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Price Stability and the Inter-industry Propagation of
Stochastic Impulse: Formulating dynamic price equation and
an application of the Langevin equation
Hitoshi Hayami
July 2001
KEO Discussion Paper No. 62
Price Stability and the Inter-industry Propagation of
Stochastic Impulse: Formulating dynamic price equation and
an application of the Langevin equation*
Hitoshi Hayami
Keio Economic Observatory
Keio University
July 2001
*This research is supported by Seimei -kai fund. i would like to thank my colleagues of Keio
Economic Observatory for their encouragement. Any errors remain the author's responsibility. This
paper is preliminary, please do not quote without the author's permission.
Abstract
Dynamic price equation has a feedback effect from the capital gain/loss of its user's cost of capital. Hayami[1993] developed the model which endoge-nously determines the capital goods prices, and this paper investigates more general characteristics of the system, and extends it into incorporating stochas-tic fluctuation due to the total factor productivity(TFP). The latter extension to stochastic system has been developed by Hayami [1999]. In a deterministic model, the system's stability depends on the degree of out-sourcing of capital goods, relatively higher out-sourcing causes the system into saddle point equi-librium. This paper presents all the possible steady states in the dynamic price
equation, and draws their phase portraits. Key words: propagation, impulse, capital goods, price equation, non-
linearity, multi-sectoral, total factor productivity.
1 Introduction
Very rapid price decrease of semi-conductor and of its products such as PC has been common phenomena in recent years.2 These products are capital goods, because they are durable and used in the industry sectors. The price of capital goods has not only instantaneous effects on the economy, but also has persistent effects through the installed equipment and the opportunity costs.
This paper investigates the fluctuation of prices including prices of capital goods through the inter-industry propagation mechanism. The most comprehensive data that are designed to incorporate the inter-industry propagation mechanism is the Leontief's input-output table. Leontief [1970] proposed the dynamic inverse model within the constant input coefficient matrix. I extend the Leontief's dynamic inverse model into the model consistent with the growth accounts.3 Hayami [1993] studies the two sector model using a numerical method, here I presents the general formula classifying characteristics of price fluctuation and its steady state .
The early attempts to investigate economic fluctuations through the inter-industry transactions are by Frisch [1933] and [1934]. Frisch [1933] summarises three types of propagation problem: (1) the time lag between production and completing the
production of capital goods. Aftalion [1913, 1927] investigates this propagation system. (2) Accumulation of erratic shocks. Slutzky [1927] and Yule [1927] de-rived several stochastic processes that have oscillation. (3) The innovation as an impulse and its reaction. Frisch sites Schumpeter's business cycle model.
2The price of PC decreased at over -17% pa. as Hulten [1990] refers. 3The growth accounts are one of the most common indices on prices and quantity, because the
Divisia index satisfies with the conditions of Fisher's tests for the ideal index numbers, Richter's invariance axiom, and Hulten's path independency.
1
Frisch [1934] considers cycles caused by a random impulse, e.g. Norway's lottery using a transaction model between shoe makers and farmers . But the model assumes fixed prices or given prices under the business cycles.
There are a lot of contributions on multi-sectoral economic dynamics in the last half of 20th century.4 I proposes the model with observable variables. "Observ-able", I mean, it is ready to obtain the parameters or the data from statistical survey . More precisely, I construct on the bases of the growth accounts equality, the Divisia index, which is now commonly used to calculate the total factor productivity. The most difficult data that I assumed is capital goods and the depreciation. I would like to avoid all the difficulty concerned with measurement of capital and depreciation, thus I adopt the simplest formulation though it is not enough to link the actual statis-tics with the model.5 I assumed that capital is a bundle of commodities that can be used over the period, and that the same commodity such as glass, or computer: can be utilised as capital goods, intermediate inputs and final demand.
2 Definitions and the growth accounts
In the introduction I point out that a rapid growth of the total factor productivity in a sector may give rise to the increase of unit cost for another sectors through capital loss of the installed equipment, and this may cause volatile price fluctuation in the economy as a whole. In this section, I present a basic equation to show how the
prices transmit across the sectors. It is not necessary to provide an optimal control model in order to describe the price interdependency between sectors, but some
justification of the definition on cost of capital that I use may be necessary. First I show one of the possible explanation of the cost of capital, and next introduce the price equations based on the growth accounts of cost.
Consider a cost minimising production sector using materials, labour inputs and capital goods. The producer of a commodity x will minimise the cost C over the
planning period (0, T) with a discount rate r.
T
C ` c(t)e-'tdt, (1)
0 where c(t) is an instant cost and defined as the sum of labour, material and invest-
4For example , Leontief [1970], I will show the difference from our model lately,describes us-ing the input-output tables. Goodwin [1953] proposed linear multi-sectoral model, and recently Antonello [1999] described Goodwin's model in terms of an optimal control problem. Nishimura [2001] surveys non-linear dynamics using optimal growth model with multi-sectoral capital goods.
5Hulten [1990] surveys problems associated with the measurement of capital, and Diewert [1980]
surveys aggregation problems on the capital.
2
ment costs:
c(t) = w'l -}- px'x -}- pI'xI, (2)
where w, px, pI denotes nL wages, n material prices , and n prices of investment goods respectively. 1, x, xI denotes nL types of labour, n kinds of materials, and n kinds of investment goods respectively. There are n kinds of commodity in the economy, each commodity can be used as either material , investment or consump-tion. The producer is subject to the current production technology , which is described as
f(y,1, x, k) =0, (3)
where y denotes an output commodity (or commodity vector) that this sector pro-duces, and k denotes capital commodity vector that forms into the sector's equip-ment and the other durables.
The producer minimise eq (1) subject to eq (3) and a capital formation equa-tion (4)
lc = xI -- S k, (4)
where k denotes the time derivative of capital goods vector dk/ dt , and S denotes a diagonal matrix of depreciation for each capital goods.6
51 0 ••• 0 0 b2 ••• 0 S=
0 0 •.. bn
The first order (necessary) conditions are derived from either the calculus of variations or the dynamic programming, expressed as follows:
bG
bl
SG
bx
bG
bk
=
e-rt
= e-rt
W-aaf =0 al
of Px - Ax` = 0
a bPi - A of d ) - ak dt
ae-rtc(t)
(5)
(6)
ak
6See Hulten and Wykoff [1981] and Hulten [1990] for problems of the formulation. Several other
types of depreciation and aggregation formula have been proposed, but many economic theorists do not consider these problems seriously, see for example Nishimura [2001].
