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Dynamic Optimization in Continuous-Time Economic Models …
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Language: english
Created: Tue Dec 15 10:27:18 1998
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Dynamic Optimization in Continuous-Time Economic Models
(A Guide for the Perplexed)
Maurice Obstfeld*
University of California at Berkeley
First Draft: April 1992
*I thank the National Science Foundation for research support.
I. Introduction
The assumption that economic activity takes place
continuously is a convenient abstraction in many applications.
In others, such as the study of financial-market equilibrium, the
assumption of continuous trading corresponds closely to reality.
Regardless of motivation, continuous-time modeling allows
application of a powerful mathematical tool, the theory of
optimal dynamic control.
The basic idea of optimal control theory is easy to grasp--
indeed it follows from elementary principles similar to those
that underlie standard static optimization problems. The purpose
of these notes is twofold. First, I present intuitive
derivations of the first-order necessary conditions that
characterize the solutions of basic continuous-time optimization
problems. Second, I show why very similar conditions apply in
deterministic and stochastic environments alike. 1
A simple unified treatment of continuous-time deterministic
and stochastic optimization requires some restrictions on the
form that economic uncertainty takes. The stochastic models I
discuss below will assume that uncertainty evolves continuously
^
according to a type of process known as an Ito (or Gaussian
------------------------------------------------------------------------------------------------------------------------------------------------------------
1When the optimization is done over a finite time horizon, the
usual second-order sufficient conditions generalize immediately.
(These second-order conditions will be valid in all problems
examined here.) When the horizon is infinite, however, some
additional "terminal" conditions are needed to ensure optimality.
I make only passing reference to these conditions below, even
though I always assume (for simplicity) that horizons are
infinite. Detailed treatment of such technical questions can be
found in some of the later references.
1
diffusion) process. Once mainly the province of finance
^
theorists, Ito processes have recently been applied to
interesting and otherwise intractable problems in other areas of
economics, for example, exchange-rate dynamics, the theory of the
firm, and endogenous growth theory. Below, I therefore include a
brief and heuristic introduction to continuous-time stochastic
processes, including the one fundamental tool needed for this
^
type of analysis, Ito's chain rule for stochastic differentials.
^
Don't be intimidated: the intuition behind Ito's Lemma is not
hard to grasp, and the mileage one gets out of it thereafter
truly is amazing.
II. Deterministic Optimization in Continuous Time
The basic problem to be examined takes the form: Maximize
8
i dt------
(1) j e U[c(t),k(t)]dt
2
0
subject to
(2) Q
k(t) = G[c(t),k(t)], k(0) given,
where U(c,k) is a strictly concave function and G(c,k) is
concave. In practice there may be some additional inequality
constraints on c and/or k; for example, if c stands for
consumption, c must be nonnegative. While I will not deal in any
detail with such constraints, they are straightforward to
2
incorporate. 2 In general, c and k can be vectors, but I will
concentrate on the notationally simpler scalar case. I call c
the control variable for the optimization problem and k the state
variable. You should think of the control variable as a flow--
for example, consumption per unit time--and the state variable as
a stock--for example, the stock of capital, measured in units of
consumption.
The problem set out above has a special structure that we
can exploit in describing a solution. In the above problem,
planning starts at time t = 0. Since no exogenous variables
enter (1) or (2), the maximized value of (1) depends only on
k(0), the predetermined initial value of the state variable. In
other words, the problem is stationary, i.e., it does not change
in form with the passage of time. 3 Let's denote this maximized
value by J[k(0)], and call J(k) the value function for the
problem. If {c*(t)}8 stands for the associated optimal path of
t=0
the control and {k*(t)}8 for that of the state, 4 then by t=0
definition,
------------------------------------------------------------------------------------------------------------------------------------------------------------
2The best reference work on economic applications of optimal
control is still Kenneth J. Arrow and Mordecai Kurz, Public
Investment, the Rate of Return, and Optimal Fiscal Policy
(Baltimore: Johns Hopkins University Press, 1970).
3Nonstationary problems often can be handled by methods
analogous to those discussed below, but they require additional
notation to keep track of the exogenous factors that are
changing.
