**1. ** Read the first 13 pages of the attached paper which discusses the effect of government intervention on recessions.

2. Locate two JOURNAL articles which discuss this topic further. You need to focus on the Abstract, Introduction, Results, and Conclusion. For our purposes, you are not expected to fully understand the Data and Methodology.

3. Summarize these journal articles. Please use your own words. No copy-and-paste. Cite your sources.

Need 500 words

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MunichPersonal RePEc Archive

Should a Government Fiscally Intervene

in a Recession and, If So, How?

Harashima, Taiji

Kanazawa Seiryo University

2 April 2017

Online at https://mpra.ub.uni-muenchen.de/78053/

MPRA Paper No. 78053, posted 31 Mar 2017 09:03 UTC

Should a Government Fiscally Intervene in a Recession and, If So, How?

Taiji HARASHIMA*

April, 2017

Abstra

ct

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. However, they will not “jump” consumption in this manner and

instead will choose a “Nash equilibrium of a Pareto inefficient path” because they dislike

unsmooth and discontinuous consumption and behave strategically. The paper concludes that

increasing government 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, 10-1 Goshomachi-Ushi,

Kanazawa-shi, Ishikawa, 920-8620, Japan.

Email: harashim@seiryo-u.ac.jp or t-harashima@mve.biglobe.ne.jp.

mailto:harashim@seiryo-u.ac.j

p

mailto:t-harashima@mve.biglobe.ne.jp

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 fiscally 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 and therefore arguments 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.

Recessions 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

downwards (i.e., to lower levels of production and consumption), households must change the

consumption path to one that diminishes gradually to the posterior steady state. Therefore,

growth rates become negative; that is, a recession begins. However, the explanation of the

mechanism of this type of recession is not perfect because an important question still needs to

be answered. If households discontinuously increase (“jump up”) their consumption from the

prior steady state to a point on the posterior saddle path and then gradually move to the posterior

steady state, Pareto efficiency is held and thereby unemployment rates do not rise. Therefore,

even in a serious and large-scale recession, unemployment does not increase. This is a very

unnatural outcome of a serious recession.

Harashima (2004, 2009, 2013a) showed a mechanism by which households do not

jump up their consumption even if the steady state is changed downward because they are

intrinsically risk averse and non-cooperative and want to smooth consumption. The

consumption jump does not give them the highest expected utility; that is, unsmooth and

discontinuous consumption is not optimal for households. Hence, instead 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 rates 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 inefficient path.

Fundamental shocks that change the steady state basically mean shocks on deep

parameters. A representative fundamental shock, an upward shock on the 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 outcomes are evaluated 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 accompanied by large and accumulating government debts, but

these 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 against 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, particularly 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 an upward RTP shock could explain the occurrence of the Great

Recession and showed evidence that the estimated RTP of the United States increased in about

2008.

RTP plays an essential role in economic activities, and its importance has been

emphasized since the era of Irving Fisher (Fisher, 1930). One of the most important equations in

economics is the steady state condition

rθ

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 changeable 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

expected RTP of the representative household (RTP RH) (Harashima, 2014, 2016). That is,

changes in RTP and the expected RTP RH can be an important source of economic fluctuations.

2.2 The model

The model in this paper is based on the models in Harashima (2004, 2009, 2013a) and assumes

non-cooperative, identical, and infinitely long living households, and that the number of

households is sufficiently large. Each of the households equally

maximizes the expected utility

dtcuθtE

t

0

0

exp

subject to

t

t

t cA,kf

dt

d

k

,

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;

tt

kAfy , is the production

function; and E

0

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 all households gain the identical amount of income

tt

kAfy , in each period. It is assumed

that

0

t

t

d

c

cdu

and

0

2

2

t

t

dc

cud

; thus, households

are risk averse. In addition,

0

,

t

t

k

kAf

and

0

2

2

t

t

k

kf

. 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 there is no depreciation of capital.

2.3 A Nash equilibrium of a Pareto inefficient path

3

t

k

The prior steady state

before the shock on θ: W

The posterior steady

state after the shock

θ

Posterior line of

0

dt

dc

t

after

the shock

on θ

Line of 0

dt

dk

t

Z

Pareto inefficient transition

path

Prior line of 0

dt

dc

t

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 before the shock. After the upward RTP shock, the vertical line 0

dt

dc

t 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 result, 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 between 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: An upward RTP shock. All terms are defined in the text.

t

c

1 If depreciation of capital is assumed to exist, the “consumption” 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

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, 2013a) and also in the Appendix. Because households are

risk averse and want to smooth consumption, and are also intrinsically non-cooperative, 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 the 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 posterior 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

INTERVENE?

