Comput Econ DOI 10.1007/s10614-017-9650-3

Fiscal Policy Design in Greece in the Aftermath of the Crisis: An Algorithmic Approach Ilias Kostarakos1 · Stelios Kotsios1

Accepted: 13 January 2017 © Springer Science+Business Media New York 2017

Abstract We present a computational approach to the design of fiscal policy that is based on algorithmic, linear feedback control methods. In particular, in the context of a linear, deterministic macro-model, we develop an algorithmic procedure which allows us to design fiscal policy rules for government expenditures so that desired target-levels for GDP are exactly met (that is, complete tracking is achieved). In order to examine the effectiveness of our method we estimate the model for the Greek economy and run some counterfactual policy experiments. These experiments indicate that, for the Greek economy in the beginning of the crisis in early 2010, expansionary fiscal policy would have been able to stimulate growth and reduce the debt-to-GDP ratio. Keywords Fiscal policy · Public debt · Linear feedback control · Algorithmic control

1 Introduction For more than 20 years, monetary policy was considered as being potent enough to manage the fluctuations of the business cycle, using the interest rate as the only policy instrument and having a low, stable level of inflation as the sole policy target. This type of policy, essentially based on the Taylor rule (see Taylor 1993) led to what has been termed as the “Great Moderation” era (see Bernanke 2004), since the volatility of the cycle was greatly reduced. During this period fiscal policy was tasked with ensuring debt sustainability, mainly via using automatic stabilizers, since any discretionary

B

Ilias Kostarakos [email protected] Stelios Kotsios [email protected]

1

Department of Economics, National and Kapodistrian University of Athens, Athens, Greece

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action was deemed as potentially destabilizing. Among the reasons cited regarding the inferiority of fiscal relative to monetary policy are the long lags in the recognition, design and implementation of fiscal policy measures, the short length of recessions and the political constraints entailed (see Blanchard et al. 2010 for a detailed analysis). However, the global financial crisis of 2008 and the ensuing debt crisis that hit countries in the southern periphery of the EU have led to a resurgence of interest in the design of fiscal policy, partly because of the binding zero lower bound constraint on interest rates which renders monetary policy, to a large extent, ineffective. A highly illustrating example of fiscal policy design comes from the case of Greece. The Greek economy suffered from persistent deficits combined with increasing debtto-GDP ratios during the period up to 2009, thus leaving no fiscal space for exercising expansionary policy when the crisis hit. In May of 2010, under heavy pressures from international markets, the Greek government agreed to implement an adjustment program designed to maximize credibility and ensure that public finances are sound, so that public debt is back on a sustainable path. The program was based on frontloaded implementation of fiscal consolidation measures; in particular, certain targets—the socalled conditionality targets—were set for the main policy variables (primary deficit etc.) and the Greek government had to design the fiscal measures necessary (i.e. the appropriate changes in government expenditures and revenues) to ensure that the targets would be reached within the specified time-frame. Due to the austerity nature of the program, the aforementioned measures included cuts in (nominal) public sector wages, layoffs in the public sector etc. What is important to note is that the program was designed on the basis of the feedback methodology: once the conditionality targets were set, the relevant measures were designed and implemented. The program was evaluated over certain intervals, e.g. quarterly, and depending on the assessment the policy measures were in many cases re-designed—this is a key element of the feedback approach. The Greek program exhibited elements of ‘positive’ feedback, meaning that if a target was missed, then the measures were intensified in the same direction; for example, if the target for the primary surplus was missed, then further decreases in government expenditures combined with tax hikes were implemented. Our aim in this paper is to propose a control-theoretic computational approach for the design of fiscal policy based on the algorithmic linear feedback methodology, utilizing a technique known as model matching (which, to the extent of our knowledge has not been applied before in the literature regarding the theory of economic policy design). Our approach is based on examining whether appropriate linear fiscal policy rules exist and, if they do exist, how they should be designed so that desired values for the policy targets are reached. Specifically, we develop appropriate symbolic algorithms and, once the (fixed) policy targets have been set, we use these algorithms in order to design linear fiscal policy rules (or, feedback laws) for the instrument at hand (government expenditures), so that the policy objectives are exactly met. These policy rules are ‘responsive’ (following Taylor 1993), in the sense that the parameters of the resulting algebraic expressions of the policy rules are not fixed. This is in contrast, for example, to the well-known Friedman k% rule, which stipulated that the money supply should be increased by a constant k percentage in every period, regardless of the state of the economy. Moreover, the value of the instrument in period t depends upon lagged values of the instrument and the target variables. Thus, the policy rules