4According to (2), these are related by
t
i G[c*(s),k*(s)]ds + k(0).
k*(t) = j
2
0
3
8
i e dtU[c*(t),k*(t)]dt.
------
J[k(0)] = j
2
0
The nice structure of this problem relates to the following
property. Suppose that the optimal plan has been followed until
a time T > 0, so that k(T) is equal to the value k*(T) given in
the last footnote. Imagine a new decision maker who maximizes
the discounted flow of utility from time t = T onward,
8
i e d(t T)U[c(t),k(t)]dt,
------ ------
(3) j
2
T
subject to (2), but with the intial value of k given by k(T) =
k*(T). Then the optimal program determined by this new decision
maker will coincide with the continuation, from time T onward, of
the optimal program determined at time 0, given k(0). You should
construct a proof of this fundamental result, which is intimately
related to the notion of "dynamic consistency."
You should also convince yourself of a key implication
of this result, that
T
i e dtU[c*(t),k*(t)]dt + e dTJ[k*(T)],
------ ------
(4) J[k(0)] = j
2
0
where J[k*(T)] denotes the maximized value of (3) given that k(T)
= k*(T) and (2) is respected. Equation (4) implies that we can
think of our original, t = 0, problem as the finite-horizon
problem of maximizing
4
T
i dt
------ dT ------
j e U[c(t),k(t)]dt + e J[k(T)]
0
subject to the constraint that (2) holds for 0 < t < T. Of
course, in practice it may not be so easy to determine the
correct functional form for J(k), as we shall see below!
Nonetheless, this way of formulating our problem--which is
known as Bellman's principle of dynamic programming--leads
directly to a characterization of the optimum. Because this
characterization is derived most conveniently by starting in
discrete time, I first set up a discrete-time analogue of our
basic maximization problem and then proceed to the limit of
continuous time.
Let's imagine that time is carved up into discrete intervals
of length h. A decision on the control variable c, which is a
flow, sets c at some fixed level per unit time over an entire
period of duration h. Furthemore, we assume that changes in k,
rather than accruing continuously with time, are "credited" only
at the very end of a period, like monthly interest on a bank
account. We thus consider the problem: Maximize
8
s e dthU[c(t),k(t)]h
------
(5) t
t=0
subject to
5
(6) k(t+h) ------ k(t) = hG[c(t),k(t)], k(0) given.
Above, c(t) is the fixed rate of consumption over period t while
k(t) is the given level of k that prevails from the very end of
period t ------ 1 until the very end of t. In (5) [resp. (6)] I have
multiplied U(c,k) [resp. G(c,k)] by h because the cumulative flow
of utility [resp. change in k] over a period is the product of a
fixed instantaneous rate of flow [resp. rate of change] and the
period's length.
Bellman's principle provides a simple approach to the
preceding problem. It states that the problem's value function is
given by
( )
(7) J[k(t)] = max {U[c(t),k(t)]h + e dhJ[k(t+h)]}, ------
c ( t )9 0
subject to (6), for any initial k(t). It implies, in particular,
that optimal c*(t) must be chosen to maximize the term in braces.
By taking functional relationship (7) to the limit as h L 0, we
will find a way to characterize the continuous-time optimum. 5
We will make four changes in (7) to get it into useful form.
First, subtract J[k(t)] from both sides. Second, impose the
------------------------------------------------------------------------------------------------------------------------------------------------------------
5All of this presupposes that a well-defined value function
exists--something which in general requires justification. (See
the extended example in this section for a concrete case.) I
have also not proven that the value function, when it does exist,
is differentiable. We know that it will be for the type of
problem under study here, so I'll feel free to use the value
function's first derivative whenever I need it below. With
somewhat less justification, I'll also use its second and third
derivatives.
6
constraint (6) by substituting for k(t+h), k(t) + hG[c(t),k(t)].
Third, replace e dh by its power-series representation, 1 dh + ------
------
(d2h2)/2 ------ (d3h3)/6 + .... Finally, divide the whole thing by
h. The result is
& [d (d2h/2) + ...]J[k + hG(c,k)]
(8) 0 = max
c
7U(c,k) ------ ------
J(k)}/h*,
+ {J[k + hG(c,k)] ------
8
where implicitly all variables are dated t. Notice that
J[k + hG(c,k)] ------ J(k) = {J[k + hG(c,k)] ------ J(k)}G(c,k) .
