HIAS-E-17

Liquidity Shocks and Asset Prices
Pablo A. Guerron-Quintana
Federal Reserve Bank of Philadelphia

Ryo Jinnai
Hitotsubashi University

December 1, 2015

Hitotsubashi Institute for Advanced Study, Hitotsubashi University
2-1, Naka, Kunitachi, Tokyo 186-8601, Japan
tel:+81 42 580 8604
http://hias.ad.hit-u.ac.jp/
HIAS discussion papers can be downloaded without charge from:http://hdl.handle.net/10086/27202
https://ideas.repec.org/s/hit/hiasdp.html

All rights reserved.

Liquidity Shocks and Asset Prices
Pablo A. Guerron-Quintana

Ryo Jinnai!

December 1, 2015

Abstract
In models of liquidity, stock market booms tend to follow adverse liquidity shocks. This
result is clearly at odds with the data. We demonstrate that allowing for endogenous productivity corrects this puzzling price dynamics. Negative growth prospects decrease equity
prices because of a long-run predictable component in dividend growth.

1

Introduction

The severity and widespread impact of the Great Recession has propelled substantial research tobetter understand the interconnection between the Önancial sector and the aggregate economy.
One strand of this literature emphasizes the role of shocks a§ecting the liquidity of assets, i.e., howeasily assets can be pledged as collateral. Yet these models of liquidity (Jermann and Quadrini(2012) and Kiyotaki and Moore (2012)) have often been criticized because of their counterfactual
stock market dynamics (Shi (2015)). More precisely, these models tend to predict a stock market
boom following an adverse liquidity shock to the underlying asset. Furthermore, the counterfactual
asset price response is robust to a wide range of speciÖcations such as nominal rigidities, habit
formation in consumption, and adjustment costs in capital or investment. This prediction istroublesome not only because it is counterfactual but also because it arises at precisely the heartof the transmission mechanism they feature. It is therefore crucial to correct these models before
we use them, for instance, as a laboratory to evaluate policies.
In this paper, we argue that this stock market anomaly is a consequence of the disconnect
between productivity and liquidity. To support our claim, we construct an endogenous growth!
Guerron-Quintana: Federal Reserve Bank of Philadelphia, email: pablo.guerron@phil.frb.org. Jinnai: Hitotsubashi University, email: rjinnai@ier.hit-u.ac.jp. We thank Thorsten Drazberg and Daniel Sanches for usefulcomments/discussions and Yang Liu for research assistance. Ryo Jinnai gratefully acknowledges the Önancial support from the Ministry of Education, Culture, Sports, Science and Technology of the Japanese Government through
JSPS KAKENHI Grant and the Hitotsubashi Institute for Advanced Study. The views expressed here are those of
the authors and do not necessarily reáect those of the Federal Reserve Bank of Philadelphia or the Federal Reserve
System.

1

model with cash strapped investors. SpeciÖcally, we augment Shi (2015)ís model with a stylized
learning-by-doing mechanism.1 In this framework, an adverse liquidity disturbance has a persistent, in fact permanent, negative e§ect on the level of macroeconomic variables. This prediction isconsistent with recent empirical work documenting that recessions associated with Önancial crises
tend to be long and painful (Cerra and Saxena (2008), Jorda, Schularick, and Taylor (2014), andReinhart and Rogo§ (2009)). Because of a long-run predictable component in dividend growth
associated with an adverse liquidity disturbance, the stock market plunges following the unfavorable shock if the intertemporal elasticity of substitution is large enough. This is essentially thesame insight as the one used in the long-run risk literature in Önance (Bansal and Yaron (2004)).

