Estimating the economic impacts of power supply interruptions

Energy Economics 80 (2019) 983–994

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Energy Economics journal homepage: www.elsevier.com/locate/eneco

Estimating the economic impacts of power supply interruptions Vinícius Botelho

A R T I C L E

I N F O

Article history: Received 2 July 2018 Received in revised form 11 January 2019 Accepted 23 February 2019 Available online 3 March 2019 Keywords: Power interruption costs Energy policy evaluation General equilibrium Energy economics

A B S T R A C T The cost of power system rationing, which is a crucial parameter for determining optimal resilience investments, is usually estimated using reduced-form linear models or ordinary input-output analysis. However, such methods do not properly address either consumers’ rational reactions to rationing policy or policy design nonlinearities. To solve this problem, this paper estimates the effects of power system rationing using a general equilibrium model. The model solution shows that the power cut distribution among industries is a critical variable for quantifying policy effects and provides insights into optimal policy design. © 2019 Elsevier B.V. All rights reserved.

1. Introduction Even though the price mechanism is one of the most efficient ways to allocate resources under scarcity, in recent years, several countries have faced the risk of having to adopt energy rationing policies. Some years ago, even the United Kingdom was threatened by the possibility of having to ration natural gas. Many cities, such as Cape Town, are now facing the risk of severe water shortages. Usually the result of natural resources scarcity, shortages may be caused by climate change, natural disasters, or even terrorism. To avoid shortages, it is necessary to improve infrastructure resilience. As usual, optimal infrastructure resilience investment decisions depend on marginal costs (the cost of improving infrastructure) and returns. Since returns depend on the critical infrastructure failure cost and failure probability, they depend on counter-factual analysis.1 This dependence makes such calculations of the optimal investment levels an open question. Counter-factual analysis requires the comparison of economic variables when supply restrictions bind and when they do not. Both refer to the future, and both are uncertain. Because of the complexity of this task, several methods for addressing this problem have

E-mail address: [email protected]. For example, how much would government revenues drop if energy consumption fell 10% next year? Such a calculation depends on the difference between revenue levels next year when the policy is implemented and when it is not. Such a difference is a counter-factual. 1

https://doi.org/10.1016/j.eneco.2019.02.015

been developed, including survey responses,2 market behavior,3 and theoretical models.4 The focus of this study is to provide insight into rationing cost estimation methods that rely on theoretical models. These are typically based on linear approaches5 and input-output analysis.6 Such models often require assumptions that are much stronger than those used in modern economic theory applications, since they do not adequately incorporate economic agents’ rational behavior and use linear systems to estimate non-linear effects. Previous papers have already

2 See, for example, Caves et al. (1992), Beenstock et al. (1998), Carlsson and Martinsson (2008), and Baik et al. (2018a). 3 See, for example, Bental and Ravid (1982), Beenstock (1991), and Baik et al. (2018b). 4 See, for example, Tishler (1993). It is noteworthy that some studies employ more than one approach. For example, Bental and Ravid (1982) use theoretical foundations to link market behavior and power supply interruption costs. For further discussion, Sanghvi (1982) provides several examples of papers using all of these methodologies. 5 For linear models estimated using econometric methods, see Munasinghe and Gellerson (1979), Cheng et al. (2013), Carpio (2014), Harish et al. (2014), and Zaman et al. (2015). For linear models using VoLL or similar approaches, see Serra and Fierro (1997), de Nooij et al. (2007), de Nooji et al. (2009), Reichl et al. (2013), Wolf and Wenzel (2014), and Wolf and Wenzel (2015). For linear models estimated to evaluate the relationship between energy consumption and GDP (thus, as will be discussed, facing the same problems presented in this paper), see Masih and Masih (1998), Cheng (1999), Hatemi-J and Irandoust (2005), Khan et al. (2008), Gbadebo and Okonkwo (2009), Aslan and Kum (2010), Belke et al. (2011), Ahmad et al. (2012), Campo and Sarmiento (2013), Adhikari and Chen (2013). 6 For example, see Chen and Vella (1994), Santos and Haimes (2004), Lian and Haimes (2006), Anderson et al. (2007), Barker and Santos (2010b), Barker and Santos (2010a), Akhtar and Santos (2013), Jonkeren and Giannopoulos (2014), and Vasconcelos and Carpio (2015).

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acknowledged these flaws. For example, Rose and Liao (2005), Rose (2015), Sanstad (2016), Ou et al. (2016), Chen et al. (2017), Prager et al. (2018), and Wing and Rose (2018) presented some of the problems discussed in this paper and proposed general equilibrium models7 as a possible solution for them. In particular, Rose (2015), Sanstad (2016), Chen et al. (2017), Prager et al. (2018), and Wing and Rose (2018) explicitly acknowledge general equilibrium modeling is the state of the art approach to Economic Consequence Analysis, which means, in this case, general equilibrium models are the best approach to evaluate economic impacts of rationing policy. Nevertheless, Rose and Liao (2005) and Ou et al. (2016) have not adequately addressed the problems they presented. Their models assume that the consumption level of every consumer is always the maximum allowed by policy restrictions. However, the constraint imposed on each consumer affects all the others, so it is possible that some consumers optimally choose to consume less than their quota because of the restrictions imposed on others.8 In other words, the optimal consumption levels used in those papers do not take into account the fact rationing restrictions may not bind for everyone. Consequently, what those papers consider as an optimal response may not be optimal. Thus, even though they are correct about the fact that general equilibrium models can solve many of the internal consistency flaws of usual rationing models, they did not derive the general equilibrium solution under rationing correctly. In fact, the correct optimal consumption levels under consumption rationing are much more difficult to calculate: in an economy with 55 industries, in principle, estimating the effects of rationing policy requires 255 simulations, since all binding possibilities must be tested for all industries. In this paper, a methodology to estimate the economic impacts of rationing policy without the need for such a large number of simulations is presented. Furthermore, the Rose and Liao (2005) approach requires calibrating many more parameters than required for input-output methods, and estimates for many of these parameters are not easily available (such as all economic activity sectors’ substitution elasticities for capital, energy, water, labor, imports, and exports). Since inputoutput matrices are handy, in spite of its well-known fragilities, the input-output method remains very popular. Additionally, the Ou et al. (2016) equilibrium is not walrasian, the mainstream of general equilibrium models in economics. Finally, Rose and Liao (2005) do not clearly state the optimization problems of main economic agents. In summary, both methods fail to adequately use ordinary economic policy evaluation tools to evaluate energy policy design. More recent approaches, such as Chen et al. (2017), Prager et al. (2018), and Chen and Rose (2018), discuss the application of general equilibrium models to estimate the economic impacts of terrorist attacks and other threats. Such events are similar to power supply interruptions in the sense both can be interpreted as physical constraints on production levels. These works estimate linear reduced-form equations to explain general equilibrium results using data from artificial shock simulations. However, as will be discussed throughout this paper, there are many situations in which such

