9.3. AN EQUILIBRIUM MODEL OF BANKRUPTCY the class i security is denoted by Pi >0 and defined by minV+C Pj D.D (9.1) Pn+1 ∑p (9.2) Let p=(Pi, . ,Pn+1) denote the vector of equilibrium security prices. A competitive equilibrium consists of the market value of the firm V, a vecto of security prices p 0 satisfying( 9. 1)and(9. 2), and an attainable allocation f such that for almost every agent a E A, f(a) maximizes r +v(ae subject to the budget constraint Ve+x≤(a)+p:d(a) and the non-negativity constraint(,e)20 An attainable allocation f is Pareto-efficient if there does not exist an attain able allocation f such that f(a)is weakly preferred to f(a) for almost every a and f'(a)is strictly preferred to f(a) for a non-negligible set of agents Theorem 1 If(, V, p)is a competitive equilibrium then f is Pareto-efficien This is just the first theorem of welfare economics, of course. The non- tandard part is the definition of security prices to allocate the value of the firm among the different creditors. These prices do not play any role in clear ing markets; they merely serve to allocate the value of the firm according to the seniority of different classes of debt and equity. An important observation is that any price vector p is consistent with efficiency. In other words, the fact that debt claims can be used to purchase the firm increases the market value of the firm v but does not affect the efficiency of the outcome 9.3.2 Lex-efficiency An attainable allocation f respects limited liability if f(a) is weakly preferred to(w(a), 0) for almost every agent a9.3. AN EQUILIBRIUM MODEL OF BANKRUPTCY 5 the class i security is denoted by pi ≥ 0 and defined by pi = minn V + C − P j<i pjDj , Di o Di , i = 1, ..., n, (9.1) pn+1 = V + C −Xn j=1 pjDj . (9.2) Let p = (p1, ..., pn+1) denote the vector of equilibrium security prices. A competitive equilibrium consists of the market value of the firm V , a vector of security prices p ≥ 0 satisfying (9.1) and (9.2), and an attainable allocation f such that, for almost every agent a ∈ A, f(a) maximizes x + v(a)e subject to the budget constraint V e + x ≤ w(a) + p · d(a) and the non-negativity constraint (x, e) ≥ 0. An attainable allocation f is Pareto-efficient if there does not exist an attain￾able allocation f0 such that f0 (a) is weakly preferred to f(a) for almost every a and f0 (a) is strictly preferred to f(a) for a non-negligible set of agents. Theorem 1 If (f,V,p) is a competitive equilibrium then f is Pareto-efficient. This is just the first theorem of welfare economics, of course. The non￾standard part is the definition of security prices to allocate the value of the firm among the different creditors. These prices do not play any role in clear￾ing markets; they merely serve to allocate the value of the firm according to the seniority of different classes of debt and equity. An important observation is that any price vector p is consistent with efficiency. In other words, the fact that debt claims can be used to purchase the firm increases the market value of the firm V but does not affect the efficiency of the outcome. 9.3.2 Lex-efficiency An attainable allocation f respects limited liability if f(a) is weakly preferred to (w(a), 0) for almost every agent a