Are Dynamically Inefficient Equilibria Learnable? John Duffy Department of Economics University of Pittsburgh Pittsburgh, PA 15260 USA E—mail: jduff
[email protected]
Wei Xiao SUNY at Binghamton P.O. Box 6000 Binghamton, NY 13902 USA E—mail:
[email protected] July 2011 Abstract
We consider the stability under adaptive learning dynamics of steady state equilibria in Diamond’s (1965) overlapping generations growth model with capital and money. Interior steady state equilibria of this model can be either dynamically inefficient or dynamically efficient. We show that a necessary condition for an equilibrium of this model to be stable under adaptive learning is that the equilibrium is dynamically efficient. In other words, adaptive learning can be used as a selection criterion to exclude dynamically inefficient equilibria. We also provide conditions under which a dynamically efficient equilibrium of this model involving the use of both capital and money will be stable under adaptive learning dynamics. JEL Codes: D83, E43, E52. Keywords: Dynamic Inefficiency, Learning, E-stability, Equilibrium Selection, Overlapping Generations Model.
1
Introduction
In lifecycle economies with productive capital, it has long been known (e.g., since Phelps (1965), Koopmans (1965), Cass (1972)) that equilibria can be dynamically inefficient; specifically, there is the possibility of capital over-accumulation so that interest rates are too low relative to the golden-rule level. In such cases, it is possible to reduce the capital stock and to permanently increase consumption for all generations. The canonical framework in which to illustrate dynamic inefficiency is Diamond (1965)’s competitive, overlapping generations economy with productive capital, a workhorse model in modern macroeconomics and monetary theory. In dynamic, competitive equilibrium models with the overlapping generations friction, the conditions of the First Welfare Theorem are violated due to the double infinity of goods and agents (see, e.g. Shell 1971). As dynamic efficiency corresponds to Pareto optimality in aggregate consumption, the question of dynamic inefficiency must be addressed in dynamic frameworks with capital and where the First Welfare theorem may fail to hold. Diamond’s (1965) model is thus an obvious choice; indeed it may be regarded as the “textbook” example for illustrating the possibility of dynamic inefficiency. An important question is whether economies are dynamically inefficient or dynamically efficient. There is some literature addressing this topic as a measurement issue, see, e.g., Abel et al. (1989). Here we ask a different but equally important and related question: are dynamically inefficient equilibria stable under adaptive learning dynamics of the type explored by Marcet and Sargent (1989) and Evans and Honkapohja (2001))? We also ask whether dynamically efficient equilibria are stable under the same adaptive learning dynamics. In other words, we propose the use of stability under adaptive learning dynamics as a selection criterion in economies where capital over-accumulation is an equilibrium possibility but where dynamically efficient equilibria may also exist. Lucas (1986) suggested that adaptive learning might be useful as an equilibrium selection device in a two-period pure exchange overlapping generations model with fiat money. He showed that if agents used a simple adaptive learning rule á la Bray (1982), they would converge upon the unique, interior monetary equilibrium of that model. Marcet and Sargent (1989) extended this finding to an environment where a longlived government finances a fixed deficit by printing money (seigniorage) and where agents learn according to a recursive least squares learning process. The environment they consider gives rise to a Laffer curve for government seigniorage revenues and the possibility of two steady state monetary equilibria. They show that the low inflation equilibrium of that model is stable while the high inflation equilibrium is unstable under a recursive least squares updating scheme. This work has been interpreted as supporting the notion that low inflation, monetary equilibria are attractors under adaptive learning processes in overlapping generations
1
models which are known to admit multiple equilibria. More recently, Van Zandt and Lettau (2003) and Adam et al. (2006) have shown in the seigniorage, pure-exchange overlapping generations monetary model that the high inflation steady state (Lettau and Van Zandt (2001)) or stationary paths near that steady state (Adam et al. (2006)) may be stable under adaptive learning dynamics under certain restrictive timing assumptions, e.g., if agents have contemporary observations of endogenous variables in the information sets they use to form future expectations. These findings cast some doubt on Lucas’s suggestion that adaptive learning dynamics might provide a means of selecting between low and high inflation equilibria of the model as it appears that under certain conditions both equilibria may be learnable. In all of this prior work involving the stability of monetary equilibria in overlapping generations economies, the models examined leave out alternative means of intertemporal savings, in particular, productive capital. Thus these papers have not considered the question that