7 For further references regarding general equilibrium models, see Arrow and Debreu (1954), McKenzie (1954), and Debreu (1959). For textbook approaches, see Mas-Collel et al. (1995), Woodford (2003), Ljungqvist and Sargent (2004), and Galí (2008). 8 When power supply cuts are not homogeneously distributed among industries, such concerns are even more relevant. Considering the evidence from Serra and Fierro (1997), de Nooij et al. (2007), de Nooji et al. (2009), Wolf and Wenzel (2014), and Wolf and Wenzel (2015) that the rationing distribution can significantly change rationing costs, methodologies for estimation of heterogeneous rationing policy effects are crucial for effective cost minimization. In such situations, assuming everyone would consume exactly the maximum possible consumption underestimates the rationing cost.

linearity does not hold, and a structural derivation of the shocks can provide a better understanding of the phenomena involved. Not only the use of economic tools to analyze energy policy evolved during the last years, but economic tools for policy analysis have also become much more robust and implementable over time. At present, most general equilibrium models in economics are DSGEs (Dynamic Stochastic General Equilibrium Models), and the walrasian equilibrium is their conceptual basis. DSGEs are an extension of the CGE (Computable General Equilibrium) that incorporates the evolution of variables over time and uncertainty. For examples of the broad applications of DSGEs, several central banks around the world currently use them for economic policy evaluation, such as the US (SIGMA), Canada (ToTEM), Japan (JEM), England (BEQM), Chile (MAS), Peru (MEGA-D), Norway (NEMO), Swiss (RAMSES), and Brazil (SAMBA). International institutions, such as the IMF (GIMF), also have their own DSGE models.9 Solution, calibration, and estimation of these models are widely performed by economic analysts all over the world, so the ‘C’, meaning ‘computable’, became obsolete in the modern terminology for general equilibrium applications.10 To adequately address all the issues highlighted above, this paper builds a general equilibrium model (based on mainstream economic theory principles) that can be calibrated using only input-output tables and national accounts data. By doing so, it calculates the cost of rationing policy taking into account the rational behavior of firms and consumers. Such an approach provides a practical and methodological discussion of the nonlinearities of imposing restrictions on the consumption of certain goods. The paper is organized as follows: after the Introduction, there is a theoretical and empirical critique of current rationing cost estimation methodologies. Subsequently, the third section presents a basic general equilibrium model, incorporating rationing policy. In the fourth section, the model results, calibrated for Brazilian data, are analyzed. The fifth section concludes. 2. Methodological critiques 2.1. First critique: linearity As Serra and Fierro (1997), de Nooij et al. (2007), de Nooji et al. (2009), Wolf and Wenzel (2014), and Wolf and Wenzel (2015) have previously shown, the economic impact of a power supply cut depends not only on its magnitude but also on its design. In other words, the rationing distribution among consumers matters significantly for policy impact evaluation. In such cases, rationing impact estimation using only total power cut information cannot adequately evaluate how different rationing designs could affect rationing costs, since the same total power cut could imply different impacts depending on its distribution among industries. Estimating linear models for each industry would not solve this problem because sectoral interdependency makes it difficult to determine, a priori, the industries that rationing policy would bind. For example, suppose there are only two industries in the economy: A and B. Additionally, wi is total power cut (in percentage points) required for industry i. pY is the proportion of total output produced by industry A, whereas 1 − pY is the proportion of total output produced by industry B. pW is the proportion of total energy consumption consumed by industry A, whereas 1 − pW is the same proportion consumed by B. Energy-output elasticity is one for all

9 For other examples of DSGE models applications, see Smets and Wouters (2003), Christiano et al. (2005), and Smets and Wouters (2007), in addition to the textbook references previously cited: Mas-Collel et al. (1995), Woodford (2003), Ljungqvist and Sargent (2004), and Galí (2008). 10 For further references concerning the historical evolution of general equilibrium models in economics, see Mankiw (2006), Goodfriend (2007), and Galí and Gertler (2007).

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industries, and c is the proportion of demand for industry B goods required for industry A production. yi is output reduction for industry i after the power cut (Y = pY ya + (1 − pY )yb ). Therefore, ya = wa and yb = max(wb , cwa ). Hence, wa , wb and yb do not obey a linear relation. For example, if wb = 0, then ya = wa and yb = cwa ; if wa = 0, then ya = 0 and yb = wb . The blue line in Fig. 1 represents combinations of wa and wb that have the same impact on Y. In Fig. 1, the more distant from the origin is the blue line, the greater is the impact of the power cut. Hence, in Fig. 2, the economic impact of wa and wb combinations lying on the blue line is greater than the impact of wa and wb combinations lying on the red line. Suppose that pW , pY , and c are known parameters and a WTARGET power cut is needed. Additionally, suppose that the policymaker wants to minimize the impact of rationing on Y. The government must choose wa and wb to define the rationing policy. Then, the policymaker problem can be written as Eq. (1). Fig. 2. Two curves of wa and wb combinations yielding different impacts on W. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

min Y(wa , wb )

wa ,wb

W

TARGET

W



W



s.t.