------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
h G(c,k)h
It follows that as h L 0, the left-hand side above approaches
J'(k)G(c,k). Accordingly, we have proved the following
PROPOSITION II.1. At each moment, the control c* optimal
for maximizing (1) subject to (2) satisfies the Bellman
equation
(9) 0 = U(c*,k) + J'(k)G(c*,k) ------ dJ(k)
( )
= max {U(c,k) + J'(k)G(c,k) ------ dJ(k)}.
c 9 0
This is a very simple and elegant formula. What is its
interpretation? As an intermediate step in interpreting (9),
define the Hamiltonian for this maximization problem as
7
(10) H(c,k) _ U(c,k) + J'(k)G(c,k).
In this model, the intertemporal tradeoff involves a choice
between higher current c and higher future k. If c is
consumption and k wealth, for example, the model is one in which
the utility from consuming now must continuously be traded off
against the utility value of savings. The Hamiltonian H(c,k) can
be thought of as a measure of the flow value, in current utility
terms, of the consumption-savings combination implied by the
consumption choice c, given the predetermined value of k. The
Hamiltonian solves the problem of "pricing" saving in terms of
Q
current utility by multiplying the flow of saving, G(c,k) = k, by
J'(k), the effect of an increment to wealth on total lifetime
utility. A corollary of this observation is that J'(k) has a
natural interpretation as the shadow price (or marginal current
utility) of wealth. More generally, leaving our particular
example aside, J'(k) is the shadow price one should associate
with the state variable k.
This brings us back to the Bellman equation, equation (9).
Let c* be the value of c that maximizes H(c,k), given k, which is
arbitrarily predetermined and therefore might not have been
chosen optimally. 6 Then (9) states that
(11) H(c*,k) = max {H(c,k)} = dJ(k).
c
------------------------------------------------------------------------------------------------------------------------------------------------------------
6It is important to understand clearly that at a given point in
time t, k(t) is not an object of choice (which is why we call it
a state variable). Variable c(t) can be chosen freely at time t
(which is why it is called a control variable), but its level
influences the change in k(t) over the next infinitesimal time
interval, k(t + dt) k(t), not the current value k(t). ------
8
In words, the maximized Hamiltonian is a fraction d of an
optimal plan's total lifetime value. Equivalently, the
instantaneous value flow from following an optimal plan divided
by the plan's total value--i.e., the plan's rate of return--must
equal the rate of time preference, d. Notice that if we were to
measure the current payout of the plan by U(c*,k) alone, we would
err by not taking proper account of the value of the current
increase in k. This would be like leaving investment out of our
measure of GNP! The Hamiltonian solves this accounting problem by
valuing the increment to k using the shadow price J'(k).
To understood the implications of (9) for optimal
consumption we must go ahead and perform the maximization that it
specifies (which amounts to maximizing the Hamiltonian). As a
by-product, we obtain the Pontryagin necessary conditions for
optimal control.
Maximizing the term in braces in (9) over c, we get 7
(12) Uc(c*,k) = ------ Gc(c*,k)J'(k).
The reason this condition is necessary is easy to grasp. At each
moment the decision maker can decide to "consume" a bit more, but
at some cost in terms of the value of current "savings." A unit
of additional consumption yields a marginal payoff of Uc(c*,k),
but at the same time, savings change by Gc(c*,k). The utility
------------------------------------------------------------------------------------------------------------------------------------------------------------
7I assume interior solutions throughout.
9
value of a marginal fall in savings thus is ------Gc(c*,k)J'(k); and
if the planner is indeed at an optimum, it must be that this
marginal cost just equals the marginal current utility benefit
from higher consumption. In other words, unless (12) holds,
there will be an incentive to change c from c*, meaning that c*
cannot be optimal.
