Economic Dynamics And the necessity of nonlinearity

Economic Dynamics: 

Economic Dynamics And the necessity of nonlinearity

Definitions: 

Definitions Economic Dynamics: the study of any economic process Huh? It’s easier to define by considering what you have already studied: Economic Statics the study of the determination of points of economic equilibrium no consideration of the time path taken to get there Nonlinearity: Realism in functional representation of a system First step towards evolutionary modelling…

The static-dynamics difference: 

The static-dynamics difference Consider standard micro supply and demand. We have a linear demand curve and a linear supply curve: The static approach: equate the two: In more detail…

The static-dynamics difference: 

Note this formula The static-dynamics difference State (timeless) supply and demand formulae; Work out equilibrium Break for lunch… Problem: agricultural markets not this stable… Draw graph:

The static-dynamics difference: 

The static-dynamics difference Classic example is “Minnesota Hog Cycle” Not “equilibrium” but irregular cycles around long-term trend price…

The static-dynamics difference: 

The static-dynamics difference Attempted dynamic explanation: cobweb model… Recast supply and demand as time-lagged (actually time-delayed ) functions: Demand now reflects prices now Supply now reflects prices last season Farmers plant based on last year’s returns: Adaptive expectations Basic formulae: Yields difference equation for prices: Gives same equilibrium result as static formula Producers expect next season’s price to be same as last season’s; or… Producers plant this season’s crop based on last season’s price

The static-dynamics difference: 

The static-dynamics difference Set P t =P t-1 =P: So eventual outcome same as statics? “Statics is long-run dynamics?” Depends on values of parameters… Note this formula

The static-dynamics difference: 

The static-dynamics difference If slope parameters b s /b d <1, “dynamics=statics” But for b s /b d >1, “dynamic instability”

The static-dynamics difference: 

The static-dynamics difference No convergence to equilibrium price; “Crazy” prices: negative, tending to +/- infinity… Randomness no help… System tends to impossible prices

The static-dynamics difference: 

The static-dynamics difference In cobweb model, dynamic answer diverges from static answer if suppliers are more responsive to price than consumers. (Which group do you think is more responsive?) “Mad” result of negative prices is result of “mad” assumption of linear functions (which allow negative supply, and negative demand!). Effect disappears with “sensible” nonlinear functions Why “sensible”? Because linearity an abstraction Nothing in the real world is really linear Not even neoclassical economics…

The static-dynamics difference: 

The static-dynamics difference Markets (and models of markets) cannot be linear Crazy results (negative prices & quantities) product of linear form for demand & supply curves Given D t =a d -b d P t , feed in high P t , you’ll get negative D t But even neoclassical theory doesn’t justify linear demand & supply curves “Non-satiation” implies D   as P 0 Ditto supply: stops at 0, reaches finite maximum as marginal cost  Nonlinear, time-delayed models give realistic cycles—no need to hypothesise “rational” expectations to tame the cobweb…

The static-dynamics difference: 

The static-dynamics difference Compare linear to nonlinear Simple nonlinear demand/supply curves Modified rectangular hyperbolas Basic hyperbola y=1/x Area under hyperbola crucial to definition of log, exponential; Used for illustration purposes only here… Generalised hyperbola formula is Used to derive S & D curves:

The static-dynamics difference: 

The static-dynamics difference Nonlinear demand: Nonlinear supply: Graphing them: More realistic even in terms of neoclassical theory than standard “linear” curves used Linear obsession mainly due to lazy pedagogy But had real impact on development of theory

The static-dynamics difference: 

The static-dynamics difference Solving for P as a function of time with these curves: Generates sustained cycles: More “interesting” deterministic dynamics possible with more complex functions Chaos can arise Impact of noise instructive:

The static-dynamics difference: 

The static-dynamics difference Any pattern at all can result, without breakdown :

The static-dynamics difference: 

The static-dynamics difference “Looks like” empirical data too, even though model incredibly simple: So statics is not “long run dynamics” Dynamics can answer questions statics can’t even pose Apparent “volatility clustering”… Very difficult to get from linear models Simple with nonlinear model—product of being “far from equilibrium”

Why the economic obsession with statics?: 

