An Equilibrium Model with Computationally Constrained Agents
aa r X i v : . [ q -f i n . E C ] N ov An Equilibrium Model with ComputationallyConstrained Agents Wolfgang KuhleMax Planck Institute for Research on Collective Goods, Kurt-Schumacher-Str. 10, 53113Bonn, Germany. Email: [email protected].
Abstract:
We study a large economy in which firms cannot compute exact solutionsto the non-linear equations that characterize the equilibrium price at which they can sellfuture output. Instead, firms use polynomial expansions to approximate prices. The preci-sion with which they can compute prices is endogenous and depends on the overall level ofsupply. At the same time, firms’ individual supplies, and thus aggregate supply, depend onthe precision with which they approximate prices. This interrelation between supply andprice forecast induces multiple equilibria, with inefficiently low output, in economies thatotherwise have a unique, efficient equilibrium. Moreover, exogenous parameter changes,which would increase output were there no computational frictions, can diminish agents’ability to approximate future prices, and reduce output. Our model therefore accommo-dates the intuition that interventions, such as unprecedented quantitative easing, can putagents into “uncharted territory”.
Keywords: Polynomial Inference, Self-Referential Equilibria, Glitch Equilibria
Few people would claim that they are able to compute future equilibrium outcomes, suchas prices, with any accuracy. Despite this, textbook models implicitly assume that, givenall relevant data, agents compute exact numeric values for future equilibrium prices, re-spectively, the entire distribution of these prices if the model involves risk. In this paper,we assume that economic agents are computationally constrained to the use of polynomialfunctions. That is, instead of being able to solve arbitrary non-linear problems, they rely I am particularly indebted to Martin Hellwig for detailed comments on an earlier draft of this paper.I also thank Dominik Grafenhofer, Sebastian Klein, Harvey Lapan, and Carl Christian von Weizs¨ackerfor discussions on equilibrium models. Finally, I received helpful questions and comments from seminarparticipants in Bonn. First draft July 2015.
1n polynomial expansions to approximate future equilibrium outcomes. Put differently,agents act just like economic researchers who use polynomials, such as the Arrow-Prattapproximation, to restate complicated non-linear problems in terms of workable polyno-mials.Using a two-period model, in which firms employ polynomial approximations to inferfuture selling prices for their output, we find that multiple equilibria emerge in well-behaved economies that would have a unique, efficient equilibrium if agents could computefuture equilibria with perfect accuracy. Moreover, exogenous parameter changes, whichwould increase economic activity were agents computationally unconstrained, can reduceeconomic activity as they make it harder for agents to approximate equilibrium prices.Our results rely on the fact that there are levels of supply where a polynomial ap-proximation to the function, which relates equilibrium supply to equilibrium price, is ofgood quality, and other levels where it is of low quality. Put differently, a firm’s ability tocompute equilibrium prices changes with the level of aggregate supply. At the same time,individual supply, and thus aggregate supply, varies with the precision with which agentscan predict prices. This interaction gives rise to two coexisting types of equilibria. In thefirst, economic activity falls into intervals where agents’ polynomial approximations are ofhigh quality and the role of the computational friction is small. These equilibria can coin-cide with the rational expectations equilibrium (REE). In the second type, computationalfrictions are important and agents find it difficult to predict prices: Aggregate supply is (i)low and (ii) falls into an interval where agents’ approximations, to the equation describingequilibrium, are of poor quality.In one interpretation, we may think of a farmer who must decide in spring how muchcorn he should plant. This farmer may know the price at which corn tends to sell inyears with “normal” supply. Moreover, he might know that small increases in aggregatesupply tend to reduce prices, i.e, that demand is locally downward sloping. Finally, hemay know that this downward slope tapers off as supply increases. The farmer, however,is unable to calculate all numeric values that the demand function takes over its entiredomain. If he wants to calculate those prices, which obtain once supply differs fromthose levels that he is familiar with, he must use a polynomial expansion to extrapolatethe new price. In a macroeconomic interpretation, we think of a large number of firmsthat have to choose production today in anticipation of future demand. These firmsknow the price at which their goods sell in “normal times”. However, if firms collectivelycut production today, it will be difficult for them to know whether future selling prices2ncrease, due to reduced supply, or fall, since the layoffs, associated with production cuts,reduce demand. This intuition extends naturally to economies where demand concernsa vector of goods, which may involve substitutes and complements. In such a setting itappears even more natural to assume that firms cannot solve for the overall equilibrium.Instead, a firm, which produces a particular good, may, if it is exceptionally well informed,use the economy’s Jacobian matrix to compute demand for its particular good in termsof a first-order polynomial approximation.Regarding parameter changes, the uncertainty that agents face in our model does notoriginate from a world with stochastically changing parameters. Instead, agents know themagnitude of the parameter change in advance; the difficulty is to predict its consequences.As an illustration, we refer to two representative comments made on the quantitativeeasing program: Stanley Druckenmiller, an accomplished investor with a thirty-year trackrecord, commented in retrospect “I didn’t know how it was going to end... I wouldhave said inflation [which] would have been dead wrong.” Similarly, taking an ex-anteperspective, Joseph Stiglitz predicted that QE2 would likely bring interest rates down “alittle”, but that it was unclear what the risks, ranging from economic growth to“a wholeset of other potential risks that - may result from this policy” , were. Likewise, thereappears to be no consensus among observers how, if at all, a UK exit from the EU willaffect the economies of the UK and the remainder EU. Similar arguments apply to morelong-term problems such as the demographic transition. As we show, agents’ inabilityto perform comparative statics alters model predictions considerably: Large parameterchanges, which would unambiguously increase output if agents could compute the model’s Diamond (1982), and Cooper and John (1988), develop models where search frictions result inupward-sloping aggregate demand functions. See Hellwig (1993) for a review of models with non-monotonic demand. More generally, such effects are important in economies where Say’s law, statingthat supply creates its own demand, is of relevance. See also Chamley (2014) for a related model ofsavings and investment. Speech given at the 2015 Dealbook conference. Interview given on the Charlie Rose show on 3 November 2010. See Soros (1994) for case studies highlighting market participants’ difficulties to anticipate the im-pact that pre-announced central bank policies have on macroeconomic equilibrium variables. Similarly,Niederhoffer (1997), p. 381, concludes his discussion on excess demand functions: “The difficulty is thatnobody knows what the equilibrium level is until at least the morning after the fact.” That is, we know from birth statistics that cohorts entering the labor market will be smaller andcohorts entering retirement will grow. At the same time, it proves difficult to predict how such changesimpact future growth paths.
