Rational Asset Price Bubbles and the Real Economy

Transcription

1 Rational Asset Price Bubbles and the Real Economy Lecture in Advanced Macroeconomics II Benjamin Larin Leipzig University Institute for Theoretical Economics Macroeconomics July 2 and 6, 2015 (updated: July ) Benjamin Larin (Leipzig University) Bubble Theory July 2 and 6, / 35

2 0. Bubbles in the news: Chinese stock market Source: The New Yorker, Eastern Exchanges, June 29, 2015 Benjamin Larin (Leipzig University) Bubble Theory July 2 and 6, / 35

3 0. Outline 1 Empirics: bubbles in the history of market economies 2 Definition of the term bubble 3 Do rational bubbles exist? 4 What rules out rational bubbles in theory? 5 The Tirole [1985] model 6 The Martin and Ventura [2012] model 7 Summary and Conclusion Benjamin Larin (Leipzig University) Bubble Theory July 2 and 6, / 35

4 1.Empirics: bubbles in the history of market economies Asset price bubbles are not a recent phenomenon! The Dutch Tulipmania from the 1630s The Mississippi bubble in 1719/1720 and the South Sea bubble in 1720 Land bubbles and bubbles in railroad bonds in the US during the 19th century Real estate and stock bubble in the US in 1920s Great Depression Real estate and stock bubble in Japan during Japanese lost decade Dotcom bubble in US house price and credit market bubble ending in 2006/2007 Nice book on the history of financial crises and bubbles: Kindleberger and Aliber [2005] Benjamin Larin (Leipzig University) Bubble Theory July 2 and 6, / 35

5 1.Empirics: bubbles in the history of market economies Bubbles and the real economy: USA Home and Share Price Index Share prices index House price index t Growth rate in percent Output growth Consumption growth Capital growth t Source: Feenstra et al. [2013] - PENN World Table; FRED; OECD Benjamin Larin (Leipzig University) Bubble Theory July 2 and 6, / 35

6 1.Empirics: bubbles in the history of market economies Bubbles and the real economy: Germany Share Price Index Share prices index t Growth rate in percent Output growth Consumption growth Capital growth t Source: Feenstra et al. [2013] - PENN World Table; OECD Benjamin Larin (Leipzig University) Bubble Theory July 2 and 6, / 35

7 2.Definition of the term bubble What exactly do we mean by bubble? Benjamin Larin (Leipzig University) Bubble Theory July 2 and 6, / 35

8 2.Definition of the term bubble Different definitions Garber [2001, 4] "Bubble is one of the most beautiful concepts in economics and finance in that it is a fuzzy word filled with import but lacking a solid operational definition. Thus, one can make whatever one wants of it." in the media: pronounced price increase of an asset followed by a sharp decline critical! Kindleberger and Aliber [2005] "[...] a bubble is an upward price movement over an extended period of fifteen to forty months that then implodes." imprecise: Monotonous price increase or can there be short declines? Why exactly months? What is the precise meaning of imploding prices? critical! Benjamin Larin (Leipzig University) Bubble Theory July 2 and 6, / 35

9 2.Definition of the term bubble Different definitions (Cont.) Shiller [2005] Irrational Exuberance "I define a speculative bubble as a situation in which news of price increases spurs investor enthusiasm, which spreads by psychological contagion from person to person, in the process amplifying stories that might justify the price increases and bringing in a larger and larger class of investors, who, despite doubts about the real value of an investment, are drawn to it partly through envy of others successes and partly through a gambler s excitement". Eugene Fama [Cassidi, 2010] Interview in The New Yorker "I don t even know what a bubble means. These words have become popular. I don t think they have any meaning." Benjamin Larin (Leipzig University) Bubble Theory July 2 and 6, / 35

