Dynamic Free Riding with Irreversible Investments : On - line Appendix

In this appendix we present the proofs omitted in “Dynamic Free Riding with Irreversible Investments” by Marco Battaglini, Salvatore Nunnari and Thomas Palfrey. Marco Battaglini Department of Economics Princeton University Princeton NJ 08544 [email protected] Salvatore Nunnari Department of Political Science Columbia University New York, NY 10027 [email protected] Thomas Palfrey Division of the Humanities and Social Sciences, California Institute of Technology, Pasadena, CA 91125 [email protected] 1 Proof of Proposition 1 Proposition 1. For any d, δ, n and y ∈ [ [u] −1 (1− δ(1− d)), [u] −1 (1− δ(1− d n)) ] , there is an equilibrium with steady state y in an irreversible economy. In all these equilibria convergence is monotonic and gradual. Define y(δ, d, n) = [u] −1 (1− δ(1− d)/n) and y(δ, d, n) = [u] −1 (1− δ (1− d/n)): these are the points at which y(g) = 1− d− n(1−u′(g)) δ 1− n (1) is, respectively, zero and one. Define y(d, δ) = [u′] (1− δ(1− d)): this is the point at which (1) is equal to 1 − d. Note that y(δ, d, n) < y(d, δ) and y(d, δ) < y(δ, d, n). Moreover, since we are assuming that the planner interior solution is feasible (y P (δ, d, n) < W/d), we have y(δ, d, n) < W/d. To construct an equilibrium with steady state y ∈ [y(δ, d), y(δ, d, n)] we proceed in 3 steps. Step 1. We first construct the strategies associated to a generic y. For a generic y ∈ [y(δ, d), y(δ, d, n)], let ỹ (g |y ) be the solution of the differential equation (1) when we require the initial condition: ỹ (y |y ) = y. Given y, moreover, let us define the two thresholds g(y) = y/(1 − d) and g(y) = max {ming≥0 {g |ỹ (g |y o ) ≤W + (1− d)g} , y(δ, d, n)}. In words, the second threshold is the largest point between the point at which ỹ (g |y ) crosses from below W + (1 − d)g, and y(δ, d, n) (see Figure 1 in the paper for an example). It is easy to verify that, by construction, g(y) ≥ y(δ, d); moreover, ỹ (g |y ) ∈ ((1− d)g,W + (1− d)g) with ỹ (g |y ) ∈ [0, 1] and ỹ (g |y ) ≥ 0 in [ g(y), y ] . For any y ∈ [y(δ, d), y(δ, d, n)], we now define the investment function as follows: y(g |y ) =    min { W + (1− d)g, ỹ ( g(y) |y )} g ≤ g(y) ỹ (g |y ) g(y) < g ≤ y