3.1 The government’s options

3.1.1 The three options

When households choose a Nash equilibrium of a Pareto inefficient path, the government

basically has three options: (1) do not intervene, (2) increase government consumption, and

(3) cut taxes.

If Option (1) is chosen, the gap between the posterior saddle path and the Pareto

inefficient transition path (initially the gap between points Z and W) is not filled by any demand.

Therefore, unemployment rates increase sharply and huge amounts of resources are destroyed or

left idle. High unemployment rates and destruction of resources will continue until the economy

reaches the posterior

steady state.

If Option (2) is chosen, government consumption is increased to fill the demand gap

between the posterior saddle path and the Pareto inefficient transition path, where government

consumption is indicated on a per capita basis similar to the other variables. Suppose for

simplicity that government consumption 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 expenditures, for example, spending on various kinds of administrative

services that households receive. Therefore, increases in government consumption can be

substituted for decreases in household consumption. Nevertheless, government consumption

will not directly generate utility 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 proceeds on a Pareto inefficient 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 increase, but if the

Ricardian equivalence holds, they will still proceed on a Pareto inefficient 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, if the Ricardian equivalence does

not hold, tax cuts may increase household consumption at least temporarily. Therefore, the

validity of Option (3) depends 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 tax cuts are effective, they will be effective only in the short run, and these

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), an increase in

the government consumption is financed by borrowing from or tax increases on households.

Nevertheless, financing 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, increases in taxes 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. Options (2) and (3) assume that necessary future tax increases are fully

implemented by the government.

In addition, it is assumed that a government borrows money only from its own people,

that is, not from foreigners because foreign borrowing means that foreigners also intervene in

addition to the government, and such intervention is beyond the scope of this paper.

3.2 Comparison among options

(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 subordinate option of Option (2), in which, as 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 growth will not necessarily be negative or zero and may even be low but positive.

(2) Household utility

Households choose a Nash equilibrium of a Pareto inefficient path equally in the three options.

Therefore, the utilities of households are basically same in the three options.

(3) Unemployment

In Options (1) and (3), unemployment rates 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 resources are destroyed or left

idle.

(4) Government debt

In Option (1), government debt does not increase because the government does not borrow

additional money, but in Options (2) and (3), government debt will increase because of

continuous financing by borrowing. However, if taxes are raised properly to pay off the debt in

the future, government debt will stabilize in some future

period.

3.3 Government debt

6

3.3.1 Is the government debt sustainable?

The usual arguments 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 an “unusual” Pareto inefficient transition path, so an alternative approach 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 assets of households, and the other household assets (other than dt) are ignored for

simplicity. In the future, dt is redeemed with interest, but the redemption takes 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),

ttt gcy , (1)

and in Option (3),

ttt gcy (2)

for any t because no new investment is made in Options (2) and (3) and the household assets

other than the government bonds are ignored; yt and ct are per capita income and consumption

of households in period t. If the condition

t

ttt

zgdr (3)

is satisfied indefinitely in a certain future period, government debt never explodes; that is, it is

sustainable where 10 tt rr is the real interest rate. By equality (1) and inequality (3), the

condition for sustainability in Option (2) is

tttttt

zdrcy . (4)

By inequalities (2) and (3), if inequality (4) is satisfied indefinitely in a certain future period,

government debt is also sustainable in Option (3).

Because the household assets other than dt are ignored, the sum of a household’s

income and assets is

ttt cyd

.

If the sum of a household’s income and assets exceeds zt, that is, if

tttt

cydz , (5)

then zt can be imposed in the sense that households have enough resources to fully pay taxes.

Hence, by inequalities (4) and (5), if

ttt ddr (6)

7

is satisfied, taxes that satisfy the condition for sustainable debts can be imposed. Here, because

10

t

r , then inequality (6) always holds. Therefore, for any dt, there always exists zt that

satisfies inequality (3) indefinitely in a certain future period. That 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 debts are eliminated completely all at

once. That is, any dt can be sustainable.