2

Model

The economy is populated by a continuum of households, with measure one. Each household has aunit measure of members. At the beginning of the period, all members of a household are identicaland share the householdís assets. During the period, members are separated from each other, andeach member receives a shock that determines the role of the member in the period. A member
will be an entrepreneur with probability ! 2 [0; 1] and will be a worker with probability 1 # !.
These shocks are i.i.d. among the members and across time.
A period is divided into four stages: householdís decisions, production, investment, and consumption. In the householdís decision stage, all members of a household are together to pool theirassets: st units of equity claims. An equity is the ownership of a unit of capital. Aggregate shocks
to exogenous state variables are realized. Because all the members of the household are identical inthis stage, the household evenly divides the assets among the members. The head of the householdalso gives contingency plans to each member as follows. If one becomes an entrepreneur, he or
she spends it units of consumption goods to invest, sets aside xet units of consumption goods forthe consumption stage, and makes necessary trades in the stock market so that he or she returns
to the household with set+1 units of equity claims. In contrast, if the member becomes a worker,
he or she supplies lt units of labor, sets aside xw
t units of consumption goods for the consumption
stage, and makes necessary trades in the stock market so that he or she returns to the householdwith sw
t+1 units of equity claims. After receiving these instructions, the members go to the market
and will remain separated from each other until the consumption stage.
At the beginning of the production stage, each member receives the shock whose realization
determines whether the individual is an entrepreneur or a worker. Competitive Örms produce Önal1

We introduce two modiÖcations to Shi (2015). First, we abstract away from the government sector. Second,
we allow perfect risk sharing within a household at the consumption stage. These two changes allow us to obtaina clean asset pricing equation which we will use extensively in the discussion of the result.

2

consumption goods yt using labor service ltD and capital service ktD with the production technology! "% ! D "1!%yt = At ktD
lt:Here, + is the capital share, the superscript D indicates demand, and At is the technology level
which both the households and the Örms take as given. After production, a worker receives wage
income, and an individual receives compensation for capital. A fraction - of capital depreciates.
The third stage in the period is the investment stage where entrepreneurs seek Önance andundertake investment projects. We assume that an entrepreneur can transform any amount it unitsof consumption goods into it units of new capital. Individuals trade assets to Önance investment
and to achieve the asset holdings instructed earlier by their households. Markets close at the endof this stage.
The individuals return to their households in the consumption stage. The head of the household
lets an entrepreneur consume cet units of consumption goods and lets a worker consume cw
t unitsof consumption goods. The feasibility constraint in this stage isew!xet + (1 # !) xw
t = !ct + (1 # !) ct :(1)

The instructions have to satisfy the intra-temporal budget constraints:!
"
xet + it = rt st + qt (1 # -) st + it # set+1(2)

for an entrepreneur where rt and qt are the rental price and the equity price respectively, and!
"w
xw
t = rt st + qt (1 # -) st # st+1 + wt lt(3)

for a worker where wt is the wage rate. We introduce the following equity-market frictions. An
entrepreneur can issue new equity on at most a fraction 2 of investment. In addition, she can sell
at most a fraction 3t of existing capital in the market. E§ectively, these constraints introduce alower bound to the capital holdings of an entrepreneur at the end of the period:
set+1 $ (1 # 2) it + (1 # 3t ) (1 # -) st :(4)

3t is an exogenous, random variable we call a liquidity shock. A similar constraint applies toworkers, i.e., sw
t+1 $ (1 # 3t ) (1 # -) st , but we omit it because it does not bind in equilibrium
for workers are net buyers of equities under our calibration. There are non-negativity constraintsew
w
for it , lt , xw
t , ct , ct , and st+1 , but we omit them too because they do not bind in equilibrium
either. An exception is the entrepreneurís intra-temporal resource transfer, whose non-negativity

3

constraint is given by
xet $ 0:(5)

The head of the household chooses instructions to its members to maximize the value function
deÖned as
(
)
' 1!&
$ !
"%
(cet )1!&[cw
t (1 # lt ) ]v (st ; Kt ; 3t ) = max !
+ (1 # !)+ 7Et v st+1 ; Kt+1 ; 3t+1
1#6
1#6
subject to (1), (2), (3), (4), (5), and the accounting identity
st+1 = !set+1 + (1 # !) sw
t+1 :As in Shi (2015), we will restrict our attention to the case in which 1 < qt < 1=2 always
hold. The Örst inequality 1 < qt implies that the marginal costs of investment are smaller thanthe marginal revenues. The second inequality qt < 1=2 implies that the entrepreneur needs extra
liquidity for down payment whenever she expands the scale of the investment project. These
conditions jointly imply that both the entrepreneurís liquidity constraint (4) and her non-negativity
constraint for intra-temporal resource transfer (5) must be binding at the optimum. The appendix
includes the proof of this claim as well as the derivations of the optimality conditions and otherimportant results.
Optimal labor supply condition requires that the marginal utility of leisure is equated to themarginal utility of receiving wage income,:cwt= wt :
1 # ltOptimal resource allocation in the consumption stage requires that the marginal utility of entrepreneurís consumption is equated to the marginal utility of workerís consumption,!&(cet )!& = (cw(1 # lt )'(1!&) :t )Finally, the following equation determines the price of equity,
" (
#
)
!
""
cet+1 !& !
qt = Et 7
rt+1 + (1 # -) qt+1 + !;et+1 rt+1 + qt+1 3t+1 (1 # -)cetwhere ;et is the variable Shi (2015) calls liquidity services;et =qt # 1:
1 # 2qt