Related Literature:
Due to their bounded computational capacity, agents work withan approximate, misspecified model. Except for special cases, they cannot form rationalexpectations in the sense of Hutchison (1937), Grunberg and Modigliani (1954), Muth(1961), Blanchard (1979) and DeCanio (1979), which are consistent with the true model.Regarding model misspecification, our approach is thus akin to the literature on learning,Bray (1982), Marcet and Sargent (1989), and Sargent (1993), where agents use a misspec-ified least squares approach to infer unknown model parameters. Rothschild (1974) andMcLennan (1984) model firms that experiment with different supply functions to learnabout stochastic demand. Firms in our model are small, and thus changing individualsupply has no influence on prices. That is, in the dynamic extension of our model, theinformation that firms learn over time is determined by overall equilibrium rather thanindividual experimentation.We interpret our baseline model as a simple A D , A S setting, as in Keynes (1936) andSamuelson (2009), which is augmented with a computational friction. In Section 4, weshow that our model may be reinterpreted as a Diamond (1982) and Cooper and John(1988) aggregate search model, where the probability of finding a trading partner de-pends on the equilibrium level of economic activity. In this interpretation, agents’ com-putational constraint makes it difficult for them to compute the equilibrium probabilityof finding a trading partner. Regarding government intervention, Diamond (1982) andCooper and John (1988) find positive multipliers, which are due to the search friction.In the current model, where the search friction is coupled with a computational friction,government intervention has a non-monotonous effect on output.Tesfatsion (2006), Gintis (2007), Farmer and Foley (2009), and Thurner et al. (2012)argue for “agent-based” models in which agents follow decision rules that do not neces-sarily coincide with rational behavior. In the current paper, agents are computationally One argument for such a departure is computational complexity: Rubinstein (1998), Maymin (2011)and Ackerman et al. (2011) emphasize that agents might be constraint in their ability to count or tocompute conditional probabilities. Nelson and Winter (1985), Ulanowicz (2008), Hofbauer and Sandholm(2011), Arthur (2015), and Kuhle (2016) for evolutionary models where biases emerge endogenously. y = f ( y ; b ), are commonly eval-uated in terms of a first-order polynomial expansion ∆ y ≈ ∆ yf y + ∆ bf b , which yields ∆ y ∆ b ≈ f b − f y , respectively, ∂y∂b = f b − f y . Hence, we argue that the current model is method-ologically consistent in the sense that the outside researcher, i.e., the paper’s reader, willuse the same method of analysis that is used by the model’s agents.Section 2 abstracts from computational frictions and identifies the unique rationalexpectations equilibrium. Section 2.1 studies equilibria that obtain with computationallyconstrained agents. Section 2.2 considers the impact of exogenous parameter changes.Section 3.1 studies the economy’s convergence to the rational expectations equilibrium ina dynamic setting, where firms accumulate empirical knowledge. Section 3.2 introducesasymmetric information. In Section 4, we suggest different interpretations of our baselinemodel. Section 5 concludes.
We study a large economy in which a mass one of firms i ∈ [0 ,
1] produce a homogenousgood. There are two periods of time. In the first period, each firm chooses to producea quantity of goods a i in anticipation of a future selling price ˆ P . In the second period,firms sell the finished products a i inelastically to consumers at a market clearing price P . For simplicity, to ensure uniqueness of the REE in the economy without computa-tional friction, we assume that aggregate demand is twice continuously differentiable and Similarly, Mas-Colell et al. (1995), pp. 599-641, use first-order polynomial expansions to examinenon-linear demand in pure exchange economies with many commodities. As mentioned earlier, Finetti(1952), Arrow (1970), Pratt (1964) use second-order expansions to study expected utility. Likewise, thefamiliar first- and second-order conditions, f ′ ( x ) = 0 and f ′′ ( x ) <
0, for a smooth function f ( x ) to havelocal maximum at point x , stem from an expansion f ( x ) = f ( x )+ f ′ ( x )( x − x )+ f ′′ ( x )( x − x ) + O ;see Chiang and Wainwright (2005), pp. 250-253, or Samuelson (1947), pp. 357-379. Related, the stabilityof differential and difference equations, Samuelson (1947), pp. 21-121, 257-349, and 380-439, or Galor(2007). Finally, DeCanio (1979), p. 52, points out that solving expectations equilibria requires eitherthat the model is assumed to be linear, or that the model’s equations have to be linearized, i.e., rewrittenin terms of a first-order polynomial, which is what our agents do. A : P = φ ( A ) , φ A < , φ AA ≷ , φ (0) > . (1)Demand (1) represents the model’s non-linearity, respectively, the computational obstaclethat agents have to overcome. In Section 4, we suggest three different interpretations of φ ()by showing that it captures the non-linearities that individual agents face when they makeforward-looking decisions in the standard workhorse models of Diamond (1965), Diamond(1982), and the A D , A S model of Samuelson (2009). That is, φ may be interpreted as (i)future returns to savings, (ii) the probability of finding a trading partner, or (iii) theselling price for output.Firm i chooses a production schedule a ∗ i to maximize expected profits a ∗ i = arg max a i n π i = a i ˆ P − a i o , a i ≥ , where ˆ P is the firm’s expectation regarding the selling price and a i is a quadratic costfunction. Hence, agent i supplies a ∗ i = ˆ P and aggregate supply is A = Z [0 , a ∗ i di = ˆ P . (2)If agents are computationally unconstrained, they can compute demand (1) over its entiredomain. That is, for each level of aggregate supply A , they form rational price expecta-tions ˆ P = φ ( A ). In turn, they combine (1) and (2) to calculate the unique equilibriumquantity A : A = φ ( A ) , (3)and, using (1), they compute equilibrium price P : P = φ ( A ) . (4)Accordingly, we have Lemma 1.
There exists a unique, rational expectations equilibrium { A , P } ∈ R . Inthis equilibrium agents forecast prices correctly ˆ P = P .Proof. Market clearing (3)-(4) determines equilibrium quantity A >
0, which is uniquesince φ (0) > φ A <
0. Using (1), the equilibrium price is P = φ ( A ) = A > P = A and thus ˆ P = P .6 .1 Polynomial Equilibria We now assume that firms cannot compute demand over its entire domain. Instead, theyare familiar with a point on the demand function, A ∗ , φ ( A ∗ ), and the demand function’sslope φ A ( A ∗ ) at this point. It is convenient to start with the assumption that this pointis the REE of Lemma 1, i.e, agents know A , φ ( A ), and the slope φ A ( A ). In turn, oncesupply differs from A , agents use polynomial expansions to extrapolate demand to esti-mate the resulting price. Polynomial equilibrium points will be those points where thepolynomial, which mimics true demand, intersects with supply. Put differently, “poly-nomial equilibria” are those points that solve the agents’ approximate model. The REEfrom the previous section will be one, but not the only, such equilibrium.Expanding demand (1) around the perfect foresight equilibrium, agents forecast theselling price (1) as:ˆ P = φ ( A ) + φ A ( A )∆ A, ∆ A = A − A . (5)Equation (5) reflects that agents cannot numerically compute the true price P = φ ( A +∆ A ) at which a supply A = A + ∆ A sells. The reliability of estimate (5) decreases themore aggregate supply A differs from A . We assume that agents choose output a i tomaximize: a ∗ i = arg max a i n π = a i ˆ P − a i − a i τ ∆ A o , τ ≥ . (6)The profit criterion (6) allows for two interpretations. In the first, a i ˆ P − a i is the firm’sprofit given the price estimate ˆ P , and − a i τ ∆ A reflects that firms, knowing their estimateis based on a first-order expansion, which neglects second-order terms, discount τ > − τ represents the demand function’s second derivative φ AA . In thiscase agents do not discount their price estimate, and rely on a second-order Taylor-seriesexpansions to estimate the selling price. Note that the model’s coefficients throughout can be chosen such that the equilibrium deviation ∆ A is arbitrarily small, respectively, such that the approximation (5) is of arbitrarily good quality. That is, if agents knew demand’s second derivative, their price estimate (5) would write ˆ P = φ ( A ) + φ A ∆ A + φ AA ∆ A . Substituting this into (6), and setting the discount rate τ = 0, yields a ∗ i = arg max a i n a i ( φ ( A ) + φ A ∆ A + φ AA ∆ A ) − a i o . Comparison indicates that the new profit cri-terion is equivalent to the old, (6), except for the second-order derivative φ AA taking the place of thediscount rate − τ . a ∗ i = ˆ P − τ ∆ A , A = Z i ∈ [0 , a i di = ˆ P − τ ∆ A . (7)Combining supply (7) and estimated demand (5), the equilibrium quantity A , wheresupply intersects with the demand estimate, is the solution to: A = φ ( A ) + φ A ( A )∆ A − τ ∆ A . (8)For convenience, we identify equilibria j = 0 , , ... , in terms of their distance ∆ A j = A j − A to the rational expectations equilibrium A . That is, ∆ A j = 0 corresponds tothe REE. Using the fact that A = φ ( A ), we rewrite (8) as: τ ∆ A + (1 − φ A )∆ A = 0 , and note: Proposition 1.
