Model Animal Spirits, Heterogeneous Expectations and the Amplification and Duration of Crises Tiziana Assenza (Universita Catolica Milano) William Brock (University of Wisconsin) Cars Hommes (University of Amsterdam) UvA, CeNDEF 14th Annual DNB Research Conference Complex Systems: Towards a Better Understanding of Financial Stability and Crises, Amsterdam, November 3-4, 2011 Cars Hommes UvA, CeNDEF Animal Spirits and Heterogeneous Expectations
Model Animal Spirits (Keynes) much of (macro)economic activity is governed by animal spirits ◮ people have non-economic motives ◮ they are not always rational in pursuit of economic interests Keynes : animal spirits are the main source of economic fluctuations ... but animal spirits disappeared from the neoclassical, rational model Cars Hommes UvA, CeNDEF Animal Spirits and Heterogeneous Expectations
Model Animal Spirits (Akerlof and Shiller, 2009) How human psychology drives the economy, and why it matters for global capitalism 5 animal spirits: confidence, fairness, corruption, money illusion and stories ◮ cornerstone animal spirit: confidence ◮ behavioral economics : how the economy really works, when people are human ◮ animal spirits difficult to conceptualize, model and measure Goal of this paper : dynamic equilibrium model of agents’ confidence Main Result : sudden collapse of confidence accelerates and amplifies downturn or crisis and slows down recovery Cars Hommes UvA, CeNDEF Animal Spirits and Heterogeneous Expectations
Model Main hypothesis: heterogeneous expectations Brock and Hommes, 1997 main tool for modeling confidence in market for loans ◮ lenders’ heterogeneous expectations about the (exogenous) probability of succes /failure of borrowers Main finding : ◮ In the presence of a (small) fraction of pessimistic beliefs , an unexpected negative shock to credit markets triggers these pessimistic beliefs to become self-fulfilling , amplifying a “crisis" and slowing down recovery Cars Hommes UvA, CeNDEF Animal Spirits and Heterogeneous Expectations
Model Model ◮ market for loans ◮ overlapping generations (OG) structure for households ◮ households savings into a riskless asset or a risky asset (productive investments) ◮ no banks , but households can lend to firms) ◮ firms borrow capital for productive investment and pay wages to households ◮ market clearing determines the “ contract rate " for loans (i.e. the interest rate, downpayments, and other loan requirements) Cars Hommes UvA, CeNDEF Animal Spirits and Heterogeneous Expectations
Model Households: 2-period OG maximize utility u t = ln c t , t + ln c t , t + 1 under the budget constraints c t , t ≤ w t − s t , ω o + s t [( 1 − δ t ) ρ + δ t λ e ≤ t + 1 ] , c t , t + 1 ◮ c t , t / c t , t + 1 : consumption when young/old; ◮ w t = w p , t + ω y , real wages + labour endowment when young; ◮ s t real savings when young; ◮ δ t fraction invested in risky asset; ◮ λ e t + 1 = p e t + 1 r t expected return on capital ◮ p e t + 1 expected probability of success Cars Hommes UvA, CeNDEF Animal Spirits and Heterogeneous Expectations
Model Households Savings function optimal saving � � s t = 1 ω o w t − . µ e 2 t , t + 1 with expected average return on investment µ e t + 1 =: ( 1 − δ t ) ρ + δ t λ e t + 1 Assuming zero endowment when old i.e., ω o = 0, savings is s t = w t 2 . Cars Hommes UvA, CeNDEF Animal Spirits and Heterogeneous Expectations
Model Firms demand for loanable funds Firms borrow to finance productive investments ◮ p t probability of success ◮ 1 − p t probability of bankruptcy Firms choose capital x t to maximize { p t g ( x t ) − r t x t } , max x t where r t = 1 + r 0 , t is contract rate for loans (i.e. “ rental rate " on capital + other conditions for loans) optimal demand for loans : � r t � x t = x ( r t ; p t ) = g ′− 1 . p t wages from the productive sector (returns to other factor): w p , t := p t − 1 g ( x t − 1 ) − r t − 1 x t − 1 . Cars Hommes UvA, CeNDEF Animal Spirits and Heterogeneous Expectations
Model Market Equilibrium under Homogeneous Expectations demand for loans ≡ total supply of loans homogeneous expectations on probability of succes: x ( r t ; p t ) = S t ( r t ; p e t + 1 ) loan supply correspondence ρ S t ( r t ) := w t 2 ¯ ı [ r t > r ∗ t = ] p e t + 1 [no arbitrage; risk-neutrality; expected loan return ] ρ λ e p e r ∗ t + 1 ( r t ) = t + 1 r t = ρ = ⇒ t = . p e t + 1 Cars Hommes UvA, CeNDEF Animal Spirits and Heterogeneous Expectations