interests us as to whether learning can be used to select among the dynamically efficient and dynamically inefficient equilibria that are possible in overlapping generations models with productive capital. Here we consider the stability under adaptive learning dynamics of various equilibria in a standard version of Diamond’s (1965) model. The version of Diamond’s model that we consider has fiat money in place of government debt (as in Diamond’s original formulation) as the sole outside asset so to maintain comparability with the prior literature on learning in overlapping generations models. The model admits three steady state equilibria: an autarkic equilibrium, and two interior equilibria -a non-monetary, “inside money” equilibrium where capital is the only means of savings and an “outside money” equilibrium where fiat money and productive capital coexist and pay the same rate of return. The latter equilibrium is only possible if the inside money equilibrium is dynamically inefficient (see, e.g., Tirole (1985) or Azariadis (1993)). Under the benchmark assumption of perfect foresight, if all three steady state equilibria exist, then the autarkic equilibrium is a “source”, the inside money equilibrium is a “sink” and the outside money equilibrium is a “saddle”. Evans and Honkapohja (2001, section 4.8.2) have provided conditions under which interior, inside money equilibria of a “scalar” Diamond model — one without any outside asset like fiat money — are learnable by adaptive agents using a constant gain learning rule. However Evans and Honkapohja do not consider whether the dynamic efficiency/inefficiency of such interior equilibria matters for stability under learning (more precisely, E-stability), a question we address in this paper. Furthermore, the question of whether the outside money equilibrium of the “planar” Diamond model is E-stable has not, to our knowledge, been addressed and is a question we answer in this paper.
2
These questions are important for several reasons. First, the Diamond model with an outside asset is a workhorse model in macroeconomics and monetary theory. If the dynamically efficient equilibrium of this model involving both money and capital is not learnable, that would call into question a large body of work that makes use of that dynamically efficient equilibrium, e.g., work in monetary theory and on pension systems. Indeed, as Abel et al. (1989) point out, dynamic efficiency is necessary for there to be an operative bequest motive and thus for the Ricardian equivalence doctrine to hold. Second, as noted earlier, an implication of prior work in the learning literature is that monetary equilibria are learnable, non-monetary equilibria are not learnable and high inflation or hyperinflationary equilibria may be learnable under certain timing conditions. It seems important to examine whether this conclusion is robust to the inclusion of an additional asset by which individuals can save intertemporally, namely capital. Perhaps most importantly, the planar Diamond model admits multiple steady state equilibria some of which are dynamically inefficient, for instance, the nontrivial, capital-only equilibrium of the model version that we study. In this equilibrium, the steady state capital stock is too high; all agents can be made better off by lowering the capital stock to the golden rule level. We wish to know whether such dynamically inefficient equilibria are learnable or not; if not then the possibility of dynamic inefficiency, which is typically illustrated using the Diamond model, might be taken less seriously. Indeed, we are not aware of any prior work exploring a connection between the stability of an equilibrium under adaptive learning behavior and the dynamic (in)efficiency of that equilibrium. Finally, this paper adds to the learning literature by considering learning in another multivariate system that differs from the Ramsey—Cass—Koopmans infinite horizon growth model that has been extensively examined by Evans and Honkapohja (2001) and others. Using a standard version of Diamond’s (1965) model we uncover a deep connection between the stability of an interior steady state equilibrium of that model under adaptive learning dynamics and the dynamic efficiency of that steady state equilibrium. In particular we show that a necessary condition for an equilibrium of this model to be stable under adaptive learning is that the equilibrium is dynamically efficient. Thus, stability under adaptive learning, (equivalently E-stability or ‘learnability’) can be used as an equilibrium selection criterion to eliminate the dynamically inefficient equilibrium of the Diamond model from consideration. We also provide conditions under which the dynamically efficient interior equilibrium of the Diamond model with both capital and money is E-stable. The rest of the paper is organized as follows. In the next section, we present the version of the Diamond model with money and capital that we study. In section three we explore the scalar version of this model with capital only and state conditions under which the interior steady state equilibrium is stable under adaptive learning dynamics. In section four we repeat this same exercise for the planar version of the model
3
with both capital and money. Section 5 provides a summary and conclusions.