= W = p wa + 1 − p max(wb , cwa )   Y Y Y = p wa + 1 − p max(wb , cwa )

makes WEST and YEST , the impact estimates, diverge from W and Y, the true impacts. (1) min Y EST (wa , wb )

wa ,wb

Graphically, the solution of problem (1) can be seen in Fig. 3. Each red line represents a set of combinations of wa and wb that yield the same impact on Y, whereas the blue line represents combinations of wa and wb such that W = WTARGET . It is easy to see that the solution for problem (1) lies on the blue line. The farther the red line is from the origin, the greater the impact that the wa and wb combination has on Y. Therefore, solution to problem (1) is the intersection between the blue and red lines in which the red line is as close as possible to the origin. In Fig. 3, one such point is (wa = 0.1, wb = 0.4). However, if the government believes the relationship between wa , wb , W, and Y is linear, the policymaker will implement a different wa and wb combination. The linearity belief can be expressed by W the equations WEST = W when W EST = mW a wa + mb wb , with EST W W ma and mb being fixed coefficients, and Y = Y when Y EST = mYa wa +mYb wb , with mYa and mYb being fixed coefficients. In such cases, problem (1) is seen by the policymaker as problem (2). This change

s.t. W TARGET = W EST Y EST = mYa wa + mYb wb W W EST = mW a wa + mb wb

(2)

The solution of Eq. (2) is quite straightforward, almost always mY mW a corner solution. If aY > aW , then only industry B should suffer the rationing. If

mYa mYb

mb

the rationing. Only if

< mYa mYb

mb mW a , mW b

=

then only industry A should suffer mW a mW b

, a very unlikely situation, are all

combinations of wa and wb optimal.11 Nevertheless, as shown in Fig. 3, the real optimal solution is not necessarily a corner one. Therefore, by linearizing effects, one loses much insight into the optimization problem and reaches false conclusions about optimal rationing design. Further assumptions about parameter values provide a better understanding of the linearization problems discussed in this section. Supposing that any combinations of wa and wb that come from a linearization of W and Y (such as WEST and YEST ) would have at least one correct value,12 and since pW , pY , and c are known parameters, it is possible to determine upper and lower bounds for WEST and YEST , as stated by Eq. (3).        pW wa + 1 − pW wb ≤ W EST ≤ pW + c 1 − pW wa + 1 − pW wb        pY wa + 1 − pY wb ≤ Y EST ≤ pY + c 1 − pY wa + 1 − pY wb (3)  Supposing that WEST is close to its upper bounds, WEST = pW +     (pW +c(1−pW )) , W EST c 1 − pW wa + 1 − pW wb ; then, wb = 1−p W − wa 1−pW and consequently, the policymaker would consider equivalent all wa

Fig. 1. Curve of wa and wb combinations yielding the same impact on W. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

11 In fact, several rationing cost estimation methods rely on corner solutions to determine optimal rationing policy, such as the methodology of the official Brazilian power supply interruption cost estimate. 12 For at least one wa and wb , WEST = W. For at least one wa and wb , YEST = Y.

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Fig. 3. Optimal wa and wb . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

EST and wb combinations on the red line in Fig. 4. Conversely,   if W EST W W is close to its lower bounds, W = p wa + 1 − p wb , then EST

W

p W wb = 1−p W − wa 1−pW , and consequently, the policymaker would consider equivalent all wa and wb combinations on the green line in Fig. 4. In practice, the estimation procedure will make the policymaker mistakenly believe lines between the green and red lines are possible combinations of wa and wb that yield similar impacts on W. W Estimation of mW a and mb using common estimation methods would result in one line between these two. The same would happen if we analyze YEST . In Fig. 4, most of the suggested combinations for wa and wb on the red cut smaller than what is neces line would cause a power  sary W ≤ WTARGET = WEST : only extreme solutions (wa = 0 or wb = 0) would result in combinations of wa and wb that correctly estimate  the total power cut that would come after wa and wb restrictions WTARGET = WEST = W only if wa = 0 or wb = 0 . Similarly, most combinations of wa and wb on the green line gen erate a power cut greater than necessary W ≥ WEST = WTARGET .  If c = 0, the blue and red lines coincide WEST = W for all wa  and wb ; conversely, the green and blue lines only coincide when c = 1. Only in these two extreme cases is it possible for a linear model to correctly estimate the effects of a wa and wb combination

Fig. 4. Indifference curves and policy options. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

on W. In all of the other cases, the linear assumption either underestimates or overestimates the policy impacts of at least one wa and wb combination. Therefore, ordinary input-output analysis is not suited to rationing policy evaluation.13 Needless to say, the linearity assumption also compromises the proper identification and estimation of p and c, making policy recommendations based on linear models even more fragile than this section demonstrates. All in all, linear models based on econometric estimates or VoLL do not adequately evaluate rationing policy effects. Sectoral interdependency makes the impact of imposing a 10% rationing on industries A and B not equal to the sum of the impacts of imposing a 10% rationing constraint on each industry separately. Therefore, the impact of rationing policies is not linear and cannot be understood from industry regression estimates that do not take into account general equilibrium issues. For this reason, estimation methods based on VoLL, such as the one implemented by de Nooji et al. (2009), have serious shortcomings. Since the type of nonlinearity involved in this case (i.e., guessing which constraints are active) is well-defined and solvable, there is no need to make such mistakes. 2.2. Second critique: absence of rationality The economy is usually divided into firms and consumers. In this case, rationality implies that firms maximize profits and consumers maximize utility, regardless of the rationing policies implemented. Thus, the rationality critique has two arguments. First, derivation of reduced-form equations for consumption and production decisions from firms and consumers optimization problems helps to ensure all relevant variables are consistently included in the estimated and simulated economic model. Second, derivation of optimal response to rationing policy helps to understand how the set of relevant variables for consumption and production decisions, in addition to the relation between them, change after policy is implemented. Without performing such an exercise, the estimated models may suffer structural breaks after the rationing policy is implemented, becoming useless for ex-ante policy evaluation. The explicit optimization of utility and profits helps to ensure that the relevant variables in the decision-making process of consumers and firms are correctly incorporated into the estimated reducedform equation that aims to determine the economy consumption and production levels. When using the most common procedure for estimating economic rationing models in the energy policy literature (ordinary least squares), omitted variables, if correlated with the explanatory variables, may cause bias in coefficient estimates. Therefore, such omission must be avoided at all costs, and optimization of objective functions is a means to determine if there are variables that are too relevant to consumers or firms to be left aside. However, there will always be a set of variables relevant to the decision-making process that are not observable. It is also very common that they are correlated with the model explanatory variables. Such a situation takes us back to the bias problem reported in the previous paragraph. To solve such a problem, we need to change the estimation procedure: the estimator that corrects estimation biases in this situation depends on instrumental variables, such as GMM (General Method of Moments) or 2SLS (Two-Stages Least Squares).14 In particular, for macroeconomic models, typically thought of as in general equilibrium, all variables depend on each other. In such situation, the only possible instruments for parameter estimation are lagged model variables. The difficulties discussed apply to the estimation of consumers and firms reaction functions in regular times, when there is no

