For this Discussion Question, complete the following. 1. Locate two JOURNAL articles which discuss this topic further. You need to focus on the Abstract, Introduction, Results, and Conclusion. For our

Munich Personal RePEc Archive Should a Government Fiscally Intervene in a Recession and, If So, How?

Taiji Harashima Kanazawa Seiryo University 2 April 2017 Online at https://mpra.ub.uni-muenchen.de/78053/ MPRA Paper No. 78053, posted 31 March 2017 09:03 UTC 0 3 5 $ Should a Government Fiscally Intervene in a Recession and , If So, How ? Taiji H ARASHIMA * April, 2017 Abstract The validity of discretionary fiscal policy in a recession will differ according to the cause and mechanism of recession. In this paper, discretionary fiscal policy in a recession caused by a fundamental shock that changes the steady state downwards is examined. In such a recession, households need to discontinuously increase consumption to a point on the saddle path to maintain Pareto efficiency . Howeve r, they will not “jump ” consumption in this manner and instead will choose a “Nash equilibrium of a Pareto inefficient path ” because they dislike un smooth and discontinuous consumption and behave strategically . The paper concludes that increasing governmen t consumption until demand meets the present level of production and maintaining this fiscal policy for a long period is the best option. Consequent government debts can be sustainable even if they become extremely large. JEL Classification code: E20, E32, E62, H20, H30, H63 Keywords: Discretionary Fiscal policy ; Recession; Government consumption ; Government debts ; Pareto inefficiency; Time preference *Correspondence: Taiji HARASHIMA, Kanazawa Seiryo University, 1 0-1 Goshomachi -Ushi, Kanazawa -shi, Ishikawa, 920 -8620, Japan. Email: harashim@seiryo -u.ac.jp or [email protected] . 1 1 INTRODUCTION Discretionary fiscal policy has been studied from many perspectives since the era of Keynes (e.g., Keynes, 1936; Kopcke et al., 2006; Chari et al., 2009; Farmer, 2009; Alesina, 2012; Benhabib et al., 2014) . An important issue is whether a government should intervene fisc ally in a recession , and if so, how . The answer will differ according to the cause and mechanism of recession. Particularly, it will be different depending on whether “disequilibrium” is generated. The concept of disequilibrium is , however , controversial a nd therefore argument s continue even now about the use of discretionary fiscal policy in a recession. In this paper, the concept of disequilibrium is not used, but instead the concept of a “Nash equilibrium of a Pareto inefficient path ” is used . Recessio ns are generated by various shocks (e.g., Rebelo, 2005; Blanchard, 2009; Ireland, 2011; Schmitt -Grohé and Uribe, 2012; McGrattan and Prescott, 2014; Hall, 2016) . Some fundamental shocks will change the steady state, and if the steady state is changed downw ards (i.e., to lower levels of production and consumption ), households must change the consumption path to one that diminish es gradually to the posterior steady state. Therefore, growth rates become negative ; that is, a recession begins. However, the expla nation of the mechanism of this type of recession is not perfect because an important question still needs to be answered . I f households discontinuously increase (“ jump up ”) their consumption from the prior steady state to a point on the posterior saddle p ath and then gradually move to the posterior steady state, Pareto efficiency is held and thereby unemployment rates do not rise. There fore , even in a serious and large -scale recession, unemployment does not increase. This is a very unnatural outcome of a s erious recession . Harashima ( 2004, 2009, 2013 a) showed a mechanism by which households do not jump up their consumption even if the steady state is change d downward because they are intrinsically risk averse and non -cooperative and want to smooth consumpt ion. Th e consumption jump does not give them the highest expected utility ; that is , u nsmooth and discontinuous consumption is not optimal for households. Hence, i nstead of choosing the posterior saddle path, they will choose a “Nash equilibrium of a Pareto inefficient path ” as the optimal consumption path. Because of its Pareto inefficiency, unemployment rate s will increase sharply and stay high during a recession . This paper examines whether discretionary fiscal policy is necessary , and if it is necessary, how it should be implemented when an economy is in a recession and proceeding on such a Pareto inefficien t path . Fundamental shocks that change the steady state basically mean shocks on deep parameters. A representative fundamental shock, an upward shock on t he rate of time preference (RTP) , is examined in this paper . Faced with this shock, a government has three options : (1) do not intervene, (2) increase government consumption, and (3) cut taxes. The consequences of these options are examined and the ou tcomes are evaluate d to determine which is the best option. I conclude that increasing government consumption until the demand meets the present level of production and maintaining this fiscal policy during the recession is the best option. Nevertheless , this option will be accompanie d by large and accumulating government debts , but the se debts can be sustained if the government properly increases taxes in the future . This option means that huge government debts will play an essential role as a buffer again st negative effects of the fundamental shock. 2 A MECHANISM OF RECESSION 2.1 An upward RTP shock There are various possible sources of recession, but in this paper, a recession caused by a fundamental shock, par ticularly by an upward shift of RTP , is examined because an upward shift of RTP seems to be most likely the cause of the Great Recession (Harashima, 2016) . A 2 technology shock was probably not the cause of the Great Recession because technology does not suddenly and greatly regress. Frictions on price adjustments are also unlikely to be the cause because the micro -foundation of friction does not seem to be sufficiently persuasive (e.g., Mankiw, 2001), particularly the micro -foundation of its persistence. On the other hand, Harashima ( 2016 ) showed that a n upward RTP shock could explain the occurrence of the Great Recession and show ed evidence that the estimated RTP of the U nited States increased in about 2008. RTP plays an essential role in economic activities, and its importance has been emphasi zed since the era of Irving Fisher (Fisher, 1930). One of the most important equations in economics is the steady state condition where θ is RTP and r is the real rate of interest. This condition is a foundation of both static and dynamic economic studies. The mechanisms of both θ and r are equally important. Particularly, RTP is an essential element in expectations of economic activities because RTP is the discount factor for future utility . In addition, RTP has been regarded as c hangeable even over short periods (e.g., Uzawa, 1968; Epstein and Hynes, 1983; Lucas and Stokey, 1984; Parkin, 1988; Obstfeld, 1990; Becker and Mulligan, 1997). Furthermore, households behave based on the expect ed RTP of the representative household (RTP R H) (Harashima, 2014 , 2016 ). That is, change s in RTP and t he expected RTP RH can be an important source of economic fluctuation s. 2.2 The model The model in this paper is based on the model s in Harashima (2004, 2009, 2013 a) and assumes non -cooperative, i dentical, and infinitely long living households , and that the number of households is sufficiently large. Each of the households equally maximizes the expected utility subject to , where yt, ct, and kt are production, consumption, and capital per capita in period t, respectively; A is technology and constant; u is the utility function; is the production function; and E0 is the expectations operator conditioned on the agents’ period 0 information set. yt, ct, and kt are monotonically continuous and differentiable in t, and u and f are monotonically continuous functions of ct and kt, respectively. All households initially have an identical amount of financial assets equal to kt, and al l households gain the identical amount of income in each period. It is assumed that and ; thus, households are risk averse. In addition, and . Both technology (A) and labor supply are assumed to be constant ; that is, there is no technological progress or population increase. It is also assumed that t here is no depreciation of capital. 2.3 A Nash equilibrium of a Pareto inefficient path r θ   dt c u θt E t    00 exp   t t t c A,k f dt dk    t t k A f y ,   t t k A f y ,    0 t t dc c du   0 2 2  t t dc c u d   0 ,    t t k k A f   0 2 2    t t k kf 3 The prior s teady state before the shock on θ: W The posterior s teady state after the shock on θ Posterior l ine of after the shock on θ Line of Z Pareto ineffi cient transition path Prior l ine of before the shock on θ Pareto efficient saddle path after the shock on θ Pareto efficient saddle path before the shock on θ The effects of an upward shift in RTP are shown in Figure 1. Suppose first that the economy is at steady state bef ore the shock. After the upward RTP shock, the vertical line moves to the left (from the solid vertical line to the dashed vertical line in Figure 1). To keep Pareto efficiency, consumption needs to jump immediately from the steady state before the shock (the prior steady state) to point Z. After the jump, consumption proceeds on the Pareto efficient saddle path (the posterior saddle path) from point Z to the lower steady state after the shock (the posterior steady state). As a resul t, negative economic growth rates continue for a long period, but unemployment rates will not increase and resources will not be destroyed or left idle. Note that an increase in household consumption means consuming the part capital indicated by the gap be tween the posterior saddle path (the thin dashed curve) and production (the bold solid curve) for each kt, which initially is the gap between point Z and W.1 Figure 1: A n upward RTP shock . All terms are defined i n the text. 1 If depreciation of capital is assumed to exist, the “cons umption” of excess capital will be achieved by a reduction of investments that correspond to depreciated capital and an increase in consumer goods and services. 