Download Simultaneous Price and Quantity Adjustment in a Single Market

Simultaneous Price and Quantity Adjustment in a Single Market
Author(s): Martin J. Beckmann and Harl E. Ryder
Source: Econometrica, Vol. 37, No. 3 (Aug., 1969), pp. 470-484
Published by: The Econometric Society
Stable URL: http://www.jstor.org/stable/1912794
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Vol. 37, No. 3 (July, 1969)
Economentrica,
SIMULTANEOUS PRICE AND QUANTITY ADJUSTMENT IN A
SINGLE MARKET
BY MARTIN J. BECKMANN AND HARL E. RYDER
This paper discusses the dynamics of disequilibrium in a single market where both price
and quantity change in response to disequilibrium. We describe the nature of the adjustment
path under a wide variety of assumptions, noting in particular the properties of stability
in the large and in the small and the existence of limit cycles.
INTRODUCTION
in disequilibrium. Then forces will operate on bo
price and quantity simultaneously. The resulting motion, described by a curve
a price-quantity diagram, which contains the usual supply and demand functior
is not completely arbitrary. Rather a small number of distinct types of moti
can occur. An analysis of these types, which have been discovered in the study
nonlinear oscillations in physical systems [5, 8, 10, 11], may be of interest in
own right and also for the light it sheds on possible commodity cycles, as well;
on the question of market stability. It is in fact rather remarkable that these simp
facts which relate to the very core of economics, supply and demand, have so f
escaped closer scrutiny by economic theorists.
The appropriate concept of stability will differ depending on the domain (
which we wish to define it. We may usefully distinguish (1) stability in the sma
(2) stability in the large, and (3) global stability. A motion defined on an unbound(
set Q is stable in some sense if there is a domain D c Q and a bounded regi(
R c D such that all paths which begin in R remain in R, and all paths which beg
in D enter R after a finite period of time. When R is contained in an arbitrari
small neighborhood of a point X of the interior of R, the motion is stable in ti
small. This concept corresponds to asymptotic stability in the sense of Lyapunc
[see 11, p. 202, or 6]. When D Q, we have stability in the large, which corresponi
to the concept of dissipativeness [9, p. 29]. Global stability requires that bo
conditions be satisfied. It is thus the strongest concept and is a special case of ea(
of the others.
In the sections that follow, we shall discuss the stability of a system of mark
adjustment equations in each of the above senses. In Section 1 we examine ti
conditions for stability in the small. Section 2 shows that reasonable assumptiol
lead to the conclusioni that the system must be stable in the large and discusses ti
conditions for global stability. Section 3 contains a few applications.
CONSIDER A SINGLE market
1. MOTION NEAR EQUILIBRIUM
Let P denote price, Q quantity, Qd = f(P) quantity demanded at price P, ar
Ps=g(Q) "supply price," i.e., marginal cost to the industry of supply quantity X
Let t, I denote speeds of adjustment.
470
PRICE AND QUANTITY
ADJUSTMENT
471
A simple model of price and quantity adjustment combining the "Walrasian"
[12] and "Marshallian" [7, 2] approaches is
(1)
= A(Qd
-
= 1(P -
(2)
Q)
Ps).
(The dot denotes derivatives with respect to time.)
Here the first equation states that price moves in the direction of and proportional
to current excess demand, i.e., demand at current price minus current supply.
The second equation says that supply moves in the direction of and proportional
to the excess of current price over supply price (or marginal cost). Thus, above the
supply curve quantity increases, and to the left of the demand curve, prices rise,
and vice versa.
Equilibrium values of price P and quantity Q are determined as the solutions of
(3)
Q
(4)
P = PS(Q)
=
Qd(F),
The proportionality assumptions in (1) and (2) are justified since we are dealing
with small deviations from equilibrium. Now let
(5)
Qd=
aO + aP + o(P - P)
where o(P) denotes terms of smaller order than P. Similarly
(6)
bo
~
Ps
b
1 + o(Q-Q).