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take into account the state of the economy when deciding on the size of the policy instruments and, overall, they represent a more discretionary approach to fiscal policy. We note here that, following the classification system in Kendrick (2005) and Kendrick and Amman (2006), the policy rules presented in this paper are ‘handcrafted’ i.e. they are not the result of some optimization process. Also, following Kendrick and Amman (2010, 2014), we assume that fiscal policy adjustments will be made on a quarterly basis, in order to examine whether the increased frequency of policy interventions will allow for a smoother path for the economy following a severe economic downturn. Finally, we consider different specifications regarding the disbursement of government expenditures in order to examine whether a frontloaded or a backloaded approach is better suited for the problem at hand. The analysis is conducted within a linear, deterministic variant of the standard multiplier–accelerator model proposed by Samuelson (1939). The reason for choosing such a simple model is its tractability; it can be manipulated analytically and will allow us to understand the workings of the system once the proposed methodology is applied. Thus, it serves as a first step in understanding the workings of the system’s behavior before extending the methodology to more complex (nonlinear and/or stochastic) models. The main advantage of using the feedback methodology for policy design is that it will help in shortening policy lags, via shortening the design and implementation lags, since these can be explicitly incorporated into the equations of the model. Another advantage of our approach is that the solution technique ensures that the predetermined sequence of policy targets will be exactly followed (or, tracked), without any deviations. Moreover, the solution technique is parameterized and thus it allows for proper symbolic algorithms to be developed. However, the most important advantage is that based on these algorithms, a whole class of fiscal policy rules for solving the policy problems at hand can be designed enabling the policymaker to choose the rules that are the most appropriate based on different criteria (eg. the costs incurred by the implementation of a particular policy rule, so that the aim would be to choose the rule that causes the lower cost). This allows us to simulate the model under different policy rules, develop criteria for choosing policy rules and obtain important insights as to which rule is more appropriate depending on the particular case at hand. Finally, this method ensures that not only is complete tracking achieved i.e. the policy targets are exactly met but, moreover, the time path of the instruments is such that we have an immediate adjustment of the system; that is, if the policy rule is implemented in period t, the targets are met in period t + 1 (and all subsequent periods) i.e. the system immediately settles on the desired trajectory. Obviously, the lags associated with fiscal policy design and implementation make the instant adjustment of the system seem unrealistic. Nonetheless, this approach serves as a guideline of what the ‘optimal’ path for the policy instruments should be (optimal in the sense that complete tracking, without delay, of the target sequence is achieved). The results presented in this paper are, to a large extent, contingent on the linear and deterministic nature of the model, which admittedly is quite restrictive. Nonlinear models exhibit much more complex dynamic behavior which closely resembles the actual workings of the economy (e.g. multiple equilibria). However, since the bulk of the analysis regarding nonlinear systems is conducted based on linearizations in

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the vicinity of the equilibrium points, a thorough examination of the linear case is necessary in order to obtain a benchmark for the analysis. We must note here that another important dimension that should be examined is that of the effects of stochastic elements that could potentially alter the time-path and the structure of the model. The introduction of such elements excludes the possibility of complete tracking of the desired target-values and gives rise to a discussion regarding alternative policies that will allow for a better response from the policymaker. Such considerations are addressed in latter sections. Our results indicate that an economy like Greece in early 2010, facing a combination of high debt-to-GDP ratios and large declines in GDP, should implement short-term expansionary fiscal policy plans to ensure positive GDP growth rates. Of course, the results are model specific and due to the simple nature of the model, we did not expect any counter-intuitive results. However, the proposed methodology allows us to quantify the results, since we obtain the exact sequence of policy instruments necessary for reaching the policy targets. Moreover, because policy design in this case is essentially rule-based, it provides a clear, contingent policy plan for the short-term period. Thus, the implications of the counterfactual experiments set a useful benchmark regarding the design of fiscal policy following the feedback (rule-based) approach. The paper is organized as follows. In Sect. 2 we present the model. In Sect. 3 we state the problem and the solution technique. Section 4 presents the relevant algorithms. In Sect. 5 we provide the counterfactual policy experiments. Section 6 concludes.

2 The Model As already stated in the introduction, we opted for a linear, deterministic model of the macroeconomy, which will allow us to thoroughly assess the effects of the proposed fiscal policy plan. In particular, we use a variant of the standard multiplier accelerator model introduced in Samuelson (1939) [for a nonlinear version of the model see, among others, Kotsios and Leventidis (2004), Hommes (1995) and Puu (2007), and for a stochastic version see Dassios et al. (2014)], coupled with the government budget constraint. The multiplier–accelerator part consists of an income identity and two behavioral equations; this particular variant is a quarterly model of the macroeconomy. Assuming a closed economy, the income identity is the following: Y (t) = C(t) + I (t) + λ0 G(t) + λ1 G(t − 1) + λ2 G(t − 2)

(1)

This identity introduces our variation of the standard model, which is related to the treatment of the government expenditures variable. Following Kendrick and Shoukry (2014) we argue that the government’s decision to spend in period t is not immediately realized into outlays; rather, actual disbursement of the funds is spread over the following periods. Thus, the λi ∈ (0, 1) parameters, where λ0 + λ1 + λ2 = 1, indicate the percentage of the government’s decision to spend in period t that is disbursed in period t + i. This ‘spending’ mechanism is used to incorporate the well-known lags of fiscal policy: the inside lag i.e. the time elapsed until the downturn of the economy

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is recognized and the time necessary for the policymaker to formulate a response, and the outside lag, i.e. the time until the policy action undertaken affects the economy. Regarding the behavioral equations, we follow Puu (2007) in assuming that consumption depends on lagged values of income: C(t) = (1 − s)Y d (t − 1) + sY d (t − 2)