0 tk 0  dt dc t 0  dt dk t 0  dt dc t 0  dt dc t tc 4 However , this discontinuous jump to Z will be uncomfortable for risk -averse households that wish to smooth consumption. Households may instead chose a shortcut and, for example, proceed on a path on which consumption is reduced continuously from the prior steady state to the posterior steady state (the bold dashed line), although this shortcut is not Pareto efficient. The mechanism for why they are very unlikely to jump consumption is explained in Harashima (2004, 2009, 2013 a) and also in the Appendix. Because households are risk averse and want to smooth consumption, and are also intrinsically non -coo perative, they behave strategically in game theoretic situations. Because of these features, when households strategically consider whether or not the jump is better for them (i.e., they are in a game theoretic situation), they will generally conclude that they obtain a higher expected utility if they do not jump. Hence, households will not actually choose this path and instead will choose a different transition path to the steady state (e.g., the bold dashed curve). Because this transition path is not on t he posterior saddle path, it is not Pareto efficient (I call this transition path a “Nash equilibrium of a Pareto inefficient path” or more simply a “Pareto inefficient transition path”). Therefore, the excess resources indicated by the gap between the pos terior saddle path (the thin dashed curve) and the Pareto inefficient transition path (the bold dashed curve) for each kt (initially, the gap between points Z and X) will be destroyed or left idle. Unemployment rates will increase sharply and stay high for a long period. 3 SHOULD THE GOVERNMENT FISCALLY INTE R VEN E? 3. 1 The government’s options 3.1.1 The t hree options When households choose a Nash equilibrium of a Pareto inefficien t path, the government basically has three options : (1) do not i ntervene , (2) increase government consumption , and (3) cut taxes. If Option (1) is chosen, the gap between the posterior saddle path and the Pareto inefficien t transition path (initially the gap between point s Z and W) is not filled by any demand . Therefore , unemp loyment rates increase sharply and huge amount s of resources are destroyed or left idle . High unemployment rates and destruction of resources will continue until the economy reach es the posterior steady state. If Option (2) is chosen, government consumpt ion is increased to fill the demand gap between the posterior saddle path and the Pareto inefficien t transition path , where government consumption is indicated on a per capita basis similar to the other variables. Suppose for simplicity that government con sumption is zero before the shock. With increases in government consumption, the path of the sum of government and household consumption (hereafter “combined consumption”) can be equal to the posterior saddle path . Conceptually, government consumption is the collective consumption of households through government expenditure s, for example , spending on various kinds of administrative services that households receive. Therefore, increases in government consumption can be substituted for decreases in househo ld consumption. Nevertheless, government consumption will not directly generate utilit y in households. In this sense, increases in government consumption may be interpreted as forced increases in household consumption. Even if households do not want these increases in government consumption, however, the increases will work to increase aggregate demand. Option (2) therefore indicates a measure to compulsorily fill the gap between aggregate demand and supply , even against households’ will, when the economy p roceeds on a Pareto inefficien t transition path. Notice that the excess resources cannot be used for investments because the economy would otherwise deviate from a path to the steady state. 5 If Option (3) is chosen, households’ disposable incomes will inc rease, but if the Ricardian equivalence holds, the y will still proceed on a Pareto inefficien t transition path . Because household consumption does not change, high unemployment rates and destruction of a huge amount of resources continue as in Option (1). Because there is a huge amount of excess capital, no additional investment will be made. Nevertheless, i f the Ricardian equivalence does not hold, tax cuts may increase household consumption at least temporarily . Therefore, the validity of Option (3) depen ds on the validity of the Ricardian equivalence. If households are sufficiently rational, the Ricardian equivalence will basically hold at least in the long run. Therefore, even if t ax cuts are effective , they will be effective only in the short run, and t hese short run effects will be reversed because the Ricardian equivalence will hold in the long run . 3.1.2 Financing In Option (3), tax cuts are financed by borrowing from households . In Option (2), a n increase in the government consumption is financed by borrowing from or tax increases on households . Nevertheless, f inancing by borrowing will be preferred in Option (2) because the Ricardian equivalence may not necessarily hold in the short run. If the Ricardian equivalence does not hold, i ncreases in tax es may increase unemployment rates and thereby the main aim of Option (2) cannot be fully achieved. Therefore, it is highly likely that an increase in government consumption will be financed by government borrowing , and therefore borrowing is assumed in this paper . However, financing by borrowing requires tax increases in the future to pay off the debt with interest. Option s (2) and (3) assume that necessary future tax increases are fully implemented by the government. In addition, i t is assumed that a go vernment borrow s money only from its own people , that is , not from foreigners because foreign b orrowing means that foreigners also intervene in addition to the government, and such intervention is beyond the scope of this paper. 3.2 Comparison among op tions (1) Economic growth rate Because production and consumption at the posterior steady state are lower than those at the prior steady state, the rate of economic growth is equally negative during the transition in the three options except for a subordin ate option of Option (2), in which, a s will be shown in Section 4, it is zero . Nevertheless , there actually still will be steady technological progress (remember that no technological progress is assumed in the model) , and thereby the actual rates of growt h will not necessarily be negative or zero and may even be low but positive. (2) Household utilit y Households choose a Nash equilibrium of a Pareto inefficien t path equally in the three options . Therefore, the utilities of households are basically same in the three options. (3) Unemployment In Option s (1) and (3) , unemployment rate s will rise sharply and stay high for a long period . In contrast, in Option (2), high unemployment rates can be avoided because the gap of demand is filled by increases in government consumption and thereby no resource s are destroyed or left idle . (4) Government debt In Option (1), government debt do es not increase because the government do es not borrow additional money, but in Option s (2) and (3) , government debt will inc rease because of continuous financing by borrowing. However, i f taxes are raised properly to pay off the debt in the future, government debt will stabilize in some future period. 3.3 G overnment debt 6 3.3.1 Is the government debt sustainable ? The u sual a rgument s on sustainable government debts (e.g., Hamilton and Flavin, 1986; Bohn, 1995) are not applicable to the government debts in Options (2) and (3) because households proceed on a n “unusual” Pareto inefficien t transition path , so a n alternative approa ch is necessary. Let dt be per capita “extra” government debts in period t that are accumulated in Option (2) or (3). Because all dt are owned by households as assumed above , dt also indicates the financial asset s of household s, and the other household asse ts (other than dt) are ignored for simplicity . In the future, dt is redeemed with interest , but the redemption tak es a long time. Because the Ricardian equivalence will hold in the long run, it is assumed that household consumption is not influenced by dt. Let zt be per capita taxes to redeem a part of dt in period t and also let gt be additional government borrowing in Option (2) or (3) in period t. In Option (2), , (1) and in Option (3), (2) for any t because no new investment is made in Options (2) and (3) and the household asset s other than the government bonds are ignored ; yt and ct are per capita income and consumption of households in period t. If the con dition (3) is satisfied indefinitely in a certain future period, government debt never explode s; that is, it is sustainable where is the real interest rate. By equality ( 1) and inequality ( 3), the condition for sustainability in Option (2) is . (4) By inequalities ( 2) and ( 3), if inequality ( 4) is satisfied indefinitely in a certain future period , government debt is also sustain able in Option (3) . Because the household asse ts other than dt are ignored , the sum of a household’s income and assets is . If the sum of a household’s