+ bQ
Here ao and a are the coefficients of the conventional demand functions,
D = ao + aP; bo and b are those of the conventional supply functions, S = bo +
bP: With (5) and (6) the equilibrium solutions (3) and (4) assume the form
P aO -bo
b-a
'
aobb-abo
b-a
(They are nonnegative provided that ao > bo and that a < b.) Let p = P -,
q = Q - Q denote deviations from equilibrium. In terms of these the equation
system (1) and (2) becomes
(7)
(8)
p=
=
(ap -q),
p-q
Before discussing the stability and type of motion of (7) and (8), we want to
consider an alternative type of adjustment process that arises under a different
institutional arrangement.
472
M. J. BECKMANN AND H. E. RYDER
In the previous model of a product market, prices were set by sellers competing
for custom in the manner of Walras' "prix crie." In a factor market by contrast,
prices are administered by buyers. In terms of deviations from equilibrium, let q
be factor employment and p the going factor price. Let q,(p) denote the factor
supply curve. Similarly let pA(q) be the marginal revenue product curve. We then
have
(9)
P=
(10)
A(q - qJ),
P)
(Pd
P
=
Equation (9) gives the effect in a competitive factor market of involuntary unemployment (overemployment) on factor price. Equation (10)describes the adjustment
of employment to marginal revenue product. In terms of the notation (5) and (6),
we have
QS=bo +bP +o(P-P),
Pd =
ao + 1
+-Q
+o(Q-Q).
a
a
The equations of motion for the factor market are
(9')
=
(10')
q=
A(q - bp),
q-p
Since we are free to choose the units of measurement for the commodity-hih
price, and time, let us make the following transformation:
7r(t) =
4LP
)
(t)
q(
t)
In the new variables, the systems (7), (8) and (9'), (10') reduce to
(11)
i= =
(12)
=
X
a
--a
3-
0a=r A- b a
14
(11')
(12')
=
=a-b
r
-
1
C-7
A-ir=a-flrr
where a = aA/[IP and /B= bAI
and supply curves respectively.
represent the transformed slopes of the demand
PRICE AND
QUANTITY
473
ADJUSTMENT
Clearly, the two models are formally equivalent with the role of prices and of
quantities interchanged and with the coefficients replaced by their reciprocals.
All cases possible in the goods market (11), (12) will also arise in the factor market
(11'), (12') and vice versa. We shall, therefore, restrict ourselves to the case of the
goods market (11), (12).
From the theory of systems of differential equations, and more particularly
of oscillations [3,5,8, 11], it is well known [10, p. 385 ff.] that the mode of approach
to or divergence from equilibrium depends on the linear parts of the function f
and g only and more specifically on the characteristic roots of the system (11), (12),
which may be obtained by solving
(13)
la-p
-1
=
1
0.
The characteristic roots are
(14)
a
Ij--
P1,2
)- 4
+4-
)-
22(-)21(
The slope of the demand curve a may be negative (normal good) or positive
(Giffen good). The slope of the supply curve ,Bmay be positive (increasing cost)
or negative (decreasing cost). Hence, a and 1/fl may take on any values, although
the normal case is a < 0 < 1/f. Table I lists all possible types of motion in terms
of the characteristic roots p and the structural coefficients a and ,B.Figure 1 shows
graphically the combinations of a and 1/,Bfor which these various kinds of motion
occur. The shaded region indicates stable motion.
So far we have assumed the motion around equilibrium to be unrestricted.
This is not true when the equilibrium falls on the P axis (Figure 2) because the
demand curve lies below the supply curve in the positive quadrant. In that case,
(15)
P
=
[ao
+ aP
-
Q]
P; + bo -bQ)
b
b
0
as before,
if Q > 0,
~
~
~
0
bo
if Q = 0 P +
-Q
< ?.
It is easily verified that the second alternative for Q applies only to the part of
the P axis below the supply curve S. On the other hand, for all points below the
demand curve, P is positive so that any point on the P axis below Do will move
vertically upwards. Thus, the segment DOSOof the P axis represents the set of
equilibrium points. It is a distended node or spiral. It is stable in the large.
474
M. J. BECKMANN AND H. E. RYDER
TABLE I
Domain
.~~~~~
Roots
Pattern of Motion
Real, distinct
Two rays
..
.2.,.