Such an extreme tax will not actually be imposed, but if dt exceeds a certain amount

such that

ttt zrdy ,

(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 the 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 used for consumption even though dt may be extremely large. In addition,

because any dt can be sustainable, the tax increase can be started even after all the excess capital

is eliminated. Hence, a huge amount of government debt can remain even if there is no excess

capital.

Finally, it is important to note that the increased tax revenues should not be used to

finance increases in government consumption for purposes other than dealing with the excess

capital. The increased taxes should be used only to pay 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 ttt zdr is

satisfied indefinitely in a certain future period. The question arises, however, when is “a certain

future period”? The time at which taxes are raised is indeterminate in the discussion in the

previous section. The tax increase can be postponed almost indefinitely if taxes will certainly be

raised eventually. This indeterminacy may generate a political struggle because people

intrinsically dislike tax increases, and opposition parties will utilize people’s anti-tax sentiment

as ammunition to attack the government. Opposition parties 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 people that the current level of government debt is unsustainable,

because any dt can be sustainable. The incentive for the government to raise taxes to reduce dt

will therefore be weak.

Is there a problem, however, 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 larger, a large dt will not actually be sustainable. One possible

factor that may change as dt becomes larger is uncertainty. If the tax increase has been

postponed for a long period, questions about the ability of the government to govern the nation

and run the economy will arise. Faced with an extremely large dt, people may begin to suspect

that their government cannot 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 performance in the future is no longer a certainty.

It has been argued that good institutions, including governments, 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. (2005) conclude that differences in economic institutions are empirically and theoretically

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 productivity, that is, levels

of production and consumption. Therefore, if uncertainty about the ability of a government

increases, household’s expected variances of production and consumption will also increase.

Larger variances of production and consumption mean more uncertainty about the entire future

economy. That is, as dt increases, household uncertainty about the entire future economy

increases.

An important consequence of increases in uncertainty about the entire future economy

is an increase in household RTP. The concept of a temporally varying RTP has a long history

(e.g., Böhm-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

where

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 decrease 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.

Considering 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 households 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 timing and scale of tax increases.

When the long run negative effect of a huge dt on the expected household utility

becomes larger than the 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, if 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 for 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 deviation 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

increases, the amount of necessary future tax increases (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 types of taxes both practically and politically.

Nevertheless, an 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, 1998; Cochrane, 2005; Woodford, 2001). However, Harashima (2006)

showed that the relation 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 government can implement Option (2) without

worrying about an outbreak 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 household 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

Options (1) and (3), but not in Option (2). As argued in Section 3.3, the extra government debts

are sustainable if the government properly increases taxes in the future. If the future tax increase

is properly implemented, therefore, Option (2) is favorable to Options (1) and (3) because

unemployment rates do not rise.

4 HOW SHOULD 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): Increase government consumption in order for the combined consumption to

jump up to point Z and then proceed on the posterior saddle path to the posterior steady state.

Option (2-2): Increase 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) indicates 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) indicates 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, as noted

in Section 3.1.1, excess resources cannot be used for investments because the economy 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 it is highly likely that households will still proceed on a

Pareto inefficient transition path because the households’ expected utilities are not affected by

the increase in government consumption. Here, a gap between the posterior saddle path (the thin

dashed curve in Figure 1) and production (the bold solid curve) for each kt indicates excess

capital. Excess capital needs 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 needs to be increased not only to

consume excess capital but also to substitute for a reduction in household consumption that is

the source of the excess capital. That is, the government needs 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 corresponds to depreciated capital inputs and an increase in consumer goods and services.

10

production and the Pareto inefficient transition path while the economy proceeds 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: Options

(2-1-a), (2-1-b), and (2-1-c).

The easiest way for a government to consume the excess 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 laborers 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 purchased 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 Option (2-2-a) is different from destroying them

in Option (1) because the 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 options, but nothing remains in the hands of the former owners or the

government in Option (1), whereas financial assets and debts remain in the hands of the former

owners and government, respectively, in Option (2-1-a).

Another way to consume the excess resources is to export them 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 countries 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 countermeasures, 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. With minor modifications, capital

inputs can be used to produce 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), it is highly likely that households also proceed on

a Pareto inefficient transition path in Option (2-2). When households proceed on this path, if the

government does nothing, a part of the capital that is used to produce products corresponding to

households’ reduction in consumption becomes excess capital and will be destroyed, but if the

government purchases and consumes these unconsumed products, the capital need not be

destroyed and the level of capital will remain the same in the next period. If the government

purchases and consumes 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 keeping so-called “zombie” firms

alive. As in Option (2-1), high 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 Options (2-1) and (2-2) is that, unlike Option (2-1),

capital is not consumed by the government in Option (2-2), but households’ reduction 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., capital).