3
of d e-rt (6pi -- A a k + e-ri rP i- dt e-rt ((TI + b - n)p1 -- A of = 0, (7)
ak
where A denotes the Lagrange multiplier, I denotes the identical matrix of n x n dimension, it denotes the diagonal matrix of n x n dimension, the diagonal element is capital gain from each capital goods pIj/pI.
dlnpit o ... 0 dt 0 dlnP[z ... 0
7C = dt
0 ... d~Pln-
. dt
Equations (5)-(7) show that this first order condition is interpreted as static cost minimisation with 1, x, k, if the cost of capital is defined by (rI + b - ir)pI .
The total cost can be defined with the user cost of capital as follows:
C = w'1 + px'x + {(rI + b - 7r)pt}'x1. (8)
Assume the producer minimise the cost (8) with respect to 1, x, and k subject to the production technology (3) under the given input prices w, pX, r, b and PI. The same system of equations as eqs. (5)-(7) can be obtained from the first order condition. The user's cost of capital is not necessary to be derived from the dynamic pro-
gramming/the calculus of variations, because the user's cost is actually an oppor-tunity cost of capital goods for a unit of period. The opportunity cost consists of the interest earnings from the same amount of investment for securities rpik, the
physical depreciation of the capital bplk, minus the capital gain from selling the capital k at the end of a period pik, assuming the existence of a complete capital market for simplicity.
There are a lot of criticism about the formulation of optimisation, and the pro-duction functions incorporating capital stock. As Leontief [1982] criticised the tran-scendental logarithmic production function, its parameters still have "not be iden-tified with those directly observable in the real world". In this paper, I would not say that the production technology (3) has directly observable parameters. But I would like to mention the equivalence of the total cost whether it is derived from an optimisation with a production function or from a definition of opportunity cost user's cost of capital.
Leontief dynamics [1970] is the system of equations that describes physical balance of demand and supply.
xt - Axt - Bt+1 (xt+i - xt) = ct, (9)
4
where At denotes the input coefficient matrix at t, and Bt+1 denotes the capital coefficient matrix, "capital goods produced in year t are assumed to be installed and put into operation in the next year t+l", and ct denotes final demand vector.
Leontief [1970] proposes the following associated cost accounting, which is dual to eq (9).
Pt = Atpt + (1 + rt-i) Btpt_1 - Bt+i Pt + vt, (10)
where rdenotes the annual money rate of interest prevailing in that year , and vt a vector of the value added per unit of its output. This equation can be rewritten as
follows:
Pt = Atpt + (rt-iI -~ it)Btpt_1 - (Bt+t - Bt)Pt +vt, (11)
This system of equations implies that the price of output pt is equal to the sum of
the unit cost of materials Atpt, the value added per output vt, which includes em-ployment cost and depreciation, and the user's cost of net capital (rt_i I-it) Btpt_1, minus the technical change (Bt+1 -- Bt)pt.
The similar procedure can be applied to the total cost (8), but I take the total derivatives of logarithms of the total cost to decompose per cent change of the total cost into per cent change of the inputs, the input prices and the total productivity change. First take total derivative of logarithm of the total cost C:
dln C = SL' (dlnw + dlnt) -+- sx' (dlnpx + dlnx) ± SK' (dlnpK + dlnk) , (12)
where SL denotes the labour's cost share vector of its element wilt/C , sx denotes the material's cost share vector of its element pxixi/C, SK denotes the capital's cost share vector of its element pKiki/C, and pK denotes the user's cost of capital (rI + S i- it)pI.
Next introduce the total factor productivity (TFP) that is the total output per unit of the total inputs, and its growth rate is defined as follows:
dlnTFP = dlnX-- (sL'dlnt+sx'dlnx+sx'dlnk), (13)
where d In X denotes the total output index that is usually expressed in differential form as
dlnX= sy'dlny.
sY denotes the value output share vector of its element piyi/ p;y3, and y denotes the output vector as before.
7Leontief [1970], and reprinted in Leontief [1986] p.295.
5
Using the definition of the total factor productivity, the equation (12) becomes as follows:
d In C = sL'd In w + sx'd In px + sK'd In pK + d In X - d In TFP
din C/X = sL'd In w + sx'd In px + sK'd In pK - d In TFP. (14)
This simply implies that growth rate of the unit cost per unit of output is equal to the sum of growth rate of the wages, the material prices and the user's cost of capital, minus the growth rate of the total factor productivity.
The growth rate of the user's cost of capital is expressed as the sum of growth rate of its components. Since rI, S, and 7r are all diagonal matrices, din pK can be expressed using :
d In pK dln(rI+6-ir)pj
(r + & - din p11 /dt)pI1 (r+ b2 - d In p12/dt)p12 di
n
(r+ 5 - d In p1n/dt)p1n
dln (r+ 61 - dlnpl1/dt) + dlnp11din (r+ b2 - dlnp12/dt) + dlnP12
din (r + 5,. - d In p1n/dt) + d In p1ndlnr+dln61-d2 1np11/dt '' d1npI1
r+6i-dlnpl1/dt dlnr+dlnb2-d21np12/dt + di
n r+62-dlnp12/dt P12
dlnr+dln5n-d2lnpln/dt + din p111 r+6 , -dlnp[n/dt
(rI + b -ir)-1 ((drI + db)1 - d2lnpl/dt) + dlnpI, (15) where 1 denotes the column vector of 1.
Before inserting eq (15) into eq (14), remember that the above formulation is for a sector, which is to be considered as the j-th sector, and introduce the following assumptions for simplicity.
Assumption: Average cost Average cost of production C; /X5 for the j-th sector is equal to the output price of that sector p5.
C- -=p5 , (j = 1, ..., n.) .
6
Assumption: No joint production The j-th sector produces the single output X; for all j.
Assumption: Notation of the commodity Each output can be used as a material and also as a capital goods, there is n kinds of commodities in the economy. Prices for investment goods pI, for materials px, and for outputs p are no longer to be distinguished.
P=Pi=Px
Using these assumptions, the balance between input cost and output price is written as follows:
d In p; = sLi d In w + sxi d In p +sx~ (r;I + b; - rrr1 ((dr;I + db;)1 - d21np/dt) (16)
+sK;dlnp - d1nTFP;, (j = 1,.. .,n).
