Multi-Agent Firms

1. Multi-Agent Firms Rob Axtell

2. Multi-Agent Firms:How does this fit in to what we have done? • Graduate microeconomics: • Markets • Games • Firms • Common criticism of general equilibrium theory: it is not strategic (e.g., Bob Anderson’s 201A, 201B,…201W) • More fundamental criticism of the theory of the firm: not even methodologicalindividualist [Winter 1993]!

3. Not Today’s Outline • Goal: Reproduce empirical data on U.S. firms • Firm sizes • Firm growth rates • Wage-size effects • … • Formulate using game theory • Conventional ‘solution concepts’ not useful • Constant adaptation at the agent level • Against the ‘Nash program’ of game theory • From firms to cities to countries...

4. Outline of Lecture 11MAS Model of Firm Formation and Dynamics(Paper + model:www.brookings.edu/dynamics/papers/firms) • Goal: Represent a firm with multiple agents • Start with a population of agents: • What economic environment induces firm formation? We want to grow firms. • Equilibrium? Stability? Dynamics? • What relevance to empirical data? • Next: Empirical validation of model output • Later: From firms to cities to countries...

5. Many Theories of the Firm • Textbook orthodoxy: Firms as black boxes • Production function specifies technology • Profit maximization specifies behavior • Winter’s critique: Not even methodologically individualist • Coase and Williamson (‘New Institutionalism’): • “Transaction cost” approach • Principal-agent (game theoretic) approaches: • Firm as nexus of contracts (incomplete contracts) • Firm as information processing network (e.g., Radner) • Evolutionary economics (Nelson and Winter): • Purposive instead of maximizing behavior • Industrial organization • Modern game theoretic orientation has little connection to data

6. Some (Old) Empirical DataFortune 1000 c. 1970s from Ijiri and Simon [1977]

7. Critique of the Neoclassical(U-Shaped) Cost Function “[T]heory says nothing about whether the same cost curves are supposed to prevail for all of the firms in an industry, or whether, on the contrary, each firm has its own cost curve and its own optimum scale. If the former then all firms in the industry should be the same size. A prediction could hardly be more completely falsified by the facts than this one is. Virtually every industry that has been examined exhibits...a highly skewed distribution of sizes with very large firms existing side by side with others of modest size. If each firm, on the other hand, has its own peculiar optimum, then the theory says nothing about what the resulting distribution of these optima for the industry should be. Thus, the theory either predicts the facts incorrectly or it makes no prediction at all.”

8. More Critique... “All these factors make static cost theory both irrelevant for understanding the size distribution of firms in the real world and empirically vacuous.” “Economics is not a discipline in which hypotheses that follow from classical assumptions, or that are necessary for classical conclusions, are quickly abandoned in the face of hostile evidence”

9. More Critique... “All these factors make static cost theory both irrelevant for understanding the size distribution of firms in the real world and empirically vacuous.” “Economics is not a discipline in which hypotheses that follow from classical assumptions, or that are necessary for classical conclusions, are quickly abandoned in the face of hostile evidence” Herbert Simon [1958]

10. Features of Agent Computation • Heterogeneous agents: replace representative agent, focus on distribution of behavior instead of average behavior; endogenous heterogeneity • Bounded rationality: essentially impossible to give agents full rationality in non-trivial environments • Local/social interactions: agent-agent interactions mediated by inhomogeneous topology (e.g., graph, social network, space) • Focus on dynamics: no assumption of equilibrium; paths to equilibrium and non-equilibrium adjustments • Each realization a sufficiency theorem

11. Synopsis of EndogenousFirm Formation Model • Heterogeneous population of agents • Situated in an environment of increasing returns (team production) • Agents are boundedly rational (locally purposive not hyper-rational) • Rules for dividing team output (compensation systems) • Agents have social networks from which they learn about job opportunities

12. An Analytical Modelof Firm Formation Set-Up:  Consider a group of N agents, each of whom supplies input (‘effort’) ei [0,1]  Total effort level: E = i{1..N}ei  Total output: O(E) = aE + bE, a, b≥ 0 b = 0 means constant returns, b > 0 is increasing returns  Agents receive equal shares of output: S(E)= O(E)/N  Agents have Cobb-Douglas preferences for income (output shares) and leisure, Ui(ei) = S(ei,E~i)i (1-ei)1-i

13. Equilibrium Proposition 1: Nash equilibrium exists and is unique

14. Equilibrium Proposition 1: Nash equilibrium exists and is unique é 2 ù 2 2 2 2 ( ) ( ) ( ) a 2 b E a 4 ab 1 E 4 b 1 E - - - q + + q + + q + ~ i i i ~ i i ~ i * ( ) e , E max 0 , q = ê ú i i ~ i ( ) 2 b 1 + q i ë û

15. q = 0.95 q = 0.90 q = 0.80 q = 0.50 Equilibrium Proposition 1: Nash equilibrium exists and is unique é 2 ù 2 2 2 2 ( ) ( ) ( ) a 2 b E a 4 ab 1 E 4 b 1 E - - - q + + q + + q + ~ i i i ~ i i ~ i * ( ) e , E max 0 , q = ê ú i i ~ i ( ) 2 b 1 + q i ë û ei* 1 0.8 0.6 0.4 0.2 E~i 5 10 15 20