(2)

where C is consumption, Y d is disposable income and s ∈ (0, 1) is the marginal propensity to save. That is, consumption in period t is the sum of consumption in period t − 1 and the delayed consumption of period t − 2. Disposable income is equal to: (3) Y d (t) = Y (t) − T (t) and the tax receipts T (t) are assumed to take a tax-on-income form: T (t) = τ Y (t − 1)

(4)

where τ ∈ (0, 1) is the (constant) tax rate. Regarding investment, we assume that it depends on lagged values of income and the accelerator, ν > 0: I (t) = ν(Y (t − 1) − Y (t − 2)) (5) The budget constraint of the government has the standard form: B(t) = (1 + r )B(t − 1) + G(t) − T (t)

(6)

where B(t) denotes debt outstanding and r is the (constant) interest rate. After all the necessary substitutions among Eqs. (1)–(6) and some algebra, we end up with the following pair of equations: Y (t) − a1 Y (t − 1) − a2 Y (t − 2) + a3 Y (t − 3) = λ0 G(t) + λ1 G(t − 1) + λ2 G(t − 2) B(t) − (1 + r )B(t − 1) − τ Y (t − 1) = G(t)

(7)

where, a1 = 1 + ν − s, a2 = s − ν − τ (1 − s), a3 = sτ . This is the input-output form of the model, with G(t) being the input, and Y (t) and B(t) being the outputs. This discrete system can be rewritten more compactly via utilizing two alternative forms: the state-space form and the algebraic form. In order to write (7) in its statespace form, we introduce the state vector: x(t) = (Y (t), Y (t − 1), Y (t − 2), B(t), G(t), G(t − 1))T

(8)

x(t) = Ax(t − 1) + bG(t)

(9)

and obtain: where A and b are appropriate matrices (see “Appendix 1”).

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For the algebraic form we need the notion of the q-operator. This is essentially a lag operator defined as: q m f (t) = f (t − m) for any sequence f (t), t = 0, 1, 2, . . . (see Astrom and Wittenmark 1996). Then, the system (7) can be written as D(q)z(t) = K (q)G(t)

(10)

where z(t) = (Y (t), B(t))T , and D, K are q-polynomial matrices (see “Appendix 1”).

3 Formulation of the Problem Our aim in this paper is to calculate linear feedback policy rules for the available instrument (government expenditures) so that predetermined (fixed) target sequences for GDP and public debt will be simultaneously, exactly met without delay. That is, we want to solve a tracking problem in which the tracking error will be equal to zero (no deviations between the target values and the actual values of the system) and the system will settle immediately on the desired paths. The policy rules will be linear functions of the form:   G(t) = f G(t − 1), . . . , G(t − k), Y (t − 1), . . . , Y (t − l), B(t − 1), . . . , B(t − m) (11) As we can see, the instrument depends on lagged values of the targets and the instrument itself; this guarantees that the rule incorporates all the relevant information regarding the state of the system up to period t (this is known as the causality property in the mathematical control theory literature). Once the policy rule is applied to the open-loop system (10), i.e. the system before the policy intervention, it will produce a closed-loop system whose dynamic behavior will be modified in such a way that the policy targets will be reached. It should be noted here that we focus our attention to the design of short-run policy interventions (that is, the next 4–6 quarters). We can now state the policy problem that we are going to be concerned with: Economic Policy Problem A Calculate linear, causal feedback policy rules for government expenditures so that GDP and public debt simultaneously match desired target trajectories. In particular, the policy rules will be of the form:

G(t) =

k  i=1

ai G(t − i) +

b  i=1

bi B(t − i) +

m 

yi Y (t − i)

i=1

and the solution of Economic Policy Problem A amounts to the determination of the parameters ai , bi , yi . Before proceeding with the solution technique, we need to examine whether such feedback policy rules do in fact exist.

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3.1 Existence The existence of feedback policy rules capable of driving the system to a desired target crucially hinges upon the controllability property, one of the fundamental properties of dynamical systems. This notion is not new in the economic policy literature, as it was extensively used in an attempt to provide a dynamic generalization of Tinbergen’s static theory of economic policy [see among others, Aoki (1975), Buiter and Gersowitz (1981, 1984), Petit (1990) and Preston (1974)]. The following definition will help in clarifying the concept of controllability (see Elaydi 2004, p. 432): Definition 1 A system x(t + 1) = Ax(t) + Bu(t) where Ak×k , Bk×m , is said to be controllable (or, point controllable) if for any n 0 ∈ Z+ , any initial state x(n 0 ) = x0 and any given final state (the desired state) x f there exists a finite time N > n 0 and a control u(n), n 0 < n ≤ N such that x(N ) = x f . As we can see from the definition, if a dynamic system is controllable then the policymaker knows that by appropriately manipulating the available instruments, he can drive the system to a desired position, in a finite time interval. Whether a system is controllable can be verified using the following theorem (see Elaydi 2004, pp. 435–436 for the proof): Theorem 1 A system x(t + 1) = Ax(t) + Bu(t) where Ak×k , Bk×m , is controllable if and only if the rank of the controllability matrix W , where Wm×km = [B, AB, A2 B, . . . Ak−1 B], is equal to k. Applying the theorem to the model at hand shows that our system is controllable, thus it can be driven to a desired target point for GDP and public debt, in finite time, even though there is only one instrument available. However, our aim is to be able to modify the dynamic behavior of the system so that it will follow a predetermined trajectory, over a finite time-interval, not simply ‘hit’ a single desired point. That is, we want to exactly track a pre-assigned sequence of target values. This is possible only if the system possesses the path controllability property (see Aoki 1975). A sufficient condition for path controllability is that the number of instruments is at least equal to the number of targets, which obviously does not hold for our dynamical system. Thus, we can conclude that by using government expenditures only, we cannot simultaneously control the trajectories of GDP and public debt; as a result, so we proceed to a reformulation of the policy problem to be solved. 3.2 Targeting GDP Levels We restrict our attention to the case where the policymaker sets as a policy target the levels of GDP, using government expenditures as an instrument. That is, the policymaker is interested in achieving certain growth rates for GDP and he wants to design feedback policy rules that will provide him with the level of government expenditures necessary for reaching those targets. Moreover, in order to ensure the sustainability of public finances, the policy rules calculated will be such that the resulting levels of public debt will exhibit the least possible increase, i.e. debt accumulation will be