A
A1
(a +1)2>4
1
1>,B>a>2-<r
-1<A2
a=3
A3
->
As
-1
Pi = 0 > P2
<
< a <-2--
B1
a<-=
B2
a=-=
c
C3
-2-a
a
node,a unstable
One ray
p< 0
node,a stable
a+ 2
p = 0
Parallel lines, constant
distance from locus
of multiple equilibria
-a
<
<
P1 > P2 > O
2
a-a
(a
-1
Parallel lines, unstable multiple equilibria
Real, double
=4
/3
-
Pi > p2 = 0
> 2-a
(a
B
C2
Parallel lines, stable, multiple equilibria
P > 0 > P2 Saddle point,a two stable paths, all others
1> ->
-2-a
node,a stable
unstable
a=/3>-
C1
> P1 >P2
-2-
1
A4
a>
O
<-
/3
B3
or
+ 2
p> 0
<4
+
< 2-a,
Complex
>a
= a < 1
< - < 2-a,
node,' unstable
< a
No ray
Real part < 0
spiral,astable
Real part = 0
center
Real part > 0
spiral,a unstable
For graphs of the various types of stationary points, see [5, 75-84; 11, pp. 115-126].
475
PRICE AND QUANTITY ADJUSTMENT
B,
~A2
A3
B,~~~~~~~~~
\A4
a
A4
A5
02
~~~~~B3
03
A3
B3
A
A5
FIGURE.-Classification of cases.
S
P
So
FIGURE2
FIGURE2
wa
476
M. J. BECKMANNAND H. E. RYDER
A similarcase arises if the demand curve intersectsthe supply curve on the
horizontalaxis becausesupply up to a point is free (Figure3). It may be shown
that the line segmentDo, SO,is the set of equilibriumpoints and that it is stable
in the large.
Pi i
FIGURE 3
2. BEHAVIOR IN THE LARGE
Excludingthe boundarycases A2, B2, and C2, the linearapproximation(7),(8)
enablesus to settlethe questionof local stabilityin the small.For the moregeneral
questionof stabilityin the large,we must examinedynamicbehaviorin the large.
But in the largeit is not safe to assumethat demandand supplycurvesare linear.
We, therefore,turn to the nonlineargeneralizationsof (1),(2) and (9),(10).
For a productmarket,we have a generalizeddynamicsystem
p = G(P, Q),
6
= F(P, Q),
where G and F are continuous, differentiablefunctions defined for all P > 0,
Q > 0.
Here, the locus G(P,Q) = 0 representsthe demandcurve.The functionG has
the followingcharacteristics:
(17)
G(P, 0) >
,
G(O,Q)>
,
whichmeansthat the axes are alwayson or to the left of the demandcurve:
G
<0
forallP>OandforP=0,G(0,Q)>0,
(18)
aG-0
aQ
forP=O,G(O,Q)=O,
PRICE AND QUANTITY
ADJUSTMENT
477
OG
< 0 for a normalgood,
for completelyinelasticdemand,
0-?
(19)
aG
-G > 0
for a Giffengood.
Condition (18) means that price will rise when there is excess demand and fall
whenthereis excesssupply.In addition,the speedof priceadjustmentwill increase
as the discrepancybetweensupplyand demandincreases.A furthercharacteristic
is that thereexists N > 0 such that
G(P, NIP) < 0 for all P > 0.
(20)
Condition(20)meansthat total expenditurefor this productis bounded.
The locus F(P, Q) = 0 representsthe supply curve. The function F has the
followingcharacteristics:
(21)
F(P,0)>0,
F(O,Q)
0,
i.e., the price axis is on or above the supply curve,and the quantityaxis is on or
below it;
OF
>0 forallQ>OandforQ=0,F(P,0)>0,
(22)
OF
-=O
OF
forQ=0,F(P,0)-0,
< 0 for increasing costs,
aQ
(23)
OF
=
0 for constantcosts,
aF
-F
> 0 for decreasing costs.
aQ
Condition(22)meansthat output will rise when priceexceedscost and fall when
cost exceeds price. In addition, the speed of quantity adjustmentwill increase
when the differencebetweenprice and cost is increased.A furthercharacteristic
is that there exist J, K, L, M > 0 such that
F(P, Q) > 0
for 0
Q < J,
P > K,
for Q L, 0 P < M.