income and assets exceed s zt, that is , if , (5) then zt can be imposed in the sense that households have enough resources to fully pay taxes . Hence, b y inequalities ( 4) and ( 5), if (6) t t t g c y   t t t g c y   t t tt z g dr    1 0   t t r r t ttt t t z dr c y    t t t c y d   t t t t c y d z    t tt d dr  7 is satisfied, taxes that satisfy the condition for sustainable debts can be imposed. Here, because , then inequality ( 6) always holds. Th erefore , for any dt, there always exists zt that satisfies inequality ( 3) indefinitely in a certain future period . Th at is, the government debt can be sustainable for any dt, and even if dt becomes extremely large, the debt can be sustainable. Consider an extreme example. If a government collects taxes that are equivalent to dt from a household’s financial assets in a period, the government’ s debt s are eliminated completely all at once . That is, any dt can be sustainable. Such an extreme tax will not actually be imposed, but i f dt exceeds a certain amount such that , (i.e., if taxes exceed income ), then they need to be collected from a part of a household’ s holdings of dt. If households well know th e possibility of a tax on dt in the future , they will not regard their accumulated financial assets corresponding to dt as their “real” assets in the sense they can be freely use d for consumption even though dt may be extremely large . In addition, because any dt can be sustainable , the tax increase can be started even a fter all the excess capital is eliminated . Hence, a huge amount of government debt can remain even if there i s no excess capital. Finally, it is i mportant to note that the increase d tax revenues should not be used to finance increases in government consumption for purposes other tha n dealing with the excess capital. The increase d taxes should be used on ly to pa y down dt (with interest) because the economy otherwise deviates from the steady state. 3. 3.2 How large can government debt be ? Any dt can be sustainable but only if a government properly raises taxes and is satisfied indefinitel y in a certain future period. The question arises, h owever, when is “a certain future period”? The tim e at which taxes are raised is indeterminate in the discussion in the previous section . The tax increase can be postponed almost indefinitely if taxes wil l certainly be raised eventually . Th is indeterminacy may generate a political struggle because people intrinsically dislike tax increase s, and opposition parties will utilize people’s anti -tax sentiment as ammunition to attack the government. Opposition pa rties will appeal to people that a tax increase is not necessary at present and that it will only generate a recession because the Ricardian equivalence will not hold in the short run . The government may not sufficiently refute this argument and persuade p eople that the current level of government debt is unsustainable , because any dt can be sustainable . The incentive for the government to raise tax es to reduce dt will therefore be weak. Is there a problem, h owever, if dt becomes extremely large? As shown in Section 3.2.1 , other things being equal, any dt can be sustainable, but if something changes and affects the sustainability as dt becomes large r, a large dt will not actually be sustainable. One possible factor that may change as dt becomes larger is u ncertainty. If the tax increase has been postponed for a long period, question s about the ability of the government to govern the nation and run the economy will arise . Faced with an extremely large dt, p eople may begin to suspect that the ir government can not do what it should do. Hence, uncertainty about the ability of the government will increase , and increased uncertainty about the government’s ability means that the government ’s perform ance in the future is no longer a certainty . It has been argued tha t good institutions, including government s, enhance economic growth (e.g., Knack and Keefer, 1995; Mauro, 1995; Hall and Jones, 1999; Acemoglu et al., 2001, 2002; Easterly and Levine, 2003; Dollar and Kraay, 2003; Rodrik et al., 2004). Acemoglu et al. (200 5) conclude that differences in economic institutions are empirically and theoretically 1 0   tr t t t z rd y   t tt z dr  8 the fundamental cause of differences in economic development. 2 It is therefore highly likely that a government’s ability is an important determinant of total factor pro ductivity , that is , levels of production and consumption . Therefore, if uncertainty about the ability of a government increases, household’s expect ed variance s of production and consumption will also increase . Larger variance s of production and consumption mean more uncertainty about the entire future economy. That is, as dt increases, household uncertainty about the entire future economy increase s. An important consequence of increases in uncertainty about the entire future economy is an increase in hous ehold RTP. The concept of a temporally varying RTP has a long history (e.g., B öhm -Bawerk, 1889; Fisher, 1930; Uzawa, 1968; Lawrance, 1991; Becker and Mulligan, 1997). In addition, uncertainty has been regarded as a key factor that changes RTP. Fisher (1930 ) argued that uncertainty, or risk, must naturally influence RTP , and higher uncertainty tends to raise RTP . Harashima ( 2004 , 2009 ) showed a mechanism of how an increase in uncertainty leads to an increase in RTP by constructing an endogenous RTP model whe re uncertainty is defined by the stochastic dominance of the distribution of steady -state consumption. Increases in uncertainty will increase RTP RH. An increase in RTP RH indicates an increase in the real interest rate at steady state and consequently a d ecrease in production and consumption at the steady state because RTP RH is equal to the real interest rate at steady state in Ramsey -type growth models. That is, it is likely that as dt increases , long -run production and consumption will decrease . Consid ering the effect of dt on RTP RH and on long run production and consumption , therefore, a government will not have to postpone the a tax increase for a long period and to accumulate an extremely large dt. Nevertheless , the scale of the effect of dt on RTP RH is unclear . It may be small and take a long period before household s clearly recognize the negative effect of a large dt on RTP RH. Hence, the exact upper limit of dt is unclear , so there will still be much room for a government with regard to the timin g and scale of tax increases. When the long run negative effect of a huge dt on the expected household utility becomes larger than th e short run effect of deviation from the Ricardian equivalence on the expected household utility , taxes should be raised. However, it may be difficult to judge which is currently larger. On the other hand, i f the negative effect of the short run deviation from the Ricardian equivalence can be controlled such that it remains very small, it will be better to raise taxes even f or small dt. In this sense, it may be a good idea to raise the tax rate by a very small percentage point amount in every period, for example , by 0.5% per year. Because this tax increase is very small in each period, the negative effect of any short run dev iation from the Ricardian equivalence can be controlled such that it is also very small in each period. There is another relatively minor problem associated with extremely large dt. As dt increase s, the amount of necessary future t ax increase s (as shown in Section 3.3.1 ) will eventually exceed income (yt). Therefore, taxes need to be imposed not only on income but also on household’s financial assets corresponding to dt. However, large taxes on financial assets may be less easy to implement than other type s of taxes both practically and politically . Nevertheless, a n inheritance tax may be relatively easy to implement , and therefore it will be important as taxes on household’s financial assets . 3.3. 3 Price stability It has been argued that a large amount of government debt will result in high inflation (Sargent and Wallace, 1981) . Fiscal theory of price level particularly emphasizes this mechanism (Leeper, 1991; Sims, 1994, 199 8; Cochrane, 200 5; Woodford, 2001) . However, Harashima ( 2006 ) showed that the re lation between the government debts and inflation is not simple and presented a model that explains the law of motion for inflation considering government debt . The model in Harashima (2006) indicates that a large amount of government debts does not result in high 2 Some economists argue the reverse causation from growth to institutional improvement (e.g., Barro, 1999) or that institutional improvement has a smaller impact on growth than human capital (Glaeser et al., 2004). 9 inflation as long as the central bank is sufficiently independent. Inflation will not be affected by temporary increases in government expenditure and consequent future taxes . As a result, if the central bank is sufficiently independent, the gover nment can implement Option (2) without worrying about an out break of high inflation. 3. 4 Evaluation As shown in Section 3.2, the rate of economic growth in the three options is equally negative until arriving at the steady state , and h ousehold utilities are basically same in the three options. On the other hand, unemployment rates will rise sharply and stay high for a long period in Option s (1) and (3) , but not in Option (2). As argued in Section 3.3, the extra government debts are sustainable if the gove rnment properly increases taxes in the future. If the future tax increase is properly implemented , therefore, Option (2) is favorable to Option s (1) and (3) because unemployment rates do not rise . 4 HOW SHOU L D THE GOVERNMENT INCREASE ITS CONSUMPTION ? 