A note on the optimal speed of transition: Aghion and Blanchard revisited

Nævdal, Eric; Wagner, Martin Article A note on the optimal speed of transition: Aghion and Blanchard revisited German Economic Review (GER) Provided in Cooperation with: Verein für Socialpolitik / German Economic Association Suggested Citation: Nævdal, Eric; Wagner, Martin (2025) : A note on the optimal speed of transition: Aghion and Blanchard revisited, German Economic Review (GER), ISSN 1468-0475, De Gruyter, Berlin, Vol. 26, Iss. 1, pp. 1-14, https://doi.org/10.1515/ger-2024-0054 This Version is available at: https://hdl.handle.net/10419/331948 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ ger 2025; 26(1): 1–14 Eric Nævdal and Martin Wagner* A Note on the Optimal Speed of Transition: Aghion and Blanchard Revisited https://doi.org/10.1515/ger-2024-0054 Received May 21, 2024; accepted September 15, 2024; published online October 17, 2024 Abstract:This note illustrates, by reconsidering the seminal optimal speed-oftransition model of Aghion, P., and O. J. Blanchard. (1994. “On the Speed of Transition in Central Europe.” NBER Macroeconomics Annual 9: 283–319), that optimal transition paths, in general, exhibit nonlinearities and discontinuities. Aghion and Blanchard consider only an approximate solution with a constant unemployment rate over the transition process. The exact solution features an increasing unemployment rate with a discontinuity when the state sector is closed down at the optimally chosen endpoint of transition. Economic transition problems bear many similarities to scrap value problems with free terminal time, often encountered in resource economics. In relation to the transition to a green economy, the discussion in this note therefore casts doubt on the optimality of a green transition discussed in, e.g., the European Union in terms of politically specified rather than optimally designed milestones for emissions reductions, i.e., by −55 % compared to 1990 levels until 2030 and net zero until 2050. Keywords: dynamic optimization; end-of-transition; optimal unemployment rate; transition JEL Classification: C61; E61; P20 1 Introduction The history of economic development has repeatedly seen and continues to see fundamental transitions between partly drastically different economic (and political *Corresponding author: Martin Wagner, Department of Economics, University of Klagenfurt, Universitätsstrasse 65–67, 9020 Klagenfurt, Austria; Bank of Slovenia, Ljubljana, Slovenia; and Institute for Advanced Studies, Vienna, Austria, E-mail: [email protected].https://orcid.org/0000-00026123-4797 Eric Nævdal, HVL Business School, Western Norway University of Applied Sciences, Sogndal, Norway. https://orcid.org/0000-0002-7325-6811 Open Access. ©2024 the author(s), published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0 International License. 2—E. Nævdal and M. Wagner or technical) regimes. In this perspective, the first major economic transition was probably the transition from nomadic societies to agrarian, non-nomadic societies. Important major transitions in more recent centuries include, of course, the (first) industrial revolution and more recently, starting in the early 1990s, the transition to market economies in (most of) the former centrally planned communist countries. This process started with the demise of the Soviet Union and has brought quite rapid (not only) economic change to numerous countries.1However, history has not come to an end. Further transition processes are ongoing and can be expected to exert major impacts. These include, e.g., new waves of technological change labelled as “digitalization”, “industry 4.0” or “internet of things” (see, e.g. Brynjolfsson and McAfee 2011), which can be considered a 21st-century version of an industrial revolution. Another important transition process – very likely the most important transition of our time – is the transition towards an (essentially greenhouse gas) emissions-free economy necessary to limit the detrimental effects of anthropogenic climate change. This so-called green transition requires, in particular, fundamental changes in the production (and consumption) of energy, i.e., a replacement of carbon-based energy sources by carbon-neutral