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 Options (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 buildup (Option (2-2-c)). However, for the same reasons as given for Options

(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,

commonalities and differences between the two options are as follows.

(1) Period of government intervention

In Option (2-1-a), excess capital decreases 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 approaches the posterior

steady state. Hence, the government intervention never ends.

(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 larger in Option (2-1-a) than in Option (2-2-a).

However, in Option (2-1-a), excess capital gradually decreases and eventually reaches 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 rates during the transition

In Option (2-1-a), the growth rates are negative, whereas in Option (2-2-a), they are zero.

(4) Household utility

In both options, 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 stream of consumption after the shock are almost the same in

both cases.

(5) Unemployment

In both options, unemployment rates do not increase.

(6) Government debt

In both options, a large amount of government debt accumulates. However, if the government

properly increases taxes in the future, the debt will stabilize at some level in both options.

Although the period and scale of government interventions differ between the two options, these

differences basically do not matter to household optimality. Therefore, because the only

difference in the evaluated criteria is that growth rates are higher 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 best, it has its drawbacks. Huge amounts of resources need to be

disposed of in the name of the government consumption forever. Although this 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 way 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, technological progress basically requires additional increases in

capital. Instead of adding capital, however, the new capital that is embedded 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 capital 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 nevertheless 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 progress is being replaced with a reduction in excess capital. The

economy will therefore grow more slowly because of the relatively slower 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

consequences 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 Option (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 points in the early 1990s, 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 chosen Option (2-2-a) and had instead chosen

Option (1), Japan would have experienced a significantly more severe recession, possibly

similar to the Great Depression of the 1930s. Production would have decreased and

unemployment rates would have increased far more than they did actually. Therefore, the

Japanese government may be praised for choosing the best option when facing a large upward

shift of RTP RH. However, the Japanese 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 causes of the Great Depression in the 1930s have been presented, but

no consensus has been 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

unemployment rates rose sharply. In addition, this agonizing situation was prolonged. Here, I

have indicated that the best option to tackle such a situation is to adopt Option (2-2-a), but large

discretionary fiscal interventions by governments were generally seen as taboo in that period.

Government expenditures were increased only to a limited extent in the United States with the

introduction of the New Deal, and the Great Depression persisted.

However, the U.S. economy recovered in 1940s after government consumption 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 rates declined and destroying or disposing of resources stopped as predicted by

both options. 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 phenomena were observed in Germany. Germany was one of the hardest-hit

economies by the Great Depression, but after the Nazis took power in 1933, the German

economy recovered quickly and sharply. The government of Nazi Germany significantly

intervened in various aspects of the German economy. This intervention eliminated 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 diminishes gradually to the posterior steady state. Because

consumption decreases, a recession begins. In this case, if households increase their

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 behave 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 inefficient path as the optimal

consumption path. Because of its Pareto inefficiency, the unemployment rate will increase

sharply and stay high for a long period.

In this paper, I examined whether discretionary fiscal policy is necessary if this 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 consumption, and (3) cut taxes. Option (2) has several subordinate options. I

compared and evaluated these options and concluded that increasing government consumption

until the demand meets the present level of production and maintaining this fiscal policy is the

best option. The accompanying huge government debts can be sustainable even though they are

14

extremely large if the government properly increases taxes in the future. In this option, 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 maximization 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 that 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 Samuelson, 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., Böhm-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 strict 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

dtcuθtE

t

0

0

exp ,

subject to

ttt

t cδkkA,f

dt

dk

,

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;

tt

kAfy , is the production function;

>θ 0 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, respectively.