4.1 Subordinate options in Option (2) Option (2) is the best choice , but how should the government increase its consumption ? There are two basic subordinate options in Option (2) . Option ( 2-1): Increas e government consumption in order for the combined co nsumption to jump up to point Z and then proceed on the posterior saddle path to the posterior steady state . Option (2 -2): Increas e government consumption for the combined consumption to jump up to point W, and then stay at point W. Remember that combined consumption indicates the sum of government and household consumptions. Option ( 2-1) i ndicates that the government intervenes so as to make the combined consumption proceed on the posterior saddle path and eventually reach the posterior steady state , and Option ( 2-2) i ndicates that it intervenes so as to make the production and combined consumption stay at the prior steady state (i.e., at point W) forever . Note that, a s noted in Section 3.1.1, excess resources cannot be used for investments because the eco nomy would otherwise deviate from the posterior saddle path in Option (2 -1) and from point W in Option (2-2). 4. 2 Option (2 -1) 4. 2.1 Basic features When a government chooses Option (2 -1), each household may change its consumption path in response to the government ’s action , but i t is highly likely that households will still proceed on a Pareto inefficient transition path b ecause the households’ expected utilities are not affected by the increase in government consumption. Here, a gap between the posterio r saddle path (the thin dashed curve in Figure 1 ) and production (the bold solid curve) for each kt indicates exces s capital. Excess capital need s to be “consumed” for the economy to be on the posterior saddle path .3 Option (2 -1) means that excess capital is consumed by the government. In addition, to be on the posterior saddle path, government consumption need s to be increased not only to consum e excess capital but also to substitut e for a reduction in household consumption that is the source of the excess capital. That is, the government need s to consume not only the gap between the posterior saddle path and production (i.e., excess capital ), but also the gap between 3 If capital depreciation is assumed to exist, consumption of excess capital will be achieved by a reduction of investments that corre spond s to depreciated capital input s and a n increase in consumer goods and services. 10 production and the Pareto inefficien t transition path while the economy proceed s from the prior steady state to the posterior steady state . Because of the increase in government consumption, the economy proceeds on the posterior saddle path and thereby high and persistent unemployment rates are avoided . 4. 2.2 Subordinate options However, how does a government “consume” such a large quantity of excess resources, most of which were originally produced as capital? There are three basic subordinate options : Option s (2-1-a), (2 -1-b), and (2 -1-c). The easiest way for a government to consume the ex cess resources is simply to buy them from firms and dispose of them (Option (2 -1-a)). “Dispose of” in this case includes not only eliminating them but also leaving them unused forever or constructing useless infrastructure. It will also mean giving laborer s busy work , including the classic example of “having workers dig holes and then fill them back up.” These activities do not generate any utility for households, but they can be interpreted as a kind of “consumption” in the broad sense that the products pu rchased are intentionally made unusable. High unemployment rates can be avoided, but huge amounts of resources are systematically and continuously disposed of and negative growth rates continue for a long period. Disposing of the excess resources in Opti on (2 -2-a) is different from destroying the m in Option (1) because t he owners of the excess resources lose them without compensation in Option (1), but sell them to the government in Option (2-1-a). The excess resources are equally eliminated in both optio ns, but nothing remains in the hand s of the former owners or the government in Option (1) , whereas financial assets and debts remain in the hand s of the former owner s and government, respectively , in Option (2 -1-a). Another way to consume the excess resou rces is to export the m to other countries at lower prices than the prevailing international prices (Option (2 -1-b). This is not “consumption” in the literal sense, but it can be interpreted as a sort of consumption in that exports are an element of demand. The government does not necessarily need to directly export the excess resources. Instead, it can indirectly support exports by directly subsidizing firms or through various kinds of regulations. An important problem with this option is that other countri es may not accept the excessive exports. This option clearly means setting prices that are far lower than the costs of production (i.e., dumping) on a large scale. Other countries would not be likely to stay silent on this issue and would likely take count ermeasures, for example, by imposing high anti -dumping customs. Therefore, Option (2 -1-b) will generally not be adopted in a democratic country. There is one more important subordinate option . W ith minor modifications, c apital inputs can be used to prod uce arms and munitions. Hence, the necessary increase in government consumption can easily be achieved by a large military buildup (Option (2 -1-c)). An important problem with this option is that a unilateral excessive military buildup will greatly worsen international relations and increase political and military tensions among countries. Therefore, in a democratic country, Option (2 -1-c) will generally not be adopted. 4. 3 Option (2 -2) 4.3.1 Basic features For the same reason as given for Option (2 -1), i t is highly likely that households also proceed on a Pareto inefficient transition path in Option (2 -2). When household s proceed on this path , if the government does nothing, a part of the capital that is used to produce product s corresponding to household s’ reduc tion in consumption become s excess capital and will be destroyed , but if the government purchases and consumes the se unconsumed products, th e capital need not be destroyed and the level of capital will remain the same in the next period . If the gov ernment purchase s and consume s the unconsumed products in every period , capital will continue to stay at the same level indicated by point W. The phenomenon where capital is prevented from being 11 reduced by government intervention may be interpreted as keep ing so-called “zombie” firms alive. As in Option (2 -1), h igh unemployment rates can be avoided, but unlike in Option (2 -1), the growth rate is not negative . Rather, it is zero because the economy stays at point W forever. An important difference between O ptions (2 -1) and (2 -2) is that, unlike Option (2 -1), capital is not consumed by the government in Option (2 -2), but households’ reduc tion in consumption is equally substituted by an increase in government consumption in both options. That is, in Option (2 -2), the government consumes only the gap between production at point W and the Pareto inefficient transition path (bold dashed curve) and does not consume the gap between the posterior saddle path (thin dashed curve) and production at point W (i.e., capita l). As a result, production and capital remain at point W forever in Option (2 -2). 4.3.2 Subordinate options Option (2 -2) also consists of three basic subordinate options depending on what path is chosen at point W: Options (2 -2-a), (2 -2-b), and (2 -2-c). As was the case with Option (2 -1-a), the easiest way for a government to consume excess resources is simply to buy them from firms and dispose of them (Option (2 -2-a)). As with Option s (2-1-b) and (2 -1-c), the necessary jump of the government consumption can be achieved by exporting the excess resources (Option (2 -2-b)) or by a military buildu p (Option (2 -2-c)). However, for the same reasons as given for Option s (2-1-b) and (2 -1-c), Options (2 -2-b) and (2 -2-c) will generally not be adopted in a democratic country. 4.4 Comparison and evaluation Section 4.3 indicates that the only feasible options are (2 -1-a) and (2 -2-a). On major issues, common alities and differences between the two options are as follows. (1) Period of government intervention In Option (2-1-a), excess capital decrease s gradually and eventually becomes zero when the economy arrives at the posterior steady state .4 Hence, the period of transition and government intervention is definite. In Option (2 -2-a), however, the economy never approac hes the posterior steady state . Hence, t he government intervention never end s. (2) Scale of government intervention Because government consumption needs to be initially increased to point Z in Option (2 -1-a), the scale of intervention is initially much l arger in Option (2 -1-a) than in Option (2 -2-a). However, in Option (2 -1-a), excess capital gradually decrease s and eventually reach es the level of the posterior steady state, and thereby the necessary increase in government consumption decreases to zero as the economy approaches the posterior steady state. On the other hand, in Option (2 -2-a), the necessary increase in government consumption increases as household consumption gradually decreases to the level at the posterior steady state . In sum, the scale of intervention is initially larger in Option (2 -1-a) than it is Option (2 -2-a), but this relation will be reversed in some future period . (3) Growth rate s during the transition In Option (2 -1-a), the growth rate s are negative , whereas in Option (2 -2-a), they are zero. (4) Household utilit y In both o ption s, household consumption proceeds on the same Pareto inefficient transition path. In addition, the Ricardian equivalence holds in the long run. Therefore, the utilities that 4 More correctly, the economy never arrives exactly at the posterior steady state , but it arrives close to it in a definite period. 