Directory UMM :Data Elmu:jurnal:E:Energy Economics:Vol22.Issue5.2000:

Energy Economics 22 Ž2000. 527]547

Price and inventory dynamics in petroleum
product markets
Timothy J. Considinea, , Eunnyeong Heob
U

a

Department of Energy, En¨ ironmental, and Mineral Economics, The Pennsyl¨ ania State
Uni¨ ersity, 203 Eric A. Walker Building, Uni¨ ersity Park, PA 16802, USA
b
School of Ci¨ il, Urban, and Geosystem Engineering, Seoul National Uni¨ ersity, San 56-1,
Shinrim, Kwanak, Seoul 151-742, South Korea

Abstract
Unlike many studies of commodity inventory behavior, this paper estimates a model with
endogenous spot and forward prices, inventories, production, and net imports. Our application involves markets for refined petroleum products in the United States. Our model is
built around the supply and demand for storage. We estimate the model using Generalized
Method of Moments and perform dynamic, simultaneous simulations to estimate the
impacts of supply and demand shocks. Supply curves for the industry are inelastic and

upward sloping. High inventory levels depress prices. Inventories fall in response to higher
sales, consistent with production smoothing. Under higher input prices, refiners reduce their
stocks of crude oil but increase their product inventories, consistent with cost smoothing. In
some cases, imports of products are more variable than production or inventories. Q 2000
Elsevier Science B.V. All rights reserved.
JEL classifications: C5; L6; Q4
Keywords: Inventories; Convenience yield; Petroleum

1. Introduction
The second half of the 1980s was a period of rapid change in US petroleum
markets. The introduction of a netback pricing system by Saudi Arabia in late 1985,
followed by the crude oil price collapse in 1986, ushered in a flexible pricing system
U

Corresponding author. Tel.: q1-814-863-0810; fax: q1-814-863-7433.
E-mail addresses: cpw@psu.edu ŽT.J. Considine., exheo@plaza.snu.ac.kr ŽE. Heo..

0140-9883r00r$ - see front matter Q 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 1 4 0 - 9 8 8 3 Ž 0 0 . 0 0 0 5 6 - 6

528

T.J. Considine, E. Heo r Energy Economics 22 (2000) 527]547

with spot and futures markets. Crude oil and petroleum product markets now have
become fully developed through the 1990s. As these new markets matured, changes
within them became an important part of the decision-making process of the US
petroleum refining industry. Market prices of petroleum products now appear quite
sensitive to supply and demand shocks, unlike the previous era of pricing under
long-term contracts.
As a result, inventories now play an important role in the transmission of these
market shocks to petroleum product prices. Petroleum inventories are assets, and
their valuation by refiners varies with market conditions. For example, as futures or
forward prices rise above spot or cash prices, the user cost of inventories decline,
and refiners may increase stock holding to reap expected profits at some point in
the future. Similarly, relatively higher spot prices raise incentives to lower inventories, which may reduce pressure on prices for current delivery. Hence, inventories
affect the adjustment of the petroleum markets to OPEC production decisions,
weather, and other market shocks.
Understanding the linkages between prices, inventories, and production in the
petroleum refining industry is our objective. We expand on the studies of these

interactions by Bacon et al. Ž1980. using a model suitable for econometric estimation and simulation. We estimate the model using monthly data from 1985 to 1996,
conduct a goodness-of-fit simulation, and perform a set of simulations to understand how petroleum markets respond to market shocks.
We model short-run petroleum price and inventory movements using a sequence
of submodels. In the first model, we assume petroleum refiners minimize cost given
exogenous output targets and prices for inputs. We consider inventories as a
quasi-fixed factor of production with rental costs that include expected prices for
petroleum products. The second model determines spot and nearby futures prices
also contingent upon production, sales, and input prices. In this component, we
formulate the model viewing prices as a stochastic state variable.
Why do we adopt this approach? The rational expectations models of commodity
price and inventory dynamics developed by Pindyck Ž1994. and Considine Ž1997.
contain equilibrium conditions for inventories that essentially equate the marginal
costs and benefits of inventories, including a convenience yield earned from the
cost and production smoothing benefits from holding inventories. These dynamic
demand functions for inventories, however, are overidentified, containing expectations of future inventories, production, and prices. While this formulation facilitates econometric estimation, it greatly complicates model simulation because
solutions for expectations are required. In contrast, this paper provides an explicit,
closed form model for price expectations. Our approach is driven by the desire to
have a model motivated by theory yet flexible enough to allow dynamic simulation
for forecasting and policy analysis.
We close the model by solving for refinery production using market balance

equations that equate sales with production, net imports, and the net change in
stocks. The inventory block determines the net change in stocks. A set of equations
determines net imports and import prices. Among other factors, net product
imports depend upon spot prices in New York relative to those in Rotterdam. We

