A Reputational Theory of Firm Dynamics

Transcription

1 A Reputational Theory of Firm Dynamics Simon Board Moritz Meyer-ter-Vehn UCLA May 6, 2014

2 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.

3 Firm dynamics with reputation This Paper Firm invests in quality. Firm & mkt. learn about quality. Firm exits if unsuccessful. Optimal investment Firm shirks near end. Incentives are hump-shaped. Reputation No Investment Baseline Work Regions Years

4 Firm dynamics with reputation This Paper Firm invests in quality. Firm & mkt. learn about quality. Firm exits if unsuccessful Observed Investment Optimal investment Firm shirks near end. Incentives are hump-shaped. Benchmarks Consumers observe investment. Reputation No Investment Baseline Work Regions Years

5 Firm dynamics with reputation This Paper Firm invests in quality. Firm & mkt. learn about quality. Firm exits if unsuccessful Observed Investment Optimal investment Firm shirks near end. Incentives are hump-shaped. Benchmarks Consumers observe investment. Firm privately knows quality. Reputation No Investment Known Quality Baseline Work Regions Years

6 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)

7 Model

8 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.

9 Model, Part II Reputation and Self-Esteem Consumers beliefs over strategy of firm, F = F ({Ãt}, T ). Self-esteem Z t = E {At} [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 V = max {A E{At} t},t 0 ] e rt (X t ca t k)dt.

10 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 ż 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.

11 Optimal Investment & Exit

12 Optimal Strategies Exist Lemma 1. Given {x t }, an optimal {a t }, τ exists with τ τ. 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 }.

13 Optimal Investment Lemma 2. Given {x t }, optimal investment {a t } satisfies a t = { 0 if λvz (t, zt ) < c, a if λv z (t, zt ) > 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 ). Optimal strategies ordered: z t > z t z t > z t for t > t.

14 Lemma 3. Γ(t) = Marginal Value of Self-Esteem Given {x t }, if V z (t, z t ) exists it equals τ t e s t r+λ+µ(1 z u )du µ(v (0, 1) V (s, z s))ds. 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. µzt dt, if breakthrough arrives. dz changes by g z (a, z t ) = (λ + µ(1 2zt )).

15 Derivation of Investment Incentives Give firm cash value of any breakthrough, V (t, z t ) = τ t e r(s t) (x s ca s k + µz s(v (0, 1) V (s, z s))ds. Apply the envelope theorem, V z (t, z t ) = τ t e r(s t) z s z t ( µ(v (0, 1) V (s, z s)) µz sv z (s, z s) The partial derivative equals, ( s ) zs/ z t = exp (λ + µ(1 2zu))du. Placing µz sv z (s, z s) into the exponent, V z (t, z t ) = Γ(t) := τ t t ) ds. e s t (r+λ+µ(1 z u))du µ(v (0, 1) V (s, z s))ds.

16 Property 1: Shirk at the End Theorem 1. Given {x t }, any optimal strategy {a t }, τ, exhibits shirking a t = 0 on [τ ɛ, τ ]. 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.

17 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 ), Differentiating again, Γ(t) = ρ(t)γ(t) µ(v (0, 1) V (t, z t )). Γ(t) = ρ(t) Γ(t) + ρ(t)γ(t) + µż t Γ(t) + µv t (t, z 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.

18 Property 3: Exit Condition Theorem 3. If {x t } is continuous, then τ satisfies V (τ, z τ ) = (x τ k) + µz }{{} τ V (0, 1) = 0. }{{} flow profit option value

19 Equilibrium

20 Definition Equilibrium beliefs Reputation x t = E F [θ t h t =, t τ] given by Bayes rule. Under point beliefs, ẋ = λ(ã 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 ).

21 Existence Theorem 4. An equilibrium exists. Idea Strategy space compact in weak topology. Bayes rule, best response correspondences u.h.c. Apply Kakutani-Fan-Glicksberg Theorem.