12 households will obtain from the st ream of consumption after the shock are almost the same in both cases . (5) Unempl oyment In both options, unemployment rates do not increase. (6) Government debt In both options, a large amount of government debt accumulate s. However, if the gov ernment properly i ncreases taxes in the future, t he debt will stabilize at some level in both options . Although t he period and scale of government interventions differ between the two options, these differences basically do not matter to household optim ality. Therefore, because the only difference in the evaluated criteria is that growth rate s are highe r in Option (2 -2-a), Option (2-2-a) is considered to be more favorable than Option (2 -1-a). 4.5 Technological progress Although Option (2 -2-a) is the be st, it has its drawbacks. Huge amounts of resources need to be disposed of in the name of the government consumption forever. Although t his is rational from an economic point of view, it may not be environmentally or ethically reasonable . If there is a way to reduce the amount of discarded resources, that is, reduce excess capital, Option (2 -2-a) could be much better. It is impossible to find that w ay within the framework discussed in the previous sections , but if the assumption on technological progress is loosened, it may be possible . Thus far, I have assumed no technological progress , but in reality, technologies steadily progress. In addition, t echnolog ical progress basically require s additional increase s in capital. Instead of addi ng capital, however, the new capital that is embed ded in new technologies can be introduced by using part of the excess capital. As a result, the amount of excess capital is gradually reduced as part of the process of technological progress . Of course, not all of the excess c apital can be easily replaced in each period, but most of it should be able to be replaced in the long run. With the gradual replacement of the excess capital through technological progress, the excess capital will eventually be fully eliminated and the government intervention will end. Note n evertheless that this elimination process will take a long time . In addition, the economic growth caused by technological progress will be slower because part of the increase in capital required by technological prog ress is being replaced with a reduction in excess capital . The economy will therefore grow more slowly because of the relatively slow er growth of capital . 5 DISCUSSION 5.1 Japan since the 1990s Japan has experienced low , occasionally negative, growth rates since the 1990s, even though the Japanese government has spent huge amounts of money to stabilize its economy by issuing similarly huge amounts of government bonds. At the same time , the debts of the Japanese government have greatly increased . Japan ’s experience seems to be very similar to the consequence s predicted when Option (2-2-a) is chosen . This similarity implies that the stagnation of the Japanese economy since the 1990s was caused by an upward RTP shock , and the Japanese government chose Opt ion (2 -2-a) as the countermeasure to the shock . Harashima (2016 ) examines this possibility theoretically and empirically and concludes that RTP RH of Japan rose 2 –3 percentage p oint s in the early 1990 s, and this upward shift of RTP RH was the cause of the stagnation of Japanese economy since the 1990s. 13 If the Japanese government had not chose n Option (2 -2-a) and had instead chose n Option (1), Japan would have experience d a significantly more severe recession , possibly similar to the Great Depression of the 1930s . Production would have decrease d and unemployment rates would have increase d far more than they did actually. Therefore, the Japanese government may be praised for ch oosing the best option when facing a large upward shift of RTP RH. However, the Ja panese government should keep in mind that Option (2 -2-a) is only the best option if the government properly increases taxes to redeem the debts at some point in the future. 5.2 The Great Depression and World War II Many hypotheses on the cause s of the Great Depression in the 1930s have been presented , but no consensus has be en reached. The phenomena observed during the Great Depression are very similar to those predicted when Option (1) is chosen ; that is, the growth rates were negative and unemploymen t rates rose sharply. In addition, this agonizing situation was prolong ed . Here, I have indicate d that the best option to tackle such a situation is to adopt Option (2 -2-a), but large discretionary fiscal interventions by government s were generally seen as taboo in th at period. Government expenditure s were increased only to a limited extent in the U nited States with the introduction of the New Deal , and the Great Depression persisted. However, the U.S. economy recovered in 1940s after government consumptio n was greatly increased to build up the military in the face of the outbreak of World War II . It is likely that the U.S. government unintentionally or compulsorily chose Option (2 -1-c) or (2 -2-c). Unemployment rate s decline d and destroying or disposing of resources stopped as predicted by both o ption s. In this case, it appears that the taboo against discretionary fiscal intervention was broken because of the threat and outbreak of a large -scale war. Similar phenomen a were observed in Germany. Germany was one of the hardest -hit economies by the Great Depression , but after the Nazi s took power in 1933 , the German economy recovered quickly and sharply. T he government of Nazi Germany significantly intervened in various aspects of the German economy. This inter vention eliminate d the large -scale Pareto inefficiency that was generated by the Great Depression. In particular, the German government greatly built up its military so it is likely that Option (2 -1-c) or (2 -2-c) was adopted to restore the German economy. 6 CONCLUDING REMARKS If the steady state is shifted downwards by a fundamental shock , each household must change its consumption path to one that diminish es gradually to the posterior steady state . Because consumption decreases, a recession begins. In this case, i f households increase t heir consumption discontinuously to a point on the posterior saddle path and then follow that to the posterior steady state, Pareto efficiency is held and unemployment rates do not rise . However, households will not behav e like this because it does not give them the highest expected utility. Households are risk averse and dislike unsmooth and discontinuous consumption. Instead, households will choose a Nash equilibrium of a Pareto inefficien t path as the optimal consumptio n path. Because of its Pareto inefficiency, the unemployment rate will increase sharply and stay high for a long period. In th is paper , I examine d whether discretionary fiscal policy is necessary if th is type of recession occurs, and if it is necessary, how it should be implemented. Particularly , the fiscal policy for a Nash equilibrium of a Pareto inefficient path caused by an upward shock on RTP was examined. In this case, a government has three options : (1) do not intervene, (2) increase government co nsumption, and (3) cut taxes. Option (2) has several subordinate options. I compare d and evaluate d these options and conclude d that increasing government consumption until the demand meets the present level of production and maintaining this fiscal policy is the best option. The a ccompanying huge government debts can be sustainable even though they are 14 extremely large if the government properly increases taxes in the future . In t his o ption , large government debts play an essential role as a buffer against the negative effects of the shock. 15 APPENDIX A Nash equilibrium of a Pareto inefficient path A1 Model with non -cooperative households 5 A1.1 The shock The model describes the utility maximizat ion of households after an upward time preference shock. This shock was chosen because it is one of the few shocks that result in a Nash equilibrium of a Pareto inefficient path. Another important reason for selecting an upward time preference shock is th at it shifts the steady state to lower levels of production and consumption than before the shock, which is consistent with the phenomena actually observed in a recession. Although the rate of time preference (RTP) is a deep parameter, it has not been regarded as a source of shocks for economic fluctuations, possibly because RTP is thought to be constant and not to shift suddenly. There is also a practical reason, however. Models with a permanently constant RTP exhibit excellent tractability (see Samuels on, 1937). However, RTP has been naturally assumed and actually observed to be time -variable. The concept of a time -varying RTP has a long history (e.g., B öhm -Bawerk, 1889; Fisher, 1930). More recently, Lawrance (1991) and Becker and Mulligan (1997) showed that people do not inherit permanently constant RTPs by nature and that economic and social factors affect the formation of RTPs . Their arguments indicate that many incidents can affect and change RTP throughout a person’s life. For example, Parkin (1988) examined business cycles in the United States, explicitly considering the time -variability of RTP , and showed that RTP was as volatile as technology and leisure preference. A1.2 Households Households are not intrinsically cooperative. Except in a stric t communist economy, households do not coordinate themselves to behave as a single entity when consuming goods and services.

