Directory UMM :Data Elmu:jurnal:J-a:Journal Of Economic Dynamics And Control:Vol24.Issue5-7.Jul2000:

Journal of Economic Dynamics & Control
24 (2000) 833}853

On information and market dynamics:
The case of the U.S. beef market
Jean-Paul Chavas
Department of Agricultural and Applied Economics, University of Wisconsin, Madison, WI 53706, USA
Accepted 30 April 1999

Abstract
The paper investigates the nature of dynamic prices and expectations in a competitive
market. The approach is applied to the U.S. beef market, which exhibits cyclical patterns
and signi"cant biological lags in the production process. Beef price equations are
estimated under di!erent expectation regimes. The empirical results indicate the presence
of heterogeneous price expectations, with a signi"cant number of market participants
neglecting information about the existence of a beef cycle. ( 2000 Elsevier Science B.V.
All rights reserved.
Keywords: Market dynamics; Price expectations; Market #uctuations

1. Introduction
Uncertainty is a pervasive characteristic of dynamic resource allocation: it

changes over time as economic agents learn about their environment. Much
research has attempted to evaluate this learning process, along with its e!ect on
resource allocation. A major research focus has been on the characterization of
expectation formation (e.g., Chow, 1989; Eckstein, 1984; Evans and Ramey,
1992; Ezekiel, 1938; Goodwin and She!rin, 1982; Holt and Johnson, 1989;
Muth, 1961; Nerlove, 1958; Nerlove et al., 1979; Nerlove and Fornari, 1993;
Orazem and Miranowski, 1986). They include naive expectations (where future
expected values are set equal to the latest observation of the corresponding
variable; see Ezekiel (1938)), adaptive expectations (where expectations are
revised over time proportionally to latest prediction error; see Nerlove (1958)),
0165-1889/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 1 6 5 - 1 8 8 9 ( 9 9 ) 0 0 0 2 7 - 5

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J.-P. Chavas / Journal of Economic Dynamics & Control 24 (2000) 833}853

quasi-rational expectations (where expectations are consistent with the time
series model of the corresponding variable; see Nerlove et al. (1979)), as well as
rational expectations (Muth, 1961).

Since its introduction by Muth, the rational expectation hypothesis has
occupied a central place in the discussion. The rational expectations hypothesis
states that decision-makers make e$cient use of information, just as they do
of other scarce resources. The issue then is to evaluate the exact meaning
of &e$cient use' of information. If obtaining and processing information is
costly, then optimal learning is expected to depend on the net bene"ts
of learning. When some new information is costly or di$cult to process, it
may not be used by decision-makers (e.g., Conlisk, 1996; Sargent, 1993). In
such situations, simple rules of thumb for expectation formation (e.g., naive
expectations) could be used. Also, the ability to obtain and process information
may vary across individuals. For example, di!erences in education or experience
could imply di!erent learning rates across individuals, ceteris paribus. The
costs and bene"ts of information being individual speci"c, di!erent individuals
may have di!erent expectations. At the aggregate level, dynamic resource
allocation would then be in#uenced by the heterogeneity of expectations among
decision-makers.
The objective of this paper is to investigate the nature of expectation formation and dynamic pricing, with an empirical application to the U.S. beef market.
Much research has focused on describing and explaining the beef cycle (e.g.,
Fisher and Munro, 1983; Foster and Burt, 1991; Jarvis, 1974,1986; Maki,
1962; Munlak et al., 1995; Mundlak and Huang, 1996; Paarsch, 1985; Rosen

et al., 1994; Rucker et al., 1989; Trapp, 1986). In some respects, the presence
of the beef cycle can be disturbing for economists. If a predictable cycle existed,
then producers responding in a countercyclical fashion could earn larger
than normal pro"ts over time. In the presence of predictable price movements,
countercyclical production response could possibly smooth out market #uctuations, causing the cycle to disappear. Recent research has shown that rational
expectations and e$cient decisions do not necessarily imply the absence
of economic cycles. In particular, Chavas and Holt (1995), Rosen (1987),
and Rosen et al. (1994) have argued that an economic cycle can be fully
consistent with the e$cient management of an animal population under
rational expectations. Also Hommes and Sorger (1998) have shown that cyclical
and chaotic market equilibria can arise under self-ful"lling expectations
(where the perceived and actual laws of motion have the same mean and
autocorrelations).
The assumption of naive or adaptive expectations, dating back to Coase and
Fowler (1937), Ezekiel (1938) and others, has been a basic premise in much of the
literature on livestock supply response (e.g., Foster and Burt, 1991). But the
dynamics of price expectations by market participants can in#uence price and
market dynamics. This raises the issue of whether the nature of rationality could

J.-P. Chavas / Journal of Economic Dynamics & Control 24 (2000) 833}853

835

play a role in explaining the continued existence of economic cycles (e.g., Brock
and Hommes, 1997; Conlisk, 1996; Evans and Ramey, 1992; Leijonhufvud, 1993;
Sargent, 1993). This suggests a need to investigate the exact nature of
market information used in forming expectations and in making production
decisions.
This paper develops and estimates an econometric model of market prices in
the U.S. beef market. Because of production lags in the beef production process,
production decisions are made ahead of marketing decisions. As a result,
production decisions are based on expectations about future market conditions.
We investigate di!erent expectation formations within the beef industry, including naive expectations, quasi-rational expectations, and rational expectations.
Under Muth's hypothesis, rational expectations would be forward looking and
based on a full understanding of market pricing. This involves the demand side
as well as the supply side of the market. The supply side is dynamic as it involves
the management of the breeding herd over time. We derive supply dynamics
from the economics of animal population management (e.g., Chavas and
Klemme, 1986; Rosen, 1987; Rosen et al., 1994). We also consider the case of
namK ve expectations (where decision-makers react to the latest known price), and

quasi-rational expectations (where a univariate time series model of the relevant
price generates expectations). We also investigate the heterogeneity of expectations among producers in supply dynamics. We propose an econometric methodology leading to the speci"cation and estimation of a model of price
determination and dynamic market allocation. When applied to the U.S. beef
market, the methodology provides evidence of heterogeneous expectations
among beef producers. We "nd that a signi"cant number of beef producers
behaves in a way consistent with Muth's forward-looking rational expectation
hypothesis. However, such behavior characterizes only 18% of the beef market.
We "nd that about 35% of the market participants behave in a way consistent
with Nerlove's quasi-rational expectations. The remaining 47% of the market is
found to be associated with namK ve expectations, where anticipated future prices
are given by the last observed price.