Why the economic obsession with statics? Neoclassical economists tend to think: “Evolution leads to optimising behaviour” “Dynamics explains movement from one equilibrium point to another” So “statics is long run dynamics” also believed by Sraffian economists implicit in Post Keynesian or Marxian analysis using comparative static or simultaneous equation methods Evolution Dynamics Statics

Why the economic obsession with statics?: 

Why the economic obsession with statics? Modern mathematics reverses this Field of evolution larger than dynamics Dynamics larger than statics Results of evolutionary analysis more general than dynamics But two generally consistent Results of dynamics more general than statics & GENERALLY INCONSISTENT Dynamic results correct if actual system dynamic/evolutionary In general, statics will give wrong answers to questions posed about economy—whether questions posed in neoclassical or Post Keynesian terms… Statics Dynamics Evolution

General Disequilibrium: 

General Disequilibrium ‘There exist known systems, therefore, in which the important and interesting features of the system are “essentially dynamic”, in the sense that they are not just small perturbations around some equilibrium state, perturbations which can be understood by starting from a study of the equilibrium state and tacking on the dynamics as an afterthought .’ ‘If it should be true that a competitive market system is of this kind, then… No progress can then be made by continuing along the road that economists have been following for 200 years. The study of economic equilibrium is then little more than a waste of time and effort …’ Blatt (1983: 5-6)

In summary…: 

In summary… Summarising validity of analytic techniques & relations between them:

Why did economics start with statics?: 

Why did economics start with statics? Because it was easier! Marshall (of Micro “fame”) The modern mathematician is familiar with the notion that dynamics includes statics. If he can solve a problem dynamically, he seldom cares to solve it statically also... But the statical solution has claims of its own. It is simpler than the dynamical; it may afford useful preparation and training for the more difficult dynamical solution; and it may be the first step towards a provisional and partial solution in problems so complex that a complete dynamical solution is beyond our attainment. (Marshall, 1907 in Groenewegen 1996: 432)

Why did economics start with statics?: 

Why did economics start with statics? Jevons (one of the founders of General Equilibrium analysis) “If we wished to have a complete solution … we should have to treat it as a problem of dynamics. But it would surely be absurd to attempt the more difficult question when the more easy one is yet so imperfectly within our power.” [Jevons, Theory of Political Economy, Ch. 4, 4th edition, p. 93] So statics regarded as easier way to reach the same answers as the more general dynamics would give. Now known to be incorrect outside economics , but still not common knowledge within economics

Why study dynamics?: 

Why study dynamics? Many real world processes do not have an equilibrium; or do not have a single equilibrium; or do not have stable equilibria globally, or locally Examples: weather patterns; animal population growth/decline; …

Why study dynamics?: 

Why study dynamics? In these systems, equilibrium values will never apply. Equilibrium (and therefore static analysis) irrelevant to system in both short and long term system will not be at equilibrium now it is not moving towards equilibrium over time Economics? When did you last see an economy at rest?… Question is whether the economy is stable subject to shocks, or unstable… Two examples of linear vs nonlinear thinking: Hicks’s trade cycle model Kaldor’s nonlinear explanation for cycle But first, the data…

The pre-1933 Trade cycle: 

The pre-1933 Trade cycle Pre-1933 trade cycle predates “Big Government” Cycles and growth performance therefore closer to “pure market economy” results than data for post-1933 Source: NBER Macrohistorical database, http://www.nber.org/databases/macrohistory/data/01/a01007a.db (Index of manufacturing production)

Growth with cycles: 

Growth with cycles

But what cycles! : 

But what cycles!

What causes these cycles?: 

What causes these cycles? 2 classes of possible explanations Exogenous shocks to stable system economy stable, but disturbed by weather patterns, wars, etc. Endogenous fluctuations generated by dynamics of the economy itself can also have exogenous shocks imposed on this class of systems, of course First interpretation dominated early work in “economic dynamics”

Propagation and impulse: 

Propagation and impulse If cycles caused by exogenous shocks then “propagation mechanism” that which keeps disturbance at time t rippling through system till time t+T, at which time impact of disturbance completely dissipated differs from “impulse mechanism” source of random shocks from outside the economy This interpretation dominated early work because economists believed (wrongly) that endogenous cycles were not possible

The exogenous shocks interpretation: 