16. Moral Hazard in Team Production Proposition 2: Agents under-supply input at Nash equilibrium

17. Moral Hazard in Team Production Proposition 2: Agents under-supply input at Nash equilibrium Consider a 2 agent team: e2 e1

18. Homogeneous Teams Utility as a function of team size and agent type

19. Homogeneous Teams Utility as a function of team size and agent type Optimal team size as a function of agent type

20. Stability, I ‘Best reply’ effort adjustment: Agents know last period’s output

21. Stability, I ‘Best reply’ effort adjustment: Agents know last period’s output

22. Stability, I ‘Best reply’ effort adjustment: Agents know last period’s output For a «b or E~i:

23. Stability, I ‘Best reply’ effort adjustment: Agents know last period’s output For a «b or E~i:

24. Stability, I ‘Best reply’ effort adjustment: Agents know last period’s output For a «b or E~i:

25. Stability, II Proposition 3: There is an upper bound on stable group size

26. Stability, II Proposition 3: There is an upper bound on stable group size Using row sums:

27. Stability, II Proposition 3: There is an upper bound on stable group size Using row sums:

28. Stability, II Proposition 3: There is an upper bound on stable group size Using row sums: Thus, the agent who most prefers income determines maximum size

29. Stability, II Proposition 3: There is an upper bound on stable group size Using row sums: Thus, the agent who most prefers income determines maximum size Using column sums:

30. Stability, II Proposition 3: There is an upper bound on stable group size Using row sums: Thus, the agent who most prefers income determines maximum size Using column sums:

31. Stability, III For homogeneous groups: Stability boundary is close to size at which individual and group utilities are maximized

32. Homogeneous Teams Utility as a function of team size and agent type Optimal team size as a function of agent type

33. Stability, III For homogeneous groups: Stability boundary is close to size at which individual and group utilities are maximized

34. Stability, III For homogeneous groups: Stability boundary is close to size at which individual and group utilities are maximized Optimal firms live on the edge of chaos!

35. Stability, III For homogeneous groups: For heterogeneous groups: Agent with largest preference for income determines maximum stable group size Stability boundary is close to size at which individual and group utilities are maximized Optimal firms live on the edge of chaos!

36. Motivations for aComputational Model Deficiencies of the analytical model: • Representative agent/representative group formulation • Exclusive focus on equilibria, which provide no information since they are unstable • Unstable equilibria not explosive • Analogy with financial markets, turbulence • Perfectly-informed, perfectly rational agents • Synchronous updating of model with equations

37. Motivations for aComputational Model Deficiencies of the analytical model: • Representative agent/representative group formulation • Exclusive focus on equilibria, which provide no information since they are unstable • Perfectly-informed, perfectly rational agents • Synchronous updating of model with equations  Agent-based computational modeling perfectly suited to by-pass these problems

38. The Computational Modelwith Agents • Preference parameter, , distributed uniformly on (0,1) • Firm output: O(E) = E + E,  ≥ 1 • Agents are randomly activated • Each computes its optimal effort level, e*, for: • staying a member of its present firm; • moving to a different firm (random graph); • starting a new firm; • The option that yields the greatest utility is selected

39. <Run Firms code>

40. Firm Size Distribution Firm sizes are Pareto distributed, fs(1+a) a ≈ -1.09

41. Productivity: Output vs. Size Constant returns at the aggregate level despite increasing returns at the local level

42. Firm Growth Rate Distribution Growth rates Laplace distributed by K-S test Stanley et al [1996]: Growth rates Laplace distributed

43. Variance in Growth Ratesas a Function of Firm Size slope = -0.174 ± 0.004 Stanley et al. [1996]: Slope ≈ -0.16 ± 0.03 (dubbed 1/6 law)

44. Wages as a Function of Firm Size:Search Networks Based on Firms Brown and Medoff [1992]: wages  size 0.10

45. Wages as a Function of Firm Size:Search Networks Based on Firms Brown and Medoff [1992]: wages  size 0.10

46. Firm Lifetime Distribution Data on firm lifetimes is complicated by effects of mergers, acquisitions, bankruptcies, buy-outs, and so on Over the past 25 years, ~10% of 5000 largest firms disappear each year

47. Effect of Model Parametrization • Importance of (locally) purposive behavior • Vary a, b, and : Greater increasing returns means larger firms • Alternative specifications of preferences • Role of social networks • Agent ‘loyalty’ is a stabilizing force in large firms • Bounded rationality: groping for better effort levels • Alternative compensation schemes • Firm founder sets hiring standards • Firm founder acts as residual claimant

48. Effect of Model Parametrization • Importance of (locally) purposive behavior • Vary a, b, and : Greater increasing returns means larger firms • Alternative specifications of preferences • Role of social networks • Agent ‘loyalty’ is a stabilizing force in large firms • Bounded rationality: groping for better effort levels • Alternative compensation schemes • Firm founder sets hiring standards • Firm founder acts as residual claimant

49. Sensitivity to Compensation Compensation proportional to input: Si(ei,E)= eiO(E)/E All firms now stable

50. Mixed Compensation Linear combination of compensation policies: Si(ei,E)= (aei/E+(1-a)/N)O(E)