2. Animal economics
Consider a competitive "rm managing an animal population. Let b denote
t
the size of the breeding herd, as measured by the number of adults at the
beginning of year t. Given a birth rate k , the number of o!spring in year t is
t
denoted by h "k b , where k is the number of o!spring per adult. Assume that
0t

t t
t
the o!spring become adults at two years of age. This assumption matches the
biological lags of beef production (see below). If it does not die of natural causes,
each o!spring becomes an adult two years later. Denote by h the number of
jt
animals of age j at the beginning of year t, j"0, 1, 2. And let d be the natural
jt

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J.-P. Chavas / Journal of Economic Dynamics & Control 24 (2000) 833}853

death rate of animals of age j in year t. This implies that
h "k b ,
(1a)
0t
t t
h
"(1!d )h

1,t`1
1t 0t
"(1!d )k b ,
(1b)
1t t t
h
"(1!d )h
2,t`1
2t 1t
"(1!d )(1!d
)k b .
(1c)
2t
1,t~1 t~1 t~1
Denote by s the number of animals slaughtered in year t. The animals slaught
tered include both two-year-old animals and older adult members of
the breeding stock. Then, the evolution of the breeding herd (b ) over time is
t
given by:
b "(1!d )b #h

!s
t`1
3t t
2,t`1
t
"(1!d )b #(1!d )(1!d
)k b !s , using Eq. (1c), (2)
3t t
2t
1t~1 t~1 t~1
t
where d is the natural death rate of adults in year t. Eq. (2) simply states that
3t
the change in the size of the breeding herd from one time period to the next
(b !b ) equals the number of adults added to the breeding herd (h
),
t`1
t
2,t`1
minus the number of animals slaughtered (s ), minus the number of adults dying

t
of natural causes (d b ). Note that this di!ers from the speci"cation used by
3t t
Rosen et al. (1994) which involves three lags (instead of two lags in Eq. (2)). In the
context of the beef industry, a "rst mating of heifers (young cows) around
15 months old, followed by a 9 months gestation period, implies that cows have
their "rst calf around 2 years old. Thus, in a way consistent with Eq. (2),
two-year old cows are treated as adults. This is also consistent with the empirical
analysis reported by Mundlak and Huang (1996) who found no statistical
evidence for a third lag in Eq. (2).
The management of the animal population is costly. Denote by
c (q , b , h , h , s ) the cost of managing the adult population b , the young
t t t 0t 1t t
t
animals (h and h ), and slaughter s , where q is the vector of input prices at
0t
1t
t
t
time t. Assume that the animals slaughtered at time t, s , are sold on a competit

tive market at a unit price p . Then, the "rm net income generated from the
t
management of the animal population at time t is
p "p s !c (q , b , h , h , s ).
(3)
t
tt
t t t 0t 1t t
The "rm faces uncertainty about future values of the market price p , the death
t
rates d , and the productivity factor k . These variables are thus treated as
jt
t
random. They are assumed to have some subjective probability distribution
re#ecting the information available to the decision-maker. The evaluation of this
uncertainty will be discussed in more detail below.

J.-P. Chavas / Journal of Economic Dynamics & Control 24 (2000) 833}853

837

Assume that the manager of the animal population is risk neutral and makes
decisions so as to maximize the expected present value of net income over
his/her planning horizon.1 This corresponds to the following optimization
problem:
Max ME[&T (1#r)~tn ]: subject to Eqs. (1a), (1b), (2) and (3)N,
t/0
t
where E is the expectation operator, ¹ is the length of planning horizon, and
(1#r)~1 is the discount factor, r being the discount rate re#ecting time preferences. Assuming that the decision-maker learns over time, this problem can be
alternatively formulated as the following stochastic dynamic programming
problem:
< (b , b )"MaxMp #(1#r)~1E < (b ,b ):
t t t~1
t
t t`1 t`1 t
subject to Eqs. (1a), (1b), (2) and (3)N,
"Max Mp s !c (q , b , k b , (1!d
)k
b ,s)
tt
t t t t t
1,t~1 t~1 t~1 t
st
#(1#r)~1E < [(1!d )b
t t`1
3t t
#(1!d )(1!d
)k
b !s , b ]N,
(4)
2t
1,t~1 t~1 t~1
t t
where < (b , b ) is the indirect objective function (or &value function') condit t t~1
tional on the size of the breeding herd b and b , and E is the expectation
t
t~1
t
operator based on the information available at time t. Learning is represented
by improvements in the information available to the decision-maker from one
period to the next. We assume that the realized value of the random variables
(p , d , k ) become observed at time t, and that the d's and k's are independently
t jt t
distributed. Some expectation formation is needed to represent the expected
future value of these random variables. This issue is addressed in the next
section.
Eq. (4) is Bellman's equation of dynamic programming de"ning recursively
the value function < (b , b ). Under di!erentiability and assuming interior
t t t~1
solutions,2 the "rst-order necessary condition for s in (4) is
t
p !Lc /Ls !(1#r)~1 E [L< /Lb ]"0.
(5)
t
t t
t
t`1 t`1
1 The assumption of risk neutrality has been commonly made in previous work on dynamic
animal economics (e.g., Rosen, 1987; Rosen et al., 1994). Note that it neglects the possible role of risk
aversion in dynamic resource allocation. Exploring such issues is a good topic for further research.
2 Although corner solutions may exist at the micro-level, they are typically not observed at the
aggregate level. Since our empirical analysis relies on market data, our assumption of &interior
solutions' then appears &reasonable'. Note that this is consistent with the analysis presented by
Rosen (1987) and Rosen et al. (1994).