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minimized. In this case, he will manage to keep the economy on a positive growth trajectory and, at the same time, try to reduce the debt-to-GDP ratio thus ensuring that debt is on a sustainable path. So, we are interested in solving the following policy problem: Economic Policy Problem B Calculate linear, causal feedback policy rules for government expenditures so that GDP matches a pre-assigned trajectory while, at the same time, public debt levels are as ‘close as possible’ to their respective desired levels, i.e. Bt∗ − Bt  is minimized, where Bt∗ denotes the desired levels of public debt. The policy rules will now be linear functions of the form:   G(t) = f G(t − 1), . . . , G(t − k), Y (t − 1), . . . , Y (t − l)

(12)

Since in this system we have only one policy instrument and a single policy target, the condition for path controllability is satisfied so the policy maker knows that he can drive the system along the desired GDP trajectory. In order to solve the problem at hand, we utilize a technique known as model matching (or, model reference) control. In brief, model matching can be described as follows: the policymaker determines the trajectory of desired values (the reference sequence) for the policy target (say, Y ∗ ). Then, a linear system is constructed having the property that its output is exactly equal to the reference sequence; this is the ‘desired’ system. It is an artificial system, representing an ‘ideal’ economy whose trajectory the policymaker aims to track. The reference sequences can be thought of as being equilibrium points of the ‘ideal’ economy and the policymaker wants to manipulate the available instrument in such a way that the economy will successively hit these points. Having constructed the desired system, the problem at hand reduces to that of calculating policy rules for the instrument which, when applied to the original system (the open-loop system) will produce a closed-loop system that is identical to the desired one. It is important to note here that due to the parametric nature of the model-matching technique, we obtain as a solution a class of feedback rules, which constitutes a set of potential policies. This provides the policymaker with the ability to simulate the model under different policy rules in order to examine which is the most appropriate. For example, an obvious criterion would be that of choosing the policy rule that is less costly, either in terms of the size of government expenditures necessary or in terms of debt sustainability (i.e. a debt-to-GDP ratio that is declining). From a mathematical point of view, the open-loop system has the following form: D1 (q)Y (t) = K 1 (q)G(t)

(13)

where D1 (q) = 1 − a1 q − a2 q 2 − a3 q 3 and K 1 (q) = λ0 + λ1 q + λ2 q 2 . The desired system is: D1d (q)Y ∗ (t) = K 1d (q)u c (t)

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(14)

Fiscal Policy Design in Greece in the Aftermath of the…

where u c (t) is the input variable representing government expenditures in the ‘ideal’ economy. Finally, the fiscal policy rule is a linear function of the form: R(q)G(t) = T (q)u c (t) − S(q)Y (t)

(15)

where R, S and T are unknown q-polynomials to be determined, and will be such that it will ensure: (16) min Bt∗ − Bt  The following theorem is central for the solution of the problem (for the proof see “Appendix 3”): Theorem 2 The feedback law of the form (15), when applied to the open-loop system (13) modifies it in such a way that the resulting closed loop system will be identical to the desired one (14) if the following set of equations holds: R(q)D1 (q) + K 1 (q)S(q) = D1d (q)

(17)

K 1d (q)

(18)

K 1 (q)T (q) = 3.3 Issues Regarding Stability

An important implication of controllability is that it is closely connected to the stability property via the following theorem (Elaydi 2004, p. 462): Theorem 3 A system x(t + 1) = Ax(t) + Bu(t) where Ak×k , Bk×m , is stabilizable if it is controllable. Then, given that our system is controllable, we can choose a feedback policy rule from the class we obtain as a solution to the model-matching problem that will stabilize the system (or, make it more ‘stable’ if it already is stable). However, stability is not of immediate importance when short-term policy objectives are concerned. This is because even if a system is stable, this does not imply that it will reach the target(s) in finite time. However, given that the system is controllable, once the model-matching problem is solved, we know that there exists a set of feedback policy rules that will drive the system, in finite time, to the desired target point (see Petit 1990, chapter 4 for an extensive discussion on the relation between controllability and stability). Moreover, since our system is path controllable, the whole sequence of targets will be exactly tracked irrespective of whether the target points are stable or not. 3.4 Issues Regarding the Non-optimizing Framework One of the main characteristics of the approach set out in our paper is that we construct ‘handcrafted’ policy rules, i.e. policy rules that are not the result of an optimization process. Although the optimizing approach would have substantial merits as regards the trade-offs between different goals, we feel that the approach followed here can essentially encompass such considerations. Our reasoning is the following: the main

