareto efficient if it is feasible and there is no other feasible allocation that raises the lifetime utility of one type with reducing the lifetime utility of the other types. To any Pareto-efficient allocation, there exists a weight function  :   such that it is the solution of the following social planner’s problem:
Social welfare function is as follows:
 
s.t (2),(3)
Wilson(1968) considered a decision problem similar to this problem. He examined a decision under uncertainty for expected-utility (EU) maximizes with heterogeneous utility functions, but heterogeneous beliefs. Except for this restriction, the additivity property made in the EU model under static uncertainty and in the time-additive model under dynamic certainty makes these two problems equivalent. Wilson (1968) proved that the optimal collective decision policy is similar to the optimal decision policy of a representative agent who also maximizes the expected value of a function of consumption per capita in the group. The EU functional of the representative agent is additively separable with respect to the states of nature. Wilson’s result can be extended to a dynamic framework, as Constantinides (1982) has shown. The existence of
a representative agent with such simple aggregative properties has become a ba本论文由英语论文网提供整理，提供论文代写，英语论文代写，代写论文，代写英语论文，代写留学生论文，代写英文论文，留学生论文代写相关核心关键词搜索。