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J.-P. Chavas / Journal of Economic Dynamics & Control 24 (2000) 833}853

From the envelope theorem applied to (4), we have
L< /Lb "!(Lc /Lh )k
(1!d
)
t t~1
t 1t t~1
1,t~1
#(1#r)~1E [(L< /Lb )k
(1!d
)(1!d )]
t
t`1 t`1 t~1
1,t~1
2t
"!(Lc /Lh )k
(1!d
)
t 1t t~1
1,t~1
#(p !Lc /Ls )k
(1!d
)(1!d ) using Eq. (5)
t
t t t~1
1,t~1
2t

(6a)

and
L< /Lb "!Lc /Lb !(Lc /Lh ) k
t t
t t
t 0t t
#(1#r)~1 E [(L< /Lb )(1!d )]
t
t`1 t`1
3t
#(1#r)~1E [(L< /Lb )]
t
t`1 t
"!Lc /Lb !(Lc /Lh )k #(p !Lc /Ls )(1!d )
t t
t 0t t
t
t t
3t
!(1#r)~1E [!(Lc /Lh
)k (1!d )
t
t`1 1,t`1 t
1t
#(p !Lc /Ls ) k (1!d )(1!d
)],
(6b)
t`1
t`1 t`1 t
1t
2,t`1
using Eq. (5) and Eq. (6a) evaluated at time t#1. Substituting (6b) evaluated at
time t#1 into Eq. (5) then yields
(p !Lc /Ls )!(1#r)~1E [!Lc /Lb !(Lc /Lh
)k
t
t t
t
t`1 t`1
t`1 0,t`1 t`1
#(p !Lc /Ls )(1!d
)]
t`1
t`1 t`1
3,t`1
#(1#r)2E [!(Lc /Lh
) k (1!d
)
t
t`2 1,t`2 t`1
1,t`1
#(p !Lc /Ls )k (1!d
)(1!d
)]"0.
(7)
t`2
t`2 t`2 t`1
1,t`1
2,t`2
Eq. (7) is Euler's equation, giving the dynamics of the animal population under
optimal management and competitive market conditions. It characterizes price
dynamics associated with optimal "rm supply conditions.

3. Market equilibrium and expectation formation
In this section, we consider the competitive market equilibrium in an industry
composed of the "rms managing the animal population. Assuming that the
animal product is not storable, the market equilibrium price is determined by
the intersection of aggregate supply and aggregate demand. As seen in the
previous section, production decisions are made based on expected prices. In
this context, we investigate the e!ects of expectation formation on prices. We
consider three possible expectation regimes: rational expectation (regime 1);

J.-P. Chavas / Journal of Economic Dynamics & Control 24 (2000) 833}853

839

quasi-rational expectation (regime 2); and namK ve expectation (regime 3). We
assume that all "rms in the industry face a similar technology, except for
idiosyncratic shocks in d's (death rates) and k's (birth rates) that are "rm speci"c.
As a result, we assume that the random variables d and k are independently
distributed of market prices in (7).
Focusing at the market level, we consider aggregate behavior. Denote by
B the aggregate breeding herd, and by S the aggregate slaughter. Then Eq. (2)
t
t
gives the following aggregate state equation:
B "(1!D )B #(1!D )(1!D
)K
B !aS ,
t`1
3t t
2t
1,t~1 t~1 t~1
t

(8)

where D is the aggregate death rate of animals of age j at time t, K is the
jt
t
aggregate birth rate at time t, and a"1. While comparing (2) and (8) clearly
implies that a"1, we will treat a as a parameter to be estimated and use its
estimate to assess the validity of the model speci"cation (see below). Eq. (8)
represents the dynamics of the aggregate breeding herd.
3.1. Rational expectation
Under rational expectation (regime 1), price expectations are consistent
with market equilibrium conditions. If each "rm in the industry holds
rational expectation and uses a similar technology, then under a quadratic cost
function c( ) ), Euler equation (7) holds at the aggregate. Let the demand
function be
D "f (p ),
t
t

(9)

where D is aggregate demand, and the demand function is downward sloping
t
Lf/Lp (0. Let c denote the slaughter weight per animal at time t. Then
t
0t
aggregate supply at time t is (c S ), where S is the aggregate number of animals
0t t
t
slaughtered at time t. Under market equilibrium, aggregate demand is equal to
aggregate supply, or
D "c S .
t
0t t

(10)

Under rational expectation, the expectation operator E in (7) is de"ned to be
t
consistent with the information generated by the market equilibrium model.
Thus, price expectations are consistent with the reduced form of the market
equilibrium model. It follows that, under rational expectation, future prices in (7)
can be interpreted as dependent variables generated by the reduced form of the
model. Assume that the econometrician does not have more information than
the industry decision-makers. Denote by E the expectation operator based on
0t
the information available to the econometrician at time t. Then Eq. (7) can be

840

J.-P. Chavas / Journal of Economic Dynamics & Control 24 (2000) 833}853

written as
(p !Lc /Ls )!(1#r)~1[!Lc /Lb !(Lc /Lh
)E (k )
t
t t
t`1 t`1
t`1 0,t`1 t t`1
#(p !Lc /Ls ) E (1!d
)]
t`1
t`1 t`1 t
3,t`1
#(1#r)2[!(Lc /Lh
)E (k (1!d
))
t`2 1,t`2 t t`1
1,t`1
#(p !Lc /Ls )E (k (1!d
)(1!d
))]"e ,
(11)
t`2
t`2 t`2 t t`1
1,t`1
2,t`2
pt
where e is an error term satisfying E E (e )"0.
pt
0t t pt
The structural market equilibrium model consists of the breeding equation
(8), the demand equation (9), the market clearing condition (10), and the pricing
equation (11). Provided that they are identi"ed, the associated structural parameters can be consistently estimated. The presence of dependent variables on
the right-hand side of the structural equations suggests using an instrumental
variable estimation method to deal with simultaneous equation bias. The
instruments should be chosen from the information set common to the econometrician and industry decision-makers. Such instruments would be orthogonal
to the error terms, and thus provide consistent parameter estimates. Below, we
propose to use Hansen's generalized method of moment (GMM) as an instrumental variable estimation method (see Hansen, 1982; Hansen and Singleton,
1982).
3.2. Quasi-rational expectation
Under quasi-rational expectation (regime 2), prices are anticipated on the
basis of their time series properties as estimated from historical data (see
Nerlove et al., 1979). We assume that expected prices in (7) under quasi-rational
expectations are obtained from the prediction of the univariate autoregressive
process for the corresponding prices. Let z follow the autoregressive process
t
z "b #b t#& b z #e , where the b 's are parameters. Then
t
00
0
jz1 j t~j
pt
j
E (z )"b #b (t#1)# + b z
t t`1
00
0
j t`1~j
jz1

