Transcript nuclear reactors

Vienna
15 February 2012
Financial and Nuclear Meltdowns:
the structural instability of critical processes
Alessandro Vercelli
DEPFID
University of Siena
Analogies
September 2008: financial “tsunami” that triggered the “meltdown” of
the financial system
March 2011: real tsunami that triggered the partial meltdown of the
nuclear reactors 1,2,3, of the Fukushima1 plant
many other terminological and factual analogies: chain reaction,
multiplier, instability, stress tests, fragility, contagion, self-regulation
vs discretionary regulation, regulatory capture..
The analogy is not restricted to the terminology:
-nuclear reactors and financial systems are characterized by
a similar sort of complex dynamics leading to similar failures and
regulation problems
-both are characterized by huge long-term investments and shortterm profit seeking
Purpose
The analogies may be inspiring to understand better:
nuclear accidents: understandable also by a layman
having some experience of financial phenomena
Causes and consequences {
financial crises: concrete perception of the
consequences and risks
passive regulation (self-regulation) mechanisms
regulation failures {
active regulation rules and their compliance
Policy {
assessment
risk {
management
Structure of the presentation
-
Complex dynamics of a nuclear reactor
-
Nuclear safety
-
The complex dynamics of a monetary economy
-
Financial safety
-
Passive and active regulation in finance and nuclear energy generation
-
Management of hard risks and precautionary policies
1st Part
The Complex dynamics of nuclear
reactors
ear plant Fukushima1
1st Part
The Complex dynamics
of nuclear reactors
Typical BWR nuclear reactor of the Fukushima1 type
Core: where the
nuclear fuel bars
are contained and
the nuclear chain
reaction occurs
Moderator (water):
to slow down the
chain reaction
Control rods: to regulate and
control the chain reaction
The nuclear fission
nuclear energy generation
Nuclear fission {
nuclear weapons
Microphysics: when a heavy nuclide or isotope is hit by a neutron its
nucleus breaks into two or more “fission fragments” releasing a
great amount of energy in the form of heat and radiation
The most important example for energy generation is that of
uranium-235: when hit by a “free” neutron
235U
+ neutron → fission fragments + 2.4 neutrons + 192.9 MeV
nuclear fissions and nuclear chain-reactions
The formula above shows:
•
a nuclear reaction releases a great amount of energy: almost 200 millions
of eV (electronvolts): hundreds of millions more than a chemical reaction
•
each of the neutrons ejected may hit nearby nuclides triggering new
fission events → nuclear chain reaction releasing a great amount of
energy in a continuous way
•
the trouble is that a nuclear chain reaction also releases dangerous
radioactive decay of fission fragments
radiation {
energy in the form of radiation: gamma rays and neutrinos
The macrophysics of nuclear reactors
The difficult challenge of nuclear engineering is the management of
produce great amounts of energy in a continuous way
nuclear plants able to {
avoid any release of radiation outside the plants
This is very difficult since the optimal equilibrium is extremely unstable:
the analysis focuses on the dynamics of a population of free neutrons N
that reproduces itself according to the “effective neutron multiplication factor”
k=Nt+1/Nt
expresses the average number of neutrons released by one fission that
bring about another fission
The dynamics of the core (discrete time)
we can express the dynamics of Nt :
Nt = kNt-1 + N’
(2)
where N’ (exogenous flow of neutrons) is assumed constant and
k is the effective multiplication factor and plays a crucial role:
when k<1, the system is subcritical and cannot sustain a chain reaction:
→ the system is stable but the energy released rapidly fades away
→ the equilibrium number of free neutrons N* is given by:
N* =N’/(1-k),
(3)
where 1/(1-k) may be defined as the” multiplier” of exogenous neutrons
that determines the equilibrium population of neutrons
The criticality of a nuclear reactor
when k>1 the system is supercritical:
the chain reaction increases exponentially the number of fissions
and thus also the population of neutrons
→ this progressively amplifies the energy released in a growingly
uncontrollable way
The chain reaction may be exploited for a sustainable production of
energy only in the borderline case:
when k=1 the system is critical: the number of free neutrons
remains constant in a stationary process of energy release
The only useful state of the core is thus a structurally unstable
bifurcation path
Graphical representation
The dynamic behaviour of the reactor’s core under the three different
hypotheses mentioned above may be represented in a simplified
way as in the graphs 1a,b,c
we measure on the ordinates axis Nt+1 and on the abscissa axis Nt
the equation ( 2 ) has a slope that depends on k,
while the locus of possible equilibrium values (stationary since the
exogenous neutron generation rate N’ is constant) is represented by
the line at 45 degrees (where Nt+1=Nt)
Fig.1: The dynamic regimes of a nuclear reactor
Nt+1
Nt+1
N*
Nt
a) subcritical case: k<1
