Agent-based modeling and General Equilibrium

Transcription

1 Agent-based modeling and General Equilibrium Lastis symposium, ETHZ, September Antoine Mandel, Centre d Economie de la Sorbonne, Université Paris 1, CNRS

2 Outline 1 Motivation 2 Asymptotic convergence to Equilibrium 3 Observational equivalence between agent-based and general equilibrium models

3 Motivation General equilibrium models, which are still standard in macro-economic modeling, not fit to analyze non-marginal changes like financial crisis or green growth. Possible ways forward (see Enrich-EM project proposal). : 1 Enhance existing models but no need to consider models of a fundamentally different structure: learn the lessons of the financial crisis" 2 Consider the current approach is basically misleading and should be replaced by an altogether different kind of models. 3 Plurality of approaches, seize opportunities for cross-fertilization, enhance existing models with tools developed in other research communities develop more comprehensive new tools."

4 Motivation Claim: if Agent-Based models are to be these more comprehensive new tools", they should embed as a particular case general equilibrium models. In this talk: 1 General equilibrium as an asymptotic property of agent-based models. 2 Agent-based models as observationally equivalent" to General equilibrium models.

5 Outline 1 Motivation 2 Asymptotic convergence to Equilibrium 3 Observational equivalence between agent-based and general equilibrium models

6 Summary Claim: if Agent-Based models are to be these more comprehensive new tools", they should embed as a particular case general equilibrium models.

7 Gintis dynamics of general equilibrium Gintis (2007, 2008) An exchange economy with L goods, N types of consumers and M consumers of each type. Agents of type i have utility u i : R L + R and initial endowment ω i R L +. jth agent of type i has private price p i,j R L +. During a trading step," agents are randomly paired and exchange according to the following process. One of the agent propose a trade according to its (utility maximizing) demand determined according to its private price. The trade is accepted if it has a non-negative value for the trade partner (and rescaled according to stock).

8 Gintis dynamics of general equilibrium After a certain number of trading step, goods are consumed and utilities are computed Imitation: agents copy the private prices of successful peers Mutation: with a small probability, a new price is adopted at random. An Ergodic Markov chain an invariant distribution. Gintis simulations show the distribution concentrates in the configuration where every agent uses the same equilibrium price (even in the Scarf economy).

9 Gintis dynamics of general equilibrium Figure: Long-term dynamics of prices in Gintis model

10 Gintis dynamics of general equilibrium Equilibrium price as a convention in the sense of Peyton-Young (1993). Analytical results: Leontief preferences: Mandel and Botta (2010) Scarf: Mandel and Gintis (2012). General case: Gintis and Mandel (in progress)

11 Lagom: extension to a setting with capital accumulation (Jaeger, Mandel, Wolf and al) L types of goods (used as intermediary inputs, fixed capital, consumption) and one kind of labor. A finite number of firms partitioned into L sectors according to the good they produce. Production from labor, heterogeneous capital goods and intermediary inputs. Constant returns to scale. A finite number of households consume goods and provide labor, they are endowed with an utility function. An endogenous growth engine: external effect of investment on labor productivity.

12 Lagom: Agents A notion of feasible path similar to this used in optimal growth theory but dynamics emerging from agents actions and interactions rather than from intertemporal optimization goods/sectors, Households, Firms, A Government, A Financial System. Each agent has a state space consisting of: stocks of goods and money, Control/Behavioral variables: prices, expectations, targets... Agents have methods to manage their stocks using their control variables and methods to update their control variables

13 Agents coordination Schedule of Actions and Interactions controlled by a central clock: Exchange of goods: bilateral interactions based on a network structure. Firms are price-setters and use mark-up pricing. Labor market: matching of firms and households. Production/consumption Accounting Behavior and expectations updating

14 Behavioral updating Genetic evolution of strategic variables on longer time-scales: mark-ups, wages, technologies: Genetic evolution of technologies according to cost-efficiency Genetic evolution of mark-ups according to profitability. Genetic evolution of wage type according to vacancies filled. Genetic evolution of consumption shares according to utility functions. Entry and exit of firms according to profitability in the sector. Mark-up of entering firms: interest rate+ fixed premium