The model in this paper assumes non -cooperative, identical , and infinitely long living households and that the number of households is sufficiently large. Each of them equally maximizes the expected utility , subject to , where yt, ct, and kt are production, consumption, and capital per capita in period t, respectively; A is techno logy and constant; u is the utility function; is the production function; is RTP ; δ is the rate of depreciation; and E0 is the expectations operator conditioned on the agents’ period 0 information set. yt, ct, and kt are monotonically continuous and differentiable in t, and u and f are monotonically continuous functions of ct and kt, re spectively. All households initially have an identical amount of financial assets equal to kt, and all households gain the identical amount of income in each period. It is assumed 5 The model in Appendix is based on the model by Harashima (2012). See also Harashima (2004, 2013b).   dt c u θt E t    0 0 exp   t t t t c δk k A, f dt dk     t t k A f y ,    > θ 0  t t k A f y ,  16 that and ; thus, households are risk averse. For simplicity, the utility function is specified to be the constant relative risk aversion utility function if if , where γ is a constant and . In addition, and . Both technology (A) and labor supply are assumed to be constant. The effects of an upward shift in RTP are shown in Figure A1. Suppose first that the economy is at steady state before the shock. After the upward RTP shock, the vertical line moves to the left (from the solid vertical line to the dashed vertical line in Fig . 1). To keep Pareto effic iency, consumption needs to jump immediately from the steady state before the shock (the prior steady state) to point Z. After the jump, consumption proceeds on the Pareto efficient saddle path after the shock (the posterior Pareto efficient saddle path) f rom point Z to the lower steady state after the shock (the posterior steady state). Nevertheless, this discontinuous jump to Z may be uncomfortable for risk -averse households that wish to smooth consumption and not to experience substantial fluctuations. H ouseholds may instead take a shortcut and, for example, proceed on a path on which consumption is reduced continuously from the prior steady state to the posterior steady state (the bold dashed line in Fig. 1), but this shortcut is not Pareto efficient. Choosing a Pareto inefficient consumption path must be consistent with each household’s maximization of its expected utility. To examine the possibility of the rational choice of a Pareto inefficient path, the expected utilities between the two options nee d be compared. For this comparison, I assume that there are two options for each non -cooperative household with regard to consumption just after an upward shift in RTP . The first is a jump option , J, in which a household’s consumption jumps to Z and then p roceeds on the posterior Pareto efficient saddle path to the posterior steady state. The second is a non -jump option , NJ , in which a household’s consumption does not jump but instead gradually decreases from the prior steady state to the posterior steady s tate, as shown by the bold dashed line in Figure A1. The household that cho oses the NJ option reaches the posterior steady state in period . The difference in consumption between the two options in each period t is bt (≥ 0). Thus, b0 indicates the difference between Z and the prior steady state. bt diminishes continuously and becomes zero in period s. The NJ path of consumption ( ct) after the shock is monotonically continuous and differentiable in t and if . In addition, if if , where is consumption when proceeding on the po sterior Pareto efficient saddle path and is consumption in the posterior steady state. Therefore, if if .   0 t t dc c du   0 2 2  t t dc c u d   γ c c u γt t    1 1 1γ    t t c c u ln 1γ  γ 0   0 ,    t t k k A f   0 2 2    t t k kf 0  dt dc t  0 s 0  dt dc t s t 0 t t c c c ˆ   s t   0 c ct t s 0 tcˆ c 0 ˆ    t t t c c b s t   0 0tb t s 0 17 It is also assumed that, when a household chooses a different option from the one the other households choose, the difference in the accumulation of financial assets resulting from the difference in consumption ( bt) before period s between that hou sehold and the other households is reflected in consumption after period s. That is, the difference in the return on financial assets is added to (or subtracted from) the household’s consumption in each period after period s. The exact functional form of t he addition (or subtraction) is shown in Section A1.4. A1.3 Firms Unutilized products because of bt are eliminated quickly in each period by firms because holding them for a long period is a cost to firms. Elimination of unutilized products is accomplish ed by discarding the goods or preemptively suspending production, thereby leaving some capital and labor inputs idle. However, in the next period, unutilized products are generated again because the economy is not proceeding on the Pareto efficient saddle path. Unutilized products are therefore successively generated and eliminated. Faced with these unutilized products, firms dispose of the excess capital used to generate the unutilized products . Disposing of the excess capital is rational for firms because the excess capital is an unnecessary cost, but this means that parts of the firms are liquidated, which takes time and thus disposing of the excess capital will also take time. If the economy proceeds on the NJ path (that is, if all households choose the NJ option), firms dispose of all of the remaining excess capital that generates bt and adjust their capital to the posterior steady -state level in period s, which also correspond s to households reaching the posterior steady state. Thus, if the economy proc eeds on the NJ path, capital kt is if if , where is capital per capita when proceeding on the posterior Pareto efficient saddle path and is capital per capita in the posterior steady state. The real interest rate it is . Because the real interest rate equals RTP at steady state, if the economy proceeds on the NJ path, if if , where is RTP before the shock and is RTP after the shock. is monotonically continuous and differentiable in t if . A1.4 Expected utility after the shock The expected utility of a household after the shock depends on its choice of the J or NJ path. Let Jalone indicate t hat the household chooses option J, but the other households choose option NJ ; NJalone indicate that the household chooses option NJ , but the other households choose option J; Jtogether indicate that all households choose option J; and NJtogether indicate that all t t k k k ˆ   s t   0 k kt  t s 0 tkˆ k   t t t k k A f i    , θ i θ t ~ s t   0 θ it  t s 0 θ~ θ ti s t 0 18 households choose option NJ . Let p (0 ≤ p ≤ 1) be the subjective probability of a household that the other households choose the J option (e.g., p = 0 indicates that all the other households choose option NJ ). With p, the expected utility of a hou sehold when it chooses option J is , (A1) and when it chooses option NJ is (NJalone )+ , (A2) where , , , and are the expected utilities of the household when choosing Jalone , NJalone , Jtogether , and NJtogether , respectively. Given the properties of J and NJ shown in Sections A1.2 and A1.3, , (A3) and , (A4) where , (A5) and , (A6) and the shock occurred in period t = 0. Figure A2 shows the paths of Jalone and NJalone . Because there is a sufficiently large number of households and the effect of an individual household on the whole economy is negligible, in the case of Jalone , the economy almost proceeds on the NJ path . Similarly , in the case of NJalone , it almost proceeds on the J path. If the other households choose the NJ option ( Jalone or NJtogether ), consumption after s is constant as and capital is adjusted to by firms in period s. In addition, at and it are constant after s such that at equals and is equals θ, because the economy is at the posterior steady state. Nevertheless, during the transition period before s, the value of it changes from the value of the prior RTP to that of the