Divide both side of the equation by dt, then the equation is expressed in terms of
the growth rate per unit of time.
dlnpi ! dlnw ! dln _ dt _ SLj dt + Sx7 dt
+SKj (rjI + sj -~)_1 ((vi +) 1 - da (17) i d 1n - d In TFPi
In the economy as a whole, the above system of equations are described as follows:
dln = Sx dln + SK dln - Srd2 `~ dt d In W dt dt d In ~p (18) +SL
dt + (pSr + Sd) 1 - dt Eq (18) of the growth account equations corresponds to the Leontief's price equa-tion (11).
Definitions of the matrices are as follows:
SL11 SL21 SLnLI
SL12 SL22 SLnL2 SL=
SL1 SL2n " ' SLn1n
SX 11 SX21 SXn1
SX 12 SX22 ... SXn2 Sx =
SXIn SX2rL ... SXnn
7
Sr =
Sd=
P=
SKtt
dt _ _S_ K 12
T2+S12-dlnpl dt
SKIn d In p r
n+bin--dt-~
db SKtt d ~- dl"p~ r1+b11- dt
SK 12 bs13~ _. ding 1
2+51 2T d
dbt~ ----51C l n d t
at 0 ... 0 ... dt
00...
SK2t dln~ r1+S21- dt
SK22 dinpz 1
2+522- dt
__2r. SK----d In p 2 Tn+b2n- dt
dlnp _dlnp Tn+bin- dt rn+b2n
_SK21_dh r1 +b21- d
_ db dt S _K 22 d z dinp~ 12+S22-
di
d
$ K 2n --_pt-.
SK11 SK21 '
SK12 SK22 '
SKIn SK2n
0
0
dt
SK
Definitions of the vectors are as follows:
din p
dt
din w
dt
din 1FP
dt
SK 1 r1+Sn1- dt
SKn2 _ nd llia r2+62_:
,,, -_ SKnn 1+&- d In pn
dbnl SKnI __ dt ... _
d ln~Tl T1+bn1- dt
... SKn2 d 2 T2+bn2- d In Un dt
SKnn dd ri. ...--
Tn+bnn- d~~.
• SKn1
SKn2
SKnn
dlnpl
d In dt
d~Pn
dt
dlnw~
dlnwz dt
dlnwnL dt
_d In TFP1
d 1n 'P, dt
d In TFPn -at
8
d21n p
d2 In p d d t .
dt2 d21n pn
d ~t
The other definitions of variables are as follows:
Stij _____ , Sxi = pjXij
SKI; = SKI; = p C
pt din p. Ci
stl; is the i-th labour's cost share in the j-th sector, likewise sxl; is the i-th material's cost share in the j-th sector, and sKI; is the i-th capital goods' cost share in the j-th sector. There are nt types of labour., n types of materials and capital goods.
In eq (18), transpose the terms with d In p/dt to the left hand side, the system becomes as follows:
Srd2 -}- (I - Sx -- SK) dP = d~nTFP ®SL alnw _ (pSr -I- Sd)1. dt dt dt dt
(19) I investigate this system of equations, which is non-linear with respect to din p/dt, because Sr includes d in p/dt, even if all the cost share matrices are assumed to be constant.8 This system is homogeneous degree zero in prices, because the equations does not change if all the prices grow at A > 0, Ap, Aw, when
(I--Sx-'SK-SL)1 =0.
If wages w adjust to cancel the effect of the total factor productivity fluctuation, while the interest rate and the depreciation keep unchanged, the system becomes autonomous.
3 An autonomous system of the price propagation
In this section, an autonomous system of the price equation eq (20) is investigated. For simplicity, assume that all the cost shares are constant through this paper.
8Hayami[1993] proposed the same equations as a different form . Although my previous formu-lation contains errors of notations, the two sector model in that paper is exactly the same as this. As to nonlinear dynamical systems see for example, Guckenheimer Holms [1990].
9
Assumption: Constant cost shares The cost shares Sx, SK, SL are constant over
time.
z
Srdd' + (I - Sx - SK) aap = 0. (20) This is, in fact, a first order differential equation system for the rate of price changes . The system is non-linear, since Sr depends on d in p/dt. Let z denote d in p/dt for simplicity, and the system is expressed as follows:
Sr(z) a = (I-Sx-SK)z,
dln _ (21) dt - Z
There are two special cases to be remarked. One of them is the singular matrix Sr (z) . If the matrix Sr (z) is singular, the system cannot describe price changes. The other is that price change of a sector z;, is equal to r; + E. In this case, one of the elements of the matrix Sr(z) becomes infinitely positive or infinitely negative. The behaviour of the price changes around these points may change drastically.
Except for these cases, the system has a fixed point z = 0. And the behaviour of the price changes around the fixed point can be described using the eigenvalues of the following Jacobian matrix of first partial derivatives at z = 0.
J(z) _ -DZ (Sr(z)_1 (I -- Sx -- SK) z) , (22) where DZ denotes a vector partial derivative operator [a/azi} that operates a vector valued function.
Ja with parameters c may have a zero eigenvalue at the fixed point z = 0, the point (0, c ) should be a bifurcation point. But the following discussion shows that this system does not have a bifurcation phenomenon. The matrix Sr (z) -1 (I--Sx-S) can be expressed by W(z) in general.
W(z)z
DZ (W(z)z) _ DZ
1 W1i(z)z; ~~ t W2i (Z)Zi
L-j=1 Wni (z)zi 1 W1i (z)zi
L=1 W2i(z)z;
1 Wni (z)zj
10
DZ (W(z)z) =
+w11 1 as '~-z; + W~2 ... ~n 1 aw2j + ~/1n n aW2 + W21 + W ... i:; = n a Z ~1=1 a ~--;=1 az2 22 1 azn~ Z7 + W2n
~ =1 as ~; + Wnl ~g 1 as n; z; + Wn2 ... ~n l awn; ~ + W 2 J aZn, nn
Thus the Jacobian matrix at z=0 is
W11(0) W12(0) ... W1(0) W21(0) Wu(0) ... W2n(0) DZ (W(z)z) Iz=o =
W1(0) Wn2(0) ... Wnn(0)
Sr(0i (I - SX - SK) , (23)
The determinant of a product of two n x n matrices is a product of two determinants of n x n matrices suggests. that
IDz (W(z)z) IZ..oI - ISr(0)-111I - SX - SKI (24)
The determinants II - SX --- SKI does not become zero because of the Hawkins-Simon's condition, and the determinants I Sr ( 0)_i i is a reciprocal of I Sr ( 0)1. I Sr ( 0)1 is possible to be zero, but not infinity large, unless both the interest rate and the de-
preciation rate are equal to zero.