The numerator is the marginal beneÖt of investment, and the denominator is the downpayment
4required to increase investment marginally. As such, ;et measures the marginal beneÖt of providing
liquidity to an entrepreneur.
The competitive equilibrium is deÖned in a standard way. Market clearing conditions are! D "% ! D "1!%!cet + (1 # !) cw
lt;t + !it = At kt
ltD = (1 # !) lt ;ktD = Kt = st ;and!set+1 + (1 # !) sw
t+1 = (1 # -) st + !it :Capital stock evolves according toKt+1 = (1 # -) Kt + !it :The aggregate stock market value Stockt is measured by
Stockt = qt Kt+1 :We assume that asset resalability 3t obeys
log

(

3t
3

)

= 6( log

(

3t!1
3

)

+ "t :(6)"t is an i.i.d. shock.
We compare results under two alternative assumptions regarding the source of growth. In one,
we assume that the technology level At grows at a constant rate:!
"t
At = A0 @ 1!% ;where A0 is a scale parameter. This is a standard assumption in the business cycle research
including Shi (2015). In the second case, we assume that the level of At is endogenous;
At = A0 (Kt )1!% :We interpret this assumption following Arrow (1962), Sheshinski (1967), and Romer (1986);
namely, knowledge is not only a by-product of investment but also a public good that anyone

5Table 1: Parameters and Calibration Targets
Parameter
7 ; discount factor
6; relative risk aversion! ; fraction of entrepreneurs: ; curvature in leisure utility+; capital share- ; capital depreciation rate
A0 ; scale parameter@ ; steady-state growth
3; steady-state resalability6( ; persistence in resalability
2; fraction of new equity

Value
EXO
END
0.994
0.994
1/1.85 1/1.85
0.06
0.06
2.66
2.66
0.36
0.36
0.015
0.015
1
0.183
1.004
0.12
0.12
0.9
0.9
0.12
0.12

Calibration Target
Exogenously chosen
Exogenously chosen
Annual fraction of investing Örms = 0.24
Hours of work = 0.25
Labor income share (1 # +) = 0:64
Annual investment/capital = 0.076
Normalization; steady-state growth = 1.004
Steady-state growth = 1.004
Capital stock/annual output = 3.32
Exogenously chosen
Set equal to 3

can access at zero cost.2 Under this assumption, the production function is written as
Yt = A0 Kt [(1 # !) lt ]1!% :The long-run tendency for capital to experience diminishing returns is eliminated. As such, theeconomy can grow in the long run.

3

Calibration

Table 1 reports our calibration, which is similar to Shi (2015) both in values and in calibration
strategies. We choose the discount factor 7, the fraction of investors !, the capital share +, and thepersistence in resalability 6( exogenously. The relative risk aversion parameter is set at 6 = 1=1:85
so that the implied inter-temporal elasticity of substitution (the inverse of 6) is consistent with thevalue used in the Önance literature (Kung and Schmid (2015)) and within the credible set estimatedby Schorfheide, Song, and Yaron (2014). We set the steady state growth rate at @ = 1:004 andnormalize the scale parameter by A0 = 1 in the exogenous growth model. We calibrate the scaleparameter A0 in the endogenous growth model so that the modelís non-stochastic steady state
growth rate is 1:004. Following Shi (2015), we assume that the fraction of new equity 2 is equal
to the steady state resalability 3. We calibrate 3 as well as the curvature in leisure in the utility
function : and the capital depreciation rate - using the following targets (all of which are used by
Shi (2015)): the aggregate hours of work in the steady state (0.25), the ratio of capital to annual
output in the steady state (3.32), and the ratio of annual investment to capital in the steady state(0.076).
2

We want to emphasize that this assumption is made for tractability. Similar results would follow from moreelaborated versions of endogenous growth.