Condition (24) means that for a sufficientlyhigh price, a positive quantity will
be suppliedand that beyond a sufficientlylarge quantity,the cost has a positive
lowerbound.
Theseconditionsare sufficientto assurethe followingtheorem.
F(P, Q) < 0
M. J. BECKMANN AND H. E. RYDER
478
THEOREM
1: The system (16) satisfying conditions (17)-(24) is stable in the large,
that is,for any P(O),Q(O)> 0, there exists a T > 0, S > 0 such that p2(t) + Q2(t) < S
for all t > T.
PROOF: We shall construct a bounded region R such that every trajectory
starting outside the region enters it within a finite time, and every trajectory
starting inside it will remain inside. (See Figure 4.)
If KJ < N, we set K1 = N/J > K,J1 = J.
If KJ > N, we set K1 = K, J1 = N/K < J.
If LM < N, we set L1 = NIM > L,M1 = M.
If LM > N, we set L1 = L, M1 = NIL < M.
Then, K1JI = L1M, = N and (24) holds with J, K, L, M replaced by J1, K1,
L1, M1 respectively. Now we are ready to trace out the boundary of the region R.
KI|
Q1(t)
P(
SXK
___
Q
)_
t
_
PP(Q)_
_
_
t
___-__-
P2 (t),
K1
Q2(t)
LIMIT CYCLE
Q=Q(P)
LI
FIGURE
4
Q
PRICE AND QUANTITY ADJUSTMENT
479
First, let us consider the Q axis. By (17), P?> 0 when P = 0. Therefore, any path
beginning on the Q axis must either remain on the Q axis, or move into the region
P > 0. Next, let us begin at the origin and move along the P axis. By (21), 0 > 0
for Q = 0 so all paths starting on the P axis either remain on the P axis or move
into the positive quadrant. In particular, Q > 0 for Q = 0, P > K1 by (24) so that
all such paths move into the positive quadrant. Since the line segment I1 defined
by 0 < Q < J1, P = K1 is compact, 0 has a lower bound ol > 0 in I,. By (22),
Q > a for all 0 < Q < J1, P > K1. By (17) and (20), G(P, 0) > 0 and G(P, J1) < 0
for all P > K1. There must, therefore, exist 0 < t1 < J1lal for which the path
P = P1(t), Q = Q1(t)beginning at P1(O)= K1, Ql(0) = 0 satisfies G[P1(tl), Q1(t1)]
= 0. Let us define P(Q) on Q > 0 by
P[Q1(t)] = P1(t) for 0 <, t < ti;
P(Q) = P1(t1) for Q > Q1(t1).
Then any path starting on the locus P = P(Q) will remain on the locus for a limited
period if Q < Q1(t1)or move into the region P < P(Q) if Q > Q1(t1).Furthermore,
every path beginning in the region P > P(Q) will move into the region P < P(Q)
within a finite period of time.
Now let us consider the path beginning at P2(0) = P1(t1), Q2(0) = L1. The line
segment I2 defined by M1 < P < P1(tl), Q = L1 is compact; therefore, P has an
upper bound a2 in I2. By (20) and (18), a2 < 0. Also by (18) P < a2 for all M1 < P
< P1(tl), Q >? L1. Since F(P,Q) < 0 for P = M1, Q > L1, there must be a
0 < t2 < (P1(tl) - M1)/a2 for which F[P2(t2),Q2(t2)] < 0 with equality if t2 > 0.
Let us define Q(P) on 0 < P < P1(tl) by
Q[P2(t)] = Q2(t) for 0
(26)
Q(P) =
Q2(t2)
< t < t2,
for 0 - P < P2(t2).
Then any path starting on the locus Q - Q(P) will remain on the locus for a
limited period if P2(t2) < P < Pl(tl), or move into the region Q < Q(P) if
P < P2(t2).Furthermore, every path beginning in the region P < P1(tl), Q > Q(P)
will move into the region Q < Q(P) within a finite period of time.