energy sources. In this process, energy usage will shift strongly towards electricity to be generated from emissionsfree sources. Clearly, the redesign of the global energy infrastructure and system (see, e.g., International Energy Agency 2023) will have profound impacts on all sectors and potentially also on the composition of the global economy.2 Notwithstanding the heterogeneity of the scopes and impacts of the transition processes mentioned, all these transitions necessitate or imply major reallocations of production factors, in particular also of labor, from old sectors to new sectors. The importance and magnitude of these processes makes an efficient design imperative. Conceptually, an optimal transition policy is hereby defined in terms of both an optimal speed of transition and an optimal endpoint or, equivalently, an optimal duration of a transition process. We illustrate these two dimensions by deriving in detail both the optimal speed as well as the optimal endpoint of a transition process by reconsidering the well-known speed-of-transition model of Aghion and Blanchard (1994) that deals with the transition from a centrally planned towards a marketbased economy. More specifically, Aghion and Blanchard (1994, Section 6.4) present a dynamic optimization model to determine the optimal speed of transition, which 1For a recent discussion concerning the partly ongoing transition processes from centrally planned to market economies, see, e.g., Dabrowski (2023). 2A pivotal contribution describing policy needs for combatting climate change is the report of Sir Nicholas Stern, see Stern (2007) and, for an assessment of the developments since the original publication, Stern (2015).Hassler, Krusell, and Smith (2016) provide an overview over macroeconomic modelling of climate change and resource scarcity. A Note on the Optimal Speed of Transition —3 in their model corresponds to finding the optimal path of the unemployment rate (see also the discussion in Roland 2000).3When solving the dynamic optimization problem, Aghion and Blanchard (1994) do not, however, derive the exact solution, but only an “approximate” solution that neglects the behavior of the economy after the state sector has been closed down. Due to the dynamic nature of the economy, however, the post-transition economic performance influences the optimal behavior already during the transition process and thus influences the optimal path also whilst the state sector still employs people.4In this respect, Aghion and Blanchard (1994, Footnote 33, p. 305) state that they are “cheating” by setting certain quantities constant, which they label “turnpike” approximation. The exact solution differs from the approximate solution in two related ways: First, the optimal unemployment rate is not constant, but increases over time and exhibits a discontinuity when the state sector is closed down at an optimally chosen endpoint of the transition. Second, we find a higher optimal unemployment rate than Aghion and Blanchard, which implies a shorter optimal duration of the transition process, i.e., an earlier endpoint. In the Aghion and Blanchard (1994) model, the inefficiency created by the approximate solution with a constant unemployment rate is that a too low unemployment rate reduces the rate of job creation in the new sector, which slows down output growth in this more productive sector and extends the duration of the transition process.5 The transition mechanism described here – or in Aghion and Blanchard (1994) – is conceptually closely related to important aspects (that need to be 3For a detailed discussion concerning labor market dynamics in transition economies see, e.g., Boeri (2000). Our contribution here is of a conceptual or methodological nature and, thus, several aspects of labor market dynamics that are found to be relevant from a labor economics perspective are neglected, as in the model of Aghion and Blanchard (1994). Also, of course, this type of model has to be interpreted in a stylized fashion with respect to the role of the state in an economy. In the model the state sector is closed down entirely, i.e., the non-trivial role of state sectors also in market economies is, for simplicity and to focus on one aspect, abstracted from. For the same reason, i.e., to focus on one aspect, we also abstract from reform uncertainty and potential reform reversal, issues discussed in Fernandez and Rodrik (1991) or Dewatripont and Roland (1995). 