3.1 The two sector case
To illustrate the eigenvalues of the Jacobian matrix Ja(z), I derive the two sector system explicitly. The equation system can be described as follows (Hayami[1993]):
Z SK11 SK21 -1 1 _ r+&1-z1 r1+521-z2 ) ( 1 ̀~ X11 - K11 '~ X21 K21 Z2 "_ S_Ki2 Sx22 -SX12 - SK12 1 - SX22 - SK22
r2+612-zl r2+&22-z2
Z1 X)) Z2 (25)
11
or
~(1`SX11-SK 11)SK.z2. + ~SX_12_+SK1~~SK~1_ Z r2+622-z2 rl +b21 -z2 ) 1 -
\ (l-SX22-SK~2)SKZt_ ,+., ~SXZ1+$K21 ~$K z Z 1 tl +b21 `z2 r2+622-z2 ) 2 1 _
i2 - -D(z) , (26) - i (l-$X 11 -SK 11 )SK 12, ~. (SX 12+SK-1,i K_11 z r2+612-z1 rt +61 1 -z1 ) 1
+ C (1-SX22-SK22)SK2j ($X21+SK21 )SK z T r1 +61 1 -zl r2+612-z1 ) 2 where D is the determinant of Sr,
_ SKi1SK22 _ SK12SK21
~(Z) - (r1 +511 - zl)(r2 + 522 - 22) (r1 +521 - z2)(r2 + 512 - zl)• 3.2 Behaviour around the fixed point z= O
Eq (26) is still complicated to calculate the Jacobian matrix, but substituting z=0 into the Jacobian matrix makes all the derivatives related with fractions disappear .
((1-SX11-SK11 )SKI + J (SX12'}"SK~_~SK1.~ 1 r2+622 11 ±621
}
a(o) D(O) (1-SX11`SK11)SK12 +121" ($X+SK12)SK1( r2+612 r1 +b1 t
(1-SK22 + ($X21 ±SK21 )SK22 (27) C 11+621 12+622 - ( (1'_$X22-SK.22)SK1-1 + ($X21_+SK21 )SK12, rt +61 1 r2+612 ) D (O) is defined as
SK11 SK22 SK12SK21 D(0) - - -- -- , - --- --- - (28) (
Ti +611)(r2+5) (r1 +521)(r2+612).
The system has two real eigenvalues due to the fact that product of the off diag-onal elements of the Jacobian matrix Ja (0) is positive.9
9The eigenvalues of a 2 x 2 matrix A can be derived as follows: A denotes an eigenvalue of A.
CAl-Al = A ̀~ al 1 -a12 -a21 A - a22 = A2-(a1,+a22)A+a11a22-a12a21=0.
The determinant D of the second order equation for A is
D = (all + a22)2 -- 4(al1 a22 - a12a21) = (all - x22)2 + 4n12a21
Thus a12 a21 < 0 is necessary for eigenvalues A to have imaginary parts.
12
It can be shown that the determinant of Ja (0) is
IJ (O)I _ 1 (1-SXiI"-SK 11)SK22, (SX±SK12)SK21 a D(Q)2 12+622 11+621
x (1-SX 2-SK22)SK11, (S- X?1'~SK21)SK12 11+611 12+612 (1-SSKZ 2)SK21„ + (SZ1 j S.K22 rl +621 r2+622
x ((1-SX11-SKJ )SK 12 + (SX12+SK12)SK~] r2+612 11+611
--
D(o)IA where
IAl = (1 - SX11 - SK11)(l SX22 ̀ SK22) -" (SX12 + SK12)(SX21 + SK21)• (29)
Thus, the sign of the determinant I Ja (0) I depends on the sign of the determinant D (O), while IAI is positive because of the Hawkins-Simon's condition'°
The following classification of the system can be obtained.
1. If I Ja (0) I > 0, i. e. D(0) > 0 the system has a fixed point of stable.
2. If I Ja (0) I <0, i. e. D(0) < 0 the system has a fixed point of saddle.
3. The eigenvalues are diverge, when determinant D (0) is zero.
D(0) > 0 can be rewrite from eq (28) as:
SK11 SK22 > (rl + 611)(r2+ 622) S
K12SK21 (r1 +621)(T2+612)
This inequality implies that relatively large input coefficient of own capital goods than of the other sector's capital goods compared to its relative cost of the own capital goods to cost of the other sector's capital goods. That is, given the relative cost, if the economic system starts to be relied more heavily on outsourced capital
goods than before, the system may hit a bifurcation point that shows saddle point instability. Substitute the definition of SKL~ into the condition, it yields the condition in terms of capital goods quantity:
k11 > k12 k
21 k22 1OThe Hawkins-Simon's condition for a two sector input-output model is as follows, where A is
a input-coefficient matrix: 1-a11 -(112
>0. --a21 1 - a22
The condition implies the system operates in positive production for all the sectors. In this case, the inputs are not only material inputs Sx,, but also includes capital goods SKI,. The sum of both inputs needs to be in production possibility.
13
This again shows that the relatively large own capital input implies saddle point
instability of the system."
The previous classification can be interpreted as follows.
1. If sK > (r,±6„)(r2±622) SK 1221 SKor Ll> , the system has a stable fixed point. (r1 +621)(r2+612) k21 k22
2 If sK'-'sue < (r1+611)(r2+622) or k11. < k1Z , the system has a fixed point of SK12SK21 (r1+&21 )(r2+612) k21 kz2
saddle.
3. If s K 1 1 s- - (r, +6, ,) (r2+622) or - the eigenvalue of the system is SK12SK21 (r1+621 )(r2+612) k21 k22 di
verged.
3.3 Checking another fixed points i = 0
Set Eq (26) equal to 0, there may be another fixed points in the system . Simi-lar discussions to the Jacobian matrix at z = 0, it can be verified that the matrix Sr (z) -1(I ® Sx - SK) must be singular to hold the equilibrium with z 0. Be-cause of I I -- S,~ -- SKI > 0, it should be Sr (z) -1 = 0. This is impossible unless Sr(z) is singular. I shall show next that there are many singular points.