10 2.Definition of the term bubble A precise definition (see, for instance, Brunnermeier [2008]) Non-arbitrage condition (derivation at blackboard) : 1+r = E t[p t+1 ]+E t [d t+1 ] p t Rearranging, iterating forward and applying the law of iterated expectations gives (blackboard): p t = i=1 f t }{{} Et[d t+i ] (1+r) i + b t }{{} lim s Et[p t+s ] (1+r) s Definition An asset price bubble b t exists when b t = p t f t > 0. Benjamin Larin (Leipzig University) Bubble Theory July 2 and 6, / 35

11 3.Do rational bubbles exist? Empirical tests of (rational) asset price bubbles Gürkaynak [2008] compares several empirical tests for rational bubbles and the results of those tests [...] for almost every paper in the literature that finds a bubble, there is another one that relaxes some assumption on the fundamentals and fits the data equally well without resorting to a bubble. Main difficulty of all tests (Variance Bounds test, West s Two-Step test,...): Rejecting the null-hypothesis (H 0 : an asset price bubbles exists): Asset price bubbles do not exist The estimation of the underlying model is misspecified Problem: At time t we do not know f t because we do not know E t [d t+j ] for all j > 0 Benjamin Larin (Leipzig University) Bubble Theory July 2 and 6, / 35

12 4.What rules out rational bubbles in theory? What rules out rational bubbles in theory? Asset with finite maturity: Think of a sequential game with subgame-perfect equilibria Representative agent economy with infinitely-lived agents: TVC lim assets t(1+r) t assets t = lim t t (1+r) t = 0 Numerator grows (asymptotically) with the rate g (In the Ramsey model Y, K,,C, and assets grow asymptotically at the same constant rate g) Denominator grows at r g < r. However, non-arbitrage implies bubbles grow at r: b t+1 b t 1 = r t+1. rational bubbles cannot exist in the standard(!) Ramsey model! [Tirole, 1982] Benjamin Larin (Leipzig University) Bubble Theory July 2 and 6, / 35

13 5.The Tirole [1985] model The Tirole [1985] model Benjamin Larin (Leipzig University) Bubble Theory July 2 and 6, / 35

14 5.The Tirole [1985] model Set-up Firms Canonical OLG model + bubbles Keep it easy: no population growth, no technological growth, Cobb-Douglas production, logarithmic utility Mass one of firms produce under perfect competition a homogeneous final output good Y t = F(K t, L t ) = K α t L 1 α t = K α t = f(k t ) = k α t. Each individual supplies one unit of labor every period and the size of the working population is normalized to one, i.e. L t = 1 x t := Xt L t = X t. Profit maximization gives wages and interest rate w t =f(k t ) f (k t )k t = (1 α)k α t (1) r t =f (k t ) δ = αk α 1 t δ. (2) Benjamin Larin (Leipzig University) Bubble Theory July 2 and 6, / 35

15 5.The Tirole [1985] model Set-up Households Demographic structure: every period t mass one of individuals enter the economy, supply inelastically one unit of labor earning labor income in t, live the following period t + 1 in retirement from previous savings and dying at the end of t + 1 Households maximize inter-temporal utility U t = ln c y,t + β ln c o,t+1 subject to c y,t =w t a t b t c o,t+1 =(1+r t+1 )a t +(1+q t+1 )b t a t, b t 0. difference: two instead of one asset: a t and b t. Benjamin Larin (Leipzig University) Bubble Theory July 2 and 6, / 35

16 5.The Tirole [1985] model Set-up Households and bubbles Households can trade an asset M t with zero dividends (d t = 0) f t = 0 and whenever the price p t of M t is strictly positive a bubble exists! Let s make it even simpler: M t = 1 B t := p t M t = p t and b t = Bt L t = B t. Rewrite the household optimization problem: st := a t + b t xt := bt s t b t = x t s t, a t = (1 x t )s t Benjamin Larin (Leipzig University) Bubble Theory July 2 and 6, / 35