13 14

Typically, the input-output matrix elements lie between 0 and 1. See Hamilton (1994) for further details.

V. Botelho / Energy Economics 80 (2019) 983–994

rationing policy. Since rationing policy is an additional constraint imposed on the optimization of agents’ objective functions, the optimal response to it often changes not only agents’ decisions but the manners in which they decide. Therefore, when not taking into account rationality, economic models for firms’ and consumers’ demand, even when correctly specified for periods in which there is no energy rationing, may suffer structural breaks after the rationing policy begins. Needless to say, a model that suffers a structural break because of an event cannot be used to evaluate the impacts of such an event on the economy.15 All in all, it is very difficult to estimate consumers and firms reaction functions, and such difficulty increases significantly when the purpose of such estimation involves the evaluation of policy shocks that change agents’ optimization solutions. Not taking into account that agents are rational may significantly mislead policy evaluations. Going back to the specific case of energy rationing models, they usually assume prices are never affected by power supply restrictions.16 Such a fact is only compatible with rational behavior when markets are competitive and the elasticity of substitution between inputs is zero:17 there is no other combination of inputs that would produce the same output level efficiently.18 In other words, this assumption means that all firms employ exactly the same technology and that this is the only technology that works. This is too strong an assumption to rely upon. For instance, no company in the world is capable of replacing machine work with human work. For consumers, the situation is similar because these models typically build the counter-factual scenario assuming that the final demand under rationing would be equal to its level before rationing. There is no guarantee that the demand would remain at such a level, even if rationing policy were not implemented. Moreover, maybe policy is needed because of excess demand, and keeping the demand constant in the counter-factual scenario would lead to significantly overestimating the rationing costs. Additionally, when residential energy consumption levels are lowered, total household expenditure decreases. Such decreases may cause demand for other goods to rise. Therefore, utility functions are needed to understand how consumers would optimally reallocate the money they previously spent on energy consumption. Altogether, since the demands of consumers and firms depend on non-observable variables that are related to other observable and relevant variables, the estimation of reduced-form impact models should only be performed using instrumental variables. Agents’ rationality makes this problem much worse since all economic variables are simultaneously determined in equilibrium, in such a manner that each variable depends on all others, including the nonobservable ones. Therefore, not only should models clearly depend only on truly behavioral parameters but also any estimation procedure must take into account potential biases from non-observable variables affected by the policy being evaluated or by other relevant variables in the system. Even when all these conditions are met and the parameters are correctly estimated, rationality may change the manner in which agents decide and change their reaction functions, causing unanticipated structural breaks in the reduced-form model that mislead the policymaker trying to evaluate the impacts of several policy possibilities. In other words, economic agents do not react to energy rationing policy the same manner in which they behaved

15

For further details about this issue, see Lucas (1976). For example, input-output models implicitly make such an assumption. Jonkeren and Giannopoulos (2014) point out the price mechanism can be a resilience measure since it can adjust the supply and demand after a supply shock. Therefore, ignoring price adjustments would tend to overestimate the cost of the rationing policy. 17 As Bental and Ravid (1982) point out, such an approach can cause distortions that suggest that the least electricity-intensive sectors have the largest loss per unit of reduced supply. 18 I.e., without wasting resources. 16

987

before the policy was implemented. Only general equilibrium models offer a solution that naturally addresses these problems. 2.3. Case study: the statistical approach ESt represents the energy supply (MWh). GDPt represents the overall gross domestic product (currency units). Then, the cost of an energy rationing in a power system (ct , in currency units per MWh) can be expressed by Eq. (4). ct DESt1 = DGDPt

(4)

If the elasticity between the energy supply and GDP (e) is a fixed parameter, it is also possible to write the relationship between these two variables using Eq. (5), in which ut is an exogenous shock. ln(GDPt ) = e ln(ESt ) + ut

(5)

By taking derivatives in Eq. (5), it is straightforward to establish a relation between ct and e, as stated in Eq. (6). In other words, the energy cost of power supply rationing can be understood as the product of the elasticity between GDP and power supply and the nominal GDP per MWh. ct = e

GDPt ESt

(6)

Eq. (5) is a reduced form that is easily estimated by OLS. In such situations, for e to represent the elasticity between GDP and energy supply, it is necessary that E(ln(ESt )ut ) = 0. However, this is not the case: there are omitted variables in Eq. (5), captured by ut , that are correlated with a rationing policy shock (ln(ESt )) and as a result cause an omitted variable bias that derails the interpretation of e as a true elasticity. For example, technological shocks that increase energy efficiency use are accounted in ut and possibly correlated with ESt , thus compromising the identification of e. Moreover, different rationing designs with same overall intensity may have different impacts on GDP. Therefore, e probably does not even exist as a stable parameter. From July 2001 to February 2002, Brazil faced a power system rationing, providing a means to empirically test this approach. Energy consumption in this period fell 16.7% compared to the same period in the previous year. Fig. 5 shows the real-time estimated values for e,19 using data from Brazilian quarterly national accounts. The y-axis shows the elasticity estimated from the first quarter of 1996 until the quarter indicated in the x-axis. It is remarkable that between the second and third quarters of 2001, e increased from 0.50 to 0.11; the elasticity coefficient changed as soon as the rationing program started.20 For all of the reasons discussed above, the Brazilian energy rationing caused a structural break in the elasticity between GDP and power supply. Such a structural break occurred because consumers and firms adjusted their behavior after the rationing policy was implemented, and this case is a practical example for how models that do not adequately address this problem may fail to estimate impacts reasonably. If GDP power supply elasticity on 2001 were that indicated by econometric models before the energy rationing (0.50), the rationing policy implemented would have reduced GDP by 8.4%. Since Brazil grew 0.7% during the four quarters after implementing the rationing

19 20

These were obtained using ordinary least squares. It started in July.