All households initially have an identical amount of financial assets equal to kt, and all

households gain the identical amount of income

tt

kAfy , in each period. It is assumed

5 The model in Appendix is based on the model by Harashima (2012). See also Harashima (2004, 2013b).

16

that

0

t

t

dc

cdu

and

0

2

2

t

t

dc

cud

; thus, households are risk averse. For simplicity, the utility

function is specified to be the constant relative risk aversion utility function

γ

c

cu

γ

t

t

1

1

if 1γ

tt

ccu ln if 1γ ,

where γ is a constant and γ0 . In addition, 0,

t

t

k

kAf

and

0

2

2

t

t

k

kf

. 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

0

dt

dc

t moves to the left (from the solid vertical line to the dashed vertical line in Fig. 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 after the shock (the posterior Pareto efficient saddle path) from 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. Households 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 need 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 proceeds 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 state, as shown by the bold dashed line in Figure A1. The

household that chooses the NJ option reaches the posterior steady state in period 0s . 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 0

dt

dc

t if st 0 . In addition,

tt

ccc ˆ if st 0

cc

t

if ts 0 ,

where

t

ĉ is consumption when proceeding on the posterior Pareto efficient saddle path and

c

is consumption in the posterior steady state. Therefore,

0ˆ

ttt

ccb if st 0

0

t

b if ts 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 household 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 the 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

accomplished 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

corresponds to households reaching the posterior steady state. Thus, if the economy proceeds on

the NJ path, capital kt is

tt

kkk ˆ if st 0

kk

t

if ts 0 ,

where tk̂ is capital per capita when proceeding on the posterior Pareto efficient saddle path

and k is capital per capita in the posterior steady state.

The real interest rate it is

t

t

t

k

kAf

i

,

.

Because the real interest rate equals RTP at steady state, if the economy proceeds on the NJ

path,

θiθ

t

~

if st 0

θi

t

if ts 0 ,

where θ

~

is RTP before the shock and θ is RTP after the shock.

t

i is monotonically

continuous and differentiable in t if st 0 .

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 that 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

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 household when it chooses option J is

JaloneEpJtogetherpEJE

0

00

1 , (A1)

and when it chooses option NJ is

00

pENJE (NJalone)+ NJtogetherEp

0

1 , (A2)

where JaloneE

0

, NJaloneE

0

,

JtogetherE

0

, and NJtogetherE

0

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,

s

t

s

tt

dtcuθtdtbcuθtpEJE ˆexpexp

0

00

s

s

tt

dtacuθtdtbcuθtEp

0

0

expexp1 , (A3)

and

s

s

ttt

dtacuθtdtcuθtpENJE

0

00

ˆ

expexp

s

s

t

dtcuθtdtcuθtEp expexp1

0

0

, (A4)

where

s s

r

qr

drdqibθ

a

0

exp , (A5)

and

s s

r

qrtt

drdqibia

0

exp , (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 c and capital is adjusted to k by firms in period s. In addition, at and it are constant

after s such that at equals a 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

Jtogether), however, consumption after s i

s

t

ĉ and capital is not adjusted to k by firms in period s

and remains at tk̂ .

As mentioned in Section A1.2, the difference in the returns on financial assets for the

household from the returns for each of the other households is added to (or subtracted from) its

19

consumption in each period after period s. This is described by at and a 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 ac , as shown in

equation (A3). a is subtracted because the income of each household,

tt

kAfy , , including

the Jalone household, decreases equally by bt. Each of the other NJ households decreases

consumption by bt at the same time, which compensates for the decrease in income; thus, its

financial assets (i.e., capital per capita; kt) are kept equal to tk̂ . The Jalone household, however,

does not decrease its consumption, and its financial assets become smaller than those of each of

the other NJ households, which results in the subtraction of a 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 inefficient 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 JtogetherEJE

00

and NJtogetherENJE

00

.

Therefore,

NJEJE

00

s

s

t

s

t

s

tt

dtcuθtdtcuθtEdtcuθtdtbcuθtE expexpˆexpexp

0

0

0

0

s t

s

ttt

dtcucuθtdtcubcuθtE ˆexpexp

0

0

> 0

because

ttt

bcc and

t

cc ˆ .

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 γγ 0 is sufficiently small, then

0

00

NJtogetherEJaloneE .

Proof: NJtogetherEJaloneE

γ 00

0

lim

s

s γtttγ

dtcuacuθtEdtcubcuθtE

0 0

0

0

0

limexplimexp

s

s

t

dtaθtEdtbθtE

0

00

expexp

s

s s s

r

qrt

dtθtdrdqibθEdtbθtE expexpexp

0 0

0

0

s s s

r

qrt

drdqibθsEdtbθtE

0 0

00

expexpexp

s st qt dtdqitsθbθsE 00 expexpexp > 0 ,

6 The idea of a rationally chosen Pareto inefficient path was originally presented by Harashima (2004).

20

because, if st 0 , then θi

t

and

s

t

q

dqitsθ expexp . Hence, because tsθ exp

s

t

q

dqiexp , 000 NJtogetherEJaloneE for sufficiently small γ. ■

Second, the opposite special case (i.e., a utility function with a sufficiently large γ) is

examined.