(12a)

and
E (z )"b #b (t#2)
t t`2
00
0
#b [b #b (t#1)# + b z
]# + b z
.
(12b)
1 00
0
j t`1~j
j t`1~j
jz1
jz2
When z is either p or q , Eqs. (12a) and (12b) generate expected prices that are
t
t
t
consistent with the observed dynamic patterns of market prices. When substituted into (7), these expected prices give the dynamic pricing conditions under
quasi-rational expectations. Then Eq. (7) (applied at the aggregate, and using

J.-P. Chavas / Journal of Economic Dynamics & Control 24 (2000) 833}853

841

(12)), Eqs. (8)}(10) give the market dynamics under quasi-rational expectations.
Such equations can be estimated using an appropriate estimation method.
3.3. NamK ve expectation
Finally, we consider the case of namK ve expectations (regime 3). Under naive
expectations, producers are assumed to expect the last observed price. This is the
standard assumption made in the cobweb model (e.g., Ezekiel, 1938). With
respect to the variable z , it implies that
t
E (z )"z , for j51.
(13)
t t`j
t
Note that this ignores the dynamic properties of the market prices if they depart
from a random walk model. When substituted into (7), these expected prices give
the dynamic pricing conditions under naive expectations. Then Eq. (7) (applied
at the aggregate, and using (13)), Eqs. (8)}(10) give the market dynamics under
naive expectations. Again, such equations can be estimated using an appropriate
estimation method.
3.4. Heterogeneous expectations
We now consider the possibility of heterogeneity among "rms in obtaining
and processing market information. As mentioned in the introduction, this can
be due to di!erences among "rms in access to information or in the cost or
ability to process information (e.g., because of di!erences in the decision-maker's
education or experience). This can generate heterogeneous expectations within
the industry.
We consider the possible presence of three expectation regimes. Let
N"M1, 2, 3N be the set of expectation regimes within the industry, i3N denoting the ith group of "rms characterized by a given expectations formation. We
allow for some "rms to exhibit rational expectation (regime 1); others quasirational expectation (regime 2); and still others namK ve expectation (regime 3). Let
E be the expectation operator re#ecting the information available to the ith
it
group of "rms at time t, i3N.
At time t, denote by b the size of the breeding herd in the ith group, by h the
it
ijt
number of animals of age j, and by s the number of animals slaughtered in the
it
ith group, i3N. Let the cost function for the ith group at time t be
c (q , b , h , h , s ), i3N. Then, from (7), the pricing equation for the ith group is
it t it i0t i1t it
(p !Lc /Ls )!(1#r)~1E [!Lc
/Lb
t
it it
it
i,t`1 i,t`1
!(Lc
/Lh
)k
#(p !Lc
/Ls
)(1!d
)]
i,t`1 i0,t`1 i,t`1
t`1
i,t`1 i,t`1
i3,t`1
!(1#r)2E [!(Lc
/Lh
)k
(1!d
)
it
i,t`2 i1,t`2 i,t`1
i1,t`1
#(p !Lc
/Ls
)k
(1!d
)(1!d
)]"0,
(14)
t`2
i,t`2 i,t`2 i,t`1
i1,t`1
i2,t`2

842

J.-P. Chavas / Journal of Economic Dynamics & Control 24 (2000) 833}853

where d denotes the ith group death rate for animals of age j, and k is the ith
ijt
it
group productivity, i3N. Note that Eqs. (12a) and (12b) apply in Eq. (14) under
quasi-rational expectation (i"2), while Eq. (13) holds under namK ve expectation
(i"3). Eq. (14) involves the expectation operator E re#ecting the information
it
available to the ith group at time t. It stresses the role of information in dynamic
resource allocation. It indicates that price expectations can in#uence pricing.
This implies that observed market price dynamics depend on the nature of price
expectations from all market participants. In that sense, the observed dynamics
of prices can provide an empirical basis to estimate and evaluate the information
processed by market participants.
The empirical implementation of the above equations requires addressing the
issue of the relevant information set involved in the formation of expectations.
Unfortunately, data are rarely available on group-speci"c information (i.e.,
b , s , h , etc., i3N). As a result, equation (14) is typically not empirically
it it ijt
tractable for each group i3N. Here, we consider the case where data are
available only at the aggregate level. We are interested in developing a model
that would be empirically tractable in this context. Such a model should have
the following desirable characteristics. It should be consistent with by the Euler
equation (14) for all groups within the industry. It should also include as
a special case the situation of homogeneous expectations just discussed. For
example, if all "rms use a single expectation regime (say the jth regime), then the
model should reduce to the Euler equation (14) for the jth group. On that basis,
we propose to represent market pricing as a weighted sum of Eq. (14) across
groups:
+ w M(p !Lc /Ls )!(1#r)~1E [!Lc
/Lb
it t
it it
it
i,t`1 i,t`1
i|N
!(Lc
/Lh
)k
#(p !Lc
/Ls
)(1!d
)]
i,t`1 i0,t`1 i,t`1
t`1
i,t`1 i,t`1
i3,t`1
!(1#r)2E [!(Lc
/Lh
)k
(1!d
)
it
i,t`2 i1,t`2 i,t`1
i1,t`1
#(p !Lc
/Ls
)k
(1!d
)(1!d
)]N"0,
t`2
i,t`2 i,t`2 i,t`1
i1,t`1
i2,t`2

