A Reputational Theory of Firm Dynamics Simon Board Moritz - PowerPoint PPT Presentation

Introduction Model Best-Response Equilibrium Observable Informed The End Appendix A Reputational Theory of Firm Dynamics Simon Board Moritz Meyer-ter-Vehn UCLA May 6, 2014

Introduction Model Best-Response Equilibrium Observable Informed The End Appendix Motivation Models of firm dynamics ◮ Wish to generate dispersion in productivity, profitability etc. ◮ Some invest in assets and grow; others disinvest and shrink. A firm’s reputation is one of its most important assets ◮ Kotler: “In marketing, brand reputation is everything”. ◮ Interbrand: Apple brand worth $98b; Coca-Cola $79b. ◮ EisnerAmper: Reputation risk is directors’ primary concern. Reputation a special asset ◮ Reputation is market belief about quality. ◮ Reputation can be volatile even if underlying quality constant.

Introduction Model Best-Response Equilibrium Observable Informed The End Appendix This Paper Firm dynamics with reputation ◮ Firm invests in quality. ◮ Firm & mkt. learn about quality. 1 ◮ Firm exits if unsuccessful. 0.8 Work Regions 0.6 Optimal investment Reputation 0.4 ◮ Firm shirks near end. 0.2 Baseline ◮ Incentives are hump-shaped. No Investment 0 0 2 4 6 8 Years

Introduction Model Best-Response Equilibrium Observable Informed The End Appendix This Paper Firm dynamics with reputation ◮ Firm invests in quality. ◮ Firm & mkt. learn about quality. 1 ◮ Firm exits if unsuccessful. Observed Investment 0.8 Work Regions 0.6 Optimal investment Reputation 0.4 ◮ Firm shirks near end. 0.2 ◮ Incentives are hump-shaped. Baseline No Investment 0 0 2 4 6 8 Years Benchmarks ◮ Consumers observe investment.

Introduction Model Best-Response Equilibrium Observable Informed The End Appendix This Paper Firm dynamics with reputation ◮ Firm invests in quality. ◮ Firm & mkt. learn about quality. 1 ◮ Firm exits if unsuccessful. Observed Investment 0.8 Work Regions 0.6 Optimal investment Reputation Known 0.4 Quality ◮ Firm shirks near end. 0.2 ◮ Incentives are hump-shaped. No Investment Baseline 0 0 2 4 6 8 Years Benchmarks ◮ Consumers observe investment. ◮ Firm privately knows quality.

Introduction Model Best-Response Equilibrium Observable Informed The End Appendix Literature Reputation Models ◮ Bar-Isaac (2003) ◮ Kovrijnykh (2007) ◮ Board and Meyer-ter-Vehn (2013) Firm Dynamics ◮ Jovanovic (1982) ◮ Hopenhayn (1992) ◮ Ericson and Pakes (1995) Moral hazard and learning ◮ Holmstrom (1982) ◮ Bonatti and Horner (2011, 2013) ◮ Cisternas (2014)

Introduction Model Best-Response Equilibrium Observable Informed The End Appendix Model

Introduction Model Best-Response Equilibrium Observable Informed The End Appendix Model, Part I Long-lived firm sells to short-lived consumers. ◮ Continuous time t ∈ [0 , ∞ ) , discount rate r . ◮ Firm invests A t ∈ [0 , a ] , a < 1 , and exits at time T . Technology ◮ Quality θ t ∈ { L, H } where L = 0 and H = 1 . ◮ Technology shocks arrive with Poisson rate λ . ◮ Quality given by Pr( θ s = H ) = A s at last shock s ≤ t . Information ◮ Breakthroughs arrive with Poisson rate µ iff θ t = H . ◮ Consumers observe history of breakthroughs, h t . ◮ Firm additionally recalls past actions.

Introduction Model Best-Response Equilibrium Observable Informed The End Appendix Model, Part II Reputation and Self-Esteem ◮ Consumers’ beliefs over strategy of firm, F = F ( { ˜ A t } , ˜ T ) . ◮ Self-esteem Z t = E { A t } [ θ t | h t ] . ◮ Reputation , X t = E F [ θ t | h t , t < ˜ T ] . Payoffs ◮ Consumers obtain flow utility X t . ◮ Firm value �� T � { A t } ,T E { A t } e − rt ( X t − cA t − k ) dt V = max . 0

Introduction Model Best-Response Equilibrium Observable Informed The End Appendix Recursive Strategies Game resets at breakthrough, X=Z=1. ◮ { A t } , T is recursive if only depend on time since breakthrough. ◮ F is recursive if only puts weight on recursive strategies. ◮ If F recursive, then optimal strategies are recursive. ◮ Notation: { a t } , τ, { x t } , { z t } , V ( t, z t ) etc. Self-esteem ◮ Jumps to z t = 1 at breakthrough. ◮ Else, drift is ˙ z t = λ ( a t − z t ) dt − µz t (1 − z t ) dt =: g ( a t , z t ) . Assumption: A failing firm eventually exits ◮ Negative drift at top, z † := λ/µ < 1 . ◮ Exit before z † reached, z † − k + µz † (1 − k ) /r < 0 .

Introduction Model Best-Response Equilibrium Observable Informed The End Appendix Optimal Investment & Exit

Introduction Model Best-Response Equilibrium Observable Informed The End Appendix Optimal Strategies Exist t } , τ ∗ exists with τ ∗ ≤ τ . Given { x t } , an optimal { a ∗ Lemma 1. Idea ◮ Drift g ( a, z ) is strictly negative for z ∈ [ z † , 1] . ◮ V ( t, z † ) < 0 for any strategy, so τ ∗ bounded. ◮ Action space compact in weak topology by Alaoglu’s theorem. ◮ Payoffs are continuous in { z t } , and hence in { a t } , τ . Notation ◮ Optimal strategies { a ∗ t } , τ ∗ . ◮ Optimal self-esteem { z ∗ t } .