Lemma A2: If γγ 0 is sufficiently large and if

1lim0

c

a

γ

, then JaloneE

0

0

0

NJtogetherE .

Proof: Because

t

b0 , then

0lim1

lim

11

1

γ

t

γ

tt

γtttγγ c

c

c

bc

cubcu

c

γ

for any period st . On the other hand, because a0 , then for any period st , if

1lim0

c

a

γ

,

γ

l i m

γγ

cuacu

c

γ

lim

1

1

１

γ

c

a

1

1 .

Thus,

γ

l i m

γ

c

γ

1

1

[E0 (Jalone) – E0 (NJtogether)]

dtcubcuθt

c

γ

tttγ

s

γγ

limexp

1

lim

01

dtcuacuθt

c

γ

γsγγ

limexp

1

lim

1

00 .

Because 0

1

1

γ

c

γ

for any γγ 1 , then if 1lim0

c

a

γ

, NJtogetherEJaloneE

00

< 0 for sufficiently large γ . ■

The condition 1lim0

c

a

γ

indicates that path NJ from c0 to c deviates sufficiently from the

posterior Pareto efficient saddle path and reaches the posterior steady state c not taking much

time. Because steady states are irrelevant to the degree of risk aversion (γ), both c0 and c are

irrelevant to γ.

By Lemmas A1 and A2, it can be proved that 0

00

NJtogetherEJaloneE is

possible.

Lemma A3: If 1lim0

c

a

γ

, then there is a γγ 0 such that if γγ ,

21

0

00

NJtogetherEJaloneE .

Proof: If 0γ is sufficiently small, then 0

00

NJtogetherEJaloneE by Lemma A1,

and if γ is sufficiently large and if 1lim0

c

a

γ

, then NJtogetherEJaloneE

00

0 by Lemma A2. Hence, if 1lim0

c

a

γ

, there is a certain γγ 0 such that, if

γγ , then 0

00

NJtogetherEJaloneE . ■

However, 0

00

NJaloneEJtogetherE 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 posterior 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 NJtogetherEJaloneE

00

varies depending on the conditions, as Lemma A3

indicates.

By Lemma A3 and the property 0

00

NJaloneEJtogetherE , the possibility of

the choice of a Pareto inefficient transition path, that is, 0

00

NJEJE , is shown.

Proposition A1: If 1lim0

c

a

γ

and γγ , then there is a 10 pp such that if

*

pp , 0

00

NJEJE , and if *pp , 0

00

NJEJE .

Proof: By Lemma A3, if γγ , then 0

00

NJtogetherEJaloneE and

JtogetherE

0

0

0

NJaloneE . By equations (A1) and (A2),

NJEJE

00

p NJaloneEJtogetherE

00

+ (1 – p) NJtogetherEJaloneE

00

.

Thus, if 1lim0

c

a

γ

and γγ , NJEJE

p

00

0

lim

0

00

NJtogetherEJaloneE and

0lim

0000

1

NJaloneEJtogetherENJEJE

p

. Hence, by the intermediate value

theorem, there is 10 pp such that if *pp , 0

00

NJEJE and if *pp ,

0

00

NJEJE . ■

Proposition A1 indicates that, if 1lim0

c

a

γ

, γγ , 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 taking 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 conjecture 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, γγ may appear 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

22

option even though γ is very small. For example, for any γγ 0 ,

0

lim

s s

1

[E0 (Jalone) – E0 (NJtogether)]

dtcuacuθt

s

dtcubcuθt

s ss

ttt

s

s

exp

1

limexp

1

lim

000

cd

cdu

bcubcu

s

bscucu

cubcu

s

0000

0

0

000

lim

1

0

111

0

1

0

1

00

0

1

0

1

00

b

γ

c

γ

bc

cccb

γ

cb

c

γγ

γγγ

γγ

,

because

0

0

0

000

1

0

1

00

1

1lnlnln

11

lim b

c

b

ccbcc

γ

c

γ

bc

c

γγ

γ

γ

and

0

1

11

lim

11

lim

1

0

0

1

0

1

0

1

00

γ

c

b

cc

γ

c

γ

bc

c

γ

γγ

γ

γγ

γ

γ

because

0

cc . That is, for

each combination of path NJ and γ, there is 0s such that, if ss , then JaloneE

0

0

0

NJtogetherE .