6

4

Results

We suppose that all the detrended variables are in the non-stochastic steady state at time t = 0.
At the beginning of t = 1, there is an unanticipated 10% drop in liquidity from the steady state.
Thereafter 3t follows the process in (6) with "t = 0 for all t $ 2. Figure 1 plots percentage
deviations of selected variables from their realizations in an alternative scenario in which no shockhits the economy for all t $ 1. As we can see, a negative liquidity shock is contractionary ingeneral with the exception of consumption deÖned as !cet + (1 # !) cw
t , which rises in the shortrun. A few ways to overturn this counterfactual prediction are known in the literature such as
nominal wage rigidities (Ajello (2014)), insight from the news shock literature (Guerron-Quintana
and Jinnai (2015)), and investment adjustment costs (Shi (2015)). Our exercise also reveals thatthe e§ect of the shock is more persistent in the endogenous growth model than in the exogenous
growth model. In fact, it is permanent in the former while it is transitory in the latter.
Our main interest is in asset prices. In the exogenous growth model, the equity price qt
counterfactually rises in response to a negative liquidity shock while it drops in the endogenous
growth model. The same pattern is true for the aggregate stock market value Stockt = qt Kt+1 .
The following equation is useful to understand the factors behind these results:" (#)
cet+1 !&
Stockt = Et 7(+yt+1 # !it+1 + Stockt+1 ) :
cet

(7)

This equation holds in both the endogenous and exogenous growth models. Taking a Örst-orderlog approximation in the endogenous growth model, we Önd
log
|

(

)
Stockt=@ t Stock0
{z
}
IRF

log
|

(

)
Kt =K0
@t
{z
}

(8)

realized change in trend
1
!
"X
!
1!&

+ 1 # 7@|!
"
+ 1 # 7@ 1!&

7@

j=1

0
(
)1
Kt+j =Kt
Et (1 # 6) log
@j
{z
}

"
1!& j!1

expected change in trend
1
X
j=1

!

7@ 1!&

"j!1

0
( e )
(
)
(
)1
c^t+j
+^
y
y^t+j
!^{
^{t+j
%Et #6 log
+
log
#
log
;
c^et
+^
y # !^{y^+^y # !^{
^{
|
{z
}
cycle

where variables with a hat indicate that they are divided by the endogenous trend term; forexample, y^t is deÖned as y^t = yt =Kt . Variables without time subscript denote non-stochastic
steady state values of the corresponding variables. @ denotes the (gross) growth rate of the7economy in the non-stochastic steady state. The left-hand side represents the impulse response
function of the stock market in period t. The right-hand side breaks it into three factors, of which
two are contributions of the endogenous trend term, while the remaining one is the contributionof áuctuations around the trend.
The top panel of Figure 2 plots the contributions of each factor to the dynamic response of thestock market. Contributions of the cyclical terms are positive throughout the simulation. However,
the endogenous trend terms more than o§set the cyclical contributions in our framework. Not
surprisingly, the contribution of the realized change in the endogenous trend term is important onlyin the medium run because the trend term Kt is a stock variable. It is therefore the expectation
about the future trend term that is crucial to generate a negative stock market response in the
short run. The mechanism is intuitive and simple. Because the endogenous trend term will be low
in the future, so are future cash áows (remember that they are cointegrated). This expectation
depresses the current value of the stock.
Yet this prediction is sensitive to some parameter values. To show this point, we plot theimpulse response function under the alternative calibration with 6 = 1:0, i.e., log utility, in the
middle panel. Contributions of the expected change in the future trend disappear, and as a result,the aggregate stock market value positively responds to a negative liquidity shock in the shortrun. To understand the mechanism, it is important to realize that a liquidity shock a§ects not
only cash áows but also the entrepreneurís consumption proÖle and hence the stochastic discountfactor. SpeciÖcally, the impulse response function of the entrepreneurís consumption (not depicted)
monotonically decreases for t $ 1, namely,
(