2
An overlapping generations model with capital and money
2.1
Households
Time is discrete and runs from period = 0 to infinity. At every date , identical agents (households) are born who live for just 2 periods. Each of these agents are endowed with a single unit of labor in the first period of life (when young) which they inelastically supply to firms. Each household’s objective is to maximize the discounted utility from consumption over his two period lifetime (for simplicity we assume there is no uncertainty): max ( ) + (+1 )
(1)
Here (+ ) denotes the utility earned by generation from consumption of amount + of the single perishable good in each of the two periods = 0 1 of his life, and is the period discount factor. The function is assumed to satisfy the standard properties, (0) = 0; for 0, 0 () 0, 00 () 0, and lim→0 0 () = ∞. Households born at time (generation are assumed to maximize (1) subject to: +1 +1
≤ − ≤
Here denotes wage income received from firms,
− +1
+ +1 +1 +1
(2) (3)
is the nominal money stock per young household,
is the price of the good in terms of money and +1 is the real capital stock available for production at date + 1 (equivalent to the household’s savings) and +1 is the gross rate of return on capital (savings) held from date to date + 1. Let us convert the model into real terms by dividing the budget constraints (2 -3) through by the price level, to obtain: +1
≤ − − +1
(4)
≤
(5)
+ +1 +1 1 + +1
where = , = and 1 + +1 = +1 denotes the gross inflation factor between dates and + 1.
4
2.2
Firms
We suppose that firms have access to a production technology that converts capital per young worker into output of ( ) per young agent of the single perishable consumption good at time . The function has the standard properties: (0) = 0; for 0, () 0, 0 () 0, and 00 () 0; lim→0 0 () = ∞ and lim→∞ 0 () = 0. We assume that capital depreciates at a constant rate of ∈ (0 1) per period. Firms are assumed to operate under conditions of perfect competition so that factors of production (capital and labor) are paid their marginal products. Letting denote the gross real return on capital received by old agents (the owners of the firms) and letting denote the real wage paid to labor, we have that in all periods,
2.3
= 1 + 0 ( ) −
= ( ) − 0 ( )
Government
We suppose there is a government that finances real expenditures of per young household member in period by printing money. Specifically, we suppose that the total supply of fiat money grows at the constant rate, ≥ 0, = (1 + )−1 . It follows that real money balances per capita at date , are characterized by: =
(1 + )−1 −1 −1 −1 −1 (1 + )−1 (1 + )(1 + )
= =
We can write the government’s budget constraint in per capita terms as: =
−1 − or −1
=
− 1+
We assume for simplicity that households do not derive any utility from government purchases.
2.4
Optimization
The representative household’s decision problem can be stated as follows: max ( ) + (+1 ) +1
≤ − − +1 ≤
+ +1 +1 1 + +1
The first order conditions are given by: 0 ( ) = 1 5
(6)
0 (+1 ) = 2
(7)
1
= +1 2
(8)
2
≤ 1 (1 + +1 )
(9)
Combining (8) and (9), we have the no-arbitrage condition 1 = +1 1 + +1 Henceforth we shall specialize preferences, to the commonly used CRRA class, () =
1− 1− ,
which
satisfies the assumptions we made for household’s utility from consumption. We will also adopt a CobbDouglas production technology, ( ) = , where 0 1 denotes capital’s share of output. These choices will enable us to derive clear analytic results for a widely studied version of the model. Specifically, from the first order conditions (6-9) we can derive the household’s savings function as: 1−
1
+1
(+1 ) =
1
1−
1 + +1
We can thus re-write the household’s consumption at date by: = [1 − (+1 )] Savings must be held in the form of capital or real money balances. Thus, total savings is given by: +1 +
(1 + )+1 +
= (+1 ) or = (1 − ) (+1 )
Equilibria for this economy must satisfy the following system of equations: +1
=
=
1 1 + +1
(1 − ) (+1 ) − 1+ (1 + ) −1 (1 + )(1 + )
(10) (11)
= +1
(12)
= 0 ( ) + 1 −
(13)
We can eliminate from the above system using equation (12). We then have that all equilibria to the Diamond (1965) model under our CRRA-Cobb-Douglas specification must be characterized by the following
6
three equation system: +1
3
( +1 ) − 1+ (1 + ) −1 = (1 + ) =
(14) (15)
= 0 ( ) + 1 −
(16)
Learning in the Capital-Only Model
We begin by considering steady state equilibria of the capital-only model. By this we mean that we consider a version of the model where the only means of intertemporal saving is to invest in capital as we set = 0 for all . We first characterize the equilibrium of the capital—only model under perfect foresight and we then turn to the case where agents are adaptive learners.