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characteristic of the model matching approach is that it amounts to the calculation of policy rules that will force the model at hand to behave exactly like a reference model that produces the desired target-values. Therefore, if the target values are the result of an optimization process then our model will exactly track the designated optimal path. Moreover, given that we obtain as a solution of class of appropriate policy rules, we can examine the trade-offs between using different rules in order to achieve the policy goals. With reference to the case of the Greek adjustment program mentioned in the introduction, the analogy is straightforward: given the (optimal) targets for the macro variables necessary for ensuring debt sustainability, do appropriate policy vectors exist and how should they be designed so that the targets are exactly reached? Both questions can be answered following the algorithmic model matching approach we utilize in this paper.

4 The Algorithmic Solution In order to design the appropriate policy rules, i.e. calculate the polynomials R, S and T , we developed two algorithmic procedures: the ‘Desired System’ algorithm and the ‘Income Matching’ algorithm. The ‘Desired System’ algorithm produces the relevant desired system for the model at hand (for a detailed presentation of such an algorithm see Kotsios and Leventidis 2004). The ‘Income Matching’ algorithm calculates the polynomials R, S and T so that when the linear, feedback policy rules of the form: Rθ (q) = Tθ (q)u c (t) − Sθ (q)Y (t)

(19)

where θ is a vector of parameters, are applied to the open-loop system (13) they will produce a closed-loop one that is identically equal, i.e. matched, to the desired system (14). The following theorem establishes this result (for a detailed proof see “Appendix 4”): Theorem 4 The non-trivial outputs Rθ , Sθ and Tθ of the ‘Income Matching’ algorithm solve Economic Policy Problem B (under the same initial conditions) by the feedback policy rule Rθ (q) = Tθ (q)u c (t) − Sθ (q)Y (t) where Rθ (q) is a monic polynomial, i.e. its constant term is equal to 1. We now present the formal “Income Matching” algorithm: Inputs: The parameters ν, s, τ, r , the delays n, m, the initial conditions Y (0), Y (1), B(0), B(1), . . ., the artificial input u c (t) and the reference sequences Y ∗ (t), B ∗ (t). Output: The polynomials Rθ , Sθ and Tθ . Step 1: Form the polynomials D11 (q) = 1 − a1 q − a2 q 2 − a3 q 3 , D12 (q) = 0, D21 (q) = τ q, D22 (q) = 1 − (1 +r )q, K 1 (q) = λ0 + λ1 q + λ2 q 2 , K 2 (q) = 1 where a1 = 1 + ν − s, a2 = s − ν − τ (1 − s), a3 = sτ . d (q) = Step 2: Using the “Desired System” algorithm construct the polynomials D11 n m d i i i=0 αi q , K 1 (q) = i=0 κi q such that the output of the resulting desired d d system D1 (q)Y (t) = K 1 (q)u c (t) is exactly equal to the reference sequence Y ∗ (t).

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Step 3: Find a family of polynomials Rθ , Sθ and Tθ , depending on the vector of parameters θ = (θ1 , θ2 , . . . , θk ) such that the following set of equations is satisfied: d Rθ (q)D11 (q) + Sθ (q)K 1 (q) = D11 (q)

K 1 (q)Tθ (q) = K 1d (q) The Rθ polynomial must be a monic. This family of polynomials has the general form: Rθ (q) = R0,θ (q) + K 1 (q)Q Sθ (q) = S0,θ (q) − D11 Q where Q is any arbitrary q-polynomial. Step 4: By assigning a specific value to the Q polynomial (usually, Q = 0), construct the feedback policy rule: Rθ (q)G(t) = Tθ (q)u c (t) − Sθ (q)Y (t) Step 5: Construct the corresponding closed-loop system for public debt: (Rθ (q)D21 q + K 2 (q)Sθ (q))B(t) = K 2 (q)Tθ (q)u c (t) Step 6: Calculate θˆ such that the error Bt∗ − Bt  is minimized. Remark 1 The vector of parameters θ consists of the αi , κi parameters used in the construction of the desired system (see step 2 of the algorithm). Therefore, depending on our choice of the delays n and m (the number of parameters of the desired system), we can have infinitely many different feedback laws as solutions. Moreover, the class of feedback laws might be augmented by choosing a non-zero Q polynomial in step 4 of the algorithm. This allows the policymaker to choose the most appropriate rule, depending on the nature of the particular policy problem. For example, he may opt for the law that provides the smoothest transition path for the policy instrument available. Remark 2 The dependence of the solution to the arbitrary q-polynomial Q indicates that there are going to be additional lags in the feedback rules for government expenditures, unless Q = 0. However, our aim is to have as a solution feedback policy rules with the least amount of lags (in control theory terms, we aim for some sort of minimum degree solutions). The economic interpretation of this decision is that, since feedback policy is designed on a quarterly basis, it is sensible to assume that the feedback rules should depend on relatively recent values of the policy instrument and policy target variables.