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V. Botelho / Energy Economics 80 (2019) 983–994

budgetary restrictions, as stated by Eq. (7). In doing so, she chooses consumption (c) levels. max ln(c) {c}

s.t. W ≥ pc

Fig. 5. Elasticity between energy production and GDP real-time estimate.

policy, the elasticity of 0.5 indicates that without such a restriction,21 the country would have grown 9.1% (8.4 % + 0.7 %), which is an absurd result.22 The elasticity estimated using data after the policy was implemented suggests that the rationing policy caused a decrease of only 1.8% on GDP, which is a much more reasonable result (such elasticity indicates that the country would have grown 2.5% if there were no energy rationing policy implemented). However, how can a single equation for the GDP power supply elasticity reliably estimate the cost of a rationing in the power supply if the rationing policy causes the elasticity to change? There is no guarantee that such elasticity would remain constant if another rationing policy were implemented. Moreover, this paper shows that it probably would not because the impact of a rationing policy depends not only on the rationing magnitude but also on the rationing design. Therefore, historical patterns must be interpreted using consistent economic theory, not only reduced forms without theoretical foundations.

3. The general equilibrium approach The model represents a closed static economy with no information frictions and no uncertainty. There is a unit-mass continuum of consumers whose collective behavior can be emulated by a representative consumer. Additionally, there are n types of goods, each exclusively produced by a different industry. Industries are composed of a set of unit-mass with identical firms producing goods that may be used for final consumption or as intermediate inputs for other firms’ production. All problems are stated in terms of the representative consumer or representative firm. Subject to a budgetary restriction, consumers choose consumption levels maximizing an utility function; similarly, firms maximize their profits. Under a rationing policy, households and firms face an additional constraint to their optimization plans: their power consumption has an upper bound. In this fashion, nonlinearities due to sectoral interdependency are naturally accounted for. Then, the effects of a rationing policy can be calculated by comparing the economy under a rationing policy with the same economy when such a policy is not implemented. Policy effects on any of the modeled variables can be measured. 3.0.1. Households Each period, taking wages (W) and consumption prices (p) as given, the representative consumer maximizes utility, respecting

(7)

The economy is composed of n sectors, each producing one type of good. The elasticity of substitution for consumption between these goods is assumed to be one; therefore, the aggregate consumption (c) canbe expressed  by a Cobb-Doublas aggregator, as stated by n  Eq. (8) bi = 1 . i=1

c=

n 

Ci b i

(8)

i=1

3.0.2. Firms Goods produced by any firm are perfect substitutes for those produced by other firms in the same sector. Further, each firm is atomistic in its own market; there are no entry costs or special market regulations. Therefore, firms operate under perfect competition. Furthermore, firms in each sector have identical technologies: Cobb-Douglas with constant returns of scale. Labor is freely allocated between sectors in a perfectly competitive labor market. Therefore, prices equal marginal costs. Since returns of scale are constant, marginal costs do not depend on the level of production Xi , only on the prices of each intermediate input. The equation for marginal costs can be derived from the solution of the cost minimization problem (9).

 min j n ,Hi Xi

Hi W +

n

j

Pj Xi

j=1

j=1

s.t. Xi = H i

aiH

n 

jai

j

Xi

(9)

j=1

Since returns are constant to scale, Eq. (10) must be true for all sectors. n

j

ai + aiH = 1

(10)

j=1

3.0.3. Market clearing conditions In equilibrium, supply and demand must be identical. Therefore, conditions (11) and (12) must be simultaneously satisfied. As implied by the consumers’ budgetary restriction, the economy has an endowment of one unit of labor production.

Xi =

n

Xji + Ci , for all i

(11)

j=1 21 Assuming energy production levels would have stagnated in the absence of energy consumption restrictions. 22 Brazilian highest pace of growth during the last 20 years was 7.5%, in 2010. The average growth is 2.4%.

n

i=1

Hi = 1

(12)

V. Botelho / Energy Economics 80 (2019) 983–994

3.0.4. Competitive equilibrium The competitive equilibrium is defined as a final consumption vector C, an aggregate consumption level (c), a production vector X, a sectoral labor allocation vector (H), a wage (W), and a price vector P, such that market-clearing conditions are met, consumers maximize utility (Eqs. (7) and (8)), and prices equal marginal production costs, optimally derived from the solution of the problem stated in Eq. (9).23 3.1. The energy rationing approach The energy rationing design can be defined as a vector N24 containing the maximum energy consumption levels allowed for consumers (nn+1 ) and representative firms on each sector ({ni }ni=1 ). Therefore, the maximum energy consumption level for the i-th representative firm is given by ni and the representative firm problem (originally stated in Eq. (9)) assumes the form stated in Eq. (13).25

 min j n ,Hi Xi

Hi W +

n

j

Pj Xi

j=1

j=1

s.t. H

X i = H i ai

n 

j

a j i

Xi

Xin

(13)

A similar structure can be applied for the representative consumer problem, as stated in Eq. (14). max ln(c) {{c}}

s.t. W ≥ pc nn+1 ≥ Cn

(14)

Given di (di > 0), the shadow price of the energy rationing restriction, and the fact that prices equal marginal costs under perfect competition, Eq. (15) determines market prices for industry i.  Pi =