(15)

where the w 's are (non-stochastic) weights re#ecting the market share of the ith
it
group at time t, with w 50, i3N, and & w "1. Note that Eq. (15) has the
it
i|N it
desirable characteristics just mentioned. It is implied by the group-speci"c Euler
equations given in (14). And if all "rms use a single expectation regime (say the
jth regime), then w "1 and w "0 for all iOj, implying that Eq. (15) reduces
jt
it
to the Euler equation (7) or (14) for the jth group. In other words, the general
case of heterogeneous expectations given in (15) nests nicely the special case of
homogenous expectations. Short of having data on the behavior of each group,
Eq. (15) provides a convenient econometric speci"cation that introduces heterogeneous expectations in a way consistent with within-group rationality.

J.-P. Chavas / Journal of Economic Dynamics & Control 24 (2000) 833}853

843

In order for Eq. (15) to be empirically tractable, the w parameters must be
identi"ed. This requires that the three expectation regimes involve di!erent
information sets. The distinction between namK ve expectation and quasi-rational
expectation requires that market prices do not follow a random walk. But is the
distinction between quasi-rational expectation and rational expectation always
meaningful? There are situations when the two can be equivalent. For example,
this would happen if all "rms exhibit rational expectation, while quasi-rational
expectations are generated from the reduced form of the rational expectation
model. However, whenever some market participants do not exhibit &full rationality', their e!ect on price dynamics will typically in#uence quasi-rational
expectations in a way that can di!er from &Muth-rationality' (e.g., Brock and
Hommes, 1997; Hommes and Sorger, 1998). To the extent that this happens, the
distinction among expectation regimes becomes empirically meaningful.
Aggregate supply is given by & c s , where c denotes the slaughter
i|N 0it it
0it
weight per animal and s is the number of animal slaughtered in the ith group at
it
time t. Given similar technology across groups, c is assumed distributed with
0it
mean c at time t. Then, market equilibrium holds when aggregate demand is
0t
equal to aggregate supply, or
D "c (& s )#e
t
0t i|N it
dt
"c S #e ,
(16)
0t t
dt
where S "& s is the aggregate slaughter, and e is an error term with mean
t
i|N it
dt
zero. Then, the structural market equilibrium model under heterogeneous
expectations consists of the breeding equation (8), the demand equation (9), the
price equation (15), and the market clearing condition (16). As in the case of
rational expectation discussed above, the structural parameters can be consistently estimated (provided that they are identi"ed) using an instrumental variable method. Again, we will rely on Hansen's GMM estimation method to
estimate the structural parameters. Note that the simplicity of GMM estimation
allows us to provide a more re"ned analysis of expectation formation than
found in previous research.3

4. Empirical analysis of the U.S. beef market
The above model is now applied to the U.S. beef market. The breeding herd
consists of cows and bulls. After a pregnancy period of 9 months, cows typically

3 For example, Rosen et al. (1994) and Anderson et al. (1995) consider only the case of
homogenous expectations. Baak (1997) considers the case of heterogeneous expectation, but with
only two expectation regimes. All three estimate their model by the maximum likelihood estimation
method in the context of a linear-quadratic optimization problem.

844

J.-P. Chavas / Journal of Economic Dynamics & Control 24 (2000) 833}853

produce one calf per year. Calves can be fed until 1.5}2 years old and then
slaughtered. Or they can join the breeding herd, with a "rst mating around
15 month old, thus producing the "rst calf at about 2 years old. Thus, using
annual data, the U.S. beef market matches well the model developed in Sections
2 and 3. First, the assumption made that o!spring become adults after two years
is fairly accurate. Second, meat production is the only "nal output obtained
from beef (at slaughter).
Aggregate data were obtained from U.S. Department of Agriculture on the
U.S. beef market between 1948 and 1992.4 The data include the size of the
breeding herd (B ) measured by the number of breeding beef cows, aggregate
t
slaughter (S ), and aggregate production. It also includes beef price received by
t
farmers in the U.S. (p ), as well as corn price (q ) representing input cost. All
t
t
prices (p , q ) are measured as real prices, de#ated by the consumer price index.
t t
The analysis relies on annual data.
Based on the model developed in the previous section, we propose the
following per-capita speci"cation for the aggregate demand function (9)
D /pop "d #d p #d t#e ,
(17)
t
t
0
1 t
2
Dt
where pop denotes U.S. population at time t, and (d , d , d ) are demand
t
0 1 2
parameters, and e is an error term with mean zero and "nite variance.
Dt
Also, we let the mean birth rate be E (k )"E(K )"1, i.e., one calf per
t t
t
breeding cow per year. And we let the mean death rate be E(d )"E(D )"0.08.
jt
jt
The appropriateness of these values will be investigated below by testing
whether a"1 in the breeding equation (8). Also, we specify the cost function as
c ( ) )"c (q )#q (c #c t)[b #0.7h #0.3h ], where the c's are parameters
it
i0t t
t 1
2
t
1t
0t
to be estimated. This assumes that the marginal cost of slaughter is negligible.5
It also treats a calf and a one-year old animal as if they were, respectively, 0.3
and 0.7 of an adult. Finally, we specify c in Eq. (10) or Eq. (16) as c "c #c t,
0t
0t
1
2
where c and c are parameters to be estimated, c re#ecting possible changes in
1
2
2
the slaughter weight over time.
Before estimating our structural model under di!erent expectation regimes, we
investigated the dynamic properties of market prices. The evolution of detrended
real beef price is illustrated in Fig. 1. Autoregressive models of beef price (p ) and
t
corn price (q ) were speci"ed and estimated. The resulting estimates will be used
t
below to represent quasi-rational expectations (as given in Eq. (12a) and (12b)).
4 Note that the sample period is shorter than the one used by Rosen et al. (1994), Anderson et al.
(1995), or Baak (1997). This was done to avoid structural change issues that arise over longer sample
periods (e.g., during the great depression). Because of di!erent sample information and di!erences in
model speci"cation, our econometric results are not strictly comparable to theirs.
5 Experimenting with alternative speci"cations of the cost function a!ected some of the empirical
results. However, the econometric estimates of the w 's were found to be fairly insensitive to these
i
alternative speci"cations.