Let's get some further insight by exploiting again
the recursive structure of the problem. It is easy to see from
(12) that for any date t the optimal level of the control,
c*(t), depends only on the inherited state k(t) (regardless of
whether k(t) was chosen optimally in the past). Let's
write this functional relationship between optimal c and k as c*
= c(k), and assume that c(k) is differentiable. (For
example, if c is consumption and k total wealth, c(k) will
be the household's consumption function.) Functions like c(k)
will be called optimal policy functions, or more simply, policy
functions. Because c(k) is defined as the solution to (9), it
automatically satisfies
0 = U[c(k),k] + J'(k)G[c(k),k] ------ dJ(k).
Equation (12) informs us as to the optimal relation between c and
k at a point in time. To learn about the implied optimal
behavior of consumption over time, let's differentiate the
preceding equation with respect to k:
10
0 = [Uc(c*,k) + J'(k)Gc(c*,k)]c'(k) + Uk(c*,k)
+ [Gk(c*,k) d]J'(k) + J"(k)G(c*,k).
------
The expression above, far from being a hopeless quagmire, is
actually just what we're looking for. Notice first that the
left-hand term multiplying c'(k) drops out entirely thanks to
(12): another example of the envelope theorem. This leaves us
with the rest,
(13) Uk(c*,k) + J'(k)[Gk(c*,k) ------ d] + J"(k)G(c*,k) = 0.
Even the preceding simplified expression probably isn't totally
reassuring. Do not despair, however. A familiar economic
interpretation is again fortunately available.
We saw earlier that J'(k) could be usefully thought of as
the shadow price of the state variable k. If we think of k as an
asset stock (capital, foreign bonds, whatever), this shadow price
corresponds to an asset price. Furthermore, we know that under
perfect foresight, asset prices adjust so as to equate the
asset's total instantaneous rate of return to some required or
benchmark rate of return, which in the present context can only
be the time-preference rate, d. As an aid to clear thinking,
let's introduce a new variable, l, to represent the shadow price
J'(k) of the asset k:
l _ J'(k).
11
Our next step will be to rewrite (13) in terms of l. The
key observation allowing us to do this concerns the last term on
the right-hand side of (13), J"(k)G(c*,k). The chain rule of
calculus implies that
dJ'(k) dl dl
= Q;
dk dk
J"(k)G(c*,k) = * = * = l
----------------------------------- ------------ ------------ ------------ ------------
dk dt dk dt dt
and with this fact in hand, it is only a matter of substitution
to express (13) in the form
Uk + lGk + Q
l
(14) = d.
l
----------------------------------------------------------------------
This is just the asset-pricing equation promised in the
last paragraph.
Can you see why this last assertion is true? To understand
it, let's decompose the total return to holding a unit of stock k
into "dividends" and "capital gains." The "dividend" is the sum
of two parts, the direct effect of an extra unit of k on utility,
Uk, and its effect on the rate of increase of k, lGk. (We must
multiply Gk by the shadow price l in order to express the
Q
physical effect of k on k in the same terms as Uk, that is, in
terms of utility.) The "capital gain" is just the increase in
the price of k, Q.
l The sum of dividend and capital gain, divided
by the asset price l, is just the rate of return on k, which, by
12
(14) must equal d along an optimal path.
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Example
Let's step back for a moment from this abstract setting to
consolidate what we've learned through an example. Consider the
standard problem of a consumer who maximizes i8e dtU[c(t)]dt ------
0
Q
subject to k = f(k) ------ c (where c is consumption, k capital, and
f(k) the production function). Now Uk = 0, G(c,k) = f(k) ------ c, Gc
= 1, and Gk = f'(k).
------ In this setting, (14) becomes the
statement that the rate of time preference should equal the
marginal product of capital plus the rate of accrual of utility
capital gains,
Q
l
d = f'(k) + .
l
------
Condition (12) becomes U'(c) = l. Since this last equality
implies that Q = U"(c)c, we can express the optimal dynamics of c
l Q
and k as a nonlinear differential-equation system:
Q U'(c) Q
f'(k) d ,
q e
(15) c = k = f(k) c.
U"(c)
------ -----------------------------2 ------ 2 ------
z c
You can see the phase diagram for this system in figure 1.
(Be sure you can derive it yourself! The diagram assumes that
limk 8f'(k) = 0, so that a steady-state capital stock exists.)
L
The diagram makes clear that, given k, any positive initial c
13
initiates a path along which the two preceding differential
equations for c and k are respected. But not all of these paths
are optimal, since the differential equations specify conditions
that are merely necessary, but not sufficient, for optimality.