3.1
Perfect foresight
Under the assumption of perfect foresight, the economy is now characterized by the following two equations: 1−
1
+1
=
(1 − ) +1 1
1−
(17)
(1 + )(1 + +1 )
= −1 + 1 −
(18)
Combining these two equations, we can express the system as a function of only:
= (+1 ) =
−1
(19) 1
(+1 − 1 + ) [
−1 −1 1 (1 + ) )] + 1 − (1 + − +1 1−
It follows that steady state equilibrium levels for the gross interest rate, are solutions to: = () Under our CRRA-Cobb Douglas version of the model, there will be a unique interior steady state equilibrium value for in the capital-only model. Following Cass (1972) we let us define the steady state equilibrium of the model to be dynamically inefficient if 0 () = − 1 + +
(20)
or
(21)
1+ 7
Intuitively, in such an equilibrium, the interest rate is too low (lower than the population growth rate); a consequence is that there is over-accumulation of capital relative to the golden-rule level as explained in Phelps (1965).
3.2
Steady state learning
We now assume that agents do not possess perfect foresight knowledge of the interest rate process, = (). Instead, they use a simple, adaptive learning process to learn about the steady state equilibrium. Here we follow the steady state learning approach pursued by Evans and Honkapohja (2001). We begin by rewriting the system under learning as: = (+1 ) where +1 denotes agents’ time expectation of +1 . We suppose that learning agents treat +1 as an
i.i.d. process with an unknown mean. Agents attempt to estimate +1 using the learning model: = + ( − ) +1
(22)
where is a decreasing gain term. At time , is assumed to be observable. Agents are assumed to use information on all past interest factors { }=0 to estimate the steady state value of . This simple adaptive learning rule (22) is one that was proposed by Evans and Honkapohja (2001) to study the scalar Diamond model (see Evans and Honkapohja (2001, section 4.8.2). The convergence of this learning process to the steady state can be assessed by checking the stability of: +1 = + [(+1 ) − ]
As Evans and Honkapohja (2001) show, the stability condition in the decreasing gain case (corresponding to recursive least squares learning) is given by: 0 () 1
(23)
where denotes a steady state equilibrium value for . From expression (20) we have that: 0 () =
1 − 1 1 1 − (1 + ) − −
(24)
Combining (24) with condition (23) (and after some manipulation), we can state the following proposition: Proposition 1 The condition for the capital-only steady state equilibrium to be learnable (E-stable under our adaptive learning dynamic) is:
1 1+−1 ( ) 1− 8
(25)
We note that the condition given in Proposition 1 does not make reference to whether the interior steady state equilibrium of the model is a dynamically efficient or inefficient equilibrium. For example, if the following condition holds, a dynamically efficient interior steady state equilibrium is learnable:
+1≤
1 1+−1 ( ) 1−
(26)
On the other hand, a dynamically inefficient equilibrium is learnable so long as (25) holds and + 1 We next show that the condition for a dynamically inefficient equilibrium is not consistent with condition (25). That is, by contrast with a dynamically efficient equilibrium, a dynamically inefficient equilibrium is not stable under adaptive learning for all empirically plausible model calibrations. This finding is established in the following two Lemmas: Lemma 1 If 0 1, the interior steady state equilibrium of the capital-only model is not learnable. Proof. See Section A of the Appendix. Note that Lemma 1 does not specify whether the interior steady state equilibrium of the capital-only model is dynamically inefficient or efficient. Lemma 2 Suppose 1 and 0 5. Then a necessary condition for the interior steady state equilibrium of the capital-only model to be stable under adaptive learning is: 0 () +
(27)
Proof. See Section A of the Appendix Note that condition (27) is precisely the condition for which the steady state of the model is dynamically efficient. This suggests that there is an important connection between dynamic efficiency and stability under adaptive learning dynamics. The restriction on is necessary to obtain analytic conditions; however, it is also empirically plausible as capital’s share of output is not thought to be greater than 50%. Nevertheless, numerical simulations indicate that condition (27) continues to be necessary for .5 1 as well. Taken together, Lemmas 1-2 allow us to state the following proposition: Proposition 2 Suppose 0 5. Then the interior equilibrium of the capital-only model cannot be both stable under adaptive learning dynamics and dynamically inefficient. 9
This is the first finding we are aware of in the learning literature where stability under adaptive learning can be used to rule out a dynamically inefficient equilibrium. We now turn to the planar version of the Diamond model where there is both capital and money.