)
( e )
ct+1
cet
log
> log
e
t
@c
@ t+1 ce0| {z 0 } |{z}
IRF in period t

for t $ 1. Rearranging terms, we Önd

IRF in period t+1

cet+1
< @:
cet

This inequality implies that the entrepreneurís consumption growth after a negative liquidity shock(cet+1 =cet for t $ 1) is lower than the growth rate in an alternative scenario in which no liquidity
shock hits the economy (@). Its e§ect on the stock market value is clear from equation (7); the
!
"!&
!
"!&
stochastic discount factor 7 cet+1 =cet
becomes larger (the discount rate 1#7 cet+1 =cet
becomessmaller) after a negative liquidity shock than before, and hence, the current stock market value
would increase even if the future cash áows are unchanged. Put it di§erently, the marginal rate of
substitution between consumption goods in the future and consumption goods in current period
rises after the shock, making a claim of a unit of consumption goods in the future more valuable
in terms of consumption goods in the current period. This e§ect is stronger when the relative risk
aversion is larger (the inter-temporal elasticity of substitution is smaller). 6 = 1 is the threshold8Liquidity0

Investment

0

-5

-5

-10

-10
0

5

10

15

20

25

30

0

Output

0

5

1

-1

0

-1.5

-1

-2

15

20

25

30

25

30

25

30

25

30

Consumption

2

-0.5

10

-2
0

5

10

15

20

25

30

0

Labor

0

5

10

15

20

Capital Stock

0
-0.5

-1

-1
-2

-1.5

-3

-2
0

5

10

15

20

25

30

0

Equity Price

1

5

0

0

-1

-0.5

15

20

Stock Market Value

1

0.5

10

-2
0

5

10

15

20

25

30

0

endogenous growth

5

10

15

20

exogenous growth

Figure 1: Impulse Response Functions to a Liquidity Shock9value at which this e§ect through the relative price (i.e., the marginal rate of substitution) perfectlyo§sets the aforementioned e§ect through cash áows. The essentially same insight is well known in
the long-run risk literature in Önance (Bansal and Yaron (2004)), emphasizing the importance ofa large inter-temporal elasticity of substitution to obtain a reasonable stock market response to achange in growth prospects. The long-run risk is an endogenous outcome in our model caused byÖnancial shocks. This implication might be of interest in the literature searching for a structural
origin of the long-run risk.3
The decomposition also sheds a light on why in the exogenous growth model stock market
responds positively to a negative liquidity shock. This is because the aggregate stock market
value is driven by the cyclical terms alone. We illustrate this point in the bottom panel of Figure
2. Clearly, the cyclical terms are positive contributors to the aggregate stock market value, andthis prediction is robust as Shi (2015) discusses in detail.

5

Conclusion

Models of liquidity have become fashionable since the onset of the Great Recession. This isso because they provide a natural connection between Önancial frictions and the depth of thecrisis. These models, however, tend to have the unappealing feature of predicting counterfactualdynamics of asset prices. In this paper, we show that this anomaly is readily Öxed once we allow
for endogenous productivity.

References
Ajello, A. (2014): ìFinancial Intermediation, Investment Dynamics and Business Cycle Fluctuations,î Board of Governors Working Paper.
Arrow, K. J. (1962): ìThe Economic Implications of Learning by Doing,î The Review of Economic Studies, 29(3), pp. 155ñ173.
Bansal, R., and A. Yaron (2004): ìRisks for the Long Run: A Potential Resolution of Asset
Pricing Puzzles,î The Journal of Finance, 59(4), 1481ñ1509.
Cerra, V., and S. C. Saxena (2008): ìGrowth Dynamics: The Myth of Economic Recovery,î
American Economic Review, 98(1), 439ñ57.
Croce, M. M. (2014): ìLong-Run Productivity Risk: A New Hope for Production-Based Asset
Pricing?,î Journal of Monetary Economics, 66, 13ñ31.
3Previous work includes Croce (2014), Jinnai (2015), Kaltenbrunner and Lochstoer (2010), and Kung andSchmid (2015).