W aiH

a H i

n−1 



Pj j

j=1

ai

aj  i

Pn + di ain

⎛ ⎞⎤ a n ⎡ n−1 i

j ⎣1 + di ⎝ a ⎠⎦ i

(15)

j=1

When there is no energy rationing (di = 0, for all i), the solution of Eq. (15) is quite straightforward: W can be chosen as the numeraire,26 and the equations represented by Eq. (15) can determine P = {Pi }ni=1 .27 However, when there is energy rationing, di > 0 for at least one i, and Eq. (15) cannot be easily written in linear form anymore. Nevertheless, if instead of defining W as the numeraire, we define Pn as so (Pn = 1), it becomes simple to solve the model and obtain the corresponding equilibrium using linear system solution methods for any di ≥ 0. Then, comparison of both equilibria (with

C = {Ci }ni=1 ; X = {Xi }ni=1 ; H = {Hi }ni=1 ; P = {Pi }ni=1 . N = {ni }n+1 , where nn+1 is the rationing policy for consumers. i=1 25 Without loss of generality, suppose that the energy sector is the n-th sector. 26 Without loss of generality, in a walrasian equilibrium, one can choose any positive value for one price, since only relative prices matter for the equilibrium determination. 27 Using logarithms, the solution for Eq. (15) can be found using linear methods. 23

and without rationing policy) would give the impact of rationing on any variable.28 The outlined solution strategy requires the identification of D.29 However, it is very difficult to map D from N. After all, from N, it is not possible to infer, without simulating the model, which N restrictions are active. To better understand the argument, suppose that whereas a great consumption restriction is imposed on sector 1, on sector 2, only a very small energy consumption restriction is imposed (i.e., X1n >>> n1 and X2n > n2 ). Since sector 1 production requires intermediate goods from sector 2, n1 affects sector 2 energy  consumption. Hence, since X2n is the energy consumption of sector 2  when only n1 is active, it is possible that n2 > X2n . In this situation, the indirect effect on sector 2 of restricting sector 1 consumption is greater than the constraint imposed on sector 2. Therefore, even though there is a rationing target for sector 2, it does not bind. As previously discussed, to evaluate whether a constraint binds, it is necessary to know the indirect impacts of the restrictions imposed on all other sectors, which requires simulating the model for all possible binding combinations. Fortunately, setting N from D is quite easy. Additionally, solving the model for D is also quite straightforward. Therefore, the model can be solved for several D; subsequently, the N closest to the one desired by the policymaker can be found. Such a procedure would allow the evaluation of several heterogeneous rationing procedures without having to implement binding checks. 3.2. Calibration and model fit

j=1

ni ≥

989

Input-output tables are usually considered as matrices containing Leontief production function parameters. However, they can be used to calibrate Cobb-Douglas production functions, also. Assume that A is an interdependency matrix, and ai,j is the element in row i and column j of matrix A. qi is the production of industry i, and qi,j is the intermediate demand of industry j for industry i s goods. pi is the price for industry i s goods. Then, the theoretical Leontief technical coefficients (bi,j ) are defined by Eq. (16). bi,j =

qi,j qj

(16)

Because of measurement restrictions, such theoretical coefficients are not available. In practice, input-output tables are estimated using Eq. (17). Therefore, the estimated interdependency matrices are contaminated by relative prices. If the production functions are Leontief, bi,j is a structural parameter, and ai,j will fluctuate depending on relative price changes. However, if the production functions are Cobb-Douglas, ai,j is a structural parameter. In fact, input-output tables are much more reliable if the production functions are CobbDouglas since in such situations, relative price changes do not affect the model parameters. ai,j =

pi qi,j pj qj

(17) j

Hence, the ai coefficients were calibrated using the Brazilian input-output table  released  by Brazil’s official statistics institute j (IBGE) for 2010 ai = ai,j . aiH and bi were calibrated using 2013 makes and uses tables. The model solution without rationing restrictions can be used to assess whether the model fits the data well. To do so, we can compare the industry production (Fig. 6) and earnings (Fig. 7) with actual

24

28 The impact can be calculated for all model variables: GDP, wages, sectoral production, etc. 29 D = {di }n+1 . i=1

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Fig. 6. Proportion of total output (local currency) produced by each industry.

Fig. 7. Proportion of earnings per industry (local currency).

data. The correlation between the data and model industry output (in currency units) is 90.7%, whereas the correlation between the distribution of data and model earnings is 72.7%. Both correlations suggest that the model adequately reproduces the behavior of the Brazilian economy.

3.4. Energy policy applications

3.3. Shadow price simulation Consumption restrictions can be imposed on firms or final consumers. Given the stochastic processes from Eqs. (18) and (19), the results shown in this paper are based on simulation of 90,000 rationing scenarios, divided into three groups: • R1 (30,000): rationing intermediate and final consumption (dn+1 = l n+1 and di = l i for all i ≤ n) • R2 (30,000): rationing only intermediate consumption (dn+1 = 0 and di = l i for all i ≤ n) • R3 (30,000): rationing only final consumption (dn+1 = l n+1 and di = 0 for all i ≤ n)

 li =

0

if wi ≤ mi

fdj

otherwise f ∼ U[0, 3] di ∼ U[0, 1] wi ∼ U[0, 1] mi ∼ U[0, 1]

 ln+1 =

0

if w ≤ m

∼ U[0, 3]

otherwise

(18)

w ∼ U[0, 1] m ∼ U[0, 1]

model assumes unitary elasticity of substitution between goods, it is much more suitable for small power cuts (when substitution tends to be easier). However, to fully explore the model properties, the chosen parametrization allows for power cuts of as high as 70%.