4
The Model with Capital and Money
4.1
Perfect foresight
In our version of the Diamond model allowing for both capital and money there exists two interior steady state equilibria. One has savings held using both capital and money and the other has all savings held in the form of capital only. We focus on the equilibrium where both money and capital coexist. A necessary condition for this equilibrium to exist is that the capital-only steady sate equilibrium of the model is dynamically inefficient (see Tirole (1985)). This capital-only interior equilibrium is the same one discussed in the previous section, that we showed to be generally unstable under adaptive learning dynamics. The model with productive capital and money can be reduced to the following system of three equations:
1−
1
+1
1 − +1 − 1 + 1 + 1 1− 1 +
=
(28)
+1
(1 + ) −1 (1 + )
=
= −1 + 1 −
(29) (30)
A steady state equilibrium of this system is characterized by: 1 (1 + ) −1 ) 1+ 1+ 1+
= (
=
1
= (1 − )
4.2
1
1−
1 +
1−
− (1 + )
Steady state learning
Under the assumption that agents are adaptive learners, we can rewrite the system as
= (+1 )
= [(
+1 −1+ 1 −1 −1 1 −1 1 + ) )] + 1 − ) −1 + ] [ (1 + − (+1 1+ 1−
= (+1 +1 )
=
1+ 1 +1 = +1 1 + +1 +1 10
The learning algorithm for this two dimensional system is specified by: +1
= + [ − ]
+1
= + [ − ]
The T-mappings from perceived law of motion to the actual law of motion (using the terminology of Evans and Honkapohja (2001)) are given by: ( ) = ( ) ( ) = ( ) E-stability of this system is evaluated by examining the stability of the associated ordinary differential equations (ODEs):
µ
¶
=
µ
( ) ( )
¶
−
µ
¶
Our analysis of the stability of these ODEs is summarized in the following: Proposition 3 A necessary condition for E-stability of the steady state equilibrium where money and capital coexist is: 1
1
−1 1 −1−1 1+ 1 − − 1 (1 + − ) + ( − 1 + ) 1 −1 1−−1+ (1 + − )
(31)
Proof. See Section B of the Appendix. This condition is difficult to interpret. However, numerical simulations confirm that when the model is calibrated such that the capital-only equilibrium is dynamically inefficient then the above condition for the dynamically efficient steady state equilibrium holds under a wide variety of empirically plausible parameter values. Specifically, we searched over values for the six model parameters: 0 5, 0 1, 0 1, 0 11, −11 0, and .1 6 that might be reasonable given an interpretation of each period of model time as corresponding to 20-30 years (see, e.g., the Technical Appendices of La Croix and Michel (2002). This numerical exercise yields a large number of parameter combinations that satisfy the condition of Proposition 3 (details available upon request). It follows from Propositions 2-3 that in a model with both capital and money, the dynamically inefficient capital-only equilibrium is unstable under adaptive learning dynamics (E-unstable) while the dynamically efficient equilibrium involving both capital and money is learnable under empirically plausible conditions.