Model Temporary Equilibria: Homogeneous Expectations temporary equilibrium A or B depending on demand curve ρ r ∗ A = p e t + 1 r t B * r B A * r A w * x t x A t 2 Cars Hommes UvA, CeNDEF Animal Spirits and Heterogeneous Expectations
Model Dynamics under Homogeneous Expectations exogenous stochastic probability process AR(1) p t + 1 = µ + a ( p t − µ ) + ǫ t , ◮ naive expectations : p e t + 1 = p t 1 � t ◮ average expectations p e t + 1 = i = 0 p i t + 1 ◮ minimum expectations : p e t + 1 = min { p t + 1 − T , p t − T , · · · , p t − 1 , p t } p e ◮ rational expectations : t + 1 = µ + a ( p t − µ ) Cars Hommes UvA, CeNDEF Animal Spirits and Heterogeneous Expectations
Model Dynamics under Homogeneous Naive Expectations 1.00 1.12 0.98 1.10 0.96 1.08 0.94 1.06 0.92 1.04 0.90 1.02 1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96 p p_exp r Cars Hommes UvA, CeNDEF Animal Spirits and Heterogeneous Expectations
Model Dynamics under Homogeneous Expectations (average) 1.00 1.12 0.98 1.10 0.96 1.08 0.94 1.06 0.92 1.04 0.90 1.02 1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96 p average r Cars Hommes UvA, CeNDEF Animal Spirits and Heterogeneous Expectations
Model Rational Expectations 1.00 1.12 0.98 1.10 0.96 1.08 0.94 1.06 0.92 1.04 0.90 1.02 1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96 p AR1 r Cars Hommes UvA, CeNDEF Animal Spirits and Heterogeneous Expectations
Model Pessimistic Expectations (minimum, 5 lags) 1.00 1.12 0.98 1.10 0.96 1.08 0.94 1.06 0.92 1.04 0.90 1.02 1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96 p min r Cars Hommes UvA, CeNDEF Animal Spirits and Heterogeneous Expectations
Model Dynamics under Homogeneous Expectations ◮ average expectations smoothes out aggregate behavior (no extremes) ◮ minimum expectations amplifies the crisis and slows down recovery Cars Hommes UvA, CeNDEF Animal Spirits and Heterogeneous Expectations
Model Heterogeneous Expectations ( Brock and Hommes, 1997) J types of lenders, with expectations on return of risky asset: λ e j , t + 1 = p e j , t + 1 r t performance measure : relative past squared errors η memory parameter � 2 + η U j , t − 1 , U j , t = r 2 p t − p e � t j , t J � u j , t = U j , t / U tot U tot t , = U j , t . t j = 1 fractions : discrete choice model with asynchronous updating n h , t = δ n h , t − 1 + ( 1 − δ ) e − β u h , t − 1 , z t − 1 where z t − 1 is a normalization factor. The parameter δ ∈ [ 0 , 1 ] – the inertia of the traders parameter β ≥ 0 – the intensity of choice Cars Hommes UvA, CeNDEF Animal Spirits and Heterogeneous Expectations
Model Market Equilibrium under Heterogeneous Expectations demand for loans ≡ total supply of loans heterogeneous expectations on probability of success: x ( r t ; p t ) = S t ( r t ; p e 1 , t + 1 ; · · · , p e J , t + 1 ) total supply of loans under heterogeneous expectations J S t ( r t ) = w t � ı [ p e n j , t ¯ j , t + 1 r t > ρ ] , 2 j = 1 Cars Hommes UvA, CeNDEF Animal Spirits and Heterogeneous Expectations
Model Temporary Equilibria: Heterogeneous Expectations 2 Types temporary equilibrium A, B, D or C depending on demand curve r t D * r D C * r 2 B * r B A * r 1 * w * w x x x t t n t 1 1 2 2 2 Cars Hommes UvA, CeNDEF Animal Spirits and Heterogeneous Expectations
Model Two type example: average versus minimum beliefs 1.00 1.0 1.12 0.98 0.8 1.10 0.96 0.6 1.08 0.94 0.4 1.06 0.92 0.2 1.04 0.90 0.0 1.02 1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96 1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96 p av min n2 r Cars Hommes UvA, CeNDEF Animal Spirits and Heterogeneous Expectations
Model Two type example: rational versus minimum beliefs 1.00 1.0 1.12 0.98 0.8 1.10 0.96 0.6 1.08 0.94 0.4 1.06 0.92 0.2 1.04 0.90 0.0 1.02 1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96 1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96 p AR1 min n2 r Cars Hommes UvA, CeNDEF Animal Spirits and Heterogeneous Expectations
Model Conclusions ◮ confidence in economy can be modeled with lenders’ heterogeneous expectations about the probability of success of firms (banks) ◮ a small fraction of pessimistic , minimum expectations amplifies a crises and slows down recovery ; ◮ other types of human behaviour, such as trend following and fear makes these effects even stronger ◮ policy makers must take boundedly rational heterogeneous expectations into account to manage a complex economy Cars Hommes UvA, CeNDEF Animal Spirits and Heterogeneous Expectations
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