T.J. Considine, E. Heo r Energy Economics 22 (2000) 527]547

529

assume total domestic sales of petroleum products are exogenous and solve for
production by adding net inventory changes and net exports to domestic consumption. Overall, our model determines prices, inventories, production, net imports for
petroleum products in the US contingent upon exogenously determined product
consumption, crude oil prices, and prices for other factor inputs.
We assume refiners are price takers in crude oil and in petroleum product
markets. A multiproduct formulation along the lines of Griffin Ž1976. is also
adopted in this study to reflect joint production in petroleum refining. Crude oil
and petroleum liquids are the two variable inputs in the model, constituting
roughly 95% of the variable cost of petroleum refining. To reduce the size and
complexity of the model, we examine five product aggregates } gasoline, distillate
fuel, residual fuel, jet fuel, and other petroleum products, and only the first four

products are considered for the price]arbitrage relations due to the unavailability
of data on prices for other petroleum products.
Section 2 contains the theoretical foundations of the empirical model. The
functional specification of the econometric equations appears in Section 3, followed by a summary of the estimation results in Section 4. The last section contains
simulations to test the model’s forecasting ability and to estimate the market
impacts of supply and demand shocks.

2. Model formulation
The primary objective of this study is to develop a structural model of refined
petroleum product markets suitable for simulation and forecasting. To provide an
overview of the model, we devised the following block structure, summarized in
Table 1. There are five major blocks in the model that determine groups of
endogenous variables contingent upon predetermined and exogenous variables.
The latter are determined outside the entire model whereas predetermined variables depend upon some other block. The exogenous variables include crude oil
and propane prices, weather conditions measured by heating and cooling degreedays, interest rates, petroleum product sales, and storage costs.
The first block essentially determines the demand for quasi-fixed and variable
inputs for US petroleum refining. In this block, we determine month-ending
inventories of five petroleum products, including gasoline, distillate, residual, jet
fuel, and other petroleum products. This block also determines inventories of
crude oil and unfinished oils, crude oil distillation capacity, and consumption of

crude oil and petroleum liquids. These quasi-fixed and variables inputs depend
upon refinery production, rental prices, beginning inventory levels, and the exogenous variables. The second block predicts spot and futures prices conditional
upon production, quasi-fixed inputs, and prices for crude oil and propane. In the
third block, we determine net imports based upon domestic prices relative to those
in Rotterdam, product stocks, capacity, sales, and weather. In the fourth section,
simple autoregressive models provide estimates of Rotterdam prices. The fifth and
final block includes a set of identities that determine inventory user costs given

530

T.J. Considine, E. Heo r Energy Economics 22 (2000) 527]547

Table 1
Block structure of the econometric model
Equation block

Endogenous

Predetermined

Exogenous

Inventory

Product stocks Ž5.
Crude oil stocks
Work-in-process stocks
Crude oil consumption
Other liquids consumption
Crude capacity

Refinery production
Rental prices
Beginning stocks

Crude prices
Propane prices
Weather
Interest rates

Price

Spot prices Ž4.
Futures prices Ž4.

Refinery production
Ending stocks
Crude capacity

Crude prices
Propane prices

Net imports

Net imports Ž5.

Spot product prices
Rotterdam spot prices
Product stocks
Crude capacity

Product sales
Weather

Rotterdam prices

Rotterdam prices Ž4.

US spot prices
Crude oil prices
Lagged Rotterdam prices

Crude prices

Identities
Sales

Production Ž5.

Stocks

Net imports
Spot prices
Futures prices

Sales

Rental prices

Rental prices Ž8.