We now have a bounded region R defined by 0 < P < P(Q) < Pl(tl),
0 -< Q < Q(P) < Q2(t2). We have shown that any path beginningoutsideR must
move into it within a finite time and that any path beginning inside R must remain
inside. This completes the proof of Theorem 1.
Now if the boundary of the region R is traversed once, the direction of the flow
field rotates once. This means that the so-called Poincare index of the flow field
in R equals one [10, p. 400].
The following corollaries are immediate:
1: The region R contains at least one "singularity," i.e., equilibrium
COROLLARY
point of index 1 (a center, spiral, or node).
480
M. J. BECKMANN AND H. E. RYDER
COROLLARY 2: If there are n saddle points (index -1) there must be n + 1
equilibriumpoints of the spiral and/or nodal type (index + 1).
COROLLARY 3: A limit cycle contains in its interior either an equilibriumpoint
(of spiral or nodal type) or another limit cycle. The innermostlimit cycle must contain
a spiral or node [10, p. 264].
COROLLARY 4: If there is a single equilibriumpoint and no limit cycle, then the
equilibriummust be stable in the large and in the small, i.e., it is globally stable.
COROLLARY 5: If there is a single equilibriumpoint and it is unstable in the small,
then there must exist a stable limit cycle.
COROLLARY 6: If there is a single equilibriumpoint and a single limit cycle, then
either the equilibriumis unstable and the limit cycle is stable, or the limit cycle is
unstable from the inside and stable from the outside and the equilibriumpoint is
stable in the small.
COROLLARY 7: If all equilibriumpoints are unstable, then there must exist a
stable limit cycle.
The proof of Corollary 7 follows from the application of Poincare's criterion
for cycles to the boundary of region R. If along a closed trajectory all flow is
inbound and if all equilibrium points in the interior are unstable, then there exists
a stable limit cycle [8, p. 76].
3. APPLICATION TO STANDARD CASES
In conclusion let us apply the theory of motion of price and quantity as developed
here to a few standard cases. (All hairline cases will, therefore, be excluded as of
doubtful realism.)
(i) If supply is upward sloping and demand is downward sloping throughout,
then the intersection is unique. Let it be a single point representing a positive
quantity and a positive price a < 0 < 1/fl. These are the cases occurring in the
Northwest quadrant of Figure 1. They are all stable, and since the equilibrium
is unique, they are globally stable unless an unstable limit cycle (which in turn is
enclosed by a stable limit cycle) surrounds this stable equilibrium. For this to occur
the supply curve would have to be downward sloping or the demand curve upward
sloping in some region. We have the following cases: A1 stable node, C1 stable
spiral. The spiral occurs when the slope of the supply curve (adjusted for speed)
lies between the bounds of -a + 2, i.e., when supply is neither too elastic nor
too inelastic.
If supply is perfectly elastic, 1/,B= 0, we have a spiral provided -2 < a < 0.
If demand is perfectly inelastic, a = 0, then we have a spiral whenever the slope
of the supply curve does not exceed 2, (1/fl) < 2.
PRICE AND QUANTITY ADJUSTMENT
481
(ii) Downward sloping supply and demand curves. The possibilities that can
arise are shown in the Southwest quadrant of Figure 1. All cases are now possible.
If and only if the supply curve is steeper than the demand curve, 1/o > 1/fl,
are we below the hyperbola in Figure 1 and have Case A3, a saddle point. In
addition to a saddle point, which is semistable (for practical purpose it is unstable),
there must then exist two other equilibria of the spiral or nodal type (see Corollary 2,
Section 2) for which the slope of the demand curve is less than that of the supply
curve, excluding hairline cases. Let us consider these other equilibria.
If demand is everywhere downward sloping, then either situation (i) arises,
for upward sloping supply, or supply is downward sloping but less steeply than
demand.
Excluding hairline cases we can then have (see Figure 1, region above the
hyperbola): A1 a stable node, C1 a stable spiral, C3 an unstable spiral, As an
unstable node. The condition of stability is that 1/fl > a, i.e., the supply curve must
not be too steep. If all three equilibria are unstable and there are no further intersections of supply and demand, then there must exist a stable limit cycle (see
Corollary 7, Section 2).