15 Short-term dynamics Non-linearities, bounded rationality and randomness of agents behavior at the micro-level give rise to short-term fluctuations akin to business cycles Figure: Output (rate of expectations change 0.2), log. scale from 10 7 to

16 Long-term dynamics (when the economy has a unique general equilibrium) In the long-run, optimization through imitation/mutation processes and competition among firms through entry and exit become dominant. Convergence to equilibrium: prices stabilize, rationing disappear. 1 1e Figure: 10*Mark-Up (orange) and Price (yellow)

17 Long-term dynamics (when the economy has a unique general equilibrium) Figure: Unfulfilled demand

18 Long-term dynamics (when the economy has a unique general equilibrium) Once, equilibrium has been established on the labor and commodities markets, the economy enters an exponential growth path 1 1e5 1e4 1e3 1e2 1e Figure: Output (blue) and Consumption (pink), log. scale

19 Scaling with the number of sectors Similar behavior as the number of sectors increase. Model with two sectors producing investment goods, two sectors producing final goods e-1 9e-1 9e-1 9e-1 8e-1 8e-1 8e-1 8e-1 7e-1 7e-1 7e-1 7e-1 6e-1 6e-1 6e-1 6e-1 5e-1 5e-1 5e-1 5e-1 4e-1 4e-1 4e-1 4e-1 4e-1 4e-1 4e-1 4e-1 2e-1 2e-1 2e-1 2e-1 1e-1 1e-1 1e-1 1e Figure: 2*Mark-up (Orange) and Price (yellow)

20 Scaling with the number of sectors e3 1e3 1e3 1e3 1e e e e Figure: Output (blue), log. scale

21 The role of time-scales The speed of evolution of expectations main determinant of the growth rate Figure: Output for expectations rate of change of 0.05 (green), 0.1(red), 0.2 (blue), log. scale

22 The role of time-scales Equilibrium convergence only if prices evolve fast enough. Here prices updated only every five periods (every period in the preceding) 1e e-2 8e e-2 6e e-2 4e e-2 1 2e-2 1e-2 9e Figure: Unemployment (magenta) and average wage reference (green)

23 Multiple equilibria and regime changes Two perfectly symmetric sectors and linear preferences. Indeterminacy Figure: Output (yellow), final consumption (magenta)

24 Multiple equilibria and regime changes Two perfectly symmetric sectors and linear preferences. Indeterminacy of equilibrium Figure: Output (yellow), final consumption (magenta)

25 Taylor rule and indeterminacy Increasing the sensitivity of the taylor rule leads to changes in the interest rate of larger amplitude, which feedback on final demand 1e18 1 1e16 1e14 1e12 1e10 1e8 1e6 1e4 1e Figure: Output, log. scale

26 Outline 1 Motivation 2 Asymptotic convergence to Equilibrium 3 Observational equivalence between agent-based and general equilibrium models

27 The neoclassical growth model The representative agent solves for max + t=0 (1 + ρ)t u RA (c t ) s.t c t + k t+1 (1 δ)k t = F(k t ) Under standard assumptions, there exists a unique solution (c t, k t ) t N characterized by the path of capital accumulation: h RA (k t ) = k t+1 And a a stable steady state characterized by k such that h RA (k ) = k

28 OLG equilibrium At each time t 0, an household borns and lives for two periods. It supplies inelastically one unit of labor at date t and solves for max u OLG (c t t, ct t+1 ) s.t p t c t + p t+1 c t+1 w t At each time t, the firm maximizes profits: max p t (F(k t ) k t+1 ) + q t+1 k t+1 w t q t k t where p t, w t, q t, are prices of date t output, labor and capital

29 OLG equilibrium Under standard assumptions, there exists a unique equilibrium (c t, t, c t 1, t, kt, pt, wt, qt ), completely characterized by the path of capital accumulation: h OLG (k t ) = k t+1 And a stable steady state characterized by k such that h OLG (k ) = k

30 Observational equivalence (Aygari JET 1985) The OLG and the representative agent models are observationaly equivalent if they lead to identical time paths for aggregate capital, output, consumption, investment, real wage, and the real interest rate. Formally, for any u RA there exists u OLG such that h RA = h OLG and conversely. Under specific assumptions, obervational equivalence holds (Aygari JET 1985). In general, the range of dynamics that could be exhibited by OLG models is much larger than that exhibited by [RA] models and observational equivalence cannot possibly hold (Aygari JET 1985).