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5 Policy Experiments In this section we present some counterfactual policy experiments that will allow us to examine the effects of the proposed methodology for quarterly fiscal policy design on the system. In particular, our focus is on the Greek economy, because the design of fiscal policy in Greece since 2010 has come under heavy scrutiny, mainly due to the recession that many economists attribute to the adjustment program implemented. To this end, we estimate the multiplier–accelerator model presented in Sect. 2 over the period 1995Q1–2009Q4 (for details regarding the estimation see “Appendix 2”), just before the implementation of the first adjustment program, and we run some counterfactual policy experiments under different specifications of the model. In what follows, we assume that the tax-rate, τ , is equal to 0.4. 5.1 The Experiments We assume that the policymaker aims for a 1% per quarter growth in GDP levels, starting from an initial value of e 62722 million which corresponds to the 4th quarter of 2009 while, at the same time, ensuring that debt accumulation is minimized. Table 1 summarizes the target values for GDP regarding this policy experiment. In these experiments, we examined different values for the λi parameters in order to gain some insight into the effects of lags in fiscal policy implementation. Let us rewrite the mathematical expression for the lag structure: λ0 G(t) + λ1 G(t − 1) + λ2 G(t − 2), λ0 + λ1 + λ2 = 1, λi ∈ (0, 1) For example, if λ0 = 0.7 and λ2 = 0.2, then 70% of the government’s decision to spend will be disbursed in period t + 2, while only 20% will be immediately spent in period t. Moreover, we used different values for the interest rate, r , ranging from 4 to 6%, in order to examine the effects of the proposed policy plans in the accumulation of debt, when the cost of borrowing and, hence, repaying debt increases (for reasons of brevity, we only present the figures for the r = 4% case). Figure 1 presents the time path of government spending necessary for achieving the GDP growth targets, under different specifications for the λi parameters (values are in e millions). In particular, we distinguish between the following three cases: (i) The ‘Backloaded Policy’ scenario: here, we assume that λ0 = 0.7, λ1 = 0.2, λ2 = 0.1 i.e. 70% of the government’s decision to spend will be disbursed after two quarters; Table 1 GDP target values

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Time

GDP

2010Q1

63,349

2010Q2

63,989

2010Q3

64,622

2010Q4

65,269

Fiscal Policy Design in Greece in the Aftermath of the…

G Frontloaded

30 000

Even 25 000

Backloaded Actual

20 000

15 000

10 000 0

2

4

6

8

t

Fig. 1 Government spending in e mil

(ii) The ‘Frontloaded Policy’ scenario: here, we assume that λ0 = 0.1, λ1 = 0.2, λ2 = 0.7 i.e. 70% of the decision to spend is immediately realized into spending in period t; (iii) The ‘Even Policy’ scenario: here λ0 = 0.4, λ1 = 0.3, λ2 = 0.3 so that spending is evenly distributed in the following quarters It is evident from the figure that government spending needs to be increased (substantially in some cases) compared to the actual values in order to achieve GDP growth. The timing of the policy intervention plays a crucial role for the size of the expenditures necessary; the more frontloaded the policy is, i.e. the higher the value of λ2 , the smaller is the necessary increase. On the contrary, under the ‘backloaded’ policy scenario, a large increase is necessary, as can be seen from the red line in Fig. 1. Of course, this result is not new; a number of economists have argued for frontloaded stimulus spending in order to mitigate the downturn of the economy and ensure its return to a growth trajectory [see among others, DeLong and Summers (2012) and Romer (2012a, b)]. Moreover, our experiments indicate that, in all cases, the increases in government spending need to take place immediately for the GDP growth targets to be achieved, before they are lowered in subsequent quarters (yet, still higher compared to the actual ones under the adjustment program). Most importantly, under the ‘backloaded’ scenario, the transition path for government spending exhibits a large spike in period 4, while under the ‘frontloaded’ and, mainly, under the ‘even’ policy scenario, the time path is much smoother. Of equal importance is to examine whether the proposed policy plans manage to keep debt accumulation at a minimum or, at least, ensure that the resulting debt levels are on a sustainable path. The evolution of public debt under the different policy scenarios can be seen in Fig. 2 (debt levels are in e billions). Again, the ‘backloaded’ policy scenario is the one that exhibits the largest increases, whereas the ‘frontloaded’ and the ‘even’ policy scenarios provide much smoother transition paths. It is important to note here that the terminal values of public debt, when r = 4%, are such that combined with the 1% per quarter GDP growth lead to

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Debt 400

Frontloaded 380

Even 360

Backloaded 340

Actual 320 300 280 260 0

2

4

6

8

t

Fig. 2 Public debt in e bil

Table 2 Debt sustainability indicators

Table 3 Debt sustainability indicators, different levels of r

Scenario

Terminal debt value (in e billions)

B/Y

Interest payments (% of GDP)