Nt+1
Nt
b) critical case: k=1
Nt
c) supercritical case: k>1
The dynamics of the core (discrete time)
The subcritical case in fig.1a is characterized by a stable equilibrium N*
that is a function of the rate of exogenous generation of neutrons N’:
N* = N’/(1-k)
(3)
The supercritical case represented in fig.1b has no realizable equilibrium
while the population of free neutrons grows exponentially
In the critical case there is no equilibrium (while it is indeterminate in the
extreme case N’=0)
the critical case is a singularity that is structurally unstable
(in the sense of Andronov): an infinitesimal perturbation of k is sufficient
to transform the system in supercritical or subcritical
→ oscillations around the critical state
The determinants of k
The fine tuning of k is very difficult since the physical processes underlying the
aggregate value of k are probabilistic and are subject to complex dynamics
The parameter k depends on the following factors:
k = Pi Pf η - Pa - Pe
(4)
where Pi is the probability that a particular neutron strikes a fuel nucleus
Pf is the probability that the stroked nucleus undergoes a fission
η is the average number of neutrons ejected from a fission event (it is between
2 and 3 for the typical fuel utilized in nuclear plants: 235U and 39Pu)
Pa is the probability of absorption by a nucleus of the reactor not belonging to
the fuel
Pe is the probability of escape from the reactor’s core.
In other words, the product of the first three variables measures the strength of the
fission chain reaction, and thus of the energy release,
while the probability of absorption and escape measure the average leakage
from the system
Fluctuations of k and their regulation
In consequence of the probabilistic nature of its underlying process, k
necessarily fluctuates off its critical (desired) value:
“the system is never in equilibrium”
K < 1 → the efficiency in energy generation declines,
K > 1 → exponential increase in energy generation in the form of
heath and radiation:
this may easily jeopardize the safety of the reactor
→ a nuclear reactor requires reliable regulation mechanisms that
keep the average fluctuations of k at the critical value while
containing their amplitude
Active regulation
The crucial active regulation instrument is given by the “control rods”:
a small shift of the control rod inward or outward the reactor core
produces a swift change in the number of fission events
Control rods are good enough for routine
however the criticality of the dynamic process implies that the reaction to
unexpected contingencies may trigger a sequence of events that make
the reactor uncontrollable
e.g. the accident occurred at the Chernobyl Nuclear Power Plant:
a system test meant to improve safety led to a rupture of the reactor
vessel and a series of explosions that destroyed reactor 4
Self-regulation: passive safety mechanisms
high probability of regulation mistakes under unexpected events:
→ passive safety mechanisms, i.e. independent of human decisions
a crucial mechanism of self-regulation is provided by the moderator:
most moderators become less effective with increasing temperature
→ if the reactor overheats the chain reaction tends to slow down
e.g. regular water, that is used as moderator in the majority of
reactors, starts to boil sizably reducing the effective multiplier
however self-regulation may fail: e.g. there may be an unexpected
leakage of water or steam or a failure of the system to pump new
water into the reactor as in the case of Fukushima1
“redundant” passive security mechanisms
in case of failure of a mechanism the same role may be played by another one
redundancy would increase safety iff failure probability of passive mechanisms
were independent;
unfortunately their failure probabilities may be not independent in
consequence of a major shock
Fukushima1, e.g., endured the earthquake in March 2011 but had its power and
back-up generators knocked out by a 7- meter tsunami
lacking electricity to pump water needed to cool the core, engineers vented
radioactive steam into the atmosphere to release pressure, leading to a
series of explosions that blew out concrete walls around the reactors
back-up diesel generators that might have averted the disaster were
positioned in a basement, where they were submerged by sea water
Major incidents
Major nuclear incident =def one that either resulted in loss of human life
or more than US$50,000 of property damage (def by the US federal
government)
More than 100 major nuclear power plant accidents have been
recorded since 1952, totalling more than US$21 billion in property
damages
Nuclear industry claims that new technology and improved oversight
made nuclear plants much safer, but 57 major accidents occurred
since 1986
It was claimed that these accidents occurred in badly managed oldfashioned nuclear plants as in Chernobyl (1986);
however two thirds of these accidents occurred in the US and the
worst of all, the Fukushima1 disaster, in the technologically
advanced Japan
After Fukushima1
The French Atomic Energy Agency (CEA) admitted that technical innovation
cannot eliminate the risk of human errors in nuclear plant operation