The exogenous shocks interpretation Frisch in 1933 (depth of Great Depression) “The majority of the economic oscillations which we encounter seem to be explained most plausibly as free oscillations. In many cases they seem to be explained by the fact that certain exterior impulses hit the economic system and thereby initiate more or less regular oscillations” (Economic essays in honour of Gustav Cassel: 171) “If you hit a rocking horse with a club, the movement of the horse [stable propagation mechanism] will be very different to that of the club [exogenous shocks]” (198)

An example: Hicks’s 2nd order model: 

An example: Hicks’s 2nd order model Investment a lagged function of change in income: Consumption a lagged … function of income: Saving equals income minus consumption: Equating I and S yields A 2 nd order difference equation:

2nd order difference equation: 

2nd order difference equation Second order multiplier-accelerator model dominates theory of cycles in economics 1950s-1960s But properties of model show all drawbacks of linear models… Unrealistic cycles Too much—or too little—instability No “goldilocks” here Zero “equilibrium” output level

2nd order difference equation: 

2nd order difference equation 5 basic patterns, none realistic

2nd order difference equation: 

2nd order difference equation Adding noise doesn’t help much: The problem is linearity! But it’s also bad mathematics…

2nd order difference equation: 

2nd order difference equation Economists stuffed around with this model for decades A mathematician would have rejected it on day one Reason? It’s only solution is “the trivial solution” Y t =Y t-1 =Y t-2 =0 Takes “elementary” mathematical analysis to show this Convert model into matrix form If matrix non-invertible, model has meaningful solutions If non-invertible, only solution is “trivial”—zero.

The trivial solution: 

The trivial solution In matrix form: Special derived form of matrix can be inverted: Means that only solution the trivial solution. Why is this—economically speaking?

Hicks’s error: 

?????????? Hicks’s error When does desired investment equal actual savings? When income equals zero! Actual investment is related to this period’s output: or Because model equates desired I and actual S:

A better (but still linear!) model: 

A better (but still linear!) model Desired investment a function of change in output Investment adds to capital: Capital determines output Capitalists carry out investment plans: 3 rd order difference equation A more interesting (but still linear!) model. Behaviour can be broken down into equilibrium + trend + cycle components:

A better (but still linear!) model: 

A better (but still linear!) model In matrix form, this is: Special derived form of matrix can’t be inverted: As a result, non-trivial solutions possible: Non-zero values for Y over time…

Economic properties: 

Economic properties Cycles with growth C:v ratio determines nature of cycles & growth Exponential with c>v Linear with c=v Damped with c<v Realistic & period-independent values for c & v feasible

Mathematical: meaningful closed form: 

Mathematical: meaningful closed form Equilibrium if c < v Growth term Cycle term

But the limitations of being linear: 

But the limitations of being linear Model itself a “quirk” Cycle size perfectly synchronised with growth of output Mathematically, eigenvalue for growth exactly same magnitude as eigenvalue for cycles Normally, these differ in linear models Cycles also symmetrical Trade cycle is not—long booms and short slumps Need nonlinearity to get asymmetry of real world First economist to realise “the importance of being nonlinear” was Kaldor:

The endogenous critique: 

The endogenous critique Kaldor 1940, “A model of the trade cycle” Considered static model based on interaction of ex-ante savings and ex-ante investment: “the basic principle underlying all these theories may be sought in the proposition … derived from Mr Keynes’s General Theory … that economic activity always tends towards a level where Savings and Investment are equal… in the ex-ante … sense.” (78) Savings and Investment both assumed to be positively sloped functions of activity level (employment as proxy). “If we assume the S and I functions as linear , we have two possibilities:” (79)

The endogenous critique: 

The endogenous critique (1) Savings function steeper than Investment (savings rises more than investment as employment rises Y Employment S I S<I, system expands S>I, system contracts Equilibrium stable

The endogenous critique: 

The endogenous critique (2) Savings function flatter than Investment (savings rises less than investment as employment rises Employment Y I S S>I, system contracts S<I, system expands Equilibrium unstable

The endogenous critique: 

The endogenous critique Kaldor In “slope of S”> “slope of I” situation “any disturbances … would be followed by the re-establishment of a new equilibrium, with a stable level of activity… this … assumes more stability than the real world, in fact, appears to possess.” (80) In I>S situation “the economic system would always be rushing either towards a state of hyper-inflation … or towards total collapse… Since recorded experience does not bear out such dangerous instabilities, this possibility can be dismissed” (80)

The endogenous critique: 

The endogenous critique Kaldor’s solution “Since thus neither of these two assumptions can be justified, we are left with the conclusion that the I and S functions cannot both be linear.” (81) Insight: nonlinear functions make endogenous fluctuations possible, and limit size to meaningful levels Endogenous fluctuations and nonlinearity are inseparable elements of dynamic analysis.