There exists the rational expectations equilibrium ∆ A = A − A = 0 in which agents’ price forecasts are correct ˆ P = P . There exists a second equilibrium ∆ A = A − A = − (1 − φ A ) τ < in which ˆ P T P . Both equilibria in Proposition 1 are self-fulfilling. In the perfect foresight equilibrium,no firm deviates from the equilibrium supply A = A , and thus there is no need for agentsto rely on polynomial approximations: Producers know the price φ ( A ). The oppositeis the case in the second equilibrium: Once firms supply A = A , they are uncertainas to the equilibrium price, φ ( A ) = φ ( A + ∆ A ), which they can only approximate as φ ( A ) + φ A ( A )∆ A . Moreover, the error of this approximation, ( A − A ) , grows themore agents deviate from supplying A . That is, once firms deviate from the rationalexpectations equilibrium, they find it harder to estimate future prices and thus they areincentivised to deviate even further until a new equilibrium is reached. In this equilibrium,firms cannot forecast prices accurately, and thus they choose to produce a small numberof goods, at a low marginal cost, which provides a margin of safety.As we argued earlier, this interdependence between aggregate output and the individ-ual firm’s ability to understand the environment that it operates in is a crucial aspect inmost crises: Once consumers and investors change their behavior, they find themselvesin an environment that is hard to understand, and they hold back on investment andconsumption decisions waiting for the “dust to settle”. In the current interpretation,by cutting production, agents put themselves into “uncharted territory”. This aspect8s, by assumption, not captured in environments where agents can compute the entiredemand function, respectively, solve the model as in Lemma 1. Before we discuss thescope for government to correct such “glitches” in output, which turns out to be lim-ited, we make one remark: The model’s coefficients φ A ( A ) , τ can be chosen such that∆ A = A − A = − (1 − φ A ) τ < O (∆ A ). We augment demand P = φ ( A ; b ) to incorporate an exogenous parameter b . This pa-rameter is assumed to increase demand φ b = φ b ( A ; b ) > A . This parametermay be seen as government demand or money supply. In this interpretation, the fol-lowing section identifies the multiplier effect that obtains once agents need to rely onapproximations to anticipate the consequences of policy interventions.We begin with a benchmark model where agents are computationally unconstrained.Second, we study the model with friction. Comparing both settings shows that parameterincreases, which increase demand and equilibrium output in a model with unconstrainedagents, can reduce economic activity if firms are computationally constrained. Put differ-ently, parameter changes, in particular if they are large, can put agents into “unchartedterritory”, and incentivise them to cut, rather than increase, output.
Recalling our augmented demand function: P = φ ( A ; b ) , φ A < , φ AA ≷ , φ b > , φ (0; b ) > , (9)firms can anticipate the equilibrium price P correctly, if they are computationally uncon-strained as in Lemma 1. Hence, they choose a production schedule a ∗ i which maximizesprofits a ∗ i = arg max a i n π i = a i P − a i o . Aggregate supply is thus A = Z [0 , a ∗ i di = P. (10) Alternatively, as we discuss in Section 4, the model may be interpreted as the capital market ofan overlapping generations economy, where a i , A , φ () are, respectively, individual savings and aggregatesavings, and φ ( A ) is the marginal product of capital that agents expect to receive on their savings. Finally, b may be seen as public debt and A as steady state capital. P , A for every given exogenousparameter b . Once the parameter changes from b to b = b + ∆ b , the price is againcorrectly anticipated as the unique solution P , A to the equations P = A and P = φ ( A ; b + ∆ b ).How would an actual human being, or an economic researcher, try to think about theimpact of the parameter change? The outside researcher, who uses textbook methods tostudy how changes in the exogenous parameter from b to b change output and price,cannot compute A and P explicitly. Instead, he will approximate the model’s comparativestatics. That is, he will differentiate (9) and (10):∆ P ≈ φ A ( A ; b )∆ A + φ b ( A ; b )∆ b, ∆ A = A − A , ∆ b = b − b , (11)∆ A = ∆ P. (12)Combining (11) and (12) yields the model’s comparative statics: Lemma 2.
Exogenous parameter variations ∆ b change output (and price) according to ∆ A ∆ b ≈ φ b − φ A > and ∂A∂b = lim ∆ b → A ∆ b = φ b − φ A > . That is, an outside observer/analyst would use a polynomial expansion of A = φ ( A ; b ) , P = φ ( A ; b ) to approximate the impact of an exogenous parameter change as in Lemma 2. Inthe following section, we assume that firms themselves make such “polynomial inference”using such an approximation to anticipate the consequences of parameter changes. Using a first-order expansion, around A = φ ( A ; b ), agents approximate the equilibriumprice: ˆ P = φ ( A , b ) + φ A ∆ A + φ b ∆ b, ∆ A = A − A , ∆ b = b − b . (13)That is, agents have to incorporate two aspects in their demand forecast: (i) the directeffect of the parameter change ∆ b and (ii) the equilibrium response of all agents whodeviate ∆ A = 0 from their usual supply choice. As before, agents discount the priceestimate since they do not know how curvature terms of demand φ AA , φ bb , and φ Ab affect10rices: π i = a i ˆ P − a i ( τ ∆ A + τ ∆ b + τ | ∆ A || ∆ b | ) − a i a ∗ i = ˆ P − τ ∆ A − τ ∆ b − τ | ∆ A || ∆ b | τ i ≥ , i = 1 , , . (14)To find the equilibria associated with (13) and (14), it is useful to distinguish caseswhere ∆ A ≥ A ≤ A ≥ , ∆ b ≥
0. If ∆ A ≥ b ≥ A = φ ( A , b ) + φ A ∆ A + φ b ∆ b − τ ∆ A − τ ∆ b − τ ∆ A ∆ b, and thus:∆ A , = − − φ A + τ ∆ b τ ± r ( φ b − τ ∆ b ) 1 τ ∆ b + (cid:16) − φ A + τ ∆ b τ (cid:17) . (15)Combining the two equilibrium candidates in (15) with our initial assumption ∆ A ≥ Lemma 3.
If and only if ∆ b < φ b τ , there exists an equilibrium in which, compared to theperfect foresight equilibrium, production (and price) are increased: ∆ A = − − φ A + τ ∆ b τ + r ( φ b − τ ∆ b ) τ ∆ b + (cid:16) − φ A + τ ∆ b τ (cid:17) > . At the margin, increasesin the parameter increase income if ∂ ∆ A∂ ∆ b = − τ τ + 1 / φ b τ − τ τ ∆ b +2 τ τ (cid:16) − φA + τ b τ (cid:17) s ( φ b − τ ∆ b ) τ ∆ b + (cid:16) − φA + τ b τ (cid:17) > .Proof. Follows directly from (15).To interpret the equilibrium in Lemma 3, we study how it corresponds to the REEof Proposition 1. That is, we note that lim ∆ b → ∆ A (∆ b ) = 0, i.e., the equilibrium quantity A converges to the REE quantity A as b → b . On the contrary, for large changes∆ b >
0, the equilibrium loses its RE character as agents do not precisely know howdemand is impacted by the exogenous change. Such large parameter changes have anambiguous effect on output, which is captured by the term ( φ b − τ ∆ b )∆ b . On theone hand, agents extrapolate the increase in demand φ b ∆ b >
0. At the same time,agents cannot rule out that too large an increase might eventually prove counterproductive In Appendix B we discuss an alternative error term max[∆ A , ∆ b , ∆ A ∆ b ], where agents are eitherconcerned about miscalculating the parameter change’s impact, ∆ b > ∆ A , or the other agents’ reaction∆ A > ∆ b to the parameter change. τ ∆ b . That is, large, unprecedented changes in the model’s structure render agents’polynomial approximations unreliable, and put them into “uncharted territory”.Lemma 3 thus features four policy regimes. First, if the parameter change is (in-finitesimally) small, the agents’ polynomial approximations are of high quality. In thisregime, policy is as effective as in the model of Section 2.2.1, Lemma 2, where agents arecomputationally unconstrained. That is, the model’s multiplier is given by ∂ ∆ A∂ ∆ b | ∆ b =0 = ∂A∂b = φ b − φ A >
0. Second, there is an intermediate region where ∆ b ∈ [0 , ∆ b ], and policychanges have a positive effect at the margin, ∂ ∆ A∂ ∆ b >
0. These marginal returns, however,are diminishing. Third, there is a region ∆ b ∈ [∆ b , ∆ b ] where agents find themselves in“uncharted territory” and start to cut production ∂ ∆ A∂ ∆ b <
0. Finally, in the extreme casewhere ∆ b ≥ φ b τ , an equilibrium ∆ A > A ≤
0, we recall (13) and (14), and notethat −| ∆ A | = ∆ A . Accordingly, there are two candidates∆ A , = − − φ A − τ ∆ b τ ± r ( φ b − τ ∆ b ) 1 τ ∆ b + (cid:16) − φ A − τ ∆ b τ (cid:17) . (16)In view of (16), if ∆ b < φ b τ , there exists exactly one equilibrium, in which ∆ A = − − φ A − τ ∆ b τ − r ( φ b − τ ∆ b ) τ ∆ b + (cid:16) − φ A − τ ∆ b τ (cid:17) < b > φ b τ , there can exist up to two equi-libria in which economic activity is low. Combining these observations with Lemma 3yields: Corollary 1.