(19)

Therefore, the distribution of rationing restrictions emulates rationing policy with active and non-active constraints. Since the

When the power supply interruption is unexpected (e.g., caused by an accident), it typically affects an entire region, indiscriminately. Therefore, nationwide impacts of individual power plants failure can be interpreted as an heterogeneous rationing that affects primarily only the industries on that power grid. Consequently, general equilibrium models help the policymaker to calculate the cost of a power grid failure accurately and understand how such impacts would spread to the whole economy. Nevertheless, when power supply interruption is planned (e.g., when it is possible to know beforehand power production will not be sufficient to meet demand), the tools for improving rationing costs measurement can also be used to find optimal rationing policies. After all, finding the optimal policy is the same as evaluating several policy scenarios ex ante and finding the one that minimizes losses. One example of planned energy rationing is the Brazilian 2001 power supply rationing, when low hydroelectric power plants’ reservoirs required the reduction of hydroelectricity production to levels that obligated users to cut their consumption for several months. Consumption reduction targets varied depending on each firms’ sector, so understanding the effects of imposing different consumption restrictions on different industries could improve the design of similar policies in Brazil and worldwide. Additionally, efficient energy production requires marginal production costs to be minimal. Needless to say, when energy production affects rationing probability, interruption costs should be taken into account when estimating marginal power production costs. Therefore, in such situation, efficient resources allocation requires knowledge on the magnitude of interruption costs. Not surprisingly, such knowledge is even more important when production from different energy sources affects rationing probability differently. This is the case of Brazil, where power supply interruption costs and probabilities are used to determine thermoelectric power

0%

GDP (market prices)

GDP (constant prices)

V. Botelho / Energy Economics 80 (2019) 983–994

− 0.5 %

−1% 0%

− 10 % − 20 % − 30 % − 40 % − 50 % − 60 % − 70 % Energy consumption

0% −5% − 10 % − 15 % − 20 %

0%

Final and intermediate demand rationing Only intermediate demand rationing Only final demand rationing Fig. 8. Impact of energy rationing on GDP (constant prices).

plants supply. The majority of Brazilian power supply comes from hydroelectric power plants, whose production reduce their reservoir levels. If power supply interruption probability rises (because reservoir levels are low), more thermoelectric power plants operate, to preserve hydroelectric power plants’ reservoir levels. However, thermoelectric power plants are inefficient, expensive, so their efficient supply depends on the comparison of their production costs to interruption costs and probability. Hence, in such case, efficient energy production requires interruption costs to be adequately estimated. Moreover, the higher the interruption costs, the more reasonable it is to have significant idle capacity in the power supply structure. So, power supply interruption costs can also provide parameters that help determine when energy production idle capacity should be extended. Naturally, interruption cost estimation must be accurate to avoid unnecessary or insufficient investments. 4. Results The impacts of rationing on GDP and welfare are commonly referred to as the rationing policy costs. The first one is usually regarded as the macroeconomic cost, whereas the other is the general equilibrium version of the willingness to pay. At first, welfare will be measured as utility function losses, in equilibrium, after rationing policy implementation. However, the indirect utility function allows the transformation of such losses in currency units. After such a transformation, welfare and GDP results can be quantitatively compared. It is worth noting that mainstream general equilibrium models are built such that consumers own the economy’s firms. Therefore, every loss in profits, in general equilibrium, impacts consumers’ assets and consumption decisions. Hence, power cuts imposed on firms also affect consumers’ utility levels. This is one additional reason to analyze welfare losses in general equilibrium models. 4.1. GDP Fig. 8 shows the impact of different energy rationing policies on GDP. Some results are counterintuitive. For example, energy rationing strategies that impose restrictions only on final demand have no impact on GDP. The idea behind such a result is that under rationing, the final consumer demand not spent on energy services is spent on other goods. On the other hand, when intermediate consumption is rationed, the impact on GDP can be significant. These results emphasize the importance of the rationing distribution among different consumers for evaluating the policy impact. The impacts found in Fig. 8 are very low. Such results indicate that the elasticity of substitution between goods is lower than the calibration supposed.30 After all, experience suggests that the impact

30 Cobb-Douglas production functions impose unitary elasticity of substitution between inputs.

991

− 10 % − 20 % − 30 % − 40 % − 50 % − 60 % − 70 % Energy consumption Final and intermediate demand rationing Only intermediate demand rationing Only final demand rationing

Fig. 9. Impact of energy rationing on GDP (market prices).

of a homogeneous rationing policy on GDP is much greater than the one suggested by these simulations.31 Thus, further explorations with alternative models for which substitution elasticity between inputs is less than one may help better understand the impact of energy rationing on GDP. Alternatively, when GDP is measured using market prices, the impact magnitudes increase significantly, as shown in Fig. 9.32 Nevertheless, the result that rationing final consumer demand can mitigate the impacts of rationing on GDP is robust to the GDP measurement: in both figures, the green line lies on 0%. Additionally, Fig. 8 indicates a nonlinear relationship between the overall power cut and impact on GDP. As expected, the optimal policy indicates that the impact of power rationing on GDP monotonically increases as the power cut magnitude increases.33 However, the impact range (considering the impacts of all possible rationing designs for each power cut level) increases until a power cut threshold is reached, after which it starts to decrease (even while the optimal policy impact levels continue to rise). This is a natural consequence of the fact that the number of degrees of freedom for distributing a power cut among industries is small if the overall power cut is either very small or very large.34 Therefore, the nonlinear shape of Fig. 8 is probably robust across a wide range of different model specifications. 4.2. Welfare Another variable affected by the energy rationing policy is consumer welfare.35 Since households are affected by final and intermediate demand restrictions,36 it is reasonable to evaluate their welfare when choosing among different energy rationing policies. In fact, welfare evaluation is the only means to measure the economic impacts of power supply restrictions on all economic agents using a single metric. Therefore, the impact of rationing on welfare should be the benchmark for policy cost estimation. However, since welfare is not directly observable and has no clear unit of measurement, policymakers often tend to approximate it by GDP. As this subsection shows, such an approximation is premature. The welfare losses from each simulated rationing policy are shown in Fig. 10. It is worth noting that although increasing the consumption restrictions imposed on households has no impact on GDP, it significantly affects welfare. Nevertheless, for very small

31 Whereas according to the model, rationing of 20% could impact the GDP by 0.5%, Brazilian experience suggests that the impact of rationing of this magnitude is approximately 1.0% to 3.0%. 32 Provided that GDP is usually measured using constant prices, Fig. 8 contains the most relevant results. 33 The optimal rationing policy, in this case, is the policy that minimizes the impact on GDP for a given overall power cut level. Therefore, the optimal policy can be expressed as the overline in Fig. 8. 34 e.g., there is only one means of cutting 0% or 100% of overall power consumption. 35 Welfare, in the model, is measured as the representative consumer utility. 36 Households own the economy’s firms.