Introduction Model Best-Response Equilibrium Observable Informed The End Appendix Optimal Investment Given { x t } , optimal investment { a ∗ Lemma 2. t } satisfies � 0 λV z ( t, z ∗ t ) < c, if a ∗ t = λV z ( t, z ∗ a if t ) > c. Investment pays off by ◮ Raising self-esteem immediately. ◮ Raising reputation via breakthroughs. Dynamic complementarity ◮ V ( t, z ) is convex; strictly so if { x t } continuous. ◮ Raising a t raises z t + dt and incentives V z ( t, z t + dt ) . t ′ for t ′ > t . ◮ Optimal strategies ordered: z ∗ t > z ∗∗ ⇒ z ∗ t ′ > z ∗∗ t

Introduction Model Best-Response Equilibrium Observable Informed The End Appendix Marginal Value of Self-Esteem Given { x t } , if V z ( t, z ∗ Lemma 3. t ) exists it equals � τ ∗ � s e − t r + λ + µ (1 − z ∗ u ) du µ ( V (0 , 1) − V ( s, z ∗ Γ( t ) = s )) ds. t Value of self-esteem over dt ◮ dz raises breakthrough by µdzdt . ◮ Value of breakthrough is V (0 , 1) − V ( s, z ∗ t ) . Discounting the dividends ◮ Payoffs discounted at rate r . ◮ dz disappears with prob. µz ∗ t dt , if breakthrough arrives. ◮ dz changes by g z ( a, z t ) = − ( λ + µ (1 − 2 z ∗ t )) .

Introduction Model Best-Response Equilibrium Observable Informed The End Appendix Derivation of Investment Incentives ◮ Give firm cash value of any breakthrough, � τ ∗ V ( t, z ∗ e − r ( s − t ) ( x s − ca ∗ s − k + µz ∗ s ( V (0 , 1) − V ( s, z ∗ t ) = s )) ds. t ◮ Apply the envelope theorem, � τ ∗ e − r ( s − t ) ∂z ∗ � � V z ( t, z ∗ s µ ( V (0 , 1) − V ( s, z ∗ s )) − µz ∗ s V z ( s, z ∗ t ) = s ) ds. ∂z ∗ t t ◮ The partial derivative equals, � s � � ∂z ∗ s /∂z ∗ ( λ + µ (1 − 2 z ∗ t = exp − u )) du . t ◮ Placing µz ∗ s V z ( s, z ∗ s ) into the exponent, � τ ∗ � s V z ( t, z ∗ e − t ( r + λ + µ (1 − z ∗ u )) du µ ( V (0 , 1) − V ( s, z ∗ t ) = Γ( t ) := s )) ds. t

Introduction Model Best-Response Equilibrium Observable Informed The End Appendix Property 1: Shirk at the End Given { x t } , any optimal strategy { a ∗ t } , τ ∗ , exhibits Theorem 1. t = 0 on [ τ ∗ − ǫ, τ ∗ ] . shirking a ∗ Idea ◮ At t → τ ∗ , so Γ( t ) → 0 . ◮ Need technology shock and breakthrough before τ ∗ for investment to pay off. ◮ Shirking accelerates the demise of the firm.

Introduction Model Best-Response Equilibrium Observable Informed The End Appendix Property 2: Incentives are Single-Peaked Theorem 2. If { x t } decreases, investment incentives Γ( t ) are single-peaked with Γ(0) > 0 , ˙ Γ(0) > 0 and Γ( τ ∗ ) = 0 . Proof ◮ Differentiating Γ( t ) with ρ ( t ) := r + λ + µ (1 − z ∗ t ) , Γ( t ) = ρ ( t )Γ( t ) − µ ( V (0 , 1) − V ( t, z ∗ ˙ t )) . ◮ Differentiating again, ¨ Γ( t ) = ρ ( t ) ˙ z ∗ t Γ( t ) + µV t ( t, z ∗ Γ( t ) + ˙ ρ ( t )Γ( t ) + µ ˙ t ) = ρ ( t ) ˙ Γ( t ) + µV t ( t, z ∗ t ) . ◮ If { x t } is decreasing V t < 0 , and ˙ Γ( t ) = 0 implies ¨ Γ( t ) < 0 . Countervailing forces: As t rises, ◮ Dividends V (0 , 1) − V ( t, z ∗ t ) grow, and incentives increase. ◮ Get close to exit and incentives decrease.

Introduction Model Best-Response Equilibrium Observable Informed The End Appendix Property 3: Exit Condition If { x t } is continuous, then τ ∗ satisfies Theorem 3. V ( τ ∗ , z τ ∗ ) = ( x τ ∗ − k ) + µz τ ∗ V (0 , 1) = 0 . � �� � � �� � flow profit option value

Introduction Model Best-Response Equilibrium Observable Informed The End Appendix Equilibrium

Introduction Model Best-Response Equilibrium Observable Informed The End Appendix Definition Equilibrium beliefs ◮ Reputation x t = E F [ θ t | h t = ∅ , t ≤ ˜ τ ] given by Bayes’ rule. ◮ Under point beliefs, ˙ x = λ (˜ a − x ) dt − µx (1 − x ) dt . ◮ Can hold any beliefs after τ ( F ) := min { t : F (˜ τ ≤ t ) = 1 } . Recursive equilibrium � � ◮ Given { x t } , any strategy { a t } , τ ∈ supp ( F ) is optimal. ◮ Reputation { x t } derived from F via Bayes’ rule for t < τ ( F ) .