Large parameter changes ∆ b > φ b τ preclude the existence of equilibria withoutput A ≥ A . So far agents were assumed to know the rational expectations equilibrium A , φ ( A ).In Section 3.1, we study a dynamic setting where agents learn different points on thedemand curve over time. In turn, we examine how the economy converges to the REE.Second, in our baseline setting, all agents know the same point on the demand curve.Price forecasts and supply decisions are therefore the same across agents. Once differentagents know different pieces of the demand curve, this is no longer true. Rather thanknowing each other’s price forecasts and supplies, agents have to estimate price and supply12imultaneously. In Section 3.2, we extend our model to incorporate such asymmetricinformation in a manner which is akin to Bayesian inference. We abstracted from the fact that agents may learn from past mistakes, i.e., suboptimalproduction choices that were based on incorrect price estimates. One would imagine thatthey memorize these mistakes, or the observation that a quantity A is associated withan observable price P = φ ( A ). Under our current assumptions on the demand function,this price differs from the estimated price ˆ P . Hence, agents would not supply A again.Second, if agents are computationally constrained to the use of polynomials, how do theyfind the perfect foresight equilibrium in the first place? This section’s main observationis that agents will learn the REE over time.Agents who sell repeatedly into the market will, over time t = 0 , , , ... , observe anincreasing number of points on the demand curve. Regarding these points, P t = φ ( A t ),we assume that agents also learn demand’s slope φ A ( A t ) once a quantity A t is marketed.Given past observations A t , t = 0 , , , , ..., T agents can refine their price estimate as:ˆ P T +1 = φ ( A ∗ ) + φ A ( A ∗ )( A T +1 − A ∗ ) , A ∗ = arg min A t n | A T +1 − A t | o t = 0 , , , ...T (17)That is, to estimate prices, they select from the set of known points { A t } Tt =0 the point A ∗ , which is closest to the future supply A T +1 . Put differently, they use the observation A ∗ from the past, which is most similar/closest to the situation they are trying to makeinference on. In turn, agents i choose supply a i = ˆ P T +1 − ( A T +1 − A ∗ ) . (18)Hence, for a given A ∗ , there are two equilibrium candidates A T +1 = A ∗ − − φ A ( A ∗ )2 ± r ( φ ( A ∗ ) − A ∗ ) + (cid:16) − φ A ( A ∗ )2 (cid:17) . (19)To show that (17) and (19) ensure that agents learn the REE equilibrium A , φ ( A ), weproceed in two steps. First, we study the case where demand φ () is a convex function. Inthis case, convergence to the REE can be studied in terms of a simple first-order differenceequation. Second, for the remaining cases, we give an indirect argument in Appendix C.13 .1.1 Convex Demand Without loss of generality, we assume that agents start with a prior µ, φ ( µ ), µ < A .Moreover, we focus on the “+” roots of (19). For convex demand, we now show that (17)and (19) imply a first-order difference equation for supply: A T +1 = A T − − φ A ( A T )2 + r ( φ ( A T ) − A T ) + (cid:16) − φ A ( A T )2 (cid:17) . (20)First, we note that (20) indicates that agents, in Marshallian fashion, increase supply, suchthat A T +1 > A T , if the marginal revenue φ ( A T ) exceeds the marginal cost of production A T . Moreover, from Lemma 1, we know that there exists only one level of supply, namely A , where A = φ ( A ). It follows that (20) has a unique steady state at the point where A T +1 = A T = A . This steady state equilibrium is stable due to our assumption that φ is downward-sloping: For A T < A , we have φ ( A T ) − A T > A T +1 > A T . At A , the system is locally stable since ∂A T ∂A T +1 = | A T = A ∈ ( − , { A t } Tt =1 is strictly increasing, and, due to our convexityassumption on φ , that A t ≤ A ∀ t = 0 , , ....T . That is, A T always adjusts towards,but not beyond, A . Hence, according to (17), agents will always use the informationthat they learned in the previous period, when a quantity A T was marketed, to thinkabout A T +1 . This last property allows us to study convergence in terms of the first-orderdifference equation (20). One may suspect that heterogeneity in information might mitigate the possibility of multi-ple equilibria that we emphasize. Moreover, one might expect that dispersion of privateinformation induces some agents to supply too little and others too much such that, onaverage, errors cancel and supply might actually be at an efficient level. Regarding To see this, recall (17) and (18), which imply A T +1 = φ ( A T )+ φ A ( A T )( A T +1 − A T ) − ( A T +1 − A T ) ≤ φ ( A T ) + φ A ( A T )( A T +1 − A T ). At the same time, convexity of φ implies: φ ( A T +1 ) = φ ( A T + A T +1 − A T ) ≥ φ ( A T ) + φ A ( A T )( A T +1 − A T ). Taking both inequalities together, we have A T +1 − φ ( A T +1 ) ≤ A T +1 ≤ A . Where A T +1 ≤ A , follows from φ being downward-sloping and A = φ ( A )at A = A . Hence, if we start at a point µ − φ ( µ ) <
0, this implies that A T +1 ≤ A ∀ T . Finally, asmentioned before, if demand is non-convex, A t can overshoot A . In that case we require an additionalargument, which we give in Appendix C. See, e.g., Morris and Shin (1998) for a coordination problem where asymmetric information selectsunique equilibria in an economy with a continuum of players. See Galton (1907), and Grossman and Stiglitz (1976) for such wisdom of the crowd effects. i ∈ [0 ,
1] is assumed to know the selling price φ ( A i ) and demand’s firstderivative φ A ( A i ) of a particular supply A i ∈ [0 , ∞ ]. Agents are distributed over thesepoints according to an integrable density function f (). For simplicity, we normalize agents’discount rate to τ = 1.Conditional on information A i , φ ( A i ) , φ A ( A i ) agent i supplies: a ∗ i | A i = ˆ P | A i − ( ˆ A | A i − A i ) , (21)where ˆ P | A i and ˆ A | A i are agent i ′ s price and supply forecasts conditional on knowingdemand φ ( A i ) at point A i . The polynomial estimates for price and quantity are:ˆ P | A i = φ ( A i ) + φ A ( A i )( ˆ A | A i − A i ) , (22)and ˆ A | A i = Z [0 , ( a j | A j ) | A i dj. (23)Where (23) reflects that agent i uses his information at point A i to infer the informationand thus the supply of the other agents who know a different point A j . That is, agent i knows that agent j observes a point on the same demand curve and thus he uses apolynomial expansion around φ ( A i ) to estimate the information φ ( A j ) , φ A ( A j ) that player j receives . Based on this reasoning, i can construct an estimate for the other players’price estimates, which he needs to calculate aggregate supply. Agent i ′ s price and supplyestimates are thus given by the simultaneous solution of (22)-(23). In turn, he can choosesupply (21). We solve the model in Appendix E using a guess-and-verify approach.These solutions yield two main insights. First, as in Proposition 1, multiple equilibriaexist due to the interaction between agents’ ability to forecast the equilibrium and aggre-gate supply. Second, unlike the earlier model, where information was symmetric, we showin Lemma 4 that output is depressed across all equilibria since agents systematicallyunderestimate demand if the true demand function is convex. Moreover, the marginalcost of production differs among producers, and thus output, in addition to being low, isproduced inefficiently. 15 Interpretations
In this section, we reinterpret our model in terms of the three macroeconomic workhorseframeworks: (i) aggregate search models of the Diamond (1982) type, (ii) Life-cycle sav-ings models of the Diamond (1965) type, and (iii) models of supply and demand as inSamuelson (2009) and Mas-Colell et al. (1995).