10

Endogenous Growth Benchmark Calibration

1

cycle
future trend
realized trend
IRF

0.5
0
-0.5
-1
-1.5
-2
0

5

10

15

20

25

30

Endogenous Growth Log Utility

2

cycle
realized trend
IRF

1.5
1
0.5
0
-0.5
-1
-1.5
0

5

10

15

20

25

30

Exogenous Growth

0.3

cycle
IRF

0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
0

5

10

15

20

25

Figure 2: Factor Decompositions of Stock Market Responses

1130

Guerron-Quintana, P. A., and R. Jinnai (2015): ìFinancial Frictions, Trends, and the GreatRecession,î Unpublished Manuscript.
Jermann, U., and V. Quadrini (2012): ìMacroeconomic E§ects of Financial Shocks,î American Economic Review, 102(1), 238ñ71.
Jinnai, R. (2015): ìInnovation, product cycle, and asset prices,î Review of Economic Dynamics,
18(3), 484 ñ 504.
Jorda, O., M. Schularick, and A. M. Taylor (2014): ìThe Great Mortgaging: Housing
Finance, Crises, and Business Cycles,î Working Paper 20501, National Bureau of Economic
Research.
Kaltenbrunner, G., and L. A. Lochstoer (2010): ìLong-Run Risk through Consumption
Smoothing,î Review of Financial Studies, 23, 3190ñ3224.
Kiyotaki, N., and J. Moore (2012): ìLiquidity, Business Cycles, and Monetary Policy,î Working Paper 17934, National Bureau of Economic Research.
Kung, H., and L. Schmid (2015): ìInnovation, Growth, and Asset Prices,î The Journal of
Finance, 70, 1001ñ1037.
Reinhart, C., and K. Rogoff (2009): This Time Is Di§erent: Eight Centuries of Financial
Folly, vol. 1. Princeton University Press, 1 edn.
Romer, P. M. (1986): ìIncreasing Returns and Long-Run Growth,î Journal of Political Economy, 94(5), pp. 1002ñ1037.
Schorfheide, F., D. Song, and A. Yaron (2014): ìIdentifying Long-Run Risks: A Bayesian
Mixed-Frequency Approach,î NBER Working Papers 20303, National Bureau of Economic Research, Inc.
Sheshinski, E. (1967): ìOptimal accumulation with learning by doing,î in Essays on the theoryof optimal economic growth, ed. by K. Shell, pp. 31ñ52. MIT Press, Cambridge, MA.
Shi, S. (2015): ìLiquidity, assets and business cycles,î Journal of Monetary Economics, 70(0),116 ñ 132.

12

6

Appendix

6.1

Solving Householdís Problem

We assume that 1 < qt < 1=2 always hold. We argue that the liquidity constraint (4) must
be binding at the optimum. Suppose otherwise. Then the head of the household can increase
it by 6 > 0 and increase xet by (qt # 1) 6 without violating any constraints as long as 6 issu¢ciently small. These changes do not alter st+1 but increase the amount of resource available inthe consumption stage !xet + (1 # !) xw
t , implying that the initial choice is not optimal. Next, we
argue that the non-negativity constraint (5) must be binding at the optimum. Suppose otherwise.
Then the head of the household can decrease xet by 6 > 0, increase set+1 by (1=qt ) 6, increasew
sw
t+1 by (!= (1 # !)) (1=qt ) 6, and decrease xt by (!= (1 # !)) 6 without violating any constraints
as long as 6 & xet . These changes alter neither st+1 nor the amount of resource available inthe consumption stage, but make the liquidity constraint (4) slack. Because the household canincrease utility if (4) is not binding, the initial choice is not optimal.
With the binding liquidity and non-negativity constraints, the householdís problem is rewritten
as follows.
(
)
' 1!&
(cet )1!&
[cw
(1
#
l
)
]
t
v (st ; Kt ; Zt ) = max !
+ (1 # !) t
+ 7Et [v (st+1 ; Kt+1 ; Zt+1 )]1#6
1#6
subject to

and

!
"
rt st + wt lt + qt (1 # -) st # sw
t+1 =
0

1
[!cet + (1 # !) cw
t ]
1#!