Frontloaded

347.1

1.35

5.25

Backloaded

357.2

1.38

5.4

Even

342.0

1.33

5.2

Actual

330.3

1.46

5.9

r

5%

6%

Scenario

B/Y Interest payments (% of GDP)

B/Y Interest payments (% of GDP)

Frontloaded 1.4

6.7

1.47 8.3

Backloaded 1.44 6.9

1.5

Even

1.41 6.8

1.46 8.4

8.6

Actual

1.46 5.9

1.46 5.9

debt-to-GDP ratio that are lower, for all scenarios, compared to the actual one for 2010 (see Table 2). Another indicator regarding the sustainability of debt concerns the size of payments necessary for servicing public debt; under all the scenarios examined for r = 4%, the cost of servicing debt is lower compared to the actual one. We note here that running these experiments using higher values for the interest rate (up to 6%) essentially yielded the same results. In particular, even though the terminal values for public debt were higher (as expected), the cost of servicing debt as a percentage of GDP was in all cases below the 10% threshold recommended by the IMF. Moreover, only for r = 0.06 the respective values of the debt-to-GDP ratio exceeded the actual one (see Table 3).

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5.2 Some Considerations and Possible Extensions of the Model The results presented here, and the accompanying policy implications, are modelspecific. In particular, they are based on a simple multiplier–accelerator model that is confined to fiscal policy. Nonetheless, this model has the advantage that it is highly tractable, allowing us to trace the effects of the implementation of the proposed methodology. We are currently working on applying this methodology to more complex versions of the model and, in particular, nonlinear and stochastic versions. Regarding the nonlinear case, if we abandon the standard linearization approach, we are faced with the tracking of a nonlinear model; this is an open problem in the mathematical control theory literature, and thus we expect to obtain important mathematical and economic insights from such a model. For example, since nonlinear models exhibit multiple equilibria, we will see whether the policymaker is able to drive the economy from a ‘bad’ to a ‘good’ equilibrium point. Regarding the extension to the stochastic case, stochastic elements played a crucial role in shaping the trajectory of the Greek economy. We believe that incorporating such elements will not lead to important changes regarding the qualitative implications of the analysis, i.e. with regards to the fact that quarterly adjustments based on the feedback framework provide a smooth transition path [this has been demonstrated in the Kendrick and Shoukry (2014) paper]. However, we believe that the most fruitful extension will be that of applying active stochastic (or, dual) control techniques; the reason is that this framework allows for time-varying parameters (since the stochastic shocks will most likely alter the structure of the model) and, at the same time, allows us to examine the effects that current policy actions have on future states of the economy.

6 Concluding Remarks Our aim in this paper was to present a computational approach to the design of quarterly fiscal policy that is based on algorithmic linear feedback methods. To this end, relevant algorithmic procedures were developed and presented, which provide us with a family of fiscal policy rules appropriate for solving the problem at hand. The feedback framework, combined with quarterly policy interventions, ensures that the policymaker can act faster and on a more frequent basis over a given time horizon, thus being able to provide a smooth transition path back to a positive growth trajectory. The results from the counterfactual experiments indicate that for a country like Greece, suffering from a severe and prolonged economic downturn, short-term expansionary fiscal policy is capable of stimulating growth and keeping debt accumulation at a minimum level. When policy is frontloaded, the path of government expenditures is quite smooth (i.e. it does not exhibit big spikes), the increases in expenditures necessary are smaller and, as a result, debt accumulation is significantly lower.

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Appendix 1: State-Space and Algebraic Form The state-space form of the model is: x(t) = Ax(t − 1) + bG(t)

(20)

where A and b are: ⎡ 1+ν−s ⎢ 1 ⎢ ⎢ 0 ⎢ A=⎢ ⎢ −τ ⎣ 0 0

s − ν − τ (1 − s) 0 1 0 0 0

−sτ 0 0 0 0 0

0 0 0 1+r 0 0

λ1 0 0 0 0 1

⎡ ⎤ ⎤ λ0 λ2 ⎢0⎥ 0⎥ ⎢ ⎥ ⎥ ⎢ ⎥ 0⎥ ⎥, b = ⎢ 0 ⎥ ⎢1⎥ ⎥ 0⎥ ⎢ ⎥ ⎣1⎦ ⎦ 0 0 0 (21)

The polynomial matrices D(q) and K (q) are the following:

D11 (q) D12 (q) 1 − a1 q − a2 q 2 − a3 q 3 = D21 (q) D22 (q) τq



2 K 11 (q) λ + λ1 q + λ2 q = 0 K (q) = K 21 (q) 1 D(q) =

0 1 − (1 + r )q

(22)

Appendix 2: Model Estimation In the estimation of the consumption and investment functions we used data for GDP, consumption expenditure, investment and government expenditures downloaded from Eurostat (data updated on: 22/07/2015 and extracted on 18/08/2015) that span the 1995Q1–2009Q4 period, thus providing us with 60 observations. First of all, we need to test whether the variables are stationary; applying the standard ADF test, we conclude that all variables contain unit roots. Then, using the standard Johansen approach for detecting the number of cointegrating vectors and, in particular, based on the trace statistic, we see that amongst these 4 variables there are 3 cointegrating vectors. Thus, instead of using the VECM approach, we can alternatively use the two-stage least squares technique in order to estimate the simultaneous equation system. This is possible, because the order and rank conditions ensure that the system is overidentified. In order to ensure the robustness of our results, we calculate the bootstrapped standard errors. Regarding consumption, the estimation yielded: Cˆ t = 0.301 Yt−1 + 0.218Yt−2 + 0.166Yt−3 (0.0517)