posterior RTP . If the other households choose option J (NJalone or Jtogethe r), however, consumption after s is and capital is not adjusted to by firms in period s and remains at . As mentioned in Section A1.2, the difference in the returns on financial assets for th e household from the returns for each of the other households is added to (or subtracted from) its         Jalone E p Jtogether pE J E 0 0 0 1     0 0 pE NJ E      NJtogether Ep 0 1   Jalone E0   NJalone E0   Jtogether E0   NJtogether E0                     s t s t t dt c u θt dt b cu θt pE J E ˆ exp exp0 0 0                       s s t t dt a cu θt dt b cu θt E p 0 0 exp exp 1                     s s t t t dt a cu θt dt c u θt pE NJ E 0 0 0 ˆ exp exp                     s s t dt cu θt dt cu θt E p exp exp 1 0 0    s s rq r dr dqi b θ a 0 exp    s s rq r t t dr dqi b i a 0 exp c k a tcˆ k tkˆ 19 consumption in each period after period s. This is described by at and in equations (A3) and (A4), and equations (A5) and (A6) indicate that the accumulated difference in financial assets resulting from bt increases by compound interest between the period r to s. That is, if the household takes the NJalone path, it accumulates more financial assets than each of the other J households, and instead of immediately consuming these extra accumulated financial assets after period s, the household consumes the returns on them in every subsequent period. If the household takes the Jalone path, however, its consumption after s is , as shown in equation (A3). is subtracted because the income of each household , , including the Jalone household, decreases equally by bt. Each of the other NJ households decreases consumption by bt at the sam e time, which compensates for the decrease in income; thus, its financial assets (i.e., capital per capita; kt) are kept equal to . The Jalone household, however, does not decrease its consumption, and its financial assets become small er than those of each of the other NJ households, which results in the subtraction of after period s. A2 Nash Equilibrium of Pareto Inefficiency Path 6 A2.1 Rational Pareto inefficient path A2.1.1 Rational choice of a Pareto ine fficient path Before examining the economy with non -cooperative households, I first show that, if households are cooperative, only option J is chosen as the path after the shock because it gives a higher expected utility than option NJ . Because there is no possibility of Jalone and NJalone if households are cooperative, then and . Therefore, > 0 because and . Next, I examine the economy with non -cooperative households. First, the special case with a utility function with a sufficiently small γ is examined. Lemma A1: If is sufficiently small, then . Proof: > 0 , 6 The idea of a rationally chosen Pareto inefficient path was originally presented by Harashima (2004). a a c  a  t t k A f y ,  tkˆ a     Jtogether E J E 0 0      NJtogether E NJ E 0 0      NJ E J E 0 0                                    s s t s t s t t dt cu θt dt cu θt E dt cu θt dt b cu θt E exp exp ˆ exp exp 0 0 0 0                             s t s t t t dt c u c u θt dt c u b c u θt E ˆ exp exp0 0 t t t b c c   tc c ˆ    γ γ 0     0 0 0   NJtogether E Jalone E       NJtogether E Jalone E γ 0 0 0 lim                                s s γ t t t γ dt cu a c u θt E dt cu b cu θt E 0 0 0 0 0 lim exp lim exp            s s t dta θt E dt b θt E 0 0 0 exp exp                      s s s s rq r t dt θt dr dqi b θ E dtb θt E exp exp exp0 0 0 0                s s s rq r t dr dqi b θs E dtb θt E 0 0 0 0 exp exp exp               s s tq t dt dqi t s θ b θs E 0 0 exp exp exp 20 because, if , then and . Hence, because , for sufficiently small γ. ■ Second, the oppo site special case (i.e., a utility function with a sufficiently large γ) is examined. Lemma A2: If is sufficiently large and if , then . Proof: Because , then for any period . On the other hand, because , then f or any period , if , . Thus, [E0 (Jalone ) – E0 (NJtogether )] . Because for any , then if , < 0 for sufficiently large . ■ The condition indicates that path NJ from c0 to deviates sufficiently from the posterior Pareto efficient saddle path and reaches the posterior steady state not taking much time. Because steady states are irrelevant to the degree of risk aversion ( γ), both c0 and are irrelevant to γ. By Lemmas A1 and A2, it can be proved that is possible. Lemma A3: If , then there is a such that if , s t 0 θ it        s tqdqi t s θ exp exp    t s θ  exp   s tqdqi exp     0 0 0   NJtogether E Jalone E   γ γ 0 1 lim 0    c a γ   Jalone E0   0 0  NJtogether E tb0       0 lim 1 lim 1 1 1                            γ t γ t t γ t t t γ γ c c c b c cu b cu c γ  s t a0  s t 1 lim 0    c a γ γlim             γ γ c u a c u c γ lim 1 1                 １ γ c a 1 1 γlim γ c γ   1 1        dt c u b c u θt c γ t t t γ s γ γ          lim exp 1 lim 0 1        dt c u a c u θt c γ γ s γ γ           lim exp 1 lim 1 0 0    0 1 1   γ c γ   γ γ1 1 lim 0    c a γ     NJtogether E Jalone E 0 0    γ 1 lim 0    c a γ c c c     0 0 0   NJtogether E Jalone E 1 lim 0    c a γ      γ γ 0   γ γ 21 . Proof: If is sufficiently small, then by L emma A1, and if is sufficiently large and if , then by Lemma A2. Hence, if , there is a certain such that, i f , then . ■ However, because both Jtogether and NJalone indicate that all the other households choose option J; thus, the values of it and kt are the same as those when all households proceed on the posterio r Pareto efficient saddle path. Faced with these it and kt, deviating alone from the Pareto efficient path ( NJalone ) gives a lower expected utility than Jtogether to the NJ household. Both Jalone and NJtogether indicate that all the other households choose option NJ and it and kt are not those of the Pareto efficient path. Hence, the sign of varies depending on the conditions, as Lemma A3 indicates. By Lemma A3 and the property , the possibility of the choice of a Pareto inefficient transition path, that is, , is shown. Proposition A1: If and , then there is a such that if , , a nd if , . Proof: By Lemma A3, if , then and . By equations (A1) and (A2), p + (1 - p) . Thus, if and , and . Hence, by the intermediate value theorem, there is such that if , and if , . ■ Proposition A1 indicates that, if , , and p < p*, then the choice of option NJ gives the higher expected utility than that of option J to a household; that is, a household may make the rational choice of takin g a Pareto inefficient transition path. The lemmas and proposition require no friction, so a Pareto inefficient transition path can be chosen even in a frictionless economy. This result is very important because it offers counter -evidence against the conje cture that households never rationally choose a Pareto inefficient transition path in a frictionless economy. A2.1.2 Conditions for a rational Pareto inefficient path The proposition requires several conditions. Among them, may app ear rather strict. If γ* is very large, path NJ will rarely be chosen. However, if path NJ is such that consumption is reduced sharply after the shock, the NJ option yields a higher expected utility than the J     0 0 0   NJtogether E Jalone E  0 γ     0 0 0   NJtogether E Jalone E   γ 1 lim 0    c a γ     NJtogether E Jalone E 0 0  0 1 lim 0    c a γ      γ γ 0   γ γ     0 0 0   NJtogether E Jalone E     0 0 0   NJalone E Jtogether E     NJtogether E Jalone E 0 0      0 0 0   NJalone E Jtogether E     0 0 0   NJ E J E 1 lim 0    c a γ   γ γ  1 0     p p *p p     0 0 0   NJ E J E *p p     0 0 0   NJ E J E   γ γ     0 0 0   NJtogether E Jalone E   Jtogether E0   0 0   NJalone E      NJ E J E 0 0       NJalone E Jtogether E 0 0        NJtogether E Jalone E 0 0  1 lim 0    c a γ   γ γ       NJ E J E p 0 0 0 lim       0 0 0    NJtogether E Jalone E           0 lim 0 0 0 0 1      NJalone E Jtogether E NJ E J E p  1 0     p p *p p     0 0 0   NJ E J E *p p     0 0 0   NJ E J E 1 lim 0    c a γ   γ γ   γ γ 22 option even though γ is very small. For exampl e, for any , [E0 (Jalone ) – E0 (NJtogether )] , because and because . That is, for each combination of path NJ and γ, there is such that, if , then . Consider