1) Search:
In the context of the Diamond (1982), p. 887, model, agents face thefollowing choice problem:max a i { a i φ ( A ; b ) − f ( a i ) } , a i ∈ R + . Where a i is individual i ′ s output choice in Period 1, and φ ( A ; b ) is the probability withwhich agents find a trading partner in Period 2. If a trading partner is found, agent i cansell/exchange goods at price one. Finally, f ( a i ) is the cost of production.The chance of finding a trading partner, φ ( A ; b ), is an increasing function in aggregateeconomic activity A = R [0 , a i di and government demand b . Diamond (1982) shows thatsuch an economy can have multiple REE equilibria, which we call, say, A , A . Supposenow that agents know one of these REE, e.g., A ; then, if they are computationallyconstrained, as in the present paper, they would need to use a polynomial expansion tocompute the probability φ ( A + ∆ A ) of finding a trading partner that would prevail onceagents collectively deviate ∆ A from producing A . The same applies to the evaluationof the exogenous policy parameter b , which may, unlike in Diamond (1982), result in anegative multiplier effect, as discussed in Section 2.2.
2) Savings and Investment:
In the context of the Diamond (1965) model, φ () maybe interpreted as a component to agents’ consumption savings problem:max s t ,c t ,c t +1 U ( c t , c t +1 ) s.t. c t = w t − s t , c t +1 = s t (1 + r t +1 ) , c t > , c t +1 > , (24)where factor prices are functions of the prevailing capital-labor ratios k t and k t +1 : r t +1 = f ′ ( k t +1 ) , w t = f ( k t ) − f ′ ( k t ) k t . To make choice (24), agents have to form expectations regarding equilibrium interest r t +1 = f ′ ( k t +1 ). In equilibrium, the life-cycle savings condition, (1 + n ) k t +1 = s t , relatessavings and capital; n representing the exogenous rate of population growth.16uppose now that the economy is initially in a steady state at k , s . To computefuture interest rates, agents have to compute r t +1 = f ′ ( k t +1 ) = f ′ ( s t n ) =: φ ( s t ; n ). Onceagain, if agents know the prevailing interest in the steady state, they have to engage inpolynomial expansions to form price expectations ˆ r t +1 to compute the interest rate thatobtains in the (temporary) equilibria that obtain once agents choose savings s t = s +∆ s t .This argument extends to the case where agents supply labor in both periods. In thatcase, to make their savings decision, agents have to (i) approximate the interest rate and(ii) the second-period wage rate. That is, they have to approximate the Samuelson (1962)neoclassical factor-price frontier, w t +1 = ξ ( r t +1 ( k t +1 ( s t ))), which relates wages to interest,interest to the capital intensity, and finally the capital intensity to savings.
3) Supply and Demand:
Our lead interpretation was that of a simplified A D , A S setting. Taking this perspective, we suggest one reason why demand analysis may becomputationally complicated for firms. Suppose demand is given by A D = ξ ( A ; L ( A )),where ξ represents demand, which is downward-sloping in supply ∂ξ∂A <
0. At the sametime, cuts in production A reduce employment L ( A ) and demand, i.e., ∂ξ∂L ∂L∂A >
0. Ac-cordingly, agents grapple with the question whether the function φ ( A ) := ξ ( A ; L ( A )) isupward or downward sloping once supply falls into regions which they do not know frompast experience.Similarly, suppose that demand concerns a vector A ∈ R L of goods, involving sub-stitutes and complements, as in Mas-Colell et al. (1995), pp. 599-641. Suppose that afirm produces a particular good a l , then, if all other firms in the economy, supplying thevarious other goods, change their behavior, it has to analyze how changes in the suppliesand prices for the other goods L \ l , influence demand, and thus price, for good l . In turn,if this firm is exceptionally well-informed, it might know the entries of the economy’s Ja-cobian matrix. However, it need not know demand’s second- and cross-derivatives, whichonce again makes it difficult to compute demand correctly. Economists’ forecasting record suggests that it is difficult to compute future economicevents. The current model recognizes this and assumes that agents cannot compute exactnumeric values for future equilibrium outcomes such as prices.The model’s key feature is that the precision with which agents can approximate fu-17ure equilibrium prices depends on the level of aggregate economic activity, and is thusendogenous. This interdependence between aggregate output and an individual firm’sability to forecast the price at which it can sell its output gives rise to equilibria in whicheconomic activity is inefficiently low. Such equilibria may be interpreted as “glitches” ofthe overall economy. During such a glitch, agents collectively reduce economic activity.This change in behavior makes it difficult to forecast the resulting equilibrium, which, inturn, justifies the initial output cut. For similar reasons we also find that the scope forgovernment to correct such “glitches” in output is limited: Interventions, which wouldunambiguously increase output in a frictionless economy, can make it harder for firmsto predict future equilibria and reduce output even further. Our model therefore cap-tures the common place observation that large parameter changes, such as unprecedentedquantitative easing, can put agents into “uncharted territory”.The particular form in which equilibria obtain depends on the assumption that agentsuse the same Taylor-series expansions that an economic researcher, who applies standardtextbook methods, would use. More sophistication on the part of agents will undoubtedlychange the specific form and number of equilibria. However, it appears unlikely that theprecision with which future equilibrium outcomes are approximated can ever be entirelyindependent of the overall level of economic activity, which is what our findings rely on.18
Second-order Expansions
