CAPITAL ASSET MARKET 773 and mean of yield. Equations(5)and(6)constitute, for each individual, n equations describing his demand for the n assets. To determine general equilibrium, we must also specify equality between demand and supply for each asset. These market clearing conditions can be writter (7) As we would suspect, one of these conditions is superfluous. This can be seen by first summing the budget equations over all individuals ∑P(x一x)+q∑(x-x=0 ∑p∑(x-x+q∑(xn-x)=0 Suppose that (7) were satisfied for all j except n. This would mean that the first term on the left of (8) vanishes, so that Hence also the nth equation of (7 )must hold. We may therefore instead write ∑x=习 (j=1,,n-1) where x denotes the given total supply of asset j: x =2i=1X This essentially completes the equations describing general equilibrium. The system consists of the m equations(4), the m(n-1)equations(5)and(6), and the (n-1)equations(7); altogether (mn+n-1)equations. The unknowns are the mn x and the(n-1)prices pi We have counted our equations and our unknowns and found them to be equal in number. But we cannot rest with this; our main task has hardly begun. We shall bypass such problems as the existence and uniqueness of a solution to the system and rather concentrate on investigating properties of the equilibrium values of the variables, assuming that they exist. We may observe, first of all, that the equilibrium allocation of assets represents allocation to increas individuals utility without at the same time reducing the utility of one or more other individuals. This should not need any explicit proof, since it is a well known general property of a competitive equilibrium where preferences are concave, We should also mention the problem of nonnegativity of the solution to which we shall return at a later stage has content downl ued stube to sT oR ems aecondtp23013020-0 AMCAPITAL ASSET MARKET 773 and mean of yield. Equations (5) and (6) constitute, for each individual, n equations describing his demand for the n assets. To determine general equilibrium, we must also specify equality between demand and supply for each asset. These market clearing conditions can be written: m (7') (x (XJ-Cj) = O Oj= 1,.I. n). As we would suspect, one of these conditions is superfluous. This can be seen by first summing the budget equations over all individuals: m n-1 m E pj (x.- j+ q E (x'- 5n)=O 0 1j= or n-1 m m (8) Epj E (x' J + q E (x' 0n j=i i=i i=i Suppose that (7') were satisfied for all j except n. This would mean that the first term on the left of (8) vanishes, so that m X i= 1 Hence also the nth equation of (7') must hold. We may therefore instead write: m (7) Z xj =xj (j=1,..., n-1), i=1 where Xj denotes the given total supply of asset j: &j = mim= lj. This essentially completes the equations describing general equilibrium. The system consists of the m equations (4), the ni (n - 1) equations (5) and (6), and the (n -1) equations (7); altogether (mn + n -1) equations. The unknowns are the mn quantities xJ and the (n-1) prices pj. We have counted our equations and our unknowns and found them to be equal in number. But we cannot rest with this; our main task has hardly begun. We shall bypass such problems as the existence and uniqueness of a solution to the system and rather concentrate on investigating properties of the equilibrium values of the variables, assuming that they exist. We may observe, first of all, that the equilibrium allocation of assets represents a Pareto optimum, i.e., it will be impossible by some reallocation to increase one individual's utility without at the same time reducing the utility of one or more other individuals. This should not need any explicit proof, since it is a well known general property of a competitive equilibrium where preferences are concave. We should also mention the problem of nonnegativity of the solution to which we shall return at a later stage. This content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 02:20:50 AM All use subject to JSTOR Terms and Conditions