5
Conclusions
The possibility of dynamically inefficient equilibrium, in particular, capital over-accumulation, is often illustrated using Diamond’s (1965) overlapping generations growth model with productive capital and government 11
bonds (or money). When an economy is dynamically inefficient, this model gives rise to multiple interior steady state equilibria. The possibility of dynamic inefficiency allows for Pareto improving equilibria involving various mechanisms, e.g., government debt, fiat money or pension systems to absorb some of the over-accumulated capital stock enabling greater consumption for all generations. In this paper we have asked which, if any, of these interior competitive equilibria —dynamically inefficient or dynamically efficient— are stable under adaptive learning dynamics. The question is of interest as the dynamically efficient equilibria of this model (those involving capital and either government debt, fiat money or pension systems) are the focus of much applied theoretical and empirical research in macroeconomics and monetary economics. While we provide in Proposition 1 a condition under which the interior equilibrium of the capital-only model is stable under adaptive learning dynamics, we point out in Proposition 2 that this condition cannot be satisfied unless the steady state is dynamically efficient. In particular we show that for a reasonable restriction on a necessary condition for the capital-only equilibrium to be learnable is that it be dynamically efficient! That is, we appear to have uncovered a previously unknown connection between dynamic efficiency and the stability of a steady state equilibrium under adaptive learning dynamics. In Proposition 3 we provide conditions under which the dynamically efficient steady state equilibria of the Diamond model with both capital and money is stable under adaptive learning dynamics. The condition given in Proposition 3 holds for a wide range of empirically plausible parameter values, including those for which the dynamically inefficient equilibrium is unstable under adaptive learning dynamics. Taken together, these findings suggest that the prospect of dynamically inefficient outcomes in a competitive, general equilibrium growth model might be viewed with some skepticism if one adopts adaptive learning as an equilibrium selection criterion. Such a conclusion is also consistent with Abel et al.’s (1989) conclusion that the U.S. and other developed economies are dynamically efficient according to their measurement criterion. The limitations of our conclusions are that they apply on to a specific parameterization of a deterministic version of Diamond’s two-period (1965) model, in particular one with CRRA preferences and Cobb-Douglas production. Nevertheless this is a widely—studied version of Diamond’s model and these restrictions enable us to state precise, analytic results. We also have not considered stochastic, multi-period or continuous-time versions of the Diamond model nor have we considered more complicated adaptive learning processes as we believe these complications would make it more difficult, if not impossible for us to derive analytic results. We leave such generalizations of our findings to future research.
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Appendix A. Proof of Lemmas 1 and 2 The critical necessary condition for an equilibrium to be learnable is that: 0 () =
1 − 1 1 1 − (1 + ) − − 1
(32)
To prove Lemma 1, we can show that if 0 1, the above condition never holds. To see this, rewrite the critical condition (32) as: 1 − 1 1 1 (1 + ) − −
If 0 1, then the right-hand-side expression is necessarily negative, and the left-hand-side is positive. This expression can never hold. A learnable equilibrium therefore requires 1. This is the statement of Lemma 1. To prove Lemma 2 we proceed as follows. Suppose the critical condition (32) holds. Rearrange terms to get: 1
(1 + ) −
1− − 1 −1
Now suppose that 0 05. In this case, the right-hand-side expression is obviously greater than 1. Then the following must be true: 1
(1 + ) −
− 1 − 1 1
We focus on the case where 0 05 as it enables us to obtain analytic results and it seems unlikely that capital’s share of output exceeds 50% in any economy. We continue to re-arrange terms: 1
1
(1 + ) − −
−1
Now, as 1 is necessary for a learnable equilibrium, the right-hand-side expression is again greater than 1. So we have that 1
1
(1 + ) − − 1 Re-arranging terms, we have 1
1+ 1 (1 + )
13
The parameter restriction for the discount factor is 0 1, which requires (1+) As = 1+ where = 0 () − , it is obvious that if , then the parameter restriction for will always be violated. A necessary condition for the critical condition to hold is therefore that 0 () +
(33)
Note that condition (33) is exactly the condition that would guarantee that the equilibrium is dynamically efficient (see 20). This is the condition stated in Lemma 2.
B. Proof of Proposition 3 E-stability requires that the matrix =
µ
−
−
1
1
¶
has eigenvalues less than 0. We calculate the relevant elements of the Jacobian matrix as follows: 1
=
=
1
−1 1 1 −1−1 1+ − − (1 + − ) + ( − 1 + ) 1 −1 1−−1+ (1 + − )
−1 1 ( − 1 + ) 1 −1+ −1 (1 + ) ( ) +
= −
1+
= 1
The characteristic polynomials required for the calculation of eigenvalues are as follows 2 + (1 −
)− = 0
The eigenvalues can be solved as =
−(1 −
)
±
q (1 − 2
2 )
− 4
For the two eigenvalues 1 and 2 to be both negative or have negative real parts, we must have 1 2 1 + 2
1 − 0 2 0
= −
(34)
=
(35)
We next show that condition (35) always holds in this model. We first compute the signs of the two coefficients:
0 −1 1 ( − 1 + ) 1 −1+ −1 (1 + ) ( ) +
= − =
14
1+
0
The second term is negative because the only negative element in the expression is − 1. It is then obvious that the product of the two coefficients must be positive. The E-stability of the system therefore depends on the first condition, equation (34), which can be explicitly written as (31).
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