Abstract
In this chapter we build on the concepts of radical uncertainty and complexity. We introduce these concepts into our policy discussion in a quantitative manner. We earlier suggested that one of the key outcomes of allowing for radical uncertainty and complexity in policy formulation, is a shift in the focus of policy towards building resilience to systemic risks, and a much greater reliance on communities in the implementation of such resilience-focused policies. In this chapter, we introduce and propose viability theory as a mathematical tool for representing, designing, and implementing a resilience-focused public policy. The chapter has four purposes. First, to introduce viability theory into the policy tool kit. Second, to explain when this tool is a useful alternative or complement to other policy tools that are based on an optimisation framework. Third, to explain viability theory, and differentiate it from optimisation theory, with the help of a few examples. Finally, to apply viability theory to the problem at hand, which is to design policies focused on building resilience to systemic risks, when the policy environment is characterised by radical uncertainty and complexity. In this context, and within the framework provided by viability theory, public policy has two (complementary) purposes. First, to expand the “viability kernel”. Second, to ensure that the system remains within the kernel when it is subjected to external shocks. To provide continuity with the rest of the book, our examples are again centred on the interaction between environmental (emissions), social (inequality), and economic influences on wellbeing.
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Notes
- 1.
This neologism was introduced by Simon (1955).
- 2.
For rigorous introductions to viability theory see Aubin (1991), Quincampoix and Veliov (1998), Veliov (1993) and Aubin et al. (2011). Viability theory has also been applied to finance (see Pujal and Saint-Pierre 2006), managerial economics (see Krawczyk et al. 2012), as well as microeconomics (see Krawczyk and Serea 2013).
- 3.
This section draws extensively from Krawczyk and Pharo (2013, 2014a) and Krawczyk and Judd (2016).
- 4.
A similar formulation could be made for a viability problem in discrete time.
- 5.
Where K itself is a viability domain for F, K will be the viability kernel of itself.
- 6.
They depend on x and can be called Markovian; Aubin (1997) calls them regulation maps.
- 7.
Unless a steady state has been reached.
- 8.
An example of this may be the actual “real world” behaviour of inflation-targetting central banks, who will often avoid changing interest rates for as long as they can.
- 9.
An optimal control problem can however be a special case of a viability problem if constraints are so tight that only one path is considered viable.
- 10.
E.g., the inflation band in New Zealand, as in several other countries, is specified by a Policy Targets Agreement signed between the Minister of Finance and the Governor of the Central Bank.
- 11.
Aubin (1997) discusses several simple viability problems. In particular, an affine and bilinear system dynamics are considered (see ibid, pp. 46–51). Our model is inspired by the former but has a similar structure to the latter, with the economic interpretation expanded.
- 12.
Linearity of the changes is not crucial for the subsequent analysis.
- 13.
E.g., high demand could cause the “clearing” price to rise faster than inflation. However, the supplier may not want to feel some kind of a social condemnation caused by the prices behaving too differently from inflation. Also, the supplier may not want to change the menu too frequently. All that means is that the price cannot jump.
- 14.
We want to mention that relationship like (6.9) (also see (6.23), (6.38) and (6.53)–(6.54)) is frequently called – in viability theory – a differential inclusion, see e.g., Aubin et al. (2011). A differential inclusion is a generalisation of the concept of an ordinary differential equation where the right hand side of the equation is a multi-valued map. In other words, for a given state-space point, a differential inclusion tells us about the many possible evolutions of the state variables from that point, rather than a unique evolution determined by an ordinary differential equation. The “non-ordinary” feature of (6.9) is that its right-hand side is an interval \([-c,\, c]\) and not a single value. The “bunch” of possible state evolutions for the system’s dynamics can be obtained by solving the system’s equations for the entire contents of interval \([-c,\, c]\).
- 15.
Maximisation of (6.10) in a Markovian (i.e., feedback) strategy would be even more difficult.
- 16.
Remember that \(\ln (x)\) is a negative number for \(x\le 1\); its product with −0.002 is positive.
- 17.
Denote the kernel’s boundary by \(\mathsf {fr}\,\mathscr {V}_{F}(K)\); hence \(\mathscr {V}_{F}(K)\backslash \mathsf {fr}\, \mathscr {V}_{F}(K)\) is the kernel’s interior.
- 18.
For example, we know that if economic capital is larger than its golden rule steady state value, then that capital level is inefficient.
- 19.
This subsection draws from Kim and Krawczyk (2017).
- 20.
Consumption is here defined as a state variable. We derive a consumption equation from a transformation of the Euler equation, obtained as a solution to private agents’ utility maximisation. See footnote 22, in Sect. 6.3.1.1, for a brief explanation and consult the publications listed there.
- 21.