1
1
e
st+1 = !
(1 + ;t ) (rt + qt (1 # -) 3t ) + (1 # 3t ) (1 # -) st + (1 # !) sw
t+1
qt

where
;et =
First order conditions are

qt # 1:
1 # qt 2(cet )!& = Et ;

(9)

!&
(cw
(1 # lt )'(1!&) = Et ;
t )

(10)

cw
t
= Et wt ;1 # lt(11)

7Et [v 0 (st+1 ; Kt+1 ; Zt+1 )] = Et qt :

(12)

!&
: (cw
(1 # lt )'(1!&)
t )

and

13

Envelope condition isv 0 (st ; Kt ; Zt )01
(1 + ;et ) (rt + qt (1 # -) 3t ) + (1 # 3t ) (1 # -)qt
+ (1 # !) Et (rt + qt (1 # -)) := 7Et [v 0 (st+1 ; Kt+1 ; Zt+1 )] !

1

(13)

Combining (9) and (10), we Önd!&(cet )!& = (cw(1 # lt )'(1!&) :t )Combining (10) and (11), we Önd:
Combining (12) and (13), we Öndcwt= wt :
1 # ltv 0 (st ; Kt ; Zt ) = Et (rt + (1 # -) qt + !;et (rt + qt 3t (1 # -))) :Combining (9), (12), and (14), we Önd
" (
#
)
!
""
cet+1 !& !
qt = E t 7
rt+1 + (1 # -) qt+1 + !;et+1 rt+1 + qt+1 3t+1 (1 # -)
:
cet

6.2

Model summary

6.2.1

Summary before Detrending:cwt= wt ;
1 # lt!&(cet )!& = (cw(1 # lt )'(1!&) ;
t )

;et =

qt # 1;
1 # qt 2

" (
#
)
!
""
cet+1 !& !
e
qt = E t 7
rt+1 + (1 # -) qt+1 + !;t+1 rt+1 + qt+1 3t+1 (1 # -);cetrt + qt (1 # -) 3t
Kt ;1 # qt 2
yt
+
= rt
Kt
yt
(1 # +)= wt ;(1 # !) lt

it =!cet + (1 # !) cw
t + !it = yt ;
14

(14)

yt = At (Kt )% [(1 # !) lt ]1!% ;Kt+1 = (1 # -) Kt + !it :6.2.2

Exogenous Growth Modelt

If At = A0 (@ 1!% ) , the variables are detrended as follows.
:c^wt= w^t ;
1 # lt!&(^cet )!& = (^cw
(1 # lt )'(1!&) ;
t )

;et =

qt # 1;
1 # qt 2

" (
#
)
!
""
c^et+1 !& !
e
qt = E t 7 @ e
rt+1 + (1 # -) qt+1 + !;t+1 rt+1 + qt+1 3t+1 (1 # -)
;
c^t
^{t =

rt + qt (1 # -) 3t ^Kt ;1 # qt 2+(1 # +)

y^t
= rt
^t
K

y^t
= w^t ;
(1 # !) lt!^
cet + (1 # !) c^w{t = y^t ;t + !^
2 3%^ t [(1 # !) lt ]1!% ;y^t = A0 K
^ t+1 = (1 # -) K
^ t + !^{t@K

where a hat variable denotes the original variable divided by @ t , for example, c^et = cet =@ t .
The following parameters are exogenously chosen. 7 = 0:994, ! = 0:06, + = 0:36, @ = 1:004,^ = 0:076.
and A0 = 1. The ratio of annual investment to capital in the steady state is set at 4!^{=K
Survival rate of capital (1 # -) is backed out from
1 4!i
1#- =@#
:
4 K}
| {z
known

Aggregate hours of work in the steady state is set at (1 # !) l = 0:25. Hours of work per worker
in the steady state is backed out froml=1
(1 # !) l:|1 # ! {z
}
known

15

^ (4^The ratio of capital to annual output is set at K=y ) = 3:32. Steady state output is backed out
from
! !
^ 1!!
K
y^ =
(1 # !) l :
y^
|
{z
}
known

Other steady state values are backed out as follows.