J.-P. Chavas / Journal of Economic Dynamics & Control 24 (2000) 833}853

845

Fig. 1. Detrended real beef price.

The following beef price equation was estimated (standard errors in parentheses below the parameter estimates):
p"
t

0.3665
(0.0891)

#0.7350p
!0.3034p
#0.0035p
t~1
t~2
t~3
(0.1430)
(0.1803)
(0.1841)

#0.2909p
!0.3654p
t~4
t~5
(0.1753)
(0.1204)
R2"0.8794.

!0.0037t,
(0.0010)
(18)

Several diagnostic tests were performed. The Godfrey test for serial correlation
of the residual indicated no statistical evidence of serial correlation (with
a p-value of 0.827). The Ramsey RESET test of functional form failed to uncover
evidence of inappropriate functional form (with a p-value of 0.997). Finally, the
Lagrange multiplier test of the regression of the squared residuals on the
squared predicted values gave no statistical evidence of heteroscedasticity (with
a p-value of 0.151). Thus, the estimated model (18) appears to provide a good
representation of the dynamics of beef prices.
The negative and signi"cant coe$cient on the time trend t in (18) re#ects the
historical decrease in the real price of beef. The roots of the estimated di!erence

846

J.-P. Chavas / Journal of Economic Dynamics & Control 24 (2000) 833}853

equation (18) were evaluated. They are two pairs of complex roots, and one real
root. The dominant root is complex, with a modulus of 0.842. The other pair of
complex root has a modulus of 0.832, while the real root is !0.744. The two
pairs of complex roots generate cyclical patterns, with periods equal to 4 and
13.146 year, respectively. In a way consistent with previous literature (e.g., Rosen
et al., 1994), this provides empirical evidence of the existence of cycles in the beef
market. It strongly suggests that namK ve expectations would fail to capture some
important aspects of market dynamics.
The following corn price equation was also estimated (standard errors in
parentheses below the parameter estimates):
0.0094 #0.4751q
!0.0001t,
t~1
q"
t (0.0023)
(0.1020)
(0.0000)
R2"0.7525.

(19)

The negative and signi"cant coe$cient on the time trend t in (19) re#ects the
historical decrease in the real price of corn. And the coe$cient on q
suggests
t~1
a signi"cant departure from a random walk model.
We then estimated the structural models discussed in the previous section.
First, we consider three models representing the three scenarios discussed
earlier: (1) rational expectations; (2) quasi-rational expectations (as given by (12),
using the estimates of the autoregressive processes (18) and (19)); and (3) namK ve
expectations (as given by (13)). This latter scenario is the standard assumption
made in the cobweb model (e.g., Ezekiel, 1938). The parameters of these three
models were estimated for the U.S. beef market, using the generalized method of
moments (GMM) proposed by Hansen (1982), and Hansen and Singleton
(1982).6 The chosen instruments were the one-period lagged dependent variables
B , p , along with the one-period lagged corn price q , an intercept, and
t~1 t~1
t~1
a time trend. The variance}covariance matrix was robustly estimated using the
Newey}West (1987) estimator, correcting for both heteroscedasticity and serial
correlation with a lag length up to three periods. Under a set of regularity
conditions, the resulting parameter estimates can be shown to be consistent and
asymptotically normal (Hansen, 1982). The estimates are presented in Table 1.
The validity of the econometric speci"cation was "rst assessed using the
Hansen test on the overidenti"cation restrictions generated by the instruments.
The Hansen test does not give statistical evidence against the orthogonality
restrictions between the overidentifying instruments and the error terms

6 Attempts to estimate the discount rate r proved di$cult. They gave imprecise results, yielding
estimates with large standard errors. The analysis presented below assumed a discount factor
(1/(1#r)) equal to 0.98, corresponding to a value of r equal to 0.02041. (Recall that all prices are
de"ned in real terms in our analysis.)

J.-P. Chavas / Journal of Economic Dynamics & Control 24 (2000) 833}853

847

Table 1
Parameter estimate under each expectation regime

a
d

0

d

1

d

2

c

1

c

2

c
c

1
2

Minimum
distance
Hansen test,
s2(12), p-value

Rational
expectation
(i"1)

Quasi-rational
expectation
(i"2)

NamK ve
expectation
(i"3)

0.9995!
(0.0092)
127.8570!
(13.3357)
!90.8898!
(22.2213)
[!0.4200]
0.0278
(0.1718)
1.1182!
(0.0040)
!0.0017!
(0.0001)
495.2811!
(5.1780)
3.6285!
(0.1734)

0.9981!
(0.0121)
163.6737!
(6.9886)
!151.4888!
(10.9422)
[!0.7000]
!0.3881a
(0.1282)
44.0084!
(0.6139)
1.5666!
(0.0691)
496.3589!
(5.4065
3.6164
(0.1635)

0.9975!
(0.0130)
150.6589!
(9.4406)
!129.1641!
(15.8748)
[!0.5969]
!0.2418
(0.1559)
10.2585!
(0.3821)
0.1079!
(0.0159)
496.2788!
(4.5096))
3.6060!
(0.1391)

10.1255

9.7277

9.7680

0.6049

0.6398

0.6363

!Indicates that the corresponding parameter is signi"cantly di!erent from zero at the 5% level. The
asymptotic standard errors are presented in parentheses below the parameter estimates. Elasticities
evaluated at mean values are presented in brackets.