Nonetheless it is necessary to solve the equilibrium equations to show the phase portraits of the systems. Solve the next equations for z1 and z2:
i1 - 0 (30) Z2 --- 0
That is
(1-Sil-SK1 _)SK22, (S12+SK12~SK21 (1-s22-SK22)SK21 (S21+SK21)SK22 r2+622-z2 + r,+62,-z2 ) Z1 - ( r, +621 -Z2 + r2+622-z2 Z2 =0
((1-st,-SK11)SK12 + (S12+SK12)SK11 Z + ((1-S22-SK22)S1K11 (S21+SK21)SKi~ Z r2+b, 2-z, r, +6 t 1-z, 1 rt +61 , -zi r2+612-z1 ) 2 =0
(31) There are two asymptotes in each equation. One of these is a vertical or hori-
zontal line.
Z - (1-S11-SK11)SK2_(rl+b21 )+(S12 ~SK12)SK2l (r2+622) dzl 2 (1-S11-SK11)SK2 +(S12+SK12)SK21 for dt = 0 (~_-S22-SK 22 )SK 1 1 (r2+61+(S21 +SK21 )SK 12 (rl +61 1) dz z (1-S22-SK22)SKll+(S21+SK21)SK12 for d = 0
The other is a straight line of which gradient does not depend on the interest rate
or the depreciation rate. But the line is too complicate to describe in terms of the
11Benhabib and Nishimura [1998] assume single interest rate and single depreciation rate, but
introduce externality. They obtain indeterminacy, i.e. multiple equilibria in the two sector model .
14
original parameters in the system S,. or SKij. It can be shown that eq (31) can be arranged into the following form:
c~ z1 = z2 -- ~z - + z for ---- = 0 di d~ d~ (c1-d1zz) dt
Z2 + (aZd?-bzcz)~2 . for =0 dz dz d2 (c2-d2z2) dt ,
where we temporally introduce the parameters at , bi, ci, di (i = 1, 2).
al = (S21 + SK21)SK22(rl + 621) + (1 - S22 - SK22)SK21 (r2 + 622) b1 = (S21 + SK21)SK22 + (1 - 522 - SK22)SK21
C1 = (S12 + SK12)SK21 (r2 + 622) + (1 - S11 - SKll)SK22(rl + 621) dl = (S12 + SK12)SK21 + (1 - S11 ̀SK11 )SK22
a2 == (S12 + SK12)SK11 (r2 + 612) + (1 -511 - SK11)SK12(r1 +6) ) b2 = (S12 + SK12)SK11 + (1 - S11 - SK11 )SK12
C2 = 521 + SK21)SK12(rl +611)+(1 `_ S22 ̀ SK22)SK11 (r2 + 612) d2 = (S21 + SK21 )SK12 + (1 - S22 - SK22)SK11
If a, .d,. -- bici = 0, the second type of asymptotes disappears. Instead, equation dzi/ dt = 0 becomes a straight line through the origin, and its gradient does not depend on the interest rate or the depreciation rate. The gradient is determined by the constants (here I assume that they are technological factors) Sij, and SKij. Thus, the determinants aid,. --- bici = 0 are another important factors in the system, and it can be shown as:
al dl - b1 c1 = (r2 - rr + 622 -' 621)SK21 SK22IAI a2d2 - b2c2 = (r1 -T2+611 -- 612)SK11 SKI2IAI,
where IAl = (1-S11 -SK11)(1 -S22-SK22) -(S12 +SK12)(S21 +51(21)
The sign of IAl is again positive because of the Hawkins-Simon's condition in terms of the cost share, and each SKij is positive. The sign of adi -- bici = 0 is determined by the magnitude of the interest rate and the depreciation rate.
3.4 Classification of the autonomous system
Summary of the above discussions provides the general classification of the two sector system. First, the system is stable at the fixed point, or saddle at the fixed
point. Second, the sign of intercept of the asymptote for each equation it = 0 and Z2 = 0, there are four cases. The other factor that is not considered here is the magnitude of gradient of the asymptote.
There are at most four set of solutions for z1 and z2 including O.The other solu-tions are from the 3rd order polynomial equation, and the determinant vanishes at
15
the points. This situation can be shown the points, where three curves intersect in Figures 3.4-3.4.
Case I The system is stable at the fixed point z = 0. The two asymptotes for i~ = 0 and z2 - 0 both have positive intercepts to the other axis: a1 d1 --- b1 c> > 0, a2d2 --- b2c2 > 0. An economic meaning of these conditions is as follows that
the nominal cost (interest rate and depreciation) of capital goods is higher in the own sector's investment that the other sector's.
Case II The system is stable at the fixed point z = 0. The two asymptotes for ±1 = 0 and i2 = 0 both have negative intercepts to the other axis: a1 d1 - b 1 c 1 <0,
a2d2 - b2c2 <0. The nominal cost (interest rate and depreciation) of capital goods is lower in the own sector's investment that the other sector's.
Case III The system is stable at the fixed point z = 0. The asymptote for ±1 = 0 has positive intercepts to z2 axis: a1 d1 - b1 c> > 0. The asymptote for i2 = 0 has negative intercepts to z1 axis: a2d2-b2c2 < 0. The nominal cost (interest
rate and depreciation) of capital goods is higher in the own sector's investment that the other sector's for the commodity 2. The nominal cost (interest rate
and depreciation) of capital goods is lower in the own sector's investment that the other sector's for the commodity 1.
Case IV The system is stable at the fixed point z = 0. The asymptote for ±1 = 0 has negative intercepts to z2 axis: ai d1 -b1 c1 <0. The asymptote for z2 = 0 has positive intercepts to z1 axis: a2d2 -b2c2 > 0. The nominal cost (interest
rate and depreciation) of capital goods is lower in the own sector's investment that the other sector's for the commodity 2. The nominal cost (interest rate
and depreciation) of capital goods is higher in the own sector's investment that the other sector's for the commodity 1.
The same classification can be applicable to the system of saddle point.
Case V The system has a saddle point at the fixed point z = 0. The two asymptotes for ±1 = 0 and i2 = 0 both have positive intercepts to the other axis: a1 d1 -
b1 c> > 0, a2d2 - b2c2 > 0. An economic meaning of these conditions is as follows that the nominal cost (interest rate and depreciation) of capital goods
is higher in the own sector's investment that the other sector's.