4The behavior of the economy at the point in time when the state sector is closed (and thereafter) is neglected also in other speed of transition models: Brixiova and Yousef (2000) assume a constant closure rate of the state sector, which may also lead to different dynamic behavior and welfare losses compared to optimal closure. Burda (1993) also finds a constant optimal unemployment rate, where again the effect of state sector closure is not analyzed in detail. Castanheira and Roland (2000) avoid the problem by assuming that there is no unemployment and that capital can be moved freely from the old to the new sector. 5As already mentioned, our analysis is of a merely conceptual and qualitative nature, but a rising unemployment rate during an optimal transition process increases the risk of costly reform reversals and backlashes. This is an issue that, however, cannot be addressed when considering a central planning solution only. 4—E. Nævdal and M. Wagner adequately detailed in fully specified models) of the other transition processes mentioned above. This is obvious, e.g., for the mentioned 21st-century industrial revolution, by simply replacing the terminology state and private sector with old and new sector. There are also close links to the green transition, when replacing the labels state and private sector with dirty and clean sector. In the context of the green transition, the government, once it internalizes the negative climate externality of the dirty sector (with a higher private but lower social marginal product than the clean sector), faces the problem of managing an optimal transition to an effectively carbon emissions free economy.6 This short paper is organized as follows: In Section 2, we set up and analyze the Aghion and Blanchard (1994) model in detail and Section 3 draws some conclusions. 2 The Aghion–Blanchard Model: Exact Solution for Normative Analysis We focus on the dynamic optimization problem used for a normative analysis of a transition process in Aghion and Blanchard (1994, Section 6.4) and present only those parts of the model presented in their paper in detail that are of immediate relevance here. Denote with E(t) the number of people employed in the state sector (with constant marginal productivity x), with N(t) the number of people employed in the emerging private sector (with constant marginal productivity y>x>0) and with U(t) the number of unemployed people at time t. Population is normalized to one, i.e., E(t)+N(t)+U(t)=1, which implies that U(t)isboththenumberofunemployed people and the unemployment rate. Aghion and Blanchard (1994) develop an efficiency wage-based explanation for costly labor adjustment between the old state sector and the new private sector. In particular, they derive the following 6The analogy to the green transition has to be considered more carefully also in terms of modelling stocks and flows: A key aspect to be included in climate-economy transition models is that they need to take into account the (uncertain) negative impacts of the accumulated stock of emissions on all sectors in the economy via a damage function of some sort, see, e.g., Hassler, Krusell, and Olovsson (2024) for a recent policy-oriented discussion. Furthermore, the different transitions listed are, of course, intertwined, with, e.g., the direction and speed of technical change not independent of climate policies, see, e.g., Hassler, Krusell, and Olovsson (2022). Furthermore, in the context of environmental problems there is a well-known discussion about instrument choice in environmentaleconomics(quantityconstraints,taxation,...). Whichever instrumentchosen,the result will be a shrinking of the dirty sector that has to be managed by choosing optimal policy paths. A Note on the Optimal Speed of Transition —5 relationship for the speed of job creation in the new private sector (see their equation (9) on page 298):7  N=f(U)=a[U U+ca][y−rc −(b 1−U)],(1) with a,b,cand rpositive constants. Here, aindicates the impact of per-worker profits in the private sector on the