agent-based modeling and general equilibrium · claim: if agent-based models are to be these...

Motivation Asymptotic convergence to Equilibrium Observational equivalence between agent-based and general equilibrium models

Agent-based modeling and GeneralEquilibrium

Lastis symposium, ETHZ, September 11 2012

Antoine Mandel, Centre d’Economie de la Sorbonne,Université Paris 1, CNRS


Outline

1 Motivation

2 Asymptotic convergence to Equilibrium

3 Observational equivalence between agent-based andgeneral equilibrium models


Motivation

General equilibrium models, which are still standard inmacro-economic modeling, not fit to analyze non-marginalchanges like financial crisis or green growth.Possible ways forward (see Enrich-EM project proposal). :

1 Enhance existing models but no need to consider models ofa fundamentally different structure: “learn the lessons of thefinancial crisis"

2 Consider the current approach is basically misleading andshould be replaced by an altogether different kind ofmodels.

3 Plurality of approaches, seize opportunities forcross-fertilization, enhance existing models with toolsdeveloped in other research communities develop morecomprehensive new tools."


Motivation

Claim: if Agent-Based models are to be these “morecomprehensive new tools", they should embed as aparticular case general equilibrium models.In this talk:

1 General equilibrium as an asymptotic property ofagent-based models.

2 Agent-based models as “observationally equivalent" toGeneral equilibrium models.


Outline

1 Motivation




Summary

Claim: if Agent-Based models are to be these “morecomprehensive new tools", they should embed as aparticular case general equilibrium models.


Gintis’ “dynamics of general equilibrium”

Gintis (2007, 2008)An exchange economy with L goods, N types ofconsumers and M consumers of each type.Agents of type i have utility ui : RL

+ → R and initialendowment ωi ∈ RL

+.

j th agent of type i has private price pi,j ∈ RL+.

During a “trading step," agents are randomly paired andexchange according to the following process.

One of the agent propose a trade according to its (utilitymaximizing) demand determined according to its privateprice.The trade is accepted if it has a non-negative value for thetrade partner (and rescaled according to stock).



After a certain number of trading step, goods areconsumed and utilities are computedImitation: agents copy the private prices of successfulpeersMutation: with a small probability, a new price is adopted atrandom.An Ergodic Markov chain⇒ an invariant distribution.Gintis’ simulations show the distribution concentrates in theconfiguration where every agent uses the same equilibriumprice (even in the Scarf economy).



Figure: Long-term dynamics of prices in Gintis’ model



Equilibrium price as a convention in the sense ofPeyton-Young (1993).Analytical results:

Leontief preferences: Mandel and Botta (2010)Scarf: Mandel and Gintis (2012).General case: Gintis and Mandel (in progress)


Lagom: extension to a setting with capitalaccumulation (Jaeger, Mandel, Wolf and al)

L types of goods (used as intermediary inputs, fixedcapital, consumption) and one kind of labor.A finite number of firms partitioned into L sectors accordingto the good they produce.Production from labor, heterogeneous capital goods andintermediary inputs.Constant returns to scale.A finite number of households consume goods and providelabor, they are endowed with an utility function.An endogenous growth engine: external effect ofinvestment on labor productivity.


Lagom: Agents

A notion of feasible path similar to this used in optimalgrowth theory but dynamics emerging from agents actionsand interactions rather than from intertemporaloptimization.1-10 goods/sectors, 103-104 Households, 102-103 Firms, AGovernment, 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 theircontrol variables and methods to update their controlvariables


Agents coordination

Schedule of Actions and Interactions controlled by acentral clock:

Exchange of goods: bilateral interactions based on anetwork structure . Firms are price-setters and use mark-uppricing.Labor market: matching of firms and households.Production/consumptionAccountingBehavior and expectations updating


Behavioral updating

Genetic evolution of strategic variables on longertime-scales: mark-ups, wages, technologies:

Genetic evolution of technologies according tocost-efficiencyGenetic evolution of mark-ups according to profitability.Genetic evolution of wage type according to vacanciesfilled.Genetic evolution of consumption shares according to utilityfunctions.Entry and exit of firms according to profitability in the sector.Mark-up of entering firms: interest rate+ fixed premium


Short-term dynamics

Non-linearities, bounded rationality and randomness ofagents’ behavior at the micro-level give rise to short-termfluctuations akin to business cycles

20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360 380 400 420 440 460 480 500 520 540 560 580 600 620 640 660 680 700 720 740 760 780 800 820 840 860 880 900 920 940 960 980 1000 1020 1040 1060 1080 1100 1120 1140 1160 1180 1200 1220 1240 1260 1280 1300 1320 1340 1360 1380 1400 1420 1440 1460 1480 1500 1520 1540 1560 1580 1600 1620 1640 1660 1680 1700 1720 1740 1760 1780 1800 1820 1840 1860 1880 1900 1920 1940 1960 1980

1

1e1

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Figure: Output (rate of expectations change 0.2) , log. scale from 107

to 7.107


Long-term dynamics (when the economy has a uniquegeneral equilibrium)

In the long-run, optimization through imitation/mutationprocesses and competition among firms through entry andexit become dominant.Convergence to equilibrium: prices stabilize, rationingdisappear.

200 400 600 800 1000 1200 1400 1600 18000

1e-1

0

1

1

Figure: 10*Mark-Up (orange) and Price (yellow)



Figure: Unfulfilled demand



Once, equilibrium has been established on the labor andcommodities markets, the economy enters an exponentialgrowth path

200 400 600 800 1000 1200 1400 1600 18001

1e1

1e2

1e3

1e4

1e5

1

Figure: Output (blue) and Consumption (pink), log. scale


Scaling with the number of sectors

Similar behavior as the number of sectors increase.Model with two sectors producing investment goods, twosectors producing final goods.