In this appendix, we derive our results for a setting where agents know of demand’s firstand second derivatives. They can thus use second-order expansions to estimate prices:ˆ P = φ ( A ) + φ A ∆ A + 12 φ AA ∆ A , (25)We study the equilibria that emerge once agents are averse τ > τ = 0, we obtain the equilibria that emerge if agents are indifferent regardingerrors. Using the estimate (25), firms choose a profit-maximizing quantity: a ∗ i = arg max a i n π = a i ˆ P − a i − a i τ | ∆ A | o , τ ≥ . Individual and aggregate supply are thus a ∗ i = ˆ P − τ | ∆ A | , A = Z [0 , a ∗ i di = ˆ P − τ | ∆ A | . (26)Combining (25) and (26), we obtain: A + τ | ∆ A | = φ ( A ) + φ A ∆ A + 12 φ AA ∆ A . (27)Recalling that A = φ ( A ), (27) writes: τ | ∆ A | − φ AA ∆ A + (1 − φ A )∆ A = 0 . (28)If τ = 0 we have: Proposition 2. If τ = 0 , there exists the perfect foresight equilibrium ∆ A = 0 and asecond equilibrium ∆ A = − φ A φ AA . According to Proposition 2, economic activity in the polynomial equilibrium exceedsactivity in the perfect foresight equilibrium if the demand function’s second derivativeindicates that demand may rebound φ AA > A . A negativesecond derivative φ AA depresses output in the same manner as the discount factor τ inProposition 1. As before, estimated demand can meet supply more than twice if agentsdiscount their price estimate: Proposition 3. If τ > , there exist at least two equilibria: ∆ A = 0 , and ∆ A = − τ φ AA − q ( τ φ AA ) + − φ A τ < , in which economic activity is depressed. If φ AA > ,and ( τ φ AA ) > − φ A τ , there may exist two additional equilibria where economic activityis elevated ∆ A , = τ φ AA ± q ( τ φ AA ) − − φ A τ > . roof. Using (28), we find the perfect foresight equilibrium ∆ A = 0. To identify theremaining equilibria, we distinguish cases (i) ∆ A <
A >
A <
0: we note that | ∆ A | = − ∆ A . Dividing by ∆ A , we find that(28) has roots: ∆ A , = − τ φ AA ± q ( τ φ AA ) + − φ A τ . However, only one root satisfies theinitial assumption ∆ A <
0. That is, ∆ A = − τ φ AA − q ( τ φ AA ) + − φ A τ <
0, regardlessof the sign of φ AA . The other root ∆ A = − τ φ AA + q ( τ φ AA ) + − φ A τ > τ φ AA ) > φ A < − φ A τ >
0. It thus violates the initial assumption ∆
A <
A >
0: we note that | ∆ A | = ∆ A and find that (28) has two realroots ∆ A , = τ φ AA ± q ( τ φ AA ) − − φ A τ if ( τ φ AA ) > − φ A τ . Both of these roots arenegative if φ AA < A >
0. Hence, φ AA > A > A as given. In equilibrium, however, aggregate supply depends on agents’ price forecasts.Hence, the price forecast itself is an equilibrium outcome. At this point, it is clear thathigher-order polynomials yield even more equilibria, and that propositions 1-4 carry overqualitatively once we introduce demand Φ( A ), for a vector A of goods. B Alternative Discounting
The price estimate is as before:ˆ P = φ ( A , b ) + φ A ∆ A + φ b ∆ b, ∆ A = A − A , ∆ b = b − b . (29)The model differs in agents’ discounting: π i = a i ˆ P − a i max[∆ A , ∆ b , ∆ A ∆ b ] − a i a ∗ i = ˆ P − max[∆ A , ∆ b ] (30)According to (30) there are two regimes. First, agents fear that they miscalculate theimpact of the exogenous parameter change in case ∆ b > ∆ A . Second, agents fear thatthey misjudge the other agents’ reaction to the exogenous parameter variation ∆ A > ∆ b .To analyze the equilibrium outcomes associated with (29) and (30), we distinguish caseswhere the parameter change is relatively large, | ∆ b | > | ∆ A | , from cases where its impactis relatively small, | ∆ b | < | ∆ A | . Note that | ∆ A | is a function of | ∆ b | . That is, we start20ith the assumption that, e.g., | ∆ b | > | ∆ A | and solve for the equilibrium ∆ A . In turn,we check whether the initial assumption | ∆ b | > | ∆ A | is correct. Proposition 4. If | φ b − ∆ b − φ A | < , there exists only one equilibrium ∆ A = (cid:16) φ b − φ A − ∆ b − φ A (cid:17) ∆ b in which | ∆ b | > | ∆ A | . In this equilibrium lim b → b ∆ A ∆ b = φ b − φ A = ∂A∂b | A ,b .Proof. Individual supply is a i = ˆ P − max[∆ A , ∆ b ] under the assumption that | ∆ b | > | ∆ A | , we have a i = ˆ P − ∆ b . Aggregate supply is thus A = R i ∈ [0 , a i = ˆ P − ∆ b .Equilibrium requires A = A + φ A ∆ A + φ b ∆ b − ∆ b , respectively, ∆ A (1 − φ A ) + ∆ b (∆ b − φ b ) = 0. Solving yields ∆ A = φ b − ∆ b − φ A ∆ b . It remains to note that | φ b − ∆ b − φ A | < | ∆ b | > | ∆ A | .As before, Proposition 4, ∆ A = (cid:16) φ b − φ A − ∆ b − φ A (cid:17) ∆ b , shows that agents extrapolate thepositive impact that a parameter change ∆ A = φ b − φ A . At the same time, they are facingincreased uncertainty as to the actual price at which their products sell ∆ A = − ∆ b − φ A ∆ b .A large parameter change where ∆ b > φ b thus reduces economic activity as the uncertaintythat it creates outweighs the expansive effect φ b > A . For cases where | ∆ b | < | ∆ A | we have: Proposition 5.
There exists an upper bound ∆ b > and an equilibrium where ∆ A = − (1 − φ A ) − q φ b ∆ b + ( (1 − φ A )) < and | ∆ b | < | ∆ A | , if ∆ b ∈ [0 , ∆ b ] . If φ b − φ A > there exists an upper bound ∆ b > and a second equilibrium ∆ A = − (1 − φ A ) + q φ b ∆ b + ( (1 − φ A )) > where | ∆ b | < | ∆ A | , if ∆ b ∈ [0 , ∆ b ] .Proof. Individual supply is a i = ˆ P − max[∆ A , ∆ b ] under the assumption that | ∆ b | < | ∆ A | , we have a i = ˆ P − ∆ A . Aggregate supply is thus A S = R i ∈ [0 , a i = ˆ P − ∆ A .Equilibrium requires A S = A D such that A = A + φ A ∆ A + φ b ∆ b − ∆ A , respectively,∆ A +(1 − φ A )∆ A + φ b ∆ b = 0. Solving yields ∆ A , = − (1 − φ A ) ± q φ b ∆ b + ( (1 − φ A )) .Both roots are real since we assumed φ b > b >