An interdisciplinary team from MIT estimated that, given the expected growth of
nuclear power from 2005 – 2055, at least four serious nuclear power
accidents would be expected in this period: Fukushima1 is only the first
After Fukushima deep and extensive revision of energy policy:
-many countries stopped construction of new plants including Germany, Italy,
Switzerland and France
-all the other countries are revising and severely downsizing the programs of
construction of new plants including Japan, USA and the UK
Although the disruptions provoked by the Great Recession are not inferior,
the financial system and regulation policies did not undergo a similar
process of radical revision
Arguments pro nuclear energy
Taking account of the high risks intrinsic in the production of nuclear energy we may
wonder why it has been developed so much and risks to be further developed
a) clean: GHGs emissions as renewables
arguments: relatively { b) cheap: less than renewables
c) safe: less casualties and radiation than with fossil fuels
Counter-arguments:
a) estimates that take account of the entire life cycle of a nuclear plant, including
its construction, its decommissioning, and waste disposal, find a much higher
average level of GHGs emissions in between renewables and fossil sources
(see, e.g., the meta-study by Sovacool, 2010)
Arguments against nuclear energy
b) the favourable cost estimates are criticized for not taking full account of
- the entire life cycle of the plant
- the scarcity of the fuel similar to that of oil
- the external diseconomies
- the crucial role of an arbitrary high rate of discount
c) the belief in nuclear safety underestimates the number of casualties brought
about by nuclear energy because:
-Accidents under-reported
-Difficult to establish the probabilistic cause-effect nexus even in the short run
-It does not take into account the long-run effects of radiation on human health
2° Part
The Complex dynamics of financial
systems
The Complex dynamics
of a monetary economy
I want to show that a monetary economy is characterized by a complex dynamics
that shows deep analogies with that of a nuclear reactor,
although the complex dynamics of a sophisticated monetary economy is more
elusive as it is characterized by a plurality of interacting criticalities
However, the complex dynamics of a nuclear reactor and its crucial criticality is
fully recognized by nuclear physics and nuclear plants engineering
on the contrary, the complex dynamics of the financial system is generally
ignored, even denied, by mainstream economics and finance
In both cases the issue of control of complex dynamics is taken lightheartedly
and the risks originating from week, late or mismanaged control of critical
processes are greatly underestimated
Nuclear and economic chain reactions
In a nuclear reactor the chain reaction is based on the
alternation between free neutrons N hitting nearby nuclides, fission
of nuclides F hit by free neutrons, and free neutrons N’ ejected from
the hit nuclides
N-F-N’…
in a monetary economy the chain reaction is based on the
alternation between flows of commodities sold C, flows of money M
from the buyer to the seller, and flows of commodities C’ bought
with the money received
C-M-C’…
The income-expenditure view
From the point of view of economic units the economic chain reaction
translates in the alternation of
income flows yit received e.g. by a certain unit m from other units
and
expenditure flows eit financed by previous income flows:
ymt – emt – ynt – ent – yqt …
we get a chain reaction whose “strength” is given by
cit = eit+1 / yit
where c may be interpreted as the marginal propensity to spend
(consume) from flows of income (with a time lag)
The aggregate real system: the multiplier
taking account of exogenous expenditure e’ we may easily derive from
the only constraint of alternation of income and expenditure in a
money economy the Kahn-Keynes “multiplier” model
(Kahn, 1931 and Keynes, 1936; derivation in Sordi-Vercelli, 2005 and 2012)
the cumulative effects of this alternation triggered by an impulse e’
representing the exogenous expenditure converge towards:
y* = e’/(1- c)
where c is the propensity to consume and
m = 1/(1- c)
is the so called Kahn-Keynes “income multiplier”
The multiplier as subcritical dynamics
In the Kahn-Keynes multiplier
y* = e’/(1- c)
c plays the same role of the “effective multiplication factor” k in the
equations describing the dynamic behaviour of a nuclear reactor:
N* = N’/(1-k)
the propagation process is dynamically stable since c < 1
The analogy with the subcritical case of a nuclear chain reactor is striking
as the propagation process has a similar dynamic structure (see fig.2a)
(Leo Szilard was probably influenced by the income multiplier)
The subcritical case and the role of saving
In this simple model of income generation the stability of the real system is
assured by a positive saving rate: normal case
However, in the last decades the saving rate greatly diminished in
developed countries progressively pushing the real economic system
towards a critical regime, so reducing its stability
In a few countries, and most notably in the US, the saving rate became