The importance of being nonlinear: 

The importance of being nonlinear Linear models can be: Cycles in linear system require Frisch/Hicks/Econometrics approach Harrod’s initial model

The importance of being nonlinear: 

The importance of being nonlinear Nonlinear systems can be: Cycles can occur because system is: Not so different from linear model Completely unlike linear model

The importance of being nonlinear: 

The importance of being nonlinear Advantages of linear systems: Easily analysed (closed form solutions exist) Powerful analytic maths (linear algebra) Proof by theorem Stable linear dynamic system’s behaviour a function of parameter values of system only Behaviour can be broken down into Equilibrium value Growth component Cyclical component Disadvantages of linear systems: Unrealistic for most open systems

The importance of being nonlinear: 

The importance of being nonlinear Disadvantages of nonlinear systems: Difficult to analyse (no closed form solutions) No analytic maths Many high level forms of maths needed to characterise, but no analytic results possible Proof by simulation rather than theorem System’s behaviour a function of both parameter values and initial conditions Path dependent behavior Behaviour cannot be broken down into growth and cyclical components Instead, magnitude of cycles a function of deviation from equilibrium; equilibria often “repellers” rather than “attractors”

The importance of being nonlinear: 

The importance of being nonlinear Advantages of nonlinear systems Realistic for most open systems Most “open systems”—ones subject to evolutionary change—are “far from equilibrium” ones Nonlinear dynamics approximate this; “Evolution” with fixed parameters Tractable compared to true evolutionary modelling

Statics vs. Dynamics: 

Statics vs. Dynamics Economics unique amongst mathematically-oriented disciplines in reliance upon static methodology (simultaneous equations rather than differential equations) Reliance on statics not limited to Neoclassicals Many Keynesian/Kaleckian theorists (including the masters) use simultaneous equations “Sraffian” economists criticise all other schools using advanced equilibrium-oriented methodology Why? Belief that economic system will settle down to equilibrium “in the long run” “Dynamics simply describes transients”

Statics vs. Dynamics: 

Statics vs. Dynamics Long ago shown to be untrue even for “general equilibrium” neoclassical models (Jorgenson 1960,61, 63; McManus 1963; Blatt 1983) Linear component of input-output system with growth must be unstable in either price or output vector Reliance on static methods a hangover from past practice and faith Dynamic answers to economic questions fundamentally different to static ones EVEN IF model “Keynesian” Example: Steedman’s critique of Kaleckian pricing theory

Steedman on Kalecki: 

Steedman on Kalecki A (mathematical/methodological) critique of Kaleckian microfoundations: A “Kalecki after Sraffa”? No consideration of macro (“capitalists get what they spend...”) Input-output analytic critique of markup pricing theory and related theory of distribution

A “brute fact”: 

A “brute fact” “the costs of any industry are constituted by the prices of industrial products and it would be ... one-sided to say that ‘prices are largely cost determined’ without saying also that ‘costs are to a significant degree price determined’” Justified attack on lack of analytic consideration of input-output relations in Kaleckian tradition... Unjustified attack on Kaleckian analysis of the process of price setting

Steedman’s Crucible: 

Steedman’s Crucible A model of price setting which takes account of input-output relations Circulating capital only; no overhead labour Equilibrium analysis, quantities taken as given, which leaves prices only:

Equilibrium Prices: 

Equilibrium Prices Reworking this equation yields: Price can be expressed as a function of markup, but Given input-output relations, price in industry j will at least depend on all 1...n industries which are basic QED I: prices in industry j cannot be set without regard to conditions in other industries (Followed by critiques of averages, vertical integration, wages share, etc.)