speed of private-sector job creation, bare unemployment benefits and cis a constant related to the “wage-premium”, derived using an efficiency-wage setting, that private firms are willing to pay (over and above unemployment benefits as outside option). Furthermore, ris the interest rate and the cost of job creation in the private sector is given by 1 2ar (f(U))2.8The government chooses the optimal speed of closure of the inefficient state sector and, thereby, unemployment. Remark 1. It may be interesting to discuss the building blocks leading to privatesector job creation as given in (1) in a bit more detail: This relationship is based on the combination of three modelling assumptions with an efficiency-wage consideration. First, Aghion and Blanchard (1994) assume that the rate of private-sector job creation is proportional to profits per worker, i.e.,  N=a(y−z−𝑤)withybeing the (constant) average product of labor in the private sector, ztaxes per worker, 𝑤 the private-sector wage and the proportionality constant a, compare (2) in Aghion and Blanchard (1994). Second, private-sector wages 𝑤depend upon labor market conditions as follows 𝑤=b+c(r+ N U), with unemployment benefits b,theinterest rate r, N Uthe ratio of new jobs to unemployment and a constant cthat scales the wage premium over unemployment benefits that firms are willing to pay, compare (3) in Aghion and Blanchard (1994).9The third element is that unemployment benefits are financed by labor taxes under the condition of a balanced budget, i.e., 7To avoid overloaded notation we sometimes skip the time index t. Furthermore,  N(t) denotes the derivative of N(t) with respect to t, with this notation also used for other variables. 8Another restriction on the parameters is y−b−rc >0. In this case, f(U) is positive for values of Ularger than zero and smaller than y−b−rc y−rc . Clearly, it cannot be optimal to consider unemployment paths that include values larger than this value with negative rates of job creation in the more productive private sector. 9The relationship for private-sector wages is developed in Aghion and Blanchard (1994) in an efficiency-wage framework under the assumptions that all hires are from unemployment and that once employed in the private sector there is no risk of future unemployment. Thus, the value of being unemployed, VU, is given, see (4) and (5) in Aghion and Blanchard (1994),by rVU= N U(VN−VU)+dVU dt ,withVNbeing the value of being employed in the private sector, itself given by rVN=𝑤+dVN dt . The efficiency-wage argument enters the considerations by postulating that private-sector firms will set a wage such that the VN=VU+c,forsomec≥0. This in turn implies dVN dt =dVU dt and the two equations given in this footnote can be easily rearranged – by simply taking the difference – to lead to the equation for private-sector wages given in the remark. 6—E. Nævdal and M. Wagner Ub =(1 −U)z,see(6) in Aghion and Blanchard (1994). The relationship (1) now follows from inserting 𝑤=b+c(r+ N U)andUb =(1 −U)zinto  N=a(y−z−𝑤). Aghion and Blanchard (1994, Section 3.1) assume that at the outset of transition, employment in the state sector drops from 1 to some E(0) =E0<1, which implies an initial unemployment rate equal to U(0) =1−E0. This value of U(0) will, in general, not correspond to the optimal choice of the unemployment path that maximizes the net present value of output. Consequently, the optimal unemployment path will have a discontinuity at t=0 and jump to the optimal value from U(0) immediately. Only, when the initial unemployment rate corresponds to the optimal choice will the optimal path of the unemployment rate be as illustrated below in Figure 1. Since our focus here is on the duration and end of a transition process, we abstract from the possibility of a discontinuity at t=0 by assuming that U(0) is optimally chosen as well, or equivalently that state sector employment drops to E(0) =1−U(0)∗,withU(0)∗denoting the optimal choice of initial unemployment. The government is only concerned with efficiency and chooses employment in the state sector to maximize the present discounted value of output. The government’s optimization problem is thus given by: max E(t) ∞ ∫ 0[E(t)x+N(t)y−1 2ar(f(U(t)))2]e−rtdt,(2) subject to:  