slightly negative just before the outbreak of the subprime crisis in 2007
The stabilizing role played by the saving rate crucially depends on the simplifying
assumption that all the investment is exogenous
however, this assumption restricts the validity of the model to the short
period as capital accumulation is ignored
Taking capital accumulation into account the plausibility of a critical
and supercritical regimes increases
Fig.2 Dynamic regimes of a monetary economy
Et+1
Et+1
Y*
a) subcritical case: c<1
Yt
Et+1
Yt
b) critical case: c=1
Yt
c) supercritical case: c>1
The acceleration principle
As soon as we consider the impact of endogenous investment on the
income-expenditure chain reaction, the potential instability of the
process becomes evident, as first pointed out by Harrod (1939)
The endogenous investment It is assumed to be given by the
“acceleration principle”: it depends on the change of income according
to a proportionality factor v generally called capital coefficient:
It = v(Yt – Yt-1)
The “chain reaction” brought about by the interaction between
multiplier and accelerator is an unstable path that we may
assimilate to the critical path of a nuclear reactor
The multiplier-accelerator model
When It=St the aggregate expenditure Et is equal to the aggregate income in
the previous period Yt-1 and the system operates under a critical regime
under these conditions the economic system is characterized by what Harrod
called a warranted rate of growth g=s/v; from the assumption of criticality:
v(Yt – Yt-1) = sYt
from which we derive immediately :
(Yt – Yt-1) / Yt = s/v
this steady state is a “critical process” or a razor’s edge (see fig 2b):
a further increase of expenditure over income, however small, would render the
system supercritical determining an unsustainable rate of growth (see fig 2c):
however, in this case the real system would soon impact on the full employment
barrier bouncing back towards a subcritical regime: self-regulation at a cost
The role of credit
The instability of the economy crucially depends on the financial side
of the income-expenditure process:
in a modern monetary economy, an excess of endogenous investment
over saving in a given period is made possible by the credit system
A persisting excess of investment over saving or, more in general, of
expenditure over income has to be financed through borrowing
→ to understand the intrinsic criticality of contemporary
financialized economies we have thus to focus on the financial side
of transactions and economic decisions
The credit multiplier
The first monetary chain reaction that has been explored in the economics literature is
rooted in the alternation between credit and bank deposits:
additional credit translates in additional bank deposits that justify the concession of
further credit and so on
According to the monetarists, this process explains the money supply M as determined
by the monetary base B believed to be under the control of monetary authorities
The credit multiplier may be expressed in the following way:
M = B(1 + γ)/(α+β+γ)
where α is the legal reserve ratio
β the excess reserves ratio
γ the currency drain ratio
when α+β=1 the system does not multiply (no chain-reaction): 100% reserve system
when α+β<1 the system is subcritical and the credit multiplier has a finite value
According to the evidence produced by Koo (2011) the system is currently in a critical
state, sometimes even supercritical
Credit multiplier and financial chain reaction
The nexus between the credit multiplier and financial crises has
been emphasized since long
For example Friedman and Schwartz observed that
“a liquidity crisis in a unit fractional reserve banking system is precisely
the kind of event that triggers -and often has triggered- a chain
reaction. And economic collapse often has the character of a
cumulative process. Let it go beyond a certain point, and it will tend
for a time to gain strength from its own development as its effects
spread and return to intensify the process of collapse”
(Friedman and Schwartz 1963: p.419)
In a fractional-reserve banking system, in the event of a bank run,
the demand depositors and note holders would attempt to withdraw
more money than the bank has in reserves, causing the bank to
suffer a liquidity crisis and, ultimately, to default
Critique to exogeneous money supply (1)
The monetarist belief that money supply is exogenous and controllable fell in
disrepute since the early 1980s:
It requires a constant velocity of money
or at least its independence of the business cycle, while
the empirical evidence suggests that these validity conditions are false:
the velocity of circulation is quite volatile and strongly pro-cyclical
Goodhart, e.g., wrote that the money multiplier model is 'such an incomplete way
of describing the process of the determination of the stock of money that it
amounts to misinstruction‘
(Goodhart 1984, p.188)
and that ‘almost all those who have worked in a [central bank] believe that
this view is totally mistaken; in particular, it ignores the implications of
several of the crucial institutional features of a modern commercial banking
system....’