What about dynamics?: 

What about dynamics? Steedman considers a once-only exogenous change (of du) in u. Then from Note this equation

“Their full effects”: 

“Their full effects” QED II: Price converges to a new equilibrium vector where initial interdependence of (each) price on many (at least basic industries) markups is restored. Steedman concludes that QED III: “‘static’ analysis does not ignore time. To the contrary, that analysis allows enough time for changes in prime costs, markups, etc., to have their full effects.” Really? Like most economists, Steedman is apparently unaware of basic methods of mathematical dynamical analysis Reworking his equation into a standard difference equation:

“Their full effects”: 

“Their full effects” Equation is As autonomous difference equation This is solved by breaking into two components: First, “homogeneous” Presume solution of the form So that Substituting:

Solving difference equation: 

Solving difference equation Only possible for non-trivial x if So that constant Second, “particular”: Presume solution of the form Dispense with Collect terms in x Factor

Solving difference equation: 

Solving difference equation Simple matrix manipulation: Particular result same as Steedman’s static solution: General result sum of homogeneous plus particular solutions: Static solution same as dynamic iff this 0 as t  Skip eigenvalues

Eigenvalues & eigenvectors: 

Eigenvalues & eigenvectors “Eigen” (German for “characteristic”) values tell you how much a matrix is stretching space If modulus of dominant eigenvalue of discrete dynamic system < 1, matrix “shrinks” space and 0 as t  If modulus of dominant eigenvalue of discrete dynamic system > 1, matrix “expands” space and  as t  How much does matrix ‘stretch space’? & in which direction? Only possible for non-trivial v if

Eigenvalues & eigenvectors: 

Eigenvalues & eigenvectors is a polynomial in l . If the modulus of the dominant root of this polynomial < 1, then this dynamic system will 0 as t  and static price vector will be the final price vector If > 1, then this dynamic system will  as t  and static price vector will be irrelevant If =1, then system “marginally unstable”

Steedman’s stability: 

Steedman’s stability Steedman’s example system used With these values & modulus of maximum eigenvalue of

Steedman’s stability: 

Steedman’s stability Convergence to equilibrium in Steedman’s example system…

Steedman’s stability: 

Steedman’s stability A different example system Which static analysis would rule out for obvious reasons, but of which the modulus of maximum eigenvalue of With these values The consequence?

Steedman’s stability: 

Steedman’s stability With different input-output matrix, instability: Permanent inflation away from the negative equilibrium price vector

Steedman’s stability: 

Steedman’s stability Continuous price inflation Negative equilibrium price vector irrelevant since equilibrium unstable and prices will always diverge from it. Static analysis does not describe the “full effects” of a dynamic system unless the dynamic system is stable In real-world systems, instability/marginal instability rather than stability seems to be the rule Complex systems/evolutionary intepretation: “evolution to the edge of chaos”

With more reality?: 

With more reality? Increased realistic complexity would introduce add quantity, banks, effective demand, nonlinear wage & investment functions, etc., to prices & markups Each additional element of reality brings increased nonlinearity (even with no explicit nonlinear functions) Full system almost certainly has unstable (multiple) equilibria, hence exhibits far-from-equilibrium dynamic behaviour

Conclusion: 

Conclusion Static equilibrium not the end-product of dynamic processes Dynamics—not statics—the true crucible of economics: Not so much “Kalecki after Sraffa” as “Sraffa after Lorenz” Kaleckian price-setting process fully consistent with dynamic input-output analysis; but Kaleckian results require nonlinear dynamic input-output analysis for full expression Kaleckian analysis insufficiently developed on this front to date; but on the other hand, Sraffians unjustifiably reliant upon statics Time for some cross-pollination…

Conclusion: 

Conclusion Non-neoclassical economists almost have as much to learn about dynamics as do neoclassicals Most Post Keynesian/Marxian/Sraffian economists still only learn maths from other economists Don’t learn basics of dynamic modelling Don’t appreciate importance of nonlinearity Next lecture: some examples of “how to be dynamically nonlinear”

References: 

References Blatt, J.M., (1983). Dynamic Economic Systems , ME Sharpe, Armonk. Jorgenson, D.W., (1960). 'A dual stability theorem', Econometrica 28: 892-899. Jorgenson, D.W., (1961). 'Stability of a dynamic input-output system', Review of Economic Studies ,  28: 105-116. Jorgenson, D.W., (1963). 'Stability of a dynamic input-output system: a reply', Review of Economic Studies , 30: 148-149. McManus, M., (1963). 'Notes on Jorgenson’s model', Review of Economic Studies , 30: 141-147.