31 Observationnal equivalence, GE and ABM Obervational equivalence relationships between ABM and neoclassical growth model: Show that agent-based can subsume general equilibrium models. A Basecamp to explore out-of-equilibrium dynamics. Also an attempt to define the simplest computational structure.

32 Agents and contracts data Good = Capital Int Labor Consumption Money type Technology = (Input,Output) -> (Input,Output) data Agent = Ag AgentId AgentKind Technology Input Output data Contract = Ctr AgentId AgentId Good Price Quantity Time Time type Economy = (Date,[Contract],Array AgentId Agent) A system of heterogeneous agents An agent is a stock, a technology to transform stocks autonomously and an identity which will also serve as position/hour. The contracts are used to represent memory, expectations, interactions, networks.

33 Agent s behavior Two kind of firms: consumption and capital producers. Firms except a steady demand and produce accordingly from labor and capital, set the wage at the minimal level sufficient to keep workers employed (despite competition from other firms), they choose their input mix so as to minimize costs, finance net investment by issuing new equity, increase (resp. decrease) their price when demand is above (resp. below) the expected level. Households supply one unit of labor every period, receive wages and dividends from money invested in firms, have a subsistence and a target income (heterogeneous): they save until they have reached their target income and always consume at least the subsistence income.

34 Dynamics step :: Economy -> Economy step eco = foldr act eco ids where ids = map identity (elems (agents eco)) > act :: Economy -> AgentId -> Economy > act eco id = eco8 where > eco1 = updatefinancialstatus ((agents eco)!id) eco > eco2 = executependingcontracts ((agents eco1)!id) eco1 > eco3 = foldl actonmarket eco2 orderedmarkets > eco4 = produce ((agents eco6)!id) eco3 > orderedmarkets = [Money, (Capital 0), Labor, Consumption] act only modifies the acting agent and the list of contracts Autonomy. A deterministic model: No instrumental randomness to ensure (e.g) trading opportunities are ex-ante equal. When it acts, each agent is in a similar situation with regards to his environment.

35 Contracts All exchanges are governed by contracts of similar (computational) types. Contract specify a good, a quantity to be delivered, a price, a payment and a delivery date. Demand, supply, price setting and regulations can be good specific. For capital and consumption: deliveries can be delayed, payment can t (firms do not ration their consumers but delay delivery). For labor: deliveries nor payment can t be delayed. For equity : payment (dividend) is conditional on profits.

36 Contracts processing and implementation demand :: Economy -> Good -> Agent -> Price -> Quantity supply:: Hour -> Economy -> Good -> Agent -> Price -> Quantity suppliers :: Economy -> Good -> Agent -> [Agent] setcontract:: Economy -> Good -> Agent -> Agent -> Contract setcontract eco g buyer seller = Ctr (identity buyer) (identity seller) g p q d1 d2 where q = min (demand eco g buyer p) (supply (hour buyer) eco g seller p) d1 = if ( elem g [Money,Labor]) then (date eco+1, hour buyer) else time eco buyer d2 = time eco buyer p = case g of... executecontract :: Contract -> Economy -> Economy executecontract ctr eco= let (buyer1,seller1) = if (paymenttime ctr == time eco buyer) then (addtooutput Money (-value) buyer, addtooutput Money (value) seller) else (buyer,seller) (newbuyer,newseller) = if (deliverytime ctr == time eco buyer) then (addtooutput (good ctr) (delivery) buyer1, addtooutput (good ctr) (-delivery) seller1) else (buyer1,seller1) newctrs = [setquantity delivery ctr, delaydelivery (setquantity ((quantity ctr)-delivery) ctr)] ++ (delete ctr (contracts eco))

37 Preliminary results Steady state if initialized at equilibrium Transient dynamics not yet consistent with RA-equilibrium

38 Steady state Good stability properties around an equilibrium steady-state Figure: Prices

39 Steady state Good stability properties around an equilibrium steady-state Figure: Quantities

40 Summary ABMs has a superclass of general equilibrium models. Some asymptotic results are established. Equivalence results: work in progress.