Case VI The system has a saddle point at the fixed point z = 0. The two asymp- totes for i~ = 0 and i2 = 0 both have negative intercepts to the other axis:
a1 d1 - b 1 c 1 < 0, a2d2 - b2c2 < 0. The nominal cost (interest rate and depreciation) of capital goods is lower in the own sector's investment that the other sector's.
16
z2
0.6
0.4
0.2
0
-0 .2
-0 .4
i
i//
i
i
i
4:L
Eigenvectors ±1=0
z2=0 D(z)=0
-0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
zi
Figure 1: Case I: Stable fixed point at z = 0
Case VII The system has a saddle point at the fixed point z o 0. The asymptote for z1 = 0 has positive intercepts to z2 axis: ai d1 -- b1 c> > 0. The asymptote for z2 = 0 has negative intercepts to zi axis: a2d2 - b2c2 < 0. The nominal cost
(interest rate and depreciation) of capital goods is higher in the own sector's investment that the other sector's for the commodity 2. The nominal cost
(interest rate and depreciation) of capital goods is lower in the own sector's investment that the other sector's for the commodity 1.
Case VIII The system has a saddle point at the fixed point z = 0. The asymptote for z~ i 0 has negative intercepts to z2 axis: a1 d1--b1 c1 <0. The asymptote for z2 = 0 has positive intercepts to z1 axis: a2d2 -- b2c2 > 0. The nominal
cost (interest rate and depreciation) of capital goods is lower in the own sec- tor's investment that the other sector's for the commodity 2. The nominal cost
(interest rate and depreciation) of capital goods is higher in the own sector's investment that the other sector's for the commodity 1.
Parameter sets for the numerical illustration can be shown in Tables 1-2.
17
Table 1: Parameter sets for the stable fixed point at z = 0
Case I Case II Case III Case IV
a1 d1 - b1 C1 > 0 a1 d1 - b1 C1 < 0 al d1 - b1 C1 > 0 a1 d1 - b1 C1 < 0
a2d2 - b2c2 > 0 a2d2 - b2c2 < 0 a2d2 - b2c2 < 0 a2d2 - b2c2 > 08x11 Sx12 8x21 8x22 SK11 SK12 SK21 SK22 rl V2 511 512 521 b22
0.04
0.29
0.19 0.20
0.42
0.12
0.15
0.40
0.05
0.05 0.25
0.15
0.15
0.25
0.04
0.29
0.19
0.20
0.42
0.12 0.15
0.40
0.05
0.05
0.15
0.25
0.25
0.15
0.04
0.29
0.19
0.20
0.42 0.12
0.15
0.40
0.03
0.06
0.15
0.15 0.25
0.25
0.04
0.29 0.19
0.20
0.42
0.12
0.15
0.40 0.06
0.03
0.15
0.15
0.25
0.25
Eigenvalue
Eigenvector at z =0
Eigenvalue
Eigenvector
at z = 0
0.0540706 -1 .22028
0.731968 0.681339 ) -0 .0443101
0.615461 l -0 .788167 /
0.01915 -0.529053
0.726971 0.686668 ) -0 .0361967 0.610981 l
-0 .791646 /
0.0283218 -0 .71621
-0 .538918 0.842358
-0 .039544
0.627869
0.778319
I) (0.0302222 -0:747259
-0 .59224
0.805762 -0 .0404441
0.61659 l 0.787285 /
This table shows that the system has a stable fixed point at z = 0 The difference between cases I-TV comes from the interest rate and the deprecia-tion rate. The technological parameters are common for all cases. a1 d1 - bl c1 = (r2 - rl + b22 - &21)SK21 SK22IAI a2d2 - b2c2 = (rt - 12+&I1 -- &12)SK11 SKI2IAI a1 d1 - b1 c1 implies the difference of nominal cost of capital goods 2 between the own sector 2 and the sector 1. a2d2 - b2c2 implies the difference of nominal cost of capital goods 1 between the own sector 1 and the sector 2. J(0) I denotes the Jacobian at the fixed point z = 0: positive means stable around the fixed point. `Eigenvalues' are of the solution 11 for the equation, IA! - Ja(0)I = 0. `Eigenvectors' are the associate vectors z* with each eigenvalue, Ja(0)z* = Az*.
18
Table 2: Parameter sets for the saddle point at z = 0
Case V Case VI Case VII Case VIII
a1 d1 - b1 c1 > 0 a1 d1 - b1 c1 < 0 a1 d1 - b1 c, > 0 a1 d1 - b1 c1 < 0
a2d2 - b2c2 > 0 a2d2 - b2c2 < 0 a2d2 - b2cz < 0 a2d2 - b2c2 > 0
Sx 11
Sx12 Sx21
Sx22
SK11
SK12
SK21
SK22 Ti
r2
b11
612
621
622
0.04
0.29
0.19
0.20
. 0.14 0.32 0.15
0.10
0.05
0.05
0.25 0.15
0.15
0.25
0.04 0.29
0.19
0.20
0.14
0.32
0.15
0.10 0.05
0.05
0.15
0.25
0.25
0.15
0.04
0.29
0.19
0.20 0.14
0.32 0.15
0.10
0.03
0.06
0.15 0.15
0.25
0.25
0.04
0.29 0.19
0.20
0.14
0.32
0.15 0.10
0.06
0.03
0.15
0.15
0.25
0.25IJ«(a)I Eigenvalue Eigenvector at z=0
Eigenvalue Eigenvector at z = 0
-0 .351 1.73559
0.508159 -0 .861263
-OZ02237
0.560947 -0.827852
-1.99964
8.77869
0.502609 k -0.864514 ) -0 .227783 0.566327l
-0 .824Y81 /
-0 .648356
3.13194 0.380055 -0.924964 )
-0 .207014
0.564695 l -0 .8253
-0 .589276
2.87207
0.358354 -0.933586 ) -0 .205174
--0 .540816 -0 .841141
This table shows that the system has a saddle point at z = 0 The difference between cases I-IV and cases V-VUI comes from the technologi-cal parameters, especially from SK. Cases V-VIII shows larger capital goods input from the other sectors than own sectors. Out sourcing of capital goods implies saddle point instability, as I explained in the text. The difference between cases I-IV comes from the interest rate and the depreciation rate. The technological parameters are common for all cases. a1 d1 - b1 C1 = (r2-r1 + 622 - 621)SK21 SK22IAI a2d2 - b2c2 = (r1 - r2 + 611 - 612)SK11 SKI2IAI a1 d1 - b1 c1 implies the difference of nominal cost of capital goods 2 between the own sector 2 and the sector 1. a2d2 - b2c2 implies the difference of nominal cost of capital goods 1 between the own sector 1 and the sector 2. I J«(0) I denotes the Jacobian at the fixed point z = 0: positive means stable around the fixed point. `Eigenvalues' are of the solution A for the equation, (AI - Ja(0)I = 0. `Eigenvectors' are the associate vectors z* with each eigenvalue
, Ja(0)z* = Az*.