22 Pure Strategy Equilibria In a pure strategy equilibrium, x t = z t. {x t } decreases and incentives are single-peaked (Theorem 2). Changes in costs High costs: Full shirk equilibrium. Intermediate costs: Shirk-work-shirk equilibrium. Low costs: Work-shirk equilibrium.

23 Simulation Parameters Restaurant accounting Revenues: $x million. Capital cost: $500k. Investment cost: $125k. Interest rate: 20%. Arrival rates Breakthroughs arrive once a year. Technology shocks arrive every 5 years.

24 Value Function and Firm Distribution % of firms survive 10 years Value % of survivors 10 5 Work Region 0.05 Work Region 0 0 x e Reputation Reputation Figure: Capital cost k = 0.5, investment cost c = 0.1, interest rate r = 0.2, max. effort a = 0.99, breakthroughs µ = 1, technology shocks λ = 0.1.

25 Investment Incentives Investment Incentives, V z Reputation, x Self Esteem, z 1 Investment Incentives, V z Work Region 0 0 x e Reputation Figure: Capital cost k = 0.5, investment cost c = 0.1, interest rate r = 0.2, max. effort a = 0.99, breakthroughs µ = 1, technology shocks λ = 0.1.

26 Typical Life-cycles Reputation Reputation Years Years Reputation Reputation Years Years Figure: Capital cost k = 0.5, investment cost c = 0.1, interest rate r = 0.2, max. effort a = 0.99, breakthroughs µ = 1, technology shocks λ = 0.1.

27 Mixed Strategy Equilibria Exit Firm shirks near exit point (Theorem 1). Firms with less self-esteem exits gradually. Firm with most self-esteem exits suddenly. Reputational dynamics {x t } decreases until firms start to exit. {x t } increases when firms gradually exit.

28 Illustration of Mixed Strategy Equilibrium Work Region z(t) z t + Shirk Region zt x t 0 τ 0 τ τ τ(f )

29 Competitive Equilibrium Agent s preferences Firm i has expected output x t,i Total output of experience good is X t = i x t,idi. Consumers have utility U(X t ) + N t. Equilibrium Competitive equilibrium yields price P t = U (X t ). Stationary equilibrium: P t independent of t. Firm i s revenue is x t,i P and value is V i (t, z t ; P ). Entry Firm pays ξ to enter and is high quality with probability ˇx. Given a pure equilibrium, let zť = ˇx. Free entry determines price level: V (ť, zť; P ) = ξ.

30 Model Variation: Observable Investment

31 Observable Investment Investment a t is publicly observed Reputation and self-esteem coincide, x t = z t. Optimal strategies Optimal investment t â t = { 0 if λ ˆVz (ẑ t ) < c 1 if λ ˆV z (ẑ t ) > c Investment incentives ˆτ ˆΓ(t) = e [ s t r+λ+µ(1 ẑu)du 1 + µ( ˆV (1) ˆV ] (ẑ s )) ds. Optimal exit ˆV (ẑ t ) = ẑˆτ k + µẑˆτ ˆV (1) = 0. }{{}}{{} flow profit option value

32 Characterizing Equilibrium Theorem 5. If investment is observed, investment incentives ˆΓ(t) are decreasing with ˆΓ(0) > 0 and ˆΓ(ˆτ) = 0. Proof Value ˆV ( ) is strictly convex. Self-esteem ẑ t strictly decreases over time. Hence ˆV z (z t ) strictly decreases with ˆV z (zˆτ ) = 0. Idea: Investment is beneficial if There is a technology shock. There is a resulting breakthrough prior to exit time.

33 Value Function and Firm Distribution % of firms survive 10 years 0.25 Value % of survivors Work Region 0.05 Work Region 0 0 x e Reputation Reputation Figure: Capital cost k = 0.5, investment cost c = 0.1, interest rate r = 0.2, max. effort a = 0.99, breakthroughs µ = 1, technology shocks λ = 0.1.