Here we depart slightly from the stylised model presented in Chap. 2 and use e.g., Lee (2012) or Krawczyk and Judd (2016) who assume that the level of technology is given and fixed. In “real life” technology improves but the improvement process is indeterministic. Therefore, our results correspond to the worst case scenario of technological progress.
- 22.
Briefly, apply the maximum principle to maximise, in c, \(\displaystyle {\int _0^\infty e^{-\rho t} \frac{c^{1-\sigma }-1}{1-\sigma } dt}\) where \(\rho >0\) is the rate of time preference and \(\sigma >0\) is the relative risk aversion parameter, subject to (6.20). Take the shadow price of capital \(\lambda >0\), which is equal to the marginal utility of consumption, and obtain \(\displaystyle {c=\frac{1}{\lambda ^{\frac{1}{\sigma }}}}\). Then, (6.21) follows from the equation for shadow price \(\dot{\lambda }\). For more details see e.g., Krawczyk and Judd (2016) or Driscoll-O’Keefe and Krawczyk (2016).
- 23.
We claim in Sect. 6.1 that an economic viability problem does not require specification of the discount rate, or time-preference. Therefore, we say, viable policy does not suffer from a generation bias. This is true in the government’s problem. The emission-control strategy, computed in Sect. 6.3.1.3, is generation independent. The time-preference rate \(\rho \) in (6.21) is entirely due to the standard formulation of the representative agent problem.
- 24.
It can also reflect the strong possibility that risk averse individuals would like to ascertain whether a shock to income is temporary or permanent, before they adjust their consumption rates in response.
- 25.
For example, for the representative agent’s time preference \(\rho =0.04\), what can happen in 50 years is about 5 times less important than what will happen in 10 years. Indeed, \(\displaystyle {\frac{e^{-0.04\cdot 10}}{e^{-0.04\cdot 50}}=\frac{0.67}{0.13} \approx 5.15} \). However, as commented on \(\sigma \) in footnote 23, the government’s viability problem is generation-neutral.
- 26.
Orbiting in K would also be acceptable.
- 27.
If the economic agents (perhaps mistakenly) treat z as a constant, then they have a standard optimal control problem. Its Markovian solution is a consumption function c(k; z), obtained from the Euler equation in the agent’s optimisation problem. The agents constantly revise their plan, in light of new information about z; however, the consumption function remains the same.
- 28.
A few explanatory remarks on the method of interpreting the boulders may be necessary. The dimensionality of our problem is 3 (i.e., there are three state variables k, c, and z). Emission X is also a variable of interest in this problem and so is, to a limited extent, economic output y. These values (i.e., k, c, z, X, and output) are computed by VIKAASA and stored in a 5D array. Hence, the 3D boulders in Fig. 6.5 are 3D “slices” through the 5D array. In the following figures, the system’s dynamics will also be discussed with the help of 2D slices. The variables, through which the 5D array is sliced, are listed in each picture’s title.
- 29.
Actually, in \({\mathscr {V}}_F(K) \subset K\), because the horizon is infinite.
- 30.
The time profile figures display the time paths not only of k, c, z, and X, but also of output y, emission control adjustments v, and “velocity”. The velocity is the Euclidean norm \(\sqrt{k^2+c^2+z^2\,}\), which informs us about the steadiness of the economy. Evidently, the closer the velocity is to zero, the closer the economy is to a steady state. Unsurprisingly, all velocities converge to near-zero in Fig. 6.7 where viable evolutions are plotted, whereas they fail to converge to zero in Fig. 6.8 where non-viable evolutions are plotted.
- 31.
These numbers result from a particular discretisation of the set K implemented by VIKAASA, necessary to obtain the viability kernel and the economic evolutions.
- 32.
This subsection draws from Krawczyk and Townsend (2015a, b, c).
- 33.
See footnote 21.
- 34.
Judd (1987) allows labour and capital income to be taxed at different rates. For simplicity, we assume that each factor income is taxed at the same rate. For a viability model with two taxation rates see Driscoll-O’Keefe and Krawczyk (2016).
- 35.
See footnote 23.
- 36.
In the following paragraph we rely on Krawczyk and Townsend (2015b) for some details.
- 37.
Remember the Occupy Movement’s slogan? We are the 99%!
- 38.
We have not tested that relationship for other countries.
- 39.
Krawczyk and Townsend (2015b) obtained data from 1972 with Statistics New Zealand’s Infoshare tool; older data were obtained from the latest revision available in the New Zealand Official Yearbooks.
- 40.
The latter is the agent’s first order optimality condition. Actually, there are two such first-order conditions. The other is \(\ell ^\eta = \lambda (1-\tau )w\).
- 41.
The adjoint state \(\lambda (t)\) will not ‘jump’ because x(t) will not be on the boundary of K.
- 42.