(see Table 1). This suggests that the model speci"cation and the choice of the
instruments appear appropriate.
All the estimated coe$cients reported in Table 1 have the expected sign, and
most are signi"cantly di!erent from zero. The hypothesis that a"1 in Eq. (8)
fails to be rejected at the 5% signi"cance level under each expectation regime.
This indicates that the assumed values for birth rate (k) and death rates (d) are
consistent with the data. The demand parameters d , d , and d in Eq. (17) show
0 1
2
a downward sloping demand function (d (0), with an elasticity varying be1
tween !0.42 in regime 1, and !0.70 in regime 2. This indicates an inelastic
demand for beef. This inelastic demand is broadly consistent with previous
empirical beef demand estimates. As expected, the cost parameters c and
1
c show that feed cost tends to increase the marginal cost of holding animals.
2
However, the estimated c varies considerably across regimes: from 1.12 in
1

848

J.-P. Chavas / Journal of Economic Dynamics & Control 24 (2000) 833}853

regime 1, to 10.26 in regime 3, to 44.01 in regime 2. Also, the parameter c
2
(re#ecting possible technological change) is negative in regime 1, but positive in
regimes 2 and 3. This empirical evidence suggests that the three expectation
regimes have clearly di!erent implications from the viewpoint of dynamic
pricing. Finally, the estimate of the parameters c and c in Eq. (10) are similar
1
2
across regimes. It indicates that the slaughter weight has increased signi"cantly
over time.
Second, we consider the case of heterogeneous expectations, allowing for the
simultaneous presence of three expectation regimes: rational expectations
(i"1), quasi-rational expectations (i"2), and naive expectations (i"3). This
corresponds to the price equation (15). We interpret the weights w as &market
it
shares' for the ith group. Also, we assume that w "w , i.e., that the proportion
it
i
of decision-makers in each expectation regime is constant over time. While
econometrically convenient, such an assumption may appear restrictive. We will
evaluate its empirical validity below. We thus treat the w 's in (15) as parameters
i
to be estimated. Such estimation provides a basis for investigating empirically
the heterogeneity of expectations of market participants in the beef industry.
Again, the parameters of the heterogeneous expectation model were estimated
for the U.S. beef market, using Hansen's GMM estimation method. The chosen
instruments are the same as above: B , p , q , an intercept, and a time
t~1 t~1 t~1
trend. The variance}covariance matrix was robustly estimated using the
Newey}West estimator, thus correcting for both heteroscedasticity and serial
correlation with lags up to three periods. The resulting estimates are presented
in Table 2.
First, the validity of the model speci"cation was assessed using the Hansen
test concerning the overidenti"cation restrictions in GMM estimation. The
Hansen test does not provide statistical evidence against the orthogonality
restrictions between the overidentifying instruments and the error terms (see
Table 2). This suggests that the model speci"cation and the choice of the
instruments appear appropriate.
Second, we evaluated the validity of assuming that the weights w's are
constant over time. This was done conducting the analysis separately for two
sub-samples: 1948}1972 and 1973}1992. The two sub-samples gave fairly similar
estimates of the weights w's. Between the two periods, the di!erence in the
estimated weight w (corresponding to the &rational expectation' group) was
1
0.001, with a standard error equal to 0.070. And the di!erence in the estimated
weight w (corresponding to the &namK ve expectation' group) was 0.071, with
3
a standard error equal to 0.076. This shows no strong evidence that the weights
w's are changing over time. On that basis, our assumption that the weights are
constant over time appears reasonable.
All the estimated coe$cients reported in Table 2 have the expected sign, and
most are signi"cantly di!erent from zero. The estimated model provides a reasonably good "t to the data: the R2 is 0.94, 0.99, 0.99 and 0.47, respectively, for

J.-P. Chavas / Journal of Economic Dynamics & Control 24 (2000) 833}853

849

Table 2
Parameter estimate for the heterogeneous expectation model
a
d

0

d

1

d

2

c

1

c

2

c
c

1
2

w (rational)
1
w (quasi-rational)
2
w (namK ve)
3
Minimum distance
R2 for Eq. (8)
R2 for Eq. (15)
R2 for Eq. (16)
R2 for Eq. (17)

0.9966!
(0.0133)
141.4721!
(14.9281)
!113.3054!
(24.5626)
[!0.5236]
!0.1447
(0.2016)
2.8587!
(0.7073)
!0.0061!
(0.0022)
495.6623!
(6.1597)
3.6291!
(0.1862)
0.1833!
(0.0865)
0.3504!
(0.0403)
0.4662!
(0.1245)
9.9600
0.9426
0.9895
0.9883
0.4711

Hansen test, s (10), p-value "0.4440
2

!Indicates that the corresponding parameter is signi"cantly di!erent from zero at the 5% level. The
asymptotic standard errors are presented in parentheses below the parameter estimates. Elasticities
evaluated at mean values are presented in brackets.

Eqs. (8), (15)}(17). The hypothesis that a"1 in Eq. (8) fails to be rejected at the
5% signi"cance level. Again, this indicates that our assumed values for birth rate
(k) and death rates (d) are consistent with the data. The demand parameters
d , d , and d in Eq. (17) show a downward sloping demand function (d (0),
0 1
2
1
with an elasticity of !0.52. The implied inelasticity of demand for beef appears
reasonable. Again, the cost parameters c and c suggest that feed cost tends to
1
2
increase the marginal cost of holding animals. The parameter c (re#ecting
2
possible technological change) is negative and signi"cantly di!erent from zero.
This shows evidence that, over the last few decades, technological progress in the
U.S. beef industry has contributed to lower production cost. Our estimates
indicate that technical change has generated about a 10% decline in production
cost over the last 40 years. Through market equilibrium pricing, these lower