Consider an example in which path NJ is such that bt is constant and bbt before s

(Figure A3); thus,

s

t

bsbE

0

0

. In this NJ path, consumption is reduced more sharply than it

is in the case shown in Figure A2. In this case, because

s

t

bθsbθEa

0

0

, γ0 , and

ts

cc for st , then

ss

ss

ttt

cubcudtθtEdtcubcuθtE

0

0

0

0

expexp

ss

cubcu

θ

θs

E

exp1

0

, and in addition,

s

dtcuacuθtE exp

0

cubθscu

θ

θs

Ecuacu

θ

θs

EcuacudtθtE

s

expexp

exp

000

.

Hence,

E0 (Jalone) – E0 (NJtogether)

s

s

ttt

dtcuacuθtEdtcubcuθtE expexp

0

0

0

cubθscu

θ

θs

Ecubcu

θ

θs

E

ss

expexp1

00

bθscucu

θs

θs

cubcu

θ

θs

E

ss

exp1

expexp1

0

.

As γ increases, the ratio

bθscucu

cubcu

ss

decreases; thus, larger values of s can satisfy

0

00

NJtogetherEJaloneE . For example, suppose that c = 10, cs = 10.2, b = 0.3, and θ

23

= 0.05. If 1γ , then s* = 1.5 at the minimum, and if 5γ , 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., bb

t

) and *pp , 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

sharply 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

households’ 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 are NΗ

identical households in the economy where H is sufficiently large (as assumed in Section A1).

Let 10 ηη qq be the probability that a household Ηη 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), ( *** ,…,, ppp ), (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 *pp , indifferent between J and NJ (i.e., any 10,qη ) if *pp , and NJ

(i.e., qη = 0) if

*

pp . Because all households are identical, the best-response correspondence

of each household is identical such that qη = 1 if

*

pp , [0,1] if *pp , and 0 if *pp for

any household Ηη . Hence, the mixed strategy profiles (1, 1,…,1), ( *** ,…,, ppp ), and

(0,0,…,0) are the intersections of the graph of the best-response correspondences of all

households. The Pareto efficient 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 equilibrium

( *** ,…,, ppp ).

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, if 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)

dtacuacuθtdtcubcuθtE s

s

ttttt

0

0

ˆexpexp

s sttt dtcuacuθtdtcubcuθtE 00 expexp

24

= E0 (Jalone) – E0 (NJtogether) < 0 , (A7)

by Lemma A3, Jalone is the worst choice in terms of the amount of payoff, followed by

NJtogether, and NJalone, and Jtogether is the best. The outcomes of choosing option J are more

dispersed than those 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 is 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, particularly 0p , 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 spirits”

(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, discarded, 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 products 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. During the

period when capital is not reduced to the posterior steady-state level, unutilized products are

successively generated. In a period, unutilized products are generated and eliminated, but in the

next period, another, new, unutilized products are generated and eliminated. This cycle is

repeated in every period throughout the transition path, and it implies that demand is lower than

supply in every period. This phenomenon may be interpreted as a general glut or a persisting

disequilibrium by some definitions of equilibrium.

25

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28

kt

ct

Steady state before the

shock on θ

Steady state after

the shock on θ

Line of 0

dt

dc

t

after the shock on θ

Line of 0

dt

dk

t

Z

Pareto inefficient transition path

Line of 0

dt

dc

t

before the shock on θ

Pareto efficient

saddle path after

the shock on θ

Pareto efficient saddle path

before the shock on θ

Figure A1: A time preference shock

0

29

Figure A2: The paths of Jalone and NJalone

ac

Path of NJalone

ac

Path of Jalone

c

0

c

0

s t

00

bc

ct

30

Figure A3: A Pareto inefficient transition path

Posterior Pareto efficient saddle path

Path of NJtogether

c

0

c

0 s t

00

bc

ct

31

Table A1 The payoff matrix

Any other household

J NJ

H

o

u

se

h

o

ld

A

J E0(Jtogether), E0(Jtogether) E0(Jalone), E0(NJtogether)

NJ E0(NJalone), E0(Jtogether) E0(NJtogether), E0(NJtogether)