N(t)=f(U(t)),(3) N(0) =0,(4) E(t)+N(t)+U(t)=1(5) and non-negativity of E(t), N(t)andU(t). Based on the identity E(t)+N(t)+U(t)=1, one immediately observes that the problem can equivalently be formulated using U(t) as control variable, thereby eliminating E(t), which leaves only U(t)andN(t) in both the objective function and the constraints.10 This equivalent formulation of the problem is given by: max U(t) ∞ ∫ 0[(1 −N(t)−U(t))x+N(t)y−1 2ar(f(U(t)))2]e−rtdt,(6) 10 We perform this substitution to have U(t), postulated to be constant along optimal paths by Aghion and Blanchard (1994), as the control variable. The benefit of this reformulation is that it allows us to highlight the differences between the approximate and the exact solutions. A Note on the Optimal Speed of Transition —7 subject to:  N(t)=f(U(t)),(7) N(0) =0,(8) N(t)∈[0,1],(9) N(t)+U(t)≤1(10) and non-negativity of U(t). Note first that an optimal, in fact any, path must necessarily fulfill exactly one of the following two properties: There exists a 𝜏<∞such that 𝜏=inft≥0(N(t)+ U(t)=1) or condition (10) is not binding for any finite t. These two cases will be discussed separately below. Before doing so, an important property of the model is derived in Proposition 1. Proposition 1. Along any path, it holds that N(t)<1for all t <∞. Proof: ForvaluesofN(t) sufficiently close to 1, the largest possible value of  N(t) is given by setting U(t)=1−N(t). The ordinary differential equation  N(t)=f(1 − N(t)) has a stable steady state at N=1, since f(0) =0anddf(1−N) dN =−f′(1 −N)<0 for N=1. Given that N(0) =0, it follows that N(t)<1fort<∞.□ Let us now investigate potential optimal paths, starting with the case that the constraint (10) becomes binding for the first time at some 𝜏<∞. Given that state sector employment is monotonically non-increasing, it follows that for t≥𝜏the control problem has a trivial optimal solution. Denote with N(t,N𝜏) the solution to the differential equation  N(t)=f(1 −N(t)), solved over (𝜏,∞), with initial condition N(𝜏)=N𝜏. Note next that it trivially holds that 𝜕N(𝜏,N𝜏) 𝜕N𝜏 =1 and also note that up to now both 𝜏and N𝜏are unspecified. The objective function of the optimization problem from 𝜏onwards is given by: V(𝜏,N𝜏)= ∞ ∫ 𝜏[N(t,N𝜏)y−1 2ar(f(1−N(t,N𝜏)))2]e−rtdt. (11) Note the following relationships for the partial derivatives of the objective function (11): 𝜕V(𝜏,N𝜏) 𝜕𝜏 =− [N(𝜏,N𝜏)y−1 2ar(f(1−N(𝜏,N𝜏)))2]e−r𝜏,(12) 8—E. Nævdal and M. Wagner 𝜕V(𝜏,N𝜏) 𝜕N𝜏 = ∞ ∫ 𝜏[y+1 ar f(1−N(t,N𝜏))f′(1−N(t,N𝜏))]e−rtdt =y re−r𝜏+ ∞ ∫ 𝜏[1 ar f(1−N(t,N𝜏))f′(1−N(t,N𝜏))]e−rtdt.(13) The optimization problem corresponding to the case considered can be rewritten as a scrap value problem with free terminal time, i.e., as a problem where 𝜏is to be chosen optimally as well: max U(t)∈[0,1],𝜏∈[0,∞)⎡⎢⎢⎣ 𝜏 ∫ 0[(1 −N(t)−U(t))x+N(t)y−1 2ar(f(U(t)))2]e−rtdt+V(𝜏,N𝜏)⎤⎥⎥⎦ , (14) subject to (7),(9) and (10). Problems of this type are studied in Seierstad and Sydsæter (1987,Theorem3 and Note 2, p. 182–184), which provide necessary conditions for optimality.11 The (current-value) Hamiltonian corresponding to this problem is given by H(N,U,𝜇,𝜏)=(1 −N−U)x+Ny−1 2ar (f(U))2+𝜇f(U),whereweignore,for brevity, the other constraints, (9) and (10), and the associated multipliers. It is straightforward but cumbersome to present the solution including these additional terms in the Lagrangean.12 Necessary conditions for optimality are given by: −x−1 ar f(U)f′(U)+𝜇f′(U)=0,(15) 𝜇 =r𝜇+x−y.(16) Furthermore, the following transversality condition has to hold: 𝜇(𝜏)e−r𝜏=𝜕V(𝜏,N𝜏) 𝜕N𝜏 .(17) 11 To be precise we arrive at this type of problem only after verifying that the additional constraints – see the following Footnote12 – are not binding. Problems with these additional constraints considered, i.e., with mixed and pure state constraints are discussed in Seierstad and Sydsæter (1987, Chapter 6). 12 We use the terminology of Seierstad and Sydsæter (1987) and refer to the Hamiltonian augmented by the additional constraints as Lagrangean. It can be shown that these constraints are not binding, except possibly at t=0andt=∞. More specifically, it can be shown that the only possible case where any other constraint than U(t)+N(t)≤1 is binding for t<∞is the case where U(0) =1, in which case 𝜏=0.