- 43.
- 44.
We assume \(\alpha =0.43, \rho =0.04, \eta =1, \sigma = 0.5, \delta =0.05\).
- 45.
This section draws from Krawczyk and Townsend (2015c).
- 46.
- 47.
Building clean capital requires time. Investing in clean technology and in labour skills takes resources away from dirty capital and thus dirty capital will diminish. In the transition, we may experience a decline big enough to push the economy outside K.
- 48.
This is a reflection of the limitations of the specific model we are using. E.g., low z could be achieved for a larger A – total factor productivity. Then low z would not necessarily come at the expense of aggregate productive investment.
- 49.
See the explanation provided in footnote 48.
- 50.
- 51.
The tiny differences come from that “slicing” through B - left panel - does not have to generate the same points that “slicing” through B and \(z=1\) - right panel - does.
- 52.
See Appendix 6.2.
- 53.
The purple slice was computed for \(z=1\) and without the constraint on emissions. So, \(X(0) > {\bar{X}}\) in the seventh panel of Fig. 6.18 is of no concern.
- 54.
Indeed, c and B appear smoother in the time profile figures, than k. Emission control z can move fast if u changes fast while taxation \(\tau \) can be kept constant.
- 55.
In the k, z-slice (right panel) almost all points in \([0.1,\, 1.25] \times [0,\, 1]\) appear viable. It is the elongated shape of the k, c-slice in the left panel which tells us about viable combinations of k and c.
- 56.
- 57.
Obviously, a larger value of A means technological improvement. Whether increasing the other model parameters means improvement may be a value judgment.
- 58.
See Krawczyk and Pharo (2011) and Krawczyk and Pharo (2014b); also Krawczyk et al. (2013).
- 59.
VIKAASA is also compatible with GNU Octave, though its GUI is not.
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Appendices
Appendix 6.1: Empirical Evidence
The Emission-Control Model
We have suggested in Sect. 6.3.1.3 that the current levels of capital and consumption may determine a country’s willingness to actively pursue an emission-control policy. An example of a high-capital high-consumption country that invests heavily in renewable energy sources (mainly wind farms) is Germany. Its neighbour, Poland, is less industrialised, consumes less and constantly requests permission (from the European Commission) to postpone the application of the tight European emission rules.
The Inequality Model
One of the conclusions of Sect. 6.3.2.3 is that high values of the relative factor share are viable with low debt, but not with high debt. By extension, this means that highly indebted economies will have neither high inequality nor low tax rates.
This does indeed seem to be the case. Japan has the highest public debt in the world, with public debt in 2010 equalling 206% of GDP (The World Bank 2015). Our model predicts that Japan will have neither high inequality nor low tax rates: this is correct, Japan’s 1% share in 2010 was 9.51% (Alvaredo et al. 2014) and its top marginal tax rate was 40% (National Tax Agency 2010). In contrast the country in the Alvaredo et al. (2014) database with the highest 1% share in 2010 was Columbia, where the wealthiest 1% take 20.45% of national income. Columbia had debt equal to only 38% of GDP (The World Bank 2015); the top tax rate was 33%.
Appendix 6.2: A Method for Finding Viability Kernels
VIKAASAFootnote 55 is a suite of MATLAB® programmes that approximate viabilitykernels. The algorithms applied by VIKAASA use truncated stabilisation and an approach suggested in Gaitsgory and Quincampoix (2009).
VIKAASA main window
VIKAASA can be used either as a set of MATLAB® functions, or via a GUI.Footnote 56 The GUI can specify the viability problem, run the kernel approximation algorithms and display the results. A detailed (though somewhat outdated) manual for VIKAASA can be found in Krawczyk and Pharo (2011). The latest version of VIKAASA is available for download at Krawczyk and Pharo (2014b). In Fig. 6.26, we show the main window of VIKAASA.
In this paper, our algorithm solves a truncated optimal stabilisation problem for each element of \(K^h \subset K\), a discretisation of K. For each \(x^h \in K^h\), VIKAASA assesses whether a dynamic evolution originating at \(x^h\) can be controlled to a (nearly) steady state without leaving the constraint set in finite time. Those points that can be brought close enough to such a state are included in the kernel while those that are not are excluded. This algorithm we use is called the inclusion algorithm, as opposed to the exclusion algorithm, both explained in Krawczyk et al. (2013) and Krawczyk and Pharo (2011). We note that this computational method will miss some viable points if they are viable only because the evolutions starting at them are large cycles. However, we did not encounter similar points in our experiments.
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Karacaoglu, G., Krawczyk, J.B., King, A. (2019). Viability Theory for Policy Formulation. In: Intergenerational Wellbeing and Public Policy. Springer, Singapore. https://doi.org/10.1007/978-981-13-6104-3_6
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