19
0.6
0.4
0.2
zz 0
-0 .2
-0 .4
~ ge
vectors i1 = 0 iz = 0
(Z) =0
-0 .2
0.6
0.4
0.2
Z2 0
-0 .2
-0 .4
-0.2
-0 .1 0 0.1 0.2 0.3 0.4
zi
Figure 2: Case II: Stable fixed point at 2 = 0
0.5
1
i
I
I
i
i
i
i
i
i
4
i
Eigenvectors - -=i1 = 0i2 = 0
I7(z) - 01 1
-0.1 0 0 .1 0.2 0.3 0.4
11
Figure 3: Case III: Stable fixed point at z = 0
20
0.5
Z2
Z2
0.6
0.4
0.2
0-
-0.2
-0.4
-0 .2
0.6
0.4
0.2
0
-0.2
-0.4
i
I
i
i
i
Eigenveetors
I
=0Z~z2=0 -~
-0.1 0 0.1 0.2 0.3 0.4 0.5
Figure 4: Case IV: Stable fixed point at z == 0
F4i
i
i
i
4
igen
D(
ctors -0
=0
-0.2 -0.1 0 0.1 0.2 0.3 0.4
z1
Figure 5: Case V: Saddle point at z = 0
21
0.5
0.6
0.4
0.2
Z2 0
-0.2
-0.4
i
i
i
i
i
Eigenvectors ii = 0
Z2 = 0 - D(z)=0 --
-0.2
0.6
0.4
0.2
Z2 0-
-0 .2
-0.4
-0 .2
-0 .1 0 0.1
Figure 6: Case VI:
0.2 0.3
Zi
Saddle point at z
0.4
0
0.5
I I f I
I
/
I
i
i
i
i
i
Eigenvectorsz1 = 0Z2 = 0
D(z)=0I
-0.1 0 0.1
Figure 7: Case VII:
0.2 0.3
z1
Saddle point at z
0.4
0
0.5
22
z2
0.6
0.4
0.2
0
-0.2
-0.4
i
i
~T
i
i
i
i
i
Eigenvectors ±i=0 0
D(z)=0
-0 .2 -0.1 0 0.1 0.2 0.3 0.4 0.5
Figure 8: Case VIII: Saddle point at z = 0
4 Price variation Langevin equation: A simple case
In this section, introduce a stochastic factor in fluctuation of the total factor produc-
tivity. It is very difficult to solve the system which incorporates non-linear differen-
tial equations and stochastic factors. First, take a single equation and next consider
two sector system.
4.1 One sector model
The definition of variables are the same as before, the rate of (aggregate) price change is denoted as dlnp/dt = z, a denotes a = (1-Sx--SK)/SK, SX = pxx/C the cost share for the intermediate input, SK = pLK/C the cost share to the capital
goods input. PK = p (r + b - z) the user's cost of capital, r the interest rate, & the depreciation rate.
Assume that the cost shares are constant as in the previous sections. The pro-ductivity change din TFP/dt includes a drift term p.-rFp and the Langevin term L(t), which is a source of stochastic fluctuation.
Assumption: Total factor productivity (TFP)
din TFP dt _ MTFP + L(t).
23
Using the above stochastic productivity specification, the one sector model for eq (19) can be expressed as follows:
_~. ' d In TFP d In w $ y • S K-- ' r+b-z Z - (1-" SX - SK) Z -- d - - $ L dt- -' r&-z r - r+&-z s Z - a(r + s - Z)Z = r+&-z d1nTFP - S~ dlnw
SsK dt S K dt r+ z (N
TFP + L (t)) - jd 1n w - r - b $K SK dt
Assume that the average productivity change µ-rFP is allocated into the wage
increase, the interest rage change, and the depreciation change , that is
1 dlnw -E--SK (r+b)µTFP"SL at ---SK(r+b) ~
where E is assumed to be close enough to zero .
Using this equation, rewrite the system as follows:
Z - a(r + b - z)z = r SK ZL(t) -. E (32)
Assumption: L(t) the Langevin term Introduce the assumption on the Langevin term:
< L(t) >= 0, < L(t)L(t') >= rs(t -
where the notation < o > denotes taking expectation, r denotes variance of L(t), b (t --- t') denotes Dirac's delta function.
The Langevin equation has multiple interpretations.12 One of the most famous interpretation is due to Ito, who interprets the time derivative eq (32) as derived from the following integral:
z(t + tit) -z(t) = f t+°t a(r + s - z(s))z(s) ds + { r+s - z(t) rt+°t L s ds + e[.1t SK SK } J t ( )
t+°t (33) It {a(r + s - z(s))z(s) + E} ds
+ r+b-z(} t) t+°t L(s) ds. SK J't The other famous interpretation is due to Stratonovich , who interprets the time
derivative of the left hand side of eq (32) as follows:
z(t + fit) -z(t) = f t+°t a(r + b - z(s))z(s)ds + { r+s - z(t+_°t)+z(t) +°t L s ds + EOt SK ~ 2SK } J'1 t ( )
t+°t (34) = f t {a(r + s - z(s))z(s) + e} ds f r+b z(t+i t)+z(t) t+°t + SK - - 2SK J ft L(s)ds.
12This part of the paper depends on van Kampen [1992].
24
The interpretation on the coefficient of L(t) differs from each other. Ito's inter-
pretation assumes that z has determined before L(t) has some value, therefore the integral can be taken as if z is a constant. But Stratonovich's interpretation assumes that z moves while L(t) takes into action, therefore the integral has the averaged coefficient. The solution of the system significantly depends on the interpretation taken into account.
Here I assume that a lot of productivity changes occur microscopically, and it is under given macroscopic price variation z. And introduce the distribution of L(t) as Gaussian.