17 5.The Tirole [1985] model Set-up Household optimization problem rewritten max {ct y, ct+1 o, s t, x t } ln c y,t + β ln c o,t+1 (3) s.t. c y t =w t s t c o,t+1 =[(1+r t+1 )(1 x t )+(1+q t+1 )x t ] s t 0 x t 1. Solution (blackboard) s t = β 1+β w t 1 if 1+r t+1 < 1+q t+1 x t = 0 if 1+r t+1 > 1+q t+1. [0, 1] if 1+r t+1 = 1+q t+1 Benjamin Larin (Leipzig University) Bubble Theory July 2 and 6, / 35

18 5.The Tirole [1985] model Set-up Non-arbitrage condition By definition, 1+q t+1 := p t+1 p t = b t+1 b t Bubbles can only exist if 1+r t+1 = 1+q t+1 (the case 1+r t+1 < 1+q t+1 is not possible, because k t 0 r t = f (k t ) δ ): b t+1 b t = 1+r t+1 (4) The share of bubbles x t adjusts such that eq. (4) holds Benjamin Larin (Leipzig University) Bubble Theory July 2 and 6, / 35

19 5.The Tirole [1985] model Equilibrium Equilibrium (I) Goods markets clearing condition: k t+1 = a t Law of motion of capital k t+1 = s t b t = β 1+β w t b t. (5) bubbles crowd-out capital (plausible?) Definition: Equilibrium A competitive equilibrium is a sequence of capital stock, bubbles, individual consumption and factor prices {k t, b t, c y,t, c o,t, r t, w t } t=0 such that {k t } evolves according to eq. (5), {b t } t=0 evolves according to eq. (4), individual consumption {c y,t, c o,t } t=0 solves the maximization problem defined in eq. (3), the factor price sequence {r t, w t } t=0 is given by eq. (2) and eq. (1), and 0 b t < β 1+β w t. Benjamin Larin (Leipzig University) Bubble Theory July 2 and 6, / 35

20 5.The Tirole [1985] model Equilibrium Equilibrium (II) Definition: Bubbly equilibrium trajectory and bubbly steady state A bubbly equilibrium trajectory is a competitive equilibrium in which the sequence {b t } t=0 is positive for some t > 0 and where the economy reaches a steady state in t. A bubbly steady state is a competitive equilibrium in which b t > 0 for t. Benjamin Larin (Leipzig University) Bubble Theory July 2 and 6, / 35

21 5.The Tirole [1985] model Equilibrium Reduced form of the model The model can be reduced to a two-dimensional, non-linear difference equation system in the variables b t and k t : β(1 α) k t+1 = 1+β kα t b t b t+1 =(1+αkt+1 α 1 δ)b t β(1 α) 0 b t < 1+β kα t b 0, k 0 given. For b 0 = 0 the model reduces to the standard Diamond OLG model without bubbles Benjamin Larin (Leipzig University) Bubble Theory July 2 and 6, / 35

22 5.The Tirole [1985] model Steady states Steady states The model has three steady states: Trivial steady state in k = b = 0 Bubble-less steady state in b = 0 and k = Bubbly steady state with b = ( α k = ( α δ ) 1 1 α δ ) 1 1 α [ (1 α)β 1+β [ (1 α)βδ ] 1 1 α ] α(1+β) 1 and Remember: the economy is dynamically inefficient if r < 0 (n = g = 0) Define the bubble-less steady state interest rate as r := αk α 1 δ Reformulation of the bubbly steady state bubble: b = (1 α)β ( α ) 1 1 α α(1+β) δ ( r ) b t has to be positive b > 0 r < 0, i.e. bubbles are only possible when the bubble-less economy is dynamically inefficient Interpretation of b t as money theoretical justification of a zero-lower bound on interest rates Benjamin Larin (Leipzig University) Bubble Theory July 2 and 6, / 35