Indeed, only one path will be optimal in general: we can
write the associated policy function as as c* = c(k) (it is
graphed in figure 1). For given k, paths with initial
consumption levels exceeding c(k) imply that k becomes negative
after a finite time interval. Since a negative capital stock is
nonsensical, such paths are not even feasible, let alone optimal.
Paths with initial consumption levels below c(k) imply that k
gets to be too large, in the sense that the individual could
raise lifetime utility by eating some capital and never replacing
it. These "overaccumulation" paths violate a sort of terminal
condition stating that the present value of the capital stock
should converge to zero along an optimal path. I shall not take
the time to discuss such terminal conditions here.
If we take
c1 (1/e) ------
1 ------
U(c) = , f(k) = rk,
(1/e)
-----------------------------------------------------------------------
1 ------
where e and r are positive constants. we can actually find
an algebraic formula for the policy function c(k).
Let's conjecture that optimal consumption is proportional to
wealth, that is, that c(k) = hk for some constant h to be
14
determined. If this conjecture is right, the capital stock k
Q
will follow k = (r ------ h)k, or, equivalently,
Q
k
= r
------ ------ h.
k
This expression gives us the key clue for finding h. If c =
hk, as we've guessed, then also
Q
c
= r
------ ------ h.
c
Q
c
But necessary condition (15) requires that = e(r ------ ------ d),
c
which contradicts the last equation unless
(16) h = (1 ------ e)r + ed.
Thus, c(k) = [(1 ------ e)r + ed]k is the optimal policy function. In
the case of log utility (e = 1), we simply have h = d. We get
the same simple result if it so happens that r and d are equal.
Equation (16) has a nice interpretation. In Milton
Friedman's permanent-income model, where d = r, people consume
the annuity value of wealth, so that h = r = d. This rule
results in a level consumption path. When d $ r, however, the
optimal consumption path will be tilted, with consumption rising
over time if r > d and falling over time if r < d. By writing
15
(16) as
h = r ------ e(r ------ d)
we can see these two effects at work. Why is the deviation from
the Friedman permanent-income path proportional to e? Recall
that e, the elasticity of intertemporal substitution, measures an
individual's willingness to substitute consumption today for
consumption in the future. If e is high and r > d, for example,
people will be quite willing to forgo present consumption to take
advantage of the relatively high rate of return to saving; and
the larger is e, certeris paribus, the lower will be h. Alert
readers will have noticed a major problem with all this. If r >
d and e is sufficiently large, h, and hence "optimal"
consumption, will be negative. How can this be? Where has our
analysis gone wrong?
The answer is that when h < 0, no optimum consumption plan
exists! After all, nothing we've done demonstrates existence:
our arguments merely indicate some properties that an optimum, if
one exists, will need to have.
No optimal consumption path exists when h < 0 for the
following reason. Because optimal consumption growth necessarily
Q
satisfies c/c = e(r d), and e(r - d) > r in this case, optimal
------
consumption would have to grow at a rate exceeding the rate of
Q
return on capital, r. Since capital growth obeys k/k = r ------
(c/k), however, and c > 0, the growth rate of capital, and hence
16
that of output, is at most r. With consumption positive and
growing at 3 percent per year, say, but with capital growing at a
lower yearly rate, consumption would eventually grow to be
greater than total output--an impossibility in a closed economy.
So the proposed path for consumption is not feasible. This means
that no feasible path--other than the obviously suboptimal path
with c(t) = 0, for all t--satisfies the necessary conditions for
optimality. Hence, no feasible path is optimal: no optimal path
exists.
Let's take our analysis a step further to see how the value
function J(k) looks. Observe first that at any time t, k(t) =
k(0)e(r h)t = k(0)ee(r d)t, where k(0) is the starting capital
------ ------
stock and h is given by (16). Evidently, the value function at t
= 0 is just
8
q e
( )
{ i e dt[hk(t)]1 (1/e)dt
-1 1
1 }
2 2
------ ------
9 j
J[k(0)] = 1 d
2 2 2 ------ ------
e 0
------ ------
2 2
z c
0
8
q e
( i )
{ j e dt[hk(0)ee(r d)t]1 (1/e)dt
-1 1
1 }
2 2
------ ------ ------
= 1 d
2 2 2 ------ ------
e 9 0
------ ------
2 2
z c
0
q e
( [hk(0)]1 (1/e)
-1 ------
1)
1 { } .