0. It remains to specify theconditions under which our initial hypothesis | ∆ b | < | ∆ A | holds. We start with ∆ A = − (1 − φ A ) − q φ b ∆ b + ( (1 − φ A )) and note (i) ∆ A (∆ b = 0) = − (1 − φ A ) such that | ∆ b | = 0 < | ∆ A | , (ii) the derivative ∂ ∆ A∂ ∆ b = − φb √ φ b ∆ b +( (1 − φ A )) vanishes as ∆ b becomeslarge. Taken together, (i) and (ii) imply that an upper bound ∆ b exists, such that21 ∆ b | < | ∆ A | as long as ∆ b ∈ [0 , ∆ b ]. Similarly, regarding the second equilibrium ∆ A = − (1 − φ A ) + q φ b ∆ b + ( (1 − φ A )) , we note that (i) ∆ A (∆ b = 0) = 0 such that | ∆ b | = | ∆ A | = 0, (ii) the derivative ∂ ∆ A∂ ∆ b = φb √ φ b ∆ b +( (1 − φ A )) = | ∆ b =0 φ b (1 − φ A ) and lim ∆ b →∞ ∂ ∆ A∂ ∆ b =lim ∆ b →∞ φb √ φ b ∆ b +( (1 − φ A )) = 0. Taken together, (i) and (ii) imply that if φ b − φ A > b such that | ∆ b | < | ∆ A | as long as ∆ b ∈ [0 , ∆ b ].The first equilibrium in Proposition 5 corresponds to the perfect foresight equilibrium,∆ A = 0 of Proposition 1: In the limit, where the parameter change becomes infinitesimallysmall, we obtain ∆ A = 0. In this equilibrium, increases in b indeed increase equilibriumsupply provided that these increases are small such that ∆ b < ∆ b < ∆ b . In the secondequilibrium, which corresponds to the crisis equilibrium in Proposition 1, output is strictlydecreasing in b .Taken together, Propositions 4 and 5 suggest that bold interventions by the govern-ment tend to reduce economic activity as such changes make it more difficult for agents toforecast prices. Moreover, in the crisis equilibrium ∆ A , government interventions, whichwould increase output were there no computational frictions, always reduce income. C Learning The REE
We have shown that agents can learn the A equilibrium if (i) demand is convex and (ii)agents always coordinate on the “+” equilibrium. We now show that agents also learnthe REE if (i) demand is not convex, and (ii) when agents, e.g., alternate between playingthe “+” and “-” root equilibria.We write the equilibrium condition as: A − φ ( A ∗ ) = φ A ( A ∗ )( A − A ∗ ) − ( A − A ∗ ) . (31)The left-hand side of (31) represents the difference between supply and demand in anequilibrium where agents use their knowledge of A ∗ , φ ( A ∗ ) to estimate the price. Thisdifference is 0 in the perfect foresight equilibrium where A = φ ( A ). Regarding theright-hand side, we define ε = ( A − A ∗ ), which yields A − φ ( A ∗ ) = φ A ε − ε , (32)over time as agents observe an increasing number of price quantity pairs { φ ( A t ) , A t } Tt =0 .The distance ε = ( A − A ∗ ) between the aggregate supply A and the point of estimation22 ∗ will go to zero. Hence, from (32) we have lim ε → ( A − φ ( A ∗ )) = 0. However, the onlypoint where A = φ ( A ) is A , the perfect foresight equilibrium quantity. Hence, as agentslearn more data on demand, they eventually move towards the efficient equilibrium. D Asymmetric Equilibria
Agents using φ ( A ∗ ) to think about a deviation from A ∗ , might choose a quantity A ( A ∗ )which is closer to A ∗∗ than to A ∗ . Thus, to forecast the selling price φ ( A ( A ∗ )), they wouldrather use A ∗∗ as the point of departure. In turn, once agents use A ∗∗ they might choosea supply A ( A ∗∗ ), which, however, is again closer to A ∗ than A ∗∗ . Thus, given (17), agentswould switch back to A ∗ , and so forth: | A ( A ∗ ) − A ∗ | > | A ( A ∗ ) − A ∗∗ | (33) | A ( A ∗∗ ) − A ∗∗ | > | A ( A ∗∗ ) − A ∗ | . (34)In such a situation, there exists no symmetric equilibrium between points A ∗ and A ∗∗ .Instead, there exists an asymmetric equilibrium in which a mass ψ ∈ (0 ,
1) of agents use A ∗ and a mass 1 − ψ use A ∗∗ .It follows from (17) that agents are only indifferent between using A ∗ and A ∗∗ if theequilibrium quantity A satisfies A = A ∗ + A ∗∗ . That is, agents see A ∗ and A ∗∗ as equallyinformative once the supply they want to learn about is equally distant from both points.To establish the existence of an equilibrium we have to prove that there exist shares ψ and 1 − ψ for which the equilibrium supply A ψ ψ + A − ψ (1 − ψ ) indeed equals ¯ A = A ∗ + A ∗∗ . To see this note that, given our assumptions on φ (), all real-valued equilibrium supplies, (20), fallinto a compact interval [0 , ˆ A ]. In turn, the sequence of equilibrium supplies { A t } Tt =0 , for which agentsknow demand, partitions this interval. As time progresses, this partition becomes finer and finer. Thatis, if we order the quantities A t such that A l < A l +1 , l = 1 , , ...T , the interval between A l and A l +1 is either filled with new equilibria, or, if there exist no equilibria between them, economic activity takesplace elsewhere on the interval [0 , ˆ A ]. Finally, we note that agents using A ∗ to think about a deviationfrom A ∗ , might choose a quantity A ( A ∗ ) which is closer to A ∗∗ than to A ∗ . Thus, to forecast the sellingprice φ ( A ( A ∗ )), agents would rather use A ∗∗ as the point of approximation. In turn, once agents use A ∗∗ they might choose a supply A ( A ∗∗ ), which, however, is again closer to A ∗ than A ∗∗ . Thus agentswould switch back to A ∗ , and so on. In such situations, there exists no symmetric equilibrium betweenpoints A ∗ and A ∗∗ . Instead, Appendix D shows that there does exist an asymmetric equilibrium, wherea fraction ψ of the agents uses point A ∗ and the remaining fraction 1 − ψ uses point A ∗∗ , resulting inequilibrium supply A = A ∗ + A ∗∗ . A ( ψ ) = ψA ( A ∗ , A ( ψ ))+(1 − ψ ) A ( A ∗∗ , A ( ψ )),we can write the equations that determine equilibrium as A ( ψ ) = ¯ A ¯ A := A ∗ + A ∗∗ A ( ψ ) := ψA ψ ( A ∗ , A ( ψ )) + (1 − ψ ) A − ψ ( A ∗∗ , A ( ψ )) (36) A ψ ( A ∗ , A ( ψ )) = ˆ P ψ − ( A ( ψ ) − A ∗ ) A ψ ( A ∗∗ , A ( ψ )) = ˆ P − ψ − ( A ( ψ ) − A ∗∗ )(37)ˆ P ψ = φ ( A ∗ ) + φ A ( A ∗ )( A ( ψ ) − A ∗ ) (38)ˆ P − ψ = φ ( A ∗∗ ) + φ A ( A ∗∗ )( A ( ψ ) − A ∗∗ ) (39)Solving (36)-(39) for supply A ( ψ ) yields: A ( ψ ) = − p ± r − q + ( p − q = ψφ ( A ∗ ) + (1 − ψ ) φ ( A ∗∗ ) − ψφ A ( A ∗ ) − (1 − ψ ) φ A A ∗∗ − ψ ( A ∗ ) − (1 − ψ )( A ∗∗ ) p = 1 + 2 ψA ∗ + 2(1 − ψ ) A ∗∗ − ψφ A ( A ∗ ) − (1 − ψ ) φ A ( A ∗∗ ) > − p <
0. Hence, there isonly one real root A ( ψ ) = − p + p − q + ( p ) with positive supply. Note in particular that A ( ψ = 0) ( A ( ψ = 1)) is the supply when all agents use A ∗ ( A ∗∗ ) as the point of reasoning.By our initial hypothesis (33)-(34), we have A ( ψ = 0) > ¯ A and A ( ψ = 1) < ¯ A . Hence,there exists an intermediate value ˜ ψ , at which A ( ˜ ψ ) = ¯ A , if p − q + ( p ) as required bythe indifference condition (35). E Asymmetric Information Equilibria
We solve the model in two steps. First, we guess the equilibrium outcome. Given thisguess, we solve the estimation problem for agent i . Second, we solve for the equilibriumand verify our guess.