Assumption: z Assume no immediate feedback or no reaction from L(t) to z.
Assumption: L(T) Assume L(t) obeys Gaussian process. Under these assumptions, the Ito's integral eq (35) is equivalent to the Fokker-
Planck equation (36).
t+ot t+ot
z(t + Lt) - z(t) = A(z(s)) ds + C(z(t)) L(s) ds (35) t t
aP(z, t) -a (A(z)P(z, t)) + r a2 (C(z)2P(z, t)) at (36) az 2 az2
This equation can be expanded as follows:
apa,-t) = {-A'(z) + r (C"C + C'2) } P(z, t) + (2rCC' A) ap(Z,t) PC a P(z t) az (37) +
r az
Introduce new parameters a and (3 as follows:
a(z) = VC(z)2
2
(3(z) = rC(z)C'(z) - A(z) (38) f3 (z) = r(C (z) + C(z) C "(z) - A (z) )
(3(z) + a'(z) = 2rC(z)C'(z) - A(z).
Eq (37) is rewritten to
aP(z,t) = R'(z)p(z, t) +. (x'(z) + (3(z)) apai,t~ + a(z) a2 ZZ ap(Z,t) _ a(R(Z)p(Z,t)) ____ __ t~} (39)
at - + aZ aZ
25
Apply these relations to eq (33), the following equations are obtained:
A(z) = a(r + b - z)z + e - __ C(Z) - SK
2
a(z) - r ____ 2 sK
13(z) _ --rrSK2zZ - a(r b Z)Z- (40) (3'(z) = -- a(r + b) + 2az
13(z) +.'(z) = _2rTS-- ; -- a(r + b - z)z - e. Thus, eq (33) is equivalent to the following Fokker-Planck equation.
+b- ) sK aP( ) a2 z, t a {(a(r + b - z)z + e) P(z, t)} r {(rz)2P(Z,t}
at - - az + 2 - -- az2
The solution for this equation can be obtained by numerical calculation. Strict solution has been found only for the very specific case, see for example Rogers
[1997].13
4.2 Stochastic formulation for the n sector model
In the multiple sector model, introduce E for unallocated productivity gain:
E1 EZ -
e =
En ~-'
( ~ i.=1 r1+&1-r. 1 -Lei =1 r1+51- µTFP1 -- L.n Lll
dlnw - L.- i1 bil I .TFP2 - SLj24-"_-~ - n. SKt2 T2 - ~n Sxt2 bit j 1 dt ia1 r2+6 2-zt i=1 T+62-Zt
LL S dlnwi -n Sxtz _ T ~n Sxi2 µTFP2 - j Lj2 dt i=1 r2+bt2-z; 2 i=1 r2+biz-zt b~ (41)
djnTFP1 NTFP1 + L1 (t) d In TFP - d 1fTFP2 f NTFP2 -+- L2(t) dt =' = (42)
d1n.TFPn NTFP11 + Ln(t) -- ____ - - - dt
13Finding strict solutions for the Fokker-Planck equation see for example, Nariboli[ 19'77], Hill [1982], and Zwillinger [1998]
26
In general, the dynamic price equation system can be expressed introducing e, the Langevin term L (t) and z as follows:
Srz+(I--SX-SK)z=L+e (43)
4.3 The system around the fixed point z= O
First, investigate the system around the fixed point z = 0,
Sr(0)i + (I - SX -R SK)z = L + e.
In this case, the system can be expressed as
dz(t)_= -Sro-1{(I - SX - SK)z(t) - e}dt + Sro-1 dL(t),
where Sro = Sr(0).
(44)
(45)
To solve the equation, take the following autonomous matrix equation first,la
dz(t) = -Sro-1{(I - SX - SK)z(t)}dt
It has a solution 4'(t, to) corresponded to an initial condition. The non-autonomous equations can be rewrite as follows:
dz(t) = -Sro-1{(I --- SX - SK)z(t) - e}dt + Sro-1 dL(t)
Using the solution 4(t, to), the solution for the non-autonomous equations are
given as follows:
J t J t z(t) = (t, to) (z(0) + (s)-' eds + + (s)-' Sro-' dL(s) to to
Stability of the system depends on the eigenvalues of the matrix --Sro-1 (I -SX -- SK). The condition is the same as that of the non-stochastic model that I have explained.
Solution method of the non-linear stochastic system is not yet investigated. But the Fokker-Planck equations for multiple variables can be determined from eq (43). And there are many numerical methods to solve the non-linear partial differential equations, see for example Zwillinger [1998].
Assume L (t) obeys Gaussian process and has the covariances < L(t)L, (s) > T'L; b (t - s), eq (43) become equivalent to the multivariate Fokker-Planck equation. Thus the Fokker-Planck equations for eq (43) can be shown as follows: i4Following part of this section depends on Hon [1977].
27
apa2'ti = ((Sr1(I-Sx-SK)z+e)P(z ,t)) -
~" n a2 +2 L..i,l=t aziaz(.QP(z,t)) where
S2 = 2Sr-1 (z)rSr-1T(z).
Sr-i T denotes transpose of the matrix, r denotes n x n matrix of the covariance of L(t). These equations remains to be investigated.
5 Concluding remarks
This paper constructs the classification for the general two sector model based on the price equations for growth accounting. As a result, degree of out-sourcing on capital goods is an important factor for the stability of the system. If each sector out-sources their capital goods for investment extensively, the fixed point of the system becomes a saddle point, rather than asymptotically stable point .
But as the phase portraits show, the system includes large instability area , which is separated by the locus of singularity of the system's matrix Sr. When the prices come across this singularity locus, the fluctuation explodes drastically . This is why my previous paper shows that the prices change numerically very fragile.
In this paper, I can correct two mistakes in my two previous papers . Although none of them are serious, the corrections have helped understanding the dynamic
price fluctuations, is As to stochastic model, there are a lot of problems to be solved. The non-linear
stochastic equation system is extremely difficult to solve, but it is necessary to derive the distribution of prices. That will explain how frequent the price changes go into unstable. region, such as deflation spiral, or hyper inflation. For that purpose , I have derived the general partial differential equation system for future investigation .
1SIn Hayami [1993] , the n sector model has errors in the notation. But the other part does not change at all. Hayami [1999] includes also minor proof reading errors, but I should reconsider the last part of the solution. At the moment, I have not yet verified the solution that provides the price distribution.
28
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