2 2
= 1
9d (e d) d0
2 2 --------------------------------------------------------------------------------------------------------- ------ ------
e
------ ------
2 2 1)(r ------ ------ ------
z c
So the value function J(k) has the same general form as the
utility function, but with k in place of c. This is not the last
17
time we'll encounter this property. Alert readers will again
notice that to carry out the final step of the last calculation,
I had to assume that the integral in braces above is convergent,
that is, that d ------ (e ------ 1)(r ------ d) > 0. Notice, however, that d ------
(e ------ 1)(r ------ d) = r ------ e(r ------ d) = h, so the calculation is valid
provided an optimal consumption program exists. If one doesn't,
the value function clearly doesn't exist either: we can't specify
the maximized value of a function that doesn't attain a maximum.
This counterexample should serve as a warning against blithely
assuming that all problems have well-defined solutions and value
functions.
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Return now to the theoretical development. We have seen how
to solve continuous-time determinstic problems using Bellman's
method of dynamic programming, which is based on the value
function J(k). We have also seen how to interpret the derivative
of the value function, J'(k), as a sort of shadow asset price,
denoted by l. The last order of business is to show that we have
actually proved a simple form of Pontryagin's Maximum Principle:8
PROPOSITION II.2. (Maximum Principle) Let c*(t) solve the
problem of maximizing (1) subject to (2). Then there exist
variables l(t)--called costate variables--such that the
Hamiltonian
------------------------------------------------------------------------------------------------------------------------------------------------------------
8First derived in L.S. Pontryagin et al., The Mathematical Theory
of Optimal Processes (New York and London: Interscience
Publishers, 1962).
18
H[c,k(t),l(t)] _ U[c,k(t)] + l(t)G[c,k(t)]
is maximized at c = c*(t) given l(t) and k(t); that is,
dH
(17) (c*,k,l) = Uc(c*,k) + lGc(c*,k) = 0
dc
-----------
-
at all times (assuming, as always, an interior
solution). Furthermore, the costate variable obeys the
differential equation
Q = ld dH
(18) l (c*,k,l) = ld [Uk(c*,k) + lGk(c*,k)]
dk
------ -----------
- ------
Q
for k = G(c*,k) and k(0) given. 9
------------------------------------------------------------------------------------------------------------------------------------------------------------
9You should note that if we integrate differential-equation
(18), we get the general solution
8
dH
l(t) = ie d(s t) [c*(s),k(s),l(s)]ds + Aedt,
j
------ ------
dk
2 -----------
-
t
where A is an arbitrary constant. [To check this claim, just
differentiate the foregoing expression with respect to t: if the
integral in the expression is I(t), we find that Q = dI
l (dH/dk) ------
+ dAe dt = dl (dH/dk).] I referred in the prior example to an ------
additional terminal condition requiring the present value of the
capital stock to converge to zero along an optimal path. Since
l(t) is the price of capital at time t, this terminal condition
usually requires that limt 8e dtl(t) = 0, or that A = 0 in the ------
L
19
You can verify that if we identify the costate variable with
the derivative of the value function, J'(k), the Pontryagin
necessary conditions are satisfied by our earlier dynamic-
programming solution. In particular, (17) coincides with (12)
and (18) coincides with (14). So we have shown, in a simple
stationary setting, why the Maximum Principle "works." The
principle is actually more broadly applicable than you might
guess from the foregoing discussion--it easily handles
nonstationary environments, side constraints, etc. And it has a
stochastic analogue, to which I now turn. 10
------------------------------------------------------------------------------------------------------------------------------------------------------------
solution above. The particular solution that remains equates the
shadow price of a unit of capital to the discounted stream of its
shadow "marginal products," where the latter are measured by
partial derivatives of the flow of value, H, with respect to k.
10For more details and complications on the deterministic
Maximum Principle, see Arrow and Kurz, op. cit.
20