(Goodhart, 1994, p.1424)
Money and the monetary base:
money supply is endogenous
Source: [Notes on Mishkin Ch.14 - P.18]
The money multiplier is endogenous
Critique to exogeneous money supply (2)
The credit multiplier has been rejected in particular by the advocates
of an endogenous money theory advanced since long and
subscribed among others by Schumpeter and later many postKeynesians (in particular Basil Moore)
Endogenous money theory states that the supply of money is
credit-driven and determined endogenously by the demand for
bank loans, rather than exogenously by monetary authorities
In this case the analogy with nuclear reactor’s instability is even
stronger
The trouble with criticality is that, even in the absence of
significant external shocks, a small change from within the
system may be sufficient to trigger an unstable chain reaction
(e.g. stability destabilizing in Minsky)
The liquidity criticality (1)
financial inflow yt
In a given period t, each economic unit is characterized by a {
financial outflow et
The ratio:
k=et/yt
that I call liquidity ratio is a significant index of a unit’s current financial
conditions as it affects both its liquidity and solvency (see Vercelli, 2011)
It is also a financial multiplication factor (the analogous of c in the multiplier):
also in this case the critical state is the only one sustainable in the long
run, while a deviation from it tends to increase up to a threshold
its value may be higher than unity and may persist in such a state for a
relatively long time → in this case the financial system is supercritical:
a supercritical financial process is often called a “bubble”
The liquidity criticality (2)
A supercritical process is made possible by credit:
creates inflows ex nihilo for the borrower in the expectation that its
consequent increase in outflows will generate in the future higher
inflows that will permit the repayment of debt with an interest
the increase in the extant credit of the private sector typically happens in
the period of vigorous expansion of the economy when the euphoria
of the agents leads them to seek a higher leverage
When the ensuing financial bubble(s) burst(s) the system becomes
suddenly subcritical in order to reduce the leverage
in this simple formalization the critical path may be identified with Minsky’s
period of tranquillity when the system is characterized by stationary
expectations (see Sordi and Vercelli, 2011)
The liquidity criticality
A simple way to explain this sudden change is to assume that during the
expansion and the boom the units adopt extrapolative expectations
E  fit 1   fit    fit  fi   1  
e
t
e
e
 f  f   f
it
i
 0
e
i
while as soon as the crisis is triggered and during the period of
depression the expectations are regressive
E  fit 1   fit    fit  fi   1  
r
t
r
r
 f  f   f
it
i
0   1
r
i
From: Sordi, S. and A.Vercelli, Heterogeneous expectations and strong
uncertainty in a Minskyian model of financial fluctuations, 2010,
DEPFID Discussion Paper, forthcoming in JEBO
The solvency criticality
In order to understand the sudden switch from a supercritical dynamics to subcritical dynamics
and vice versa, we have to introduce a second source of criticality that interacts with the first
one
The current value of the multiplying factor affects its expected values the sum of which
determines the solvency of the economic unit:
n
kit* 
 E  eit  s  / 1  r 
s 0
n
s
 E  yit  s  / 1  r 
s
s 0
where r is the discount factor and k* is the solvency ratio
When k* < 1 the unit has a positive net worth and is solvent;
k* = 1 is the critical value beyond which the unit becomes insolvent since its net worth
is negative and is going to be bankrupt unless it is very rapidly bailed out
To avoid this fate, the economic units have a desired value of the insolvency ratio
k* = 1 – μ