23 5.The Tirole [1985] model Dynamics How to draw the phase diagram (I) (blackboard) Rewrite the difference equations in first differences ( x t+1 := x t+1 x t ) β(1 α) k t+1 = 1+β kα t b t k t b t+1 =(αk α 1 t+1 δ)b t Draw the k t+1 = 0 and b t+1 = 0 loci β(1 α) k t+1 = 0 : b t = 1+β kα t k t b t+1 = 0 : b t = 0 b t+1 = 0 : b t = (1 α)β 1+β kα t ( α δ ) 1 1 α Benjamin Larin (Leipzig University) Bubble Theory July 2 and 6, / 35

24 5.The Tirole [1985] model Dynamics How to draw the phase diagram (II) The k t+1 = 0 and b t+1 = 0 loci divide the positive k b space into four regions In each region the direction of trajectories is similar How to get the directions in the regions Take for example the kt+1 = 0 locus On the kt+1 locus k t is constant, while b t can change Looking at the differential equation kt+1 and assuming that b t is slightly below the k t+1 = 0 locus shows that k t is increasing The opposite holds for bt slightly above the locus Do the same for the bt = 0 locus Benjamin Larin (Leipzig University) Bubble Theory July 2 and 6, / 35

25 5.The Tirole [1985] model Dynamics Phase Diagram: r < 0 b t b t= b k k t=0 ˆb t(k t) b t= kt k Benjamin Larin (Leipzig University) Bubble Theory July 2 and 6, / 35

26 5.The Tirole [1985] model Dynamics Phase Diagram: r > 0 b t k t=0 b t=0 b t=0 k kt Benjamin Larin (Leipzig University) Bubble Theory July 2 and 6, / 35

27 5.The Tirole [1985] model Numerical solution Numerical solution The model cannot be solved analytically (a solution would be functions of k t depending only on parameters and time t, i.e. k t = h(t, b 0, k 0, α, β, δ) However, we can make use of the recursive structure of the model and simply solve it in a spreadsheet (see accompanying spreadsheet)! Benjamin Larin (Leipzig University) Bubble Theory July 2 and 6, / 35

28 6.The Martin and Ventura [2012] model Set Up Set Up (I) Contribution: Contrary to Tirole [1985], bubbles can be expansionary Main mechanism: bubbles loosen borrowing constraints Firm sector is unchanged Households maximize end-of-life-consumption: E t U t = E t [c t+1 ] s t = w t = (1 α)k α t When are households indifferent between investing in bubbles or productive capital? When the expected return to bubbles is equal to the interest rate: E t b t+1 b t + b N t = 1+r t+1. b N t are newly created bubbles (b t := p t M t, M t = M t 1 + M N t 1, such that Etb t+1 b t+b N t = Etp t+1 p t ) Benjamin Larin (Leipzig University) Bubble Theory July 2 and 6, / 35

29 6.The Martin and Ventura [2012] model Set Up Set Up (II) Households differ wrt investment efficiency: ǫ of the households can convert one unit of there income into one unit of capital, while 1 ǫ of the households can convert one unit of their income into only 0 < σ < 1 units of capital Law of motion of capital when b t = 0: where A := ǫ+(1 ǫ)σ. k t+1 =ǫ(1 α)k α t +(1 ǫ)σ(1 α)k α t =A(1 α)k α t, Productive individuals cannot borrow from unproductive individuals (think of a default probability greater than σ) aggregate inefficiency of investments Benjamin Larin (Leipzig University) Bubble Theory July 2 and 6, / 35

30 6.The Martin and Ventura [2012] model Set Up Set Up (III) Simplifying assumption: δ = 1 Law of motion/non-arbitrage-condition for bubbles = σαk α 1 t+1 (b t + b N b t + bt N t ) if (1 ǫ)(1 α)kt α E t [b t+1 ] [ σαk α 1 t+1 (b t + bt N ), αk α 1 t+1 (b t + bt N ) ] b t + bt N if (1 ǫ)(1 α)kt α = αk α 1 t+1 (b t + bt N b t + bt N ) if (1 ǫ)(1 α)kt α and 0 b t < (1 α)k α t. < 1 = 1, > 1 Benjamin Larin (Leipzig University) Bubble Theory July 2 and 6, / 35

