5.2. RENEGOTIATION 5.2.4 The spanning condition The firm makes all the offers, so has all the bargaining power. The venture capitalist can guarantee that he will get at least V*(dt-1, mt-1 ht)at date t, where(dt-1, mt-1)is the pre-existing contract and the information set is ht. In equilibrium, renegotiation of the contract(dt-1, mt-1) will give him exactly V"(dt-1, mt-1ht) Definition 1 If we can choose a contract(dt-1, mi-1) to satisfy V*(di-1, mt-llht=Eu((wr)Iht for each date t and history ht, then we say that the spanning condition for implementation of the first-best risk -sharing allocation is satisfied It turns out that this condition is sufficient as well as necessary for im- sementation of the first best. If the spanning condition is satisfied, then the bargaining game has a subgame perfect equilibrium that implements the first best 5.2.5 Subgame Perfect Equilibrium Proposition 2 If the spanning condition is satisfied, there exists a Pareto- efficient SPE of the renegotiation game, that is, a SPe that results in the implementation of first-best risk sharing 5.2.6 An Example a parametric example illustrates the requirements of the theory and also al- lows us to see whether the spanning condition will be satisfied in a reasonable case. Suppose that both the firm and the venture capitalist have constant absolute risk aversion and suppose that wt follows a random walk ut+aw.pr.丌 t+1 ut2+bw.pr.1-丌,5.2. RENEGOTIATION 7 5.2.4 The spanning condition The firm makes all the offers, so has all the bargaining power. The venture capitalist can guarantee that he will get at least V ∗(dt−1, mt−1|ht) at date t, where (dt−1, mt−1) is the pre-existing contract and the information set is ht. In equilibrium, renegotiation of the contract (dt−1, mt−1) will give him exactly V ∗(dt−1, mt−1|ht). Definition 1 If we can choose a contract (dt−1, mt−1) to satisfy V ∗ (dt−1, mt−1|ht) = E[v(y(wT )|ht] (5.1) for each date t and history ht, then we say that the spanning condition for implementation of the first-best risk-sharing allocation is satisfied. It turns out that this condition is sufficient as well as necessary for im￾plementation of the first best. If the spanning condition is satisfied, then the bargaining game has a subgame perfect equilibrium that implements the first best. 5.2.5 Subgame Perfect Equilibrium Proposition 2 If the spanning condition is satisfied, there exists a Pareto￾efficient SPE of the renegotiation game, that is, a SPE that results in the implementation of first-best risk sharing. 5.2.6 An Example A parametric example illustrates the requirements of the theory and also al￾lows us to see whether the spanning condition will be satisfied in a reasonable case. Suppose that both the firm and the venture capitalist have constant absolute risk aversion: u(x) = −e−Ax v(y) = −e−By and suppose that {wt} follows a random walk: wt+1 = ½ wt + a w. pr. π wt + b w. pr. 1 − π