Storage costs
Interest rates
Crude prices
Propane prices

predictions of spot and futures prices from the price block and refinery production
given inventories and net imports from the first and third blocks, respectively.
We formulate the inventory block using a dynamic dual formulation. Many
researchers have exploited the duality between cost and production functions to
derive input demand functions consistent with either cost minimization or profit

maximization. Following the earlier work by Lucas Ž1967. and Treadway Ž1971. on
the flexible accelerator model, McLaren and Cooper Ž1980. and Epstein and
Denny Ž1983. have extended the dual approach to dynamic problems.
A dynamic problem often embodies the distinction between the short run when
some factors of production are fixed and the long run when all factors can adjust to
equilibrium levels. Adjustment costs are often claimed as the reason for dichotomy.
These costs are generally unobservable and are often expressed as the reduction in
output resulting from diverting resources to accommodate changes in quasi-fixed
factors of production.
In petroleum refining, there may be significant costs associated with changing

T.J. Considine, E. Heo r Energy Economics 22 (2000) 527]547

531

capacity levels. In addition, adjusting inventories may be costly due to logistical
differences in transport and scheduling. Consequently, we consider the following
short-run restricted cost function for a petroleum product firm:
G Ž w, y, x, ˙
x . s minimize wz ,

Ž1.

where G is the variable cost function; w is a vector of prices for crude oil and
propane; y the vector of refinery production levels; x the vector of quasi-fixed
factors; ˙
x the vector of changes in levels of quasi-fixed factors; and z the vector of
variable inputs.
The vector of variable inputs z is the choice variable in the cost minimization
problem and is subject to a production function:
F Ž y, z, x, ˙
x. s 0 .

Ž2.

Note that Fx ) 0, Fz ) 0 and F˙x - 0, in which the last derivative reflects internal
adjustment costs.
Assume that the firm attempts to minimize the present discounted value of
future costs over an infinite horizon at some point in time } say period t s 0, by
adjusting variable input purchases and net accumulations of quasi-fixed factors. In
this case, given initial states for the quasi-fixed factors, x0 , exogenous prices for
variable and quasi-fixed inputs, and predetermined levels of output, the discounted
value function J is:
`

H0 e

J Ž x0 , y,w,¨ . s min

yr t

w G Ž w, y, x, ˙
x . q ¨ 9 x x dt ,

Ž3.

subject to positive starting values for the quasi-fixed factors, and where ¨ is the
vector of rental costs for quasi-fixed factors and r is a fixed discount factor.
The discounted value function is assumed to be real, non-negative, twice continuously differentiable, non-decreasing, and concave in w and ¨ and decreasing in x
ŽEpstein and Denny, 1983.. Under these conditions, the Hamilton]Jacobi]Bellman
expression ŽKamien and Schwartz, 1991. } often referred to as the dynamic
programming equation, is as follows:
`

H0 e

J Ž x0 , y,w,¨ . s min

yr t

w G Ž w, y, x, ˙
x. q ¨9x q ˙
x9Jx x dt .

Ž4.

This equation simply states that the discounted long-run cost is equal to the sum of
total variable costs, total rental costs of the quasi-fixed factors ¨ 9 x, and the implicit
value of additional fixed factors ˙
x9Jx ŽStefanou, 1989..
With this restatement of the objective function, we derive a set of first-order
conditions for the dynamic problem. Consider one such condition involving the rate
of change in quasi-fixed factors ˙
x:
G˙x s yJx ,

Ž5.

532

T.J. Considine, E. Heo r Energy Economics 22 (2000) 527]547

which states that the marginal adjustment costs of the quasi-fixed factors in any
period must equal their respective shadow values. Also, consider the first-order
condition with respect to the level of quasi-fixed factors x:
¨ 9 s y Ž Gx q Jx x ˙
x. ,

Ž6.

which implies that rental values should reflect the shadow value and net capital
gains from stock holding. If we differentiate Eq. Ž4. with respect to ¨ , we obtain:
rJ¨ s x q ˙
xU Jx ¨ q Ž G˙x q Jx .

­˙
xU
­¨

q Ž ¨ 9 q Gx q Jx x .

­ xU
­¨

,

Ž7.

where the asterisks indicate evaluation at optimal values. Using Eqs. Ž5. and Ž6.
and solving Eq. Ž7. for ˙
xU , we obtain the following expression for optimal net
investment in quasi-fixed factors:
Ž
.
˙xU s Jy1
x ¨ rJ¨ y x .