Under the global restrictions spelled out in Theorem 1 a single equilibrium
point for downward sloping demand and supply can exist only for the case of
demand steeper than supply. (The converse is a saddle point, and it cannot exist
by itself.) (See Corollary 2.) If this single equilibrium is unstable-cases As and
C3-then there must exist a stable limit cycle.
A limit cycle surrounding an unstable spiral C3 is generated for instance by
the system
P = (-1
+E)P - Q,
1 >?>O,
P +(1 _6Q2)Q,
(2=
> O.
Assume that speeds of adjustment A =,
and downward sloping,
(28)
Q = (-1
= 1. Here the demand curve is linear
+ 8)Pd,
and the supply curve is a cubic,
(29)
PS=
-Q
First let us prove the following lemma.
LEMMA
1: The system (27) has a unique equilibriumpoint, which is an unstable
spiral.
PROOF:
(30)
From (28) and (29) we have
Pd - Ps
Q - Q3.
The right-hand side of (30) is zero only for Q = 0. Thus, P = 0, Q = 0 is the only
equilibrium point.
482
M. J. BECKMANN AND H. E. RYDER
Instability in the small is established by examining the roots of the characteristic
polynomial (see Section 1):
+
|
(31)
-1
|
0
2
The characteristic roots of (31), given by
(32)
2
p
are complex with positive real parts, since 0 < e < 1 < 4. Thus, the motion in
the small is an unstable spiral. This completes the proof of Lemma 1.
Next, we use a criterion for dissipativeness due to Pliss [9, p. 42] to prove
a second lemma.
LEMMA 2:
The system (27) is stable in the large.
PROOF:
p2
V =
+ Q2
2
This function is positive except at the equilibrium point and approaches infinity
as the distance of (P, Q)approaches infinity. Differentiating with respect to time,
we have
=
PP + QQ
= Q2_
1
6Q4
_
_
(1
)p2
(1 _
21(Q2
-26
4-
C)p2
+(-
Both of the terms inside the brackets are nonnegative, so that if either of them is
> 1/ 6, and
greater than 1/46, then v < 0. But 6(Q2 - (1/26))2 > 1/46 if IQI
G
>
if
>
pP2
<
if
1/46
(1-_
IPI 1/2V 6(1
Thus, v
'~
Q2.!
2
5-48
b(1 - c)
The function v satisfies the conditions of Theorem 2.5 in Pliss [9, p. 423. Therefore,
the system is stable in the large. This completes the proof of Lemma 2.
Applying Corollary 5 to Lemmas 1 and 2, we obtain at once:
THEOREM2:
The system (27) has a stable limit cycle.
PRICE AND QUANTITY ADJUSTMENT
483
P
3-
2
FIGURE 5
For the case 5 = = 1/10 the limit cycle is shown in Figure 5.1
(iii) Upward sloping supply and demand curves. If the demand cuts the supply
curve from below, 0 < 1/a < 1/f, we have a saddle point A3 and at least two other
intersections must exist. If the demand curve intersects the supply curve from
above, 0 < 1/fl < 1/a, then either a stable node A1, a stable spiral C1, an unstable
node A5, or an unstable spiral C3 occurs. If all intersections of supply and demand
curves are saddle points or unstable, a stable limit cycle must exist.
While limit cycles have been at the core of some well known business cycle
models [1, 4], they have not, to our knowledge, been introduced before as a possible
1 We are indebted to D. Hochstaedter, formerly of the Institute of Econometrics and Operations
Research, University of Bonn, for carrying out the calculations on the IBM 7090 computer of the
University of Bonn.
484
M. J. BECKMANN AND H. E. RYDER
explanation of the persistenceof fluctuationsin certain commodity markets.
We see that persistentcycles may occur even when no fixed periodic lags are
involved,as in the cobwebphenomenon,whensupplyanddemandcurvesintersect
in a singlepoint at whicheitherboth are upwardsloping,or both are downward
sloping. A downwardsloping supply curve may occur in industriesthat have
economiesor in agriculturalproductionwhen the incomeeffectis dominant,and
an upwardsloping demandcurve may occur when the income effect is predominant.
University of Bonn
and
Brown University
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