34 Impact of Moral Hazard Theorem 6. If investment is observed, the firm works longer than in any baseline equilibrium. Idea When observed firm increases investment, belief also rises. Such favorable beliefs are good for the firm. Optimal investment choice higher for observed firm. With observable investment, No shirk region at the top. Work until lower reputation. Value higher, so exit later.

35 Model Variation: Privately Known Quality

36 Firm knows θ t Privately Known Quality Investment a t still unknown, so there is moral hazard. Recursive strategies Firm knows quality and time since breakthrough. Chooses investment a t and exit time τ. Value function V (t, θ t ). Optimal investment a t (θ) Independent of quality a t (θ) = a t and given by: { 1 if λ (t) > c a t = 0 if λ (t) < c where (t) := V (t, 1) V (t, 0) is value of quality. Tech. shock has probability λdt, yielding benefit (t) of work.

37 Exit Choice Assuming {x t } continuously decreases Low quality firm exits gradually when t > τ L. In equilibrium, high quality firm never exits. Assuming firm works at end, Exit condition becomes V (t, 0) = (x t c k) + λv (t, 1) = 0. }{{}}{{} flow profit option value

38 Equilibrium Characterization Theorem 7. If quality is privately observed, investment incentives (t) are increasing with (0) > 0. Proof The value of quality is present value of dividends: (t) = t e (r+λ)(s t) µ[v (0, 1) V (s, 1)]ds. Investment incentives λ (t) increase in t.

39 Impact of Private Information Known quality Work pays off if tech. shock (prob. λdt). Fight to bitter end. Low firm gradually exits; high never does. Unknown quality Work pays off if tech. shock & breakthrough (λdt µdt). Coast into liquidation. Firm exits after τ periods without breakthrough.

40 Value Function and Firm Distribution 71.8% of firms survive 10 years Value V H V L % of survivors 5 Work Region x e Work Region Reputation Reputation Figure: Capital cost k = 0.5, investment cost c = 0.1, interest rate r = 0.2, max. effort a = 0.99, breakthroughs µ = 1, technology shocks λ = 0.1.

41 Typical Life-cycles Reputation Reputation Years Years Reputation Reputation Years Years Figure: Capital cost k = 0.5, investment cost c = 0.1, interest rate r = 0.2, max. effort a = 0.99, breakthroughs µ = 1, technology shocks λ = 0.1.

42 Conclusion Model Firm dynamics in which main asset is firm s reputation. Characterize investment and exit dynamics over life-cycle. Equilibrium characterization Incentives depend on reputation and self-esteem. Shirk-work-shirk equilibrium. Benchmarks Observed investment: Work-shirk equilibrium. Privately known quality: Shirk-work equilibrium.

43 Appendix

44 Imperfect Private Information Private good news signals arrive at rate ν Self-esteem jumps to 1 when private/public signal arrive. Else, drift is ż t = λ(a t z t ) (µ + ν)z t (1 z t ). Equilibrium Γ(t) = Model is recursive since time of last public breakthrough. Investment incentives equal τ t e s t ρ(u)du [µ(v (0, 1) V (s, z s)) + ν(v (s, 1) V (s, z s))] ds where ρ(u) = r + λ + (µ + ν)(1 z u). Shirk at the end, t τ.

45 Brownian Motion Market observes signal Y t Y t evolves according to dy = µ B θ t dt + dw. Investment incentives are V z (x t, z t ) = E [ τ t where ρ u = r + λ µ2 B (1 2z u) 2 e s t ρudu+ s t (1 2zu)µ BdW u D(x s, z s )ds and D(x, z) = µ B ( x(1 x)vx (x, z) + z(1 z)v z (x, z) ). ] Results similar to good news case Shirk at end, as t τ. Shirk at start if a 1. Work in the middle if c not too large.