sufficiently far from the critical value to withstand unexpected contingencies
Financial instability hypothesis: a model
We are now in a position to restate the core of the FIH with the aid of a simple model
interaction between the liquidity ratio and the solvency ratio (cash-flow approach),
(1)
kit
*

  i 
k
it  1  i  

kit
α>0
(2)
kit*
 i  kit  1
*
kit
β>0
This elementary Lotka-Volterra model is based on:
Vercelli, A., A perspective on Minsky moments: Revisiting the Core of the Financial
Instability Hypothesis, in Review of Political Economy, vol.23(1), 2011, pp.49-67
Definition of Minsky moment and Minsky process
This Lotka-Volterra model produces clockwise cycles that have
properties very similar to those described by Minsky in the FIH
kt
2
3
5
A
B
1
ω
A Minsky moment
A-B Minsky process
1
4
1-μ
6
1
k*t
Financial fluctuations: dynamic and structural instability
kt
P
1
ω ω’
1-μt
1-μt’
1
k*t
Sequence of financial cycles (→long cycle)
The degree of instability and fragility reached in the final stage of a
financial cycle depends on the characteristics of preceding cycles
gravity and length of the last crisis
Tends to grow in proportion to {
time distance from the last great crisis
The average safety margin tends to grow progressively:
germs of a successive “great crisis”:
In the last century long financial cycles of about 30 years:
trough-to-trough: 1930-1950, 1950-1982, 1982- 2010?
Long cycles in finance, USA, 1900-2008
the most recent one: 1980-2010?
US Stock market valuation
Source: Martin Wolf FT, 26.11.08 from Robert Shiller et al.
CAPE = Cyclically adjusted price-earning ratio
Q = cyclically adjusted Tobin Q
Summing up: analogies and differences
in dynamics (1)
Significant analogies between the
nuclear reactor
dynamic behavior
{
real transactions
economic system
{
financial transactions
inflows of neutrons – fission – outflows of neutrons
Alternation
{
income – exchange – expenditure
{
financial inflows – exchange – financial outflows
Summing up: analogies and differences
in dynamics (2)
alternation → chain reaction → criticality:
< 1 subcritical regime
→ stable equilibrium
outflows/inflows { = 1 critical regime
→ unstable path
> 1 supercritical regime → exponential divergence
Kahn / Kalecki multiplier
Subcritical regime in economics {
credit multiplier
Critical regime in economics: Harrod’s warranted rate of growth
I>S or E>Y
Supercritical regime
{
net B > 0
Crucial difference in dynamics
We have seen that:
•
the alternation M-C-M’ and the ensuing ‘chain reaction’ presupposes a fully
developed monetary economy also in the subcritical case
•
a supercritical regime (I>S or E>Y) presupposes Credit
It is possible to show that:
•
In a financialized economy the dynamics of the system depends on the
liquidity
interaction of two basic criticalities {
solvency
→ dynamics even more complex: more difficult to understand and control
however the implications for regulation and policy are quite similar
Analogies in control failures: excessive
confidence in applied technology and science
a) passive safety mechanisms
Excessive confidence in self-regulation {
b) invisible hand of the market
subject to failure→ redundancy→ independent probabilities
a)
{
excessive confidence→ lax active regulation
gap from ideal markets
b)
{
excessive confidence→ lax active regulation
Analogies in control failures
undervaluation of risks: hard uncertainty→ hard risks
In both cases {
defective risk management of private and public DMs
The usual criterion of max of EU leads to excessively risky decisions
maximin (radical uncertainty)
alternative approaches {
“Choquet rule” (EU+MM) (intermediate cases)
would lead to a much more precautionary policy:
- shifting investment towards renewables and the real side of the economy
- strengthening active regulation in energy generation and finance