Ž8.

The demand for the variable input can be obtained by differentiating Eq. Ž4. with
respect to w and using Shephard’s lemma Ž zU s ­Gr­w . to obtain:
zU s rJw y ˙
x9Jx w .

Ž9.

The above problem involves the optimal choice of variable and quasi-fixed factors
to minimize cost. Eq. Ž8. determines the demand for storage while Eq. Ž9. constitutes the demand for variable inputs.
To determine equilibrium spot and futures prices, we develop a model of the
supply for storage. As Brennan Ž1958. notes, this supply is not the supply of storage
space but the supply of commodities as inventories. A supplier of storage is anyone
who holds title to stocks for future sale, including hedgers and speculators. We will
collectively call these agents oil traders. We assume that price is a stochastic state
variable analogous to the study by Considine and Larson Ž1998. and that traders
choose sales and production to maximize their expected present value of profits:
V Ž p0 , x0 , y,w,k ,t . s max
yt , s t

`

H0

Et  ps y C Ž x, y,w,k,h .4 eyr tdt ,

Ž 10 .

subject to:
d x s Ž y y s q n . dt ,

Ž 11 .

d p s m Ž p . dt q s Ž p . d b ,

Ž 12 .

where d b is the standard Weiner process,1 s is total domestic sales, d x is ending
1

A Weiner process has three important properties. First, it is a Markov process so that the current
value provides the best forecast of future values. The second property is that it has independent
increments. Finally, these increments are normally distributed with a variance that increases linearly
with time.

T.J. Considine, E. Heo r Energy Economics 22 (2000) 527]547

533

less beginning inventories, and n is imports less exports, C Ž x, y,w,k,h. represents
the cost of production, marketing, and transportation of refined petroleum products,
k is capacity, and h represents technological change.2 Eq. Ž11. simply rewrites the
market balance equation, which states that sales must come from current production, net imports, and the net change in stocks. We assume the price process wEq.
Ž12.x evolves stochastically over time, following geometric Brownian motion with a
drift component, mŽ p ., and a variance, s Ž p .. This assumption allows us to derive
an expression for the supply of storage. This derivation begins with the Bellman
equation for this problem:

½

rV s max ps y C Ž x, y,w,k,h . q Vx Ž y y s . q m Ž p . Vp q
y, s

1
2

5

s Ž p . 2 Vp p .

Ž 13 .

The partial differentials with respect to sales and production imply the following
first-order conditions:
rVs s p y Vx s 0 « p s Vx
,
rVy s yCy q Vx s 0 « p s Cy

Ž 14.

indicating that equilibrium prices reflect marginal production costs and the shadow
value of inventories. An expression for the expected change in prices follows from
differentiating Eq. Ž14. using Ito’s lemma:
E Žd p .
dt

s Vx x Ž y y s . q m Ž p . Vp x q

1
2

s Ž p . 2 Vp p x .

Ž 15 .

Differentiating Eq. Ž13. with respect to x and solving for Cx provides the
convenience yield:
Cx s Vx x Ž y y s . q m Ž P . Vp x q

1
2

s Ž P . 2 Vp p x y rVx ,

Ž 16.

which equals the expected capital gain on inventories wEq. Ž15.x net of interest
costs. Notice that for the convenience yield to reflect the variability in prices, the
value function must have non-zero third-order derivatives.
2

The introduction of a cost function for traders separates the inventory block from the price block.
Considine Ž1992. derives a supply function from the dual profit maximization problem associated with
Eq. Ž4.. This model, however, did not converge in out-of-sample simulations due to the complex cross
equation parameter restrictions between the inventory and marginal cost equations. This formulation
also required static rental price terms, which precluded the forward-looking formulation of rental prices
used in this paper. For these reasons, we chose to separate the inventory demand problem from the
price determination process.

T.J. Considine, E. Heo r Energy Economics 22 (2000) 527]547

534

Substituting Eqs. Ž14. and Ž16. into Eq. Ž15., we obtain the supply of storage
relation, which is a simple relation for the expected change in prices:
E Žd p .
dt

s Cx y rp .

Ž 17 .