500 1000 15000

1e-1

2e-1

4e-1

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1.11

500 1000 15000

1e-1

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500 1000 15000

1e-1

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500 1000 15000

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Figure: 2*Mark-up (Orange) and Price (yellow)


Scaling with the number of sectors

500 1000 1500

1e1

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500 1000 1500

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1e3

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500 1000 1500

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500 1000 1500

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Figure: Output (blue), log. scale


The role of time-scales

The speed of evolution of expectations main determinantof the growth rate

0 250 500 750 1 000 1 250 1 500 1 750 2 000

100

1000

10000

100000

1000000

10000000

100000000

Figure: Output for expectations rate of change of 0.05 (green),0.1(red), 0.2 (blue), log. scale


The role of time-scales

Equilibrium convergence only if prices evolve fast enough.Here prices updated only every five periods (every periodin the preceding)

200 400 600 800 1000 1200 1400 1600 18000

1e-2

2e-2

3e-2

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Figure: Unemployment (magenta) and average wage reference(green)


Multiple equilibria and regime changes

Two perfectly symmetric sectors and linear preferences.Indeterminacy

Figure: Output (yellow), final consumption (magenta)


Multiple equilibria and regime changes

Two perfectly symmetric sectors and linear preferences.Indeterminacy of equilibrium

Figure: Output (yellow), final consumption (magenta)


Taylor rule and indeterminacy

Increasing the sensitivity of the taylor rule leads to changesin the interest rate of larger amplitude, which feedback onfinal demand

1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 110001e2

1e4

1e6

1e8

1e10

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1e181

Figure: Output, log. scale


Outline

1 Motivation




The neoclassical growth model

The representative agent solves for

max∑+∞

t=0 (1 + ρ)tuRA(ct)s.t ct + kt+1 − (1− δ)kt = F (kt)

Under standard assumptions, there exists a uniquesolution (c∗t , k

∗t )t∈N characterized by the path of capital

accumulation:hRA(k∗t ) = k∗t+1

And a a stable steady state characterized by k∗ such that

hRA(k∗) = k∗


OLG equilibrium

At each time t ≥ 0, an household borns and lives for twoperiods. It supplies inelastically one unit of labor at date tand solves for

max uOLG(ctt , c

tt+1)

s.t ptct + pt+1ct+1 ≤ wt

At each time t , the firm maximizes profits:

max pt(F (kt)− kt+1) + qt+1kt+1 − wt − qtkt

where pt ,wt ,qt , are prices of date t output, labor andcapital


OLG equilibrium

Under standard assumptions, there exists a uniqueequilibrium (ct ,∗∗

t , ct−1,∗∗t , k∗∗t ,p∗∗t ,w

∗∗t ,q∗∗t ), completely

characterized by the path of capital accumulation:

hOLG(kt) = kt+1

And a stable steady state characterized by k∗∗ such that

hOLG(k∗∗) = k∗∗


Observational equivalence (Aygari JET 1985)

The OLG and the representative agent models areobservationaly equivalent if they “lead to identical timepaths for aggregate capital, output, consumption,investment, real wage, and the real interest rate.”Formally, for any uRA there exists uOLG such that

hRA = hOLG

and conversely.Under specific assumptions, obervational equivalenceholds (Aygari JET 1985).“In general, the range of dynamics that could be exhibitedby OLG models is much larger than that exhibited by [RA]models and observational equivalence cannot possiblyhold” (Aygari JET 1985).


Observationnal equivalence, GE and ABM

Obervational equivalence relationships between ABM andneoclassical growth model:

Show that agent-based can subsume general equilibriummodels.A Basecamp to explore out-of-equilibrium dynamics.

Also an attempt to define the simplest computationalstructure.


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 agentsAn agent is a stock, a technology to transform stocksautonomously and an identity which will also serve asposition/hour .The contracts are used to represent memory, expectations,interactions, networks.


Agent’s behavior

Two kind of firms: consumption and capital producers.Firms except a steady demand and produce accordinglyfrom labor and capital, set the wage at the minimal levelsufficient to keep workers employed (despite competitionfrom other firms), they choose their input mix so as tominimize costs, finance net investment by issuing newequity, increase (resp. decrease) their price when demandis above (resp. below) the expected level.Households supply one unit of labor every period, receivewages and dividends from money invested in firms, have asubsistence and a target income (heterogeneous): theysave until they have reached their target income andalways consume at least the subsistence income.


Dynamics

step :: Economy -> Economystep 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 ofcontracts⇒ Autonomy.A deterministic model:

No instrumental randomness to ensure (e.g) tradingopportunities are ex-ante equal.When it acts, each agent is in a similar situation withregards to his environment.


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 goodspecific.For capital and consumption: deliveries can be delayed,payment can’t (firms do not ration their consumers butdelay delivery).For labor: deliveries nor payment can’t be delayed.For equity : payment (dividend) is conditional on profits.


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 -> ContractsetContract eco g buyer seller =

Ctr (identity buyer) (identity seller) g p q d1 d2where

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 buyerd2 = time eco buyerp = case g of ...

executeContract :: Contract -> Economy -> EconomyexecuteContract 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))


Preliminary results

Steady state if initialized at equilibriumTransient dynamics not yet consistent with RA-equilibrium


Steady state

Good stability properties around an equilibriumsteady-state

Figure: Prices


Steady state

Good stability properties around an equilibriumsteady-state

Figure: Quantities


Summary

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

agent-based modeling and general equilibrium · claim: if agent-based models are to be these...

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