31 6.The Martin and Ventura [2012] model Set Up Set Up (IV) A(1 α)kt α +(1 σ)bt N σb t if k t+1 = (1 α)kt α b t if b t + bt N (1 ǫ)(1 α)kt α b t + bt N (1 ǫ)(1 α)kt α < 1 1. Two opposing effects of bubbles on capital classic crowding-out effect of bubbles via bt ( ) new bubble creation loosens borrowing constraints such that investment is more efficient via b N t (+) When the second effects dominates bubbles are expansionary Martin and Ventura [2012] show under which parameter constellations bubbles are expansionary Benjamin Larin (Leipzig University) Bubble Theory July 2 and 6, / 35

32 6.The Martin and Ventura [2012] model Numerical Solution Simulation bt bt Et 1 bt kt t t yt ct t t rt bt P t t Benjamin Larin (Leipzig University) Bubble Theory July 2 and 6, / 35

33 7.Summary and Conclusion Summary and Conclusion Asset price bubbles are a intrinsic feature of market economies Different, partly vague definitions of asset price bubbles exist, but rational asset price bubbles can be precisely defined Empirical evidence on the existence of rational asset price bubbles is mixed Asset price bubbles can be incorporated into the OLG model (Tirole [1985] model), but not into the Ramsey model In the canonical OLG model bubbles can only exist when the bubbleless economy is dynamically inefficient according to Abel et al. [1989] modern economies are dynamically efficient, while more recent evidence by Geerolf [2013] suggests the opposite Implausible feature of the Tirole [1985] model: contractionary bubbles More recent models like the Martin and Ventura [2012] and Hillebrand et al. [2013] are able to depict expansionary bubbles Open questions: How are bubble initiated? How to depict bubbles on assets with a positive fundamental value? Should government, and if yes, how should government intervene when bubbles build up and burst? Benjamin Larin (Leipzig University) Bubble Theory July 2 and 6, / 35

34 9.References References I Andrew B Abel, N Gregory Mankiw, Lawrence H Summers, and Richard J Zeckhauser. Assessing dynamic efficiency: Theory and evidence. The Review of Economic Studies, 56(1): 1 19, Markus K. Brunnermeier. Bubbles. In Lawrence Blume and Steven Durlauf, editors, The new Palgrave dictionary of economics, volume 2. Palgrave MacMillan, John Cassidi. The New Yorker. Condé Nast, New Yorks, URL Accessed: Robert C. Feenstra, Robert Inklaar, and Marcel P. Timmer. The Next Generation of the Penn World Table, URL P.M. Garber. Famous First Bubbles: The Fundamentals of Early Manias. The MIT Press, Cambridge, Massachusetts, London, England, François Geerolf. Reassessing dynamic efficiency. manuscript, Toulouse School of Economics, Refet S Gürkaynak. ECONOMETRIC TESTS OF ASSET PRICE BUBBLES: TAKING STOCK*. Journal of Economic Surveys, 22(1): , Marten Hillebrand, Tomoo Kikuchi, and Masaya Sakuragawa. Bubbles and crowding-in of capital via a savings glut. Technical report, Working Paper Series in Economics, Karlsruher Institut für Technologie (KIT), Benjamin Larin (Leipzig University) Bubble Theory July 2 and 6, / 35

35 9.References References II Charles P. Kindleberger and Robert Z. Aliber. Manias, Panics and Crashes: A History of Financial Crises. John Wiley & Sons, Hoboken, NJ, 5 edition, Alberto Martin and Jaume Ventura. Economic Growth with Bubbles. American Economic Review, 102(6): , Robert J. Shiller. Irrational Exuberance. Princeton University Press, New York, 2 edition, Jean Tirole. On the possibility of speculation under rational expectations. Econometrica: Journal of the Econometric Society, pages , Jean Tirole. Asset bubbles and overlapping generations. Econometrica: Journal of the Econometric Society, pages , Benjamin Larin (Leipzig University) Bubble Theory July 2 and 6, / 35