850

J.-P. Chavas / Journal of Economic Dynamics & Control 24 (2000) 833}853

costs translated into lower beef prices, which bene"ted consumers. Again, the
estimate of the parameters c and c in Eq. (10) indicates that the slaughter
1
2
weight has increased over time during the sample period.
Finally, the estimated market share parameters (w , w , w ) provide useful
1 2 3
information on the heterogeneity of expectations. The estimate w suggests that
1
18.3% of farmers exhibit rational expectation, which is signi"cantly di!erent
from zero. This provides evidence that a signi"cant number of farmers understand well the dynamics of the beef markets, and use this information to
anticipate market prices. The estimate w is also signi"cantly di!erent from zero.
2
It shows that 35% of farmers behave in a way consistent with Nerlove's
quasi-rational expectations. These decision-makers understand that there is beef
cycle, and use this information to generate backward-looking expectations (in
contrast with the forward-looking expectations under Muth's rationality). Finally, the estimate w suggest that 46.7% of farmers exhibit namK ve expectation,
3
which is signi"cantly di!erent from zero. This provides evidence that a large
number of farmers use the latest price information as a basis for anticipating
market prices. But they fail to understand the dynamics of the beef market, or
the existence of a beef cycle.
These results provide statistical evidence of heterogeneous expectations
among participants in the U.S. beef market. Note that such a "nding is consistent with the empirical results obtained by Baak. Our analysis shows that
a number of market participants understand beef market dynamics, and use this
information in production decisions. However, it also suggests that a fairly large
number of producers exhibit signi"cant allocative ine$ciency in the sense that
their expectation formation is &namK ve' and fails to use information related to the
dynamics of supply}demand conditions in the beef market.

5. Implications and conclusion
We have investigated the nature of price expectations in a competitive market.
The approach applied to the U.S. beef market indicates the presence of heterogeneous expectations among beef producers. The empirical evidence shows that
a large proportion of beef producers (accounting for 46.7% of production)
behaves &naively', i.e., basing production decision only on the most recently
observed market prices. This supports the basic assumption underlying the
cobweb model (e.g., Ezekiel, 1938). A signi"cant but small proportion of beef
production (18.3%) comes from producers having forward-looking price expectations formed according to the rational expectation hypothesis. And about
35% of beef production comes from farmers using quasi-rational expectations
and anticipating future prices using their observed historical patterns.
These results can be interpreted in terms of the cost and bene"t to market
participants of obtaining and processing information about market prices.

J.-P. Chavas / Journal of Economic Dynamics & Control 24 (2000) 833}853

851

Indeed, while we did not measure such cost and bene"t directly, our "ndings can
shed some light on their relative magnitude. Consider that market participants
would decide to obtain price information only if they perceive receiving positive
net bene"t from it. The "nding that 18.3% of beef production is made by
farmers exhibiting rational expectations suggest that the perceived net bene"t
of being &rational' in anticipating future prices can be positive. This is interpreted
to mean that, for these producers, the gross bene"t from understanding
the dynamics of the beef market is greater than the cost of obtaining the
associated information. But these producers provide only a small share of
the market. A large proportion of producers is found to use &backward-looking'
expectations, either quasi-rational expectations, or the simpler namK ve
expectations assumed in the cobweb model. But when would decision-makers
choose to use &backward-looking' expectations? Such expectations neglect some
relevant information about market price determination (e.g., the dynamic characteristics of supply and demand conditions). As a result, one expects the gross
bene"t of backward-looking rational expectations to be less than the gross
bene"t of forward-looking rational expectations for a particular producer.
And for the majority of beef producers using backward-looking expectations,
the net bene"t of their expectation formation must be larger than the alternatives. This implies that the cost of backward-looking expectations must be
lower. Intuitively, simple forms of expectations would be used because of their
lower cost.
This has several implications. First, our analysis can be interpreted as indirect
evidence that the cost of obtaining and processing market information is
positive and signi"cant. Second, this cost can provide incentives for decisionmakers to save on information by exhibiting bounded rationality and using
simple expectation rules (such as the namK ve expectation of the cobweb model).
This in#uences prices and the dynamics of markets. At this point, further
research is needed to investigate the linkages between bounded rationality,
expectation formation and the existence of market cycles. Third, we found
empirical evidence of heterogeneity of expectations among beef market participants. This shows that the ability to obtain and process information varies
signi"cantly among market participants. It suggests a need to research further
this heterogeneity and its role in market dynamics. Finally, the relative
importance of simple backward-looking expectations indicates the possibility of
signi"cant dynamic allocative ine$ciency in the beef market. For example,
namK ve expectations neglect information about the existence of a beef cycle. Is it
possible to improve human capital so as to reduce information cost and increase
the quality of expectation formation? At this point, there appears to be signi"cant possibilities to improve the &market intelligence' of industry decisionmakers, leading to better use of information and improved dynamic allocative
e$ciency. Exploring such possibilities appears to be a good topic for further
research.

852

J.-P. Chavas / Journal of Economic Dynamics & Control 24 (2000) 833}853

Acknowledgements
I would like to thank the editor and reviewer for useful comments on an
earlier draft of the paper. This research was supported in part from a Hatch
grant from the College of Agricultural and Life Sciences, University of Wisconsin, Madison.

References
Anderson, E.W., Hansen, L.P., McGrattan, E.R., Sargent, T.J., 1995. Mechanics of forming and
estimating dynamic linear economies. Working Paper, University of Chicago.
Baak, S., 1997. Tests for bounded rationality: an application to the U.S. cattle market. Working
paper, University of Wisconsin, Madison.
Brock, W., Hommes, C., 1997. A rational route to randomness. Econometrica 65, 1059}1095.
Chavas, J.P., Klemme, R., 1986. Aggregate milk supply response and investment behavior on U.S.
dairy farms. American Journal of Agricultural Economics 68, 55}66.
Chavas, J.-P., Holt, M.T., 1995. Nonlinear dynamics and economic instability: the optimal management of a biological population. Journal of Agricultural and Resource Economics 20, 231}246.
Chow, G.C., 1989. Rational versus adaptive expectations in present value models. Review of
Economics and Statistics 71, 376}384.
Coase, R.H., Fowler, R.F., 1937. The pig cycle in Great Britain: an explanation. Economica 4, 55}82.
Conlisk, J., 1996. Why bounded rationality? Journal of Economic Literature 44, 669}700.
Eckstein, Z., 1984. A rational expectations model of agricultural supply. Journal of Political
Economy 92, 1}19.
Ezekiel, M., 1938. The cobweb theorem. Quarterly Journal of Economics