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Welfare

0% − 0.5 % −1% − 1.5 % 0%

− 10 % − 20 % − 30 % − 40 % − 50 % − 60 % − 70 % Energy consumption Final and intermediate demand rationing Only intermediate demand rationing Only final demand rationing

Fig. 10. Impact of energy rationing on welfare (welfare units).

Welfare

0% − 0.5 % −1% − 1.5 % −1%

− 0.8 %

− 0.6 % − 0.4 % GDP

− 0.2 %

0%

Final and intermediate demand rationing Only intermediate demand rationing Only final demand rationing Fig. 11. Comparison of GDP and welfare costs for each rationing simulation.

power cuts, it is possible the imposition of energy rationing restrictions on only residential energy demand would minimize the impacts on both GDP and welfare simultaneously. For larger impacts, a GDPwelfare trade-off arises. Therefore, even though the impact shapes indicated in Figs. 8 and 10 are similar, the optimal policy design can change significantly if the policymaker objective function is based on household welfare or GDP. 4.3. GDP and welfare comparison To adequately compare impact on welfare and GDP, it is necessary to measure both using the same measurement unit. Fortunately, welfare losses can be translated into monetary units using the indirect utility function. Since consumer utility is logarithmic, a utility loss of x represents the cost of ex − 1 times the overall economy payroll. After this transformation, it is possible to compare the GDP and welfare monetary costs of rationing policy, as done in Fig. 11. The comparison allows us to conclude the following: • For every possible rationing policy, the welfare cost is always greater or equal to the GDP rationing cost. • The welfare cost almost equals the GDP cost when rationing is imposed only on firms (the blue line has a slope of 45◦ ). • The welfare cost greatly exceeds the GDP cost when rationing is imposed only on households. • The impact on GDP is always minimized when rationing is concentrated on households (final consumption). • The impact on welfare is always minimized when rationing is concentrated on firms (intermediate consumption). 4.4. Other variables Until now, the discussion of the results has focused on the interpretation of random policy impacts on GDP and welfare. However, the simulations also allow the interpretation of the average, maximum and minimum impacts of all rationing strategies on any model variables for each possible rationing intensity. The results are presented

in Fig. 12, and they reinforce the empirical basis for several statements made in the previous sections. As previously discussed, the impact range of rationing policy on all variables is very small in two situations: when the overall power cut is either very small or very large. When it is small, the damage is not significant because the shock is irrelevant. When it is very large, the damage is so generalized that the rationing distribution does not matter: since the rationing restriction imposed on any consumer cannot exceed 100%, there are fewer ways of implementing a 98% overall power cut than a 50% one. In the first case, not many rationing designs are possible. For this reason, the rationing design is very important for power cuts of average magnitude. Additionally, Fig. 12 (a) shows that there is no such thing as a fixed elasticity between power supply cuts and GDP: the elasticity not only depends on the rationing magnitude but also significantly depends on the design. Fig. 12 (b) provides a very good example of rationing policy nonlinearities. Energy rationing of 20% can cause consumer prices to decrease more than 10% or not at all. A similar phenomenon is apparent in Fig. 12 (c) and (e). Although they show impacts of much smaller magnitudes, Fig. 12 (a), (d), (f), and (h) reinforce this evidence. Fig. 12 (b) indicates that consumer prices decrease due to rationing intermediate energy demand. The result is not surprising, inasmuch as the price of energy is numeraire: consumption prices are falling relative to the price of energy after a rationing policy that reflects energy scarcity. However, when final energy demand is rationed, consumer prices tend to increase.37 The result indicates that final consumption rationing makes the consumer spend its income on other goods (since they are substitutes). Therefore, in this case, the demand for all goods increases, and prices go up. This result explains the difference between the impact of rationing on constant-price GDP, shown in Fig. 12 (a), and that on market-price GDP, shown in Fig. 12 (c). As previously discussed, since the model supposes a high degree of substitution between intermediate inputs, the impacts of rationing policy on GDP are small. It is worth noting that the impacts on GDP (constant prices), welfare, and consumption are very similar. Although the optimal policy depends on the policymaker objective function, the results shown in Fig. 12 indicate that some power cut strategies are strictly dominated by others. For example, the region between the lower green line and the lower blue line in Fig. 12 (d) features greater losses in terms of both GDP and welfare than other strategies that yield the same overall power cut. 5. Conclusion The paper applies mainstream economic theory to a very important public policy problem, which had previously only been analyzed using controversial tools. General equilibrium models provide theoretical and empirical bases to estimate the economic impacts of rationing policies. They also provide new insight into how to understand such effects: they depend not only on the rationing intensity but also on its design. In many cases, the rationing policy design is much more important than its intensity to determine the impact of rationing policy. Such features cannot be properly understood when evaluating policy using classical input-output analysis (not suited to the type of restrictions imposed by a rationing program) or econometric models (that extrapolate past correlations). Therefore, there is no fixed power supply and GDP elasticity that could estimate the policy impact correctly. Only general equilibrium models can adequately address all the particularities associated with rationing policy effects

37

Again, this is relative to the price of energy since it is numeraire.

V. Botelho / Energy Economics 80 (2019) 983–994

993

Fig. 12. Rationing policy impact on selected variables. Green: only final consumption rationed, Blue: only intermediate consumption rationed, Red: final and intermediate consumption rationed, x-axis: overall energy consumption reduced. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

evaluation, and this paper presented a method to adequately implement such an approach. The exploration of different rationing policy scenarios using general equilibrium models presented in this work has revealed some interesting patterns. For example, in some cases, policies that minimize the impact on GDP can maximize the impact on welfare. Further, there exist non-efficient rationing designs: energy rationing programs that have the same intensity but cause more losses than

others in terms of both GDP and welfare simultaneously. Such results may serve as tools for designing practical responses to power supply restrictions in the future. To further explore the general equilibrium properties of rationing policy, other production functions must be tested. The CobbDouglas high degree of substitutability between inputs probably underestimated the true rationing impact values. Additionally, the method presented in this paper can be applied to estimate the

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costs of several types of infrastructure failures, since all of them are associated with physical restrictions on the production of goods and services.

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