an example in which path NJ is such that bt is constant and before s (Figure A3); thus , . In this NJ path, consumption is reduced more sharply than it is in the case shown in Figure A2. In this case, because , , and for , then , and in addition, . Hence, E0 (Jalone ) – E0 (NJtogether ) . As γ increases , the ratio decreases ; thus, larger values of s can satisfy . For example, suppose that = 10, cs = 10.2, = 0.3, and θ   γ γ 0 0 lims s 1                dt c u a c u θt s dt c u b c u θt s s s t t t s s              exp 1 lim exp 1 lim 0 0 0               cd c du b c u b c u s b s c u c u c u b c u s 0 0 0 0 0 0 0 0 0 lim1                  0 1 1 1 0 10 10 0 0 10 10 0                              b γ c γ b c c c cb γ c b c γ γ γ γ γ γ γ         0 0 0 0 0 0 10 10 0 1 1 ln ln ln 1 1 lim b c b c c b c c γ c γ b c c γ γ γ γ                          0 1 1 1 lim 1 1 lim 1 0 0 10 10 10 0                                         γ c b c c γ c γ b c c γ γ γ γ γ γ γ γ 0c c  0s  s s   Jalone E0   0 0   NJtogether E b bt   s t bs b E 0 0    s t b θs b θ E a 0 0 γ0 t s c c  s t                          s s s s t t t c u b cu dt θt E dt cu b cu θt E 0 0 0 0 exp exp        s s c u b cu θ θs E    exp 1 0               s dt cu a cu θt E exp 0                         cu b θs cu θ θs E cu a cu θ θs E cu a cu dt θt E s              exp exp exp 0 0 0                            s s t t t dt cu a cu θt E dt cu b c u θt E exp exp 0 0 0                 cu b θs cu θ θs E cu b cu θ θs E s s          exp exp 1 0 0                                    b θs cu cu θs θs cu b cu θ θs E s s exp 1 exp exp 1 0        b θs cu c u c u b cu s s         0 0 0   NJtogether E Jalone E c b 23 = 0.05. If , the n s* = 1.5 at the minimum, and if , then s* = 6.8 at the minimum. This result implies that, if option NJ is such that consumption is reduced relatively sharply after the shock (e.g., ) and , option NJ will usually be chosen. Choosing option NJ is not a special case observed only if γ is very large, but option NJ can normally be chosen when the value of γ is within usually observed values. Conditions for generating a rational Pareto inefficient transition path therefore are not strict. In a recession, consumption usually declines shar ply after the shock, which suggests that households have chosen the NJ option. A3 Nash equilibrium A3.1 A Nash equilibrium consisting of NJ strategies A household strategically determines whether to choose the J or NJ option, considering other househol ds’ choices. All households know that each of them forms expectations about the future values of its utility and makes a decision in the same manner. Since all households are identical, the best response of each household is identical. Suppose that there a re identical households in the economy where H is sufficiently large (as assumed in Section A1). Let be the probability that a household chooses option J. The average utility of the other households almost equals that of all households because H is sufficiently large. Hence, the average expected utilities of the other households that choose the J and NJ options are E0(Jtogether ) and E0(NJtogether ), respectively. Hence, the payoff matrix of the Η-dimensional symmetric mixed strategy game can be described as shown in Table A1. Each identical household determines its behavior on the basis of this payoff matrix. In this mixed strategy game, the strategy profiles (q1,q2,…, qH) = {(1,1,…,1), ( ), (0,0,…,0)} are Nash equilibria for the following reason. By Proposition A1, the best response of household η is J (i.e., qη = 1) if , indifferent between J and NJ (i.e., any ) if , and NJ (i.e., qη = 0) if . Because all households are identical, the best -response correspondence of each household is identical such that qη = 1 if , [0,1] if , and 0 if for any household . Hence, the mixed strategy profiles (1, 1,…,1), ( ), and (0,0,…,0) are the intersections of the graph of the best -response correspondences of all households. The Pareto effi cient saddle path solution (1,1,…,1 ) ( i.e., Jtogether ) is a pure strategy Nash equilibrium, but a Pareto inefficient transition path (0,0,…,0 ) ( i.e., NJtogether ) is also a pure strategy Nash equilibrium. In addition, there is a mixed strategy Nash equilib rium ( ). A3.2 Selection of equilibrium Determining which Nash equilibrium, either NJtogether (0,0,…,0) or Jtogether (1,1,…,1), is dominant requires refinements of the Nash equilibrium, which necessitate additional criteria.

Here, i f households have a risk -averse preference in the sense that they avert the worst scenario when its probability is not known, households suppose a very low p and select the NJtogether (0,0,…,0) equilibrium. Because E0 (Jalone ) – E0 (NJalone ) 1γ 5γ b bt *p p  N Η   1 0   η η q q   Η η  * * * ,..., , p p p *p p  10, qη *p p *p p *p p *p p *p p Η η * * * ,..., , p p p * * * ,..., , p p p                   dt a cu a cu θt dt cu b cu θt E s s t t t t t             0 0 ˆ exp exp                              s s t t t dt c u a c u θt dt c u b c u θt E 0 0 exp exp 24 = E0 (Jalone ) – E0 (NJtogether ) < 0 , (A7) by Lemma A3, Jalone is the worst choice in terms of the amount of payoff, followed by NJtogethe r, and NJalone , and Jtogether is the best. The outcome s of choosing option J are more dispersed than th ose of option NJ . If households have a risk -averse preference in the above -mentioned sense and avert the worst scenario when they have no information on its probability, a household will prefer the less dispersed option ( NJ ), fearing the worst situation that the household alone substantially increases consumption while the other households substantially decrease consumption after the shock. This behavior i s rational because it is consistent with preferences. Because all households are identical and know inequality (A7), all households will equally suppose that they all prefer the less dispersed NJ option; therefore, all of them will suppose a very low p, pa rticularly , and select the NJtogether (0,0,…,0) equilibrium, which is the Nash equilibrium of a Pareto inefficient path. Thereby, unlike most multiple equilibria models, the problem of indeterminacy does not arise, and “animal spirit s” (e.g., pessimism or optimism) are unnecessary to explain the selection. A4 Amplified generation of unutilized resources A Nash equilibrium of a Pareto inefficient path successively generates unutilized products because of bt. They are left unused, dis carded, or preemptively not produced during the path. Unused or discarded goods and services indicate a decline in sales and an increase in inventory for firms. Preemptively suspended production results in an increase in unemployment and idle capital. As a result, profits decline and some parts of firms need to be liquidated, which is unnecessary if the economy proceeds on the J path (i.e., the posterior Pareto efficient path). If the liquidation is implemented immediately after the shock, unutilized produc ts because of bt will no longer be generated, but such a liquidation would generate a tremendous shock. The process of the liquidation, however, will take time because of various frictions, and excess capital that generates unutilized products because of bt will remain for a long period. 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Wolfe ed.), University of Edinburgh Press, Edinburgh, Scotland. 28 kt ct Steady state before the shock on θ Steady state after the shock on θ Line of after the shock on θ Line of Z Pareto inefficient transition path Line of before the shock on θ Pareto efficient saddle path after the shock on θ Pareto efficient saddle pat h before the shock on θ Figure A1: A time preference shock 0 0  dt dc t 0  dt dk t 0  dt dc t 29 Figure A2: The paths of Jalone and NJalone Path of NJalone Path of Jalone 0 s t ct a c a c c 0c 0 0 b c  30 Figure A3: A Pareto inefficient transition path Posterior Pareto efficient saddle path Path of NJtogether 0 s t ct c 0c 0 0 b c  31 Table A1 The payoff matrix Any other household J NJ Household A J E0(Jtogether ), E 0(Jtogether ) E0(Jalone ), E 0(NJtogether ) NJ E0(NJalone ), E 0(Jtogether ) E0(NJtogether ), E 0(NJtogether )