4) Problem of an individual agent:
To find the optimal supply of an individualagent, we must solve (22)-(23). To do so, we start with the guess that agent i beliefsthat agents j hold the same believe over aggregate supply that he holds himself, i.e.,ˆ A | A i = ( ˆ A | A j ) | A i = ˆ A . Given this guess, we show that:ˆ P | A i = ( ˆ P | A j ) | A i . (40) The root p − q + ( p ) is never complex as ψ runs from 0 to 1. To see this we recall − q ( ψ = 0) > − q ( ψ = 1) > ∂ − q∂ψ is not a function of ψ itself and thus cannot change signs as ψ varies. Accordingly, − q > ∀ ψ ∈ [0 , p − q + ( p ) is always real. i believes that agents j observe points,which lie on his estimated demand curve. That is, agent i knows that agent j estimatesthe price asˆ P | A j = φ ( A j ) + φ A ( A j )( ˆ A | A j − A j ) , and thus i estimates ˆ P | A j as:( ˆ P | A j ) | A i = φ ( A j ) | A i + φ A ( A j ) | A i (( ˆ A | A j ) | A i − A j ) , (41)where φ ( A j ) | A i = φ ( A i ) + φ A ( A i )( A j − A i ) , (42) φ A ( A j ) | A i = φ A ( A i ) , ˆ A | A i = ( ˆ A | A j ) | A i = ˆ A. (43)Taken together, (41)-(43) mean that agent i uses polynomials to approximate the demandcurve upon which the other player observes a point A j , φ ( A j ) , φ A ( A j ). Using (42)-(43),(41) rewrites:( ˆ P | A j ) | A i = φ ( A i ) + φ A ( A i )( A j − A i ) + φ A ( A i )( ˆ A − A j ) , = φ ( A i ) + φ A ( A i )( ˆ A | A i − A i ) = | ( ) ˆ P | A i . Agent i thus believes that agent j will work with a price estimate that is identical to theone he uses himself and thus he will conclude that j ′ s supply forecast is identical to hisown, such that ˆ A | A i = ( ˆ A | A j ) | A i = ˆ A , which confirms our initial guess. It remains tosolve (22)-(23) for player i ′ s forecast in the two equilibria A , :ˆ A , | A i = A i − − φ A ( A i )2 ± r ( φ ( A i ) − A i ) + (cid:16) − φ A ( A i )2 (cid:17) ,a i = ˆ P | A i − ( ˆ A | A i − A i ) = φ ( A i ) + φ A ( A i )( ˆ A | A i − A i ) − ( ˆ A | A i − A i ) (44) a i ;1 , = φ ( A i ) + φ A ( A i ) (cid:16) − − φ A ( A i )2 ± r ( φ ( A i ) − A i ) + (cid:16) − φ A ( A i )2 (cid:17) (cid:17) − (cid:16) − − φ A ( A i )2 ± r ( φ ( A i ) − A i ) + (cid:16) − φ A ( A i )2 (cid:17) (cid:17) . (45)
5) Equilibrium:
From (45) we calculate equilibrium supply as: A , = Z [0 , a , ( A i ) f ( A i ) di. (46)Concerning the equilibrium quantity A k , k = 1 ,
2, we note25 emma 4.
If demand φ is quasi-convex, then equilibrium output across both equilibria A k , k = 1 , falls short of efficient output A .Proof. To compare equilibrium output (46) to efficient output A , we recall that A = φ ( A ). We also recall that, for convex functions f , we have f ′ ( u ) ≤ f ( v ) − f ( u ) v − u . In thecurrent context, this means that agents tend to underestimate convex demand functions: φ ( A i ) + φ A ( A i )( A − A i ) ≤ φ ( A )recalling (44) we have: a i = φ ( A i ) + φ A ( A i )( ˆ A | A i − A i ) − ( ˆ A | A i − A i ) ≤ φ ( A ) − ( ˆ A | A i − A i ) ≤ φ ( A ) = A . Put differently, agents supply less than the efficient quantity since (i) they underestimatedemand and (ii) since they know that their price estimate is inaccurate.The equilibria in (46), feature two sources of inefficiency. First, aggregate output fallsshort of the efficient level A . Second, since price estimates vary, output and the marginalcost of output differ across firms. Aggregate output is thus produced inefficiently.26 eferences Ackerman, N., Freer, C., and Roy, D. (2011). Noncomputable conditional distributions.
Proceedings - Symposium on Logic and Computer Science, Toronto , pages 31–36.Arrow, K. J. (1970).
Essays in the Theory of Risk Bearing . Amsterdam: North-HollandPublishing.Arthur, B. (2015).
Complexity and the Economy . Oxford University Press.Blanchard, O. J. (1979). Backward and forward solutions for economies with rtionalexpectations.
American Economic Review Papers and Proceedings , 69:114–118.Bray, M. (1982). Learning, estimation, and stability of rational expectations.
Journal ofEconomic Theory , 26:318–339.Chamley, C. (2014). When demand creats its own supply: savings traps.
Review ofEconomic Studies , 81(2):651–680.Chiang, A. C. and Wainwright, K. (2005).
Fundamental Methods of Mathematical Eco-nomics . Boston [u.a.]: McGraw-Hill, 4. internat. edition.Cooper, R. and John, A. (1988). Coordinating coordination failures in Keynesian models.
Quarterly Journal of Economics , 103(4):441–463.DeCanio, S. (1979). Rational expectations and learning from experience.
Quarterly Jour-nal of Economics , 93:47–57.Diamond, P. A. (1965). National debt in a neoclassical growth model.
American EconomicReview , 55(5):1126–1150.Diamond, P. A. (1982). Aggregate demand management in search equilibrium.
Journalof Political Economy , 90(5):881–894.Farmer, D. and Foley, D. (2009). The economy needs agent-based modelling.
Nature,Opinion , 460(6):685–686.Finetti, B. D. (1952). Sulla preferibilita.
Annali di Economica , (11):685–709.Galor, O. (2007).
Discrete Dynamical Systems . Springer Verlag.Galton, F. (1907). Vox populi.
Nature , 75:450–451.27intis, H. (2007). The dynamics of general equilibrium.
Economic Journal , 117:1280–1309.Grossman, S. and Stiglitz, J. (1976). Information and competitive price systems.
AmericanEconomic Review Papers and Proceedings , 66(2):246–253.Grunberg, E. and Modigliani, F. (1954). The predictability of social events.
Journal ofPolitical Economy , 62:465–478.Hellwig, M. (1993). The conceptual structure of macroeconomic models: I the incomeequation.
WWZ Working paper 9308 University Basel .Hofbauer, J. and Sandholm, W. (2011). Survival of dominated strategies under evolution-ary dynamics.
Theoretical Economics , 6(3):341–377.Hutchison, T. (1937). Expectation and rational conduct.
Journal of Economics , 8(3):636–653.Keynes, J. M. (1936).
The General Theory of Employment, Interest and Money . NewYork: Harcourt-Brace Inc.Kuhle, W. (2016). Darwinian adverse selection.
Algorithmic Finance , 5:31–36.Marcet, A. and Sargent, T. (1989). Convergence of least squares learning mechanisms inself-refrential linear stochastic models.
Journal of Economic Theory , 48:337–368.Mas-Colell, A., Whinston, M. D., and Green, J. R. (1995).
Microeconomic Theory . OxfordUniversity Press.Maymin, P. (2011). Markets are efficient if and only if p=np.
Algorithmic Finance ,1(1):1–11.McLennan, A. (1984). Price dispersion and incomplete learning in the long run.
Journalof Economic Dynamics and Control , 7:331–347.Morris, S. and Shin, H. S. (1998). Unique equilibrium in a model of self-fullfilling currencyattacks.
American Economic Review , 88(3):578–597.Muth, J. (1961). Rational expectations and the theory of price movements.
Econometrica ,29:315–335. 28elson, R. and Winter, S. (1985).
An Evolutionary Theory of Economic Change . HarvardUniversity Press.Niederhoffer, V. (1997).
The Education of a Speculator . John Wiley.Pratt, J. W. (1964). Risk aversion in the small and in the large.
Econometrica , 32(1):122–136.Rothschild, M. (1974). A two-armed bandit theory of market pricing.
Journal of EconomicTheory , 9:185–202.Rubinstein, A. (1998).
Modeling Bounded Rationality . MIT press.Samuelson, P. A. (1947).
Foundations of Economic Analysis . Cambridge Ma.: HarvardUniversity Press.Samuelson, P. A. (1962). Parable and realism in capital theory: The surrogate productionfunction.
Review of Economic Studies , (29):193–206.Samuelson, P. A. (2009).
Economics: An Introductory Analysis . McGraw-Hill, 19th.edition.Sargent, T. (1993).
Bounded Rationality in Macroeconomics . Oxford University Press.Soros, G. (1994).
The Alchemy of Finance . John Wiley.Tesfatsion, L. (2006). Agent based computational economics: A constructive approachto economic theory. In
Handbook of Computational Economics , pages 831–880. ed. L.Tesfatsion, K.Judd, Elsevier.Thurner, S., Farmer, D., and Geanakoplos, J. (2012). Leverage causes fat tails andclustered volatility.
Quantitative Finance , 12(5):695–707.Ulanowicz, R. (2008). The dual nature of ecosystem dynamics.