^K^ = (4^Ky) :4^y| {z }known^{ =^ 4!^{K:
4! ^| {zK}knowny^w^ = (1 # +):(1 # !) l
|
{z
}
known

y^
r= + :
^
K
|{z}
known

e

The following equations are solved for ; , q, and 3:
;e =

q#1
;
1 # q3q = 7@ !& (r + (1 # -) q + !;e (r + q3 (1 # -))) ;^{ =r + q (1 # -) 3 ^K:
1 # q3

The following equations are solved for c^e , c^w , and :.
:

c^w
= w;
^
1#l

(^
ce )!& = (^cw )!& (1 # l)'(1!&) ;!^ce + (1 # !) c^w + !^{ = y^:
6.2.3

Endogenous growth model

If At = A0 (Kt )1!% , variables are detrended as follows.
:c^wt= w^t ;
1 # lt16!&(^cet )!& = (^cw
(1 # lt )'(1!&) ;
t )

;et =

qt # 1;
1 # qt 2

" (
#
)
!
""
c^et+1 !& !
qt = E t 7 @ t e
rt+1 + (1 # -) qt+1 + !;et+1 rt+1 + qt+1 3t+1 (1 # -)
;
c^t
^{t =

rt + qt (1 # -) 3t;
1 # qt 2+^
y t = rt

(1 # +)y^t= w^t ;
(1 # !) lt!^
cet + (1 # !) c^w{t = y^t ;t + !^
y^t = A0 [(1 # !) lt ]1!% ;@ t = (1 # -) + !^{t :where @ t = Kt+1 =Kt and a hat variable denotes the original variable divided by Kt , for example,wc^w
t = ct =Kt .
The following parameters are exogenously chosen. 7 = 0:994, ! = 0:06, + = 0:36, and@ = 1:004. The ratio of annual investment to capital in the steady state is set at 4!it =Kt js:s: =4!^{ = 0:076. Hence
0:076
^{ =
:
| 4!
{z }
known

(1 # -) is found as

1 # - = @ # !^{:
| {z }
known

Aggregate hours of work in the steady state is set at (1 # !) l = 0:25. Steady state individual
hours of work in the steady state is backed out froml=1
(1 # !) l:
1
#
!
|
{z
}
known

The ratio of capital to annual output is set at Kt = (4yt ) js:s: = 1= (4^yt ) js:s: = 3:32. Hence
( )(
)
1
1
y^ =
:
4
3:32

A0 = y^ [(1 # !) l]%!1 :
|
{z
}
known

17

y^
w^ = (1 # +):(1 # !) l
|
{z
}
known

r = +^
y :
|{z}
known

e

The following equations are solved for ; , q, and 3:
;e =

q#1
;
1 # q3q = 7@ !& (r + (1 # -) q + !;e (r + q3 (1 # -))) ;^{ =r + q (1 # -) 3:
1 # q3

The following equations are solved for c^e , c^w , and :.
:

c^w
= w;
^
1#l

(^
ce )!& = (^cw )!& (1 # l)'(1!&) ;!^ce + (1 # !) c^w + !^{ = y^:

6.3

Asset pricing equation

Pricing equation for equity is given by
" (
#
)
!
""
cet+1 !& !
qt = E t 7
rt+1 + (1 # -) qt+1 + !;et+1 rt+1 + qt+1 3t+1 (1 # -):cetMultiply Kt+1 to both sides,
" (
#
)
!
"
"
cet+1 !& !
Stockt = Et 7
+yt+1 + (1 # -) qt+1 Kt+1 + !;et+1 rt+1 + qt+1 3t+1 (1 # -) Kt+1 :cet
Because it =Kt = (rt + qt 3t (1 # -)) = (1 # qt 2) in our model economy,
Stockt" (#)"
cet+1 !& !
e
= Et 7+yt+1 + (1 # -) qt+1 Kt+1 + !;t+1 (1 # qt+1 2) it+1
cet
" (
#
)
cet+1 !&= Et 7(+yt+1 + (1 # -) qt+1 Kt+1 + ! (qt+1 # 1) it+1 )
cet
" (
#
)
cet+1 !&= Et 7(+yt+1 # !it+1 + Stockt+1 ) :cet18