(0.066)

(0.058)

(23)

Regarding investment, the estimation yielded: Iˆt = 0.8 (Yt−1 − Yt−2 ) (0.515)

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Fiscal Policy Design in Greece in the Aftermath of the…

Appendix 3: Proof of Theorem 3.2 In this appendix, we provide the proof for Theorem 3.2. The open-loop system is described by: D1 (q)Y (t) = K 1 (q)G(t) while the linear feedback policy rule is given by: R(q)G(t) = T (q)u c (t) − S(q)Y (t) Then, pre-multiplying Eq. (13) by R and taking into account that the R, K 1 polynomials are linear we have: R(q)D1 (q)Y (t) = R(q)K 1 (q)G(t) ⇔ R(q)D1 (q)Y (t) = K 1 (q)R(q)G(t) ⇔ R(q)D1 (q)Y (t) = K 1 (q)T (q)u c (t) − K 1 (q)S(q)Y (t) ⇔ (R(q)D1 (q) + K 1 (q)S(q))Y (t) = K 1 (q)T (q)u c (t) In order for the above closed-loop system to be identical to the desired one: D1d (q)Y ∗ (t) = K 1d (q)u c (t) the following set of equations must hold: R(q)D1 (q) + K 1 (q)S(q) = D1d (q) K 1 (q)T (q) = K 1d (q) This completes the proof.

Appendix 4: Proof of Theorem 4.1 In this appendix, we provide the proof for Theorem 4.1. First, we will prove that the Rθ , Sθ and Tθ q-polynomials produced by the algorithm are such that Y (t) = Y ∗ (t) and min Bt∗ − Bt . From step 3 of the algorithm we know that the linear feedback policy rule produces a closed-loop system identical to the desired one (compare with the proof of Theorem 3.1), thus ensuring Y (t) = Y ∗ (t). Then, from steps 5 and 6 of the algorithm, we know that the calculated policy rules are such that min Bt∗ − Bt . We are now going to prove the inverse problem. That is, given the open-loop system (13), the desired system (14), the q-polynomial R, S and T such that the set of equations (17) holds and Y (k) = Y ∗ (k), k = 0, 1, . . . t − 1 then we will prove that Y (t) = Y ∗ (t).

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Let d0 be the constant term of the polynomial D11 , i.e. D11 = d0 + D¯ 11 , where ¯ D11 is a polynomial without a constant term. Then: D11 (q)Y (t) = K 1 (q)G(t) ⇔ (d0 + D¯ 11 )Y (t) = K 1 (q)G(t) ⇔ Y (t) = d −1 (K 1 (q)G(t) − D¯ 11 Y (t))

(25)

0

¯ Then, Now, let r0 be the constant term of the polynomial R, so that: R = r0 + R. the feedback law becomes ¯ R(q)G(t)T (q)u c (t) − S(q)Y (t) ⇔ (r0 + R)G(t) = K 1 (q)G(t) −1 ¯ ⇔ u(t) = r (T (q)u c (t) − S(q)Y (t) − RG(t))

(26)

0

Substituting this back to (25), and dropping subscripts for ease of notation, we obtain: ¯ Y (t) = d0−1 [K 1 (r0−1 (T u c − SY − RG)) − D¯ 11 ] −1 −1 ¯ − D¯ 11 Y ] ⇔ Y (t) = d [r (K 1 T u c − K 1 SY − K 1 RG) 0

(27)

0

From the open-loop system (13), by pre-multiplying with R¯ we get: ¯ R¯ D11 Y (t) = R¯ K 1 G(t) ⇔ R¯ D11 Y (t) = K 1 RG(t)

(28)

Replacing this result back to (27) yields, combined with (14): Y (t) = d0−1 [r0−1 (K 1 T u c − K 1 SY − R¯ D11 Y ) − D¯ 11 Y ] ⇔ Y (t) = d −1 [r −1 (K 1d u c − D1d Y + R D11 Y − R¯ D11 Y ) − D¯ 11 Y ] 0

(29)

0

Finally, using the fact that Y (k) = Y ∗ (k), k = 0, 1, . . . t − 1, D11 Y d = D11 = d0 + D¯ 11 and R = r0 + R¯ we obtain:

K 1d u c ,

Y (t) = d0−1 [r0−1 (K 1d u c − D1d Y ∗ + r0 D1 Y ∗ + R¯ D11 Y ∗ − R¯ D11 Y ∗ ) − D¯ 11 Y ∗ ] ⇔ Y (t) = d −1 [r −1 (r0 d0 Y ∗ + r0 D¯ 11 Y ∗ ) − D¯ 11 Y ∗ ] ⇔ Y (t) =

0 0 −1 d0 (d0 Y ∗ )

⇔ Y (t) = Y ∗ (t) (30)

which completes the proof.

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