This relation states that the expected capital gain reflects the convenience yield net
of interest costs. During so-called market backwardations when EŽd p . is negative,
Cx could be large and negative, indicating that another barrel of inventory could
substantially lower costs or provide a high con¨ enience yield. Eqs. Ž14. and Ž17.
constitute the basis of an empirical model once we specify a cost function and its
derivatives. Estimation of the supply of storage model also could be accomplished
with an approximation of the value function. We adopt the cost function approach
because it does not require estimation of Eq. Ž15..
The demand for storage wEq. Ž8.x is contingent upon a sequence of production
targets. In other words, vertically integrated petroleum refining firms minimize cost
subject to production levels. The supply of storage wEq. Ž17.x relation defines an
equilibrium relation between expected prices and inventories. Together these
relations determine the supply and demand for inventories and the equilibrium
price for storage given by Eq. Ž17..
If sales are given in the short-run by weather, income, and other exogenous
factors, then production is simply sales plus the net inventory change less net
imports. This implies that with a model of net foreign trade of petroleum products,
we can solve for production using the market balance equation:
yssqdxyn .

Ž 18 .

Conditional upon market prices, we assume traders minimize their net import
product expenditures for each product subject to capacity, ending stocks, and final
product shipments. We assume these decisions are strongly separable from inventory and production decisions to simplify the model. With the assumption that both
imports and exports sell at the same price, we obtain the demand for net imports
by the first-order condition of this problem with respect to the import price, using
Shephard’s lemma:
­NE Ž pr ,s, x, ˙
x.
­ pr

sn,

Ž 19 .

where NE is the net import expenditure function, and pr is a vector of prices for
imported petroleum products.

3. Empirical model
This part of the paper discusses the functional specification of the component
models developed in the previous section. Most empirical applications of dynamic

T.J. Considine, E. Heo r Energy Economics 22 (2000) 527]547

535

dual models utilize a second-order linear quadratic approximation ŽEpstein and
Denny, 1983; Stefanou et al., 1992.. Another approach is to use translog or
generalized Leontief approximations as in Luh and Stefanou Ž1996.. Such forms,
however, must be modified to accommodate the linear adjustment mechanism in
the optimal net investment equation of quasi-fixed factors wEq. Ž8.x. As a result, this
study uses a quadratic approximation of the value function for the inventory and
demand module to avoid potential convergence problems with these mixed functional forms.
A normalized quadratic formulation is typically used for the approximation so
that linear homogeneity in prices can be imposed on the cost function. However,
Mahmud et al. Ž1986. finds that parameter estimates from normalized models vary
depending upon the selection of the normalizing numerator. Therefore, this study
utilizes an unrestricted model that is not homogeneous in factor prices. Following
Miron and Zeldes Ž1988., heating and cooling degree-day deviations are included
in the cost function to random weather shocks. The quadratic value function for
the inventory and demand module then takes the following form:
w
J s a9w1 x 2

q

1
2

a9¨ 1 x 8

w9

¨9

a9x1 x 8

x9

¨

a9y1 x5

y9

x
y

gw2 x 2

g¨ 2 x 8

gx2 x 8

gy2 x5

g 9¨ 8 x 2

B8 x 8

My1
8 x8

H8 x5

g 9x8 x 2
g 9y5 x 2

M8y1
x8
H95 x 8

D8 x 8
T 95 x 8

T8 x5
G5 x5

w
¨

x
y

w
q u9 f 9w2 x 2

f 9¨ 2 x 8

f 9x2 x 8

f 9y2 x5

¨

x
y

w
q e9w

e9¨

e9x

e9y

¨

x
y

,
Ž 20 .

where the variables are defined as follows:
v

v

v

v

v

w is a 2 = 1 vector of real prices for the refiners’ acquisition cost of crude oil
and propane;
y is a 5 = 1 vector of refinery production of gasoline, distillate, residual fuel,
jet fuel and kerosene, and other petroleum products;
x is an 8 = 1 vector of seven inventory categories, including crude oil and
liquids, unfinished oils, the five products, and crude distillation capacity;
¨ is an 8 = 1 vector of rental values Žor user costs. for the quasi-fixed factors;
and
u is a 2 = 1 vector of deviations of heating and cooling degree-days from their
30-year means.

536

T.J. Considine, E. Heo r Energy Economics 22 (2000) 527]547

The e vectors are random error terms that reflect errors in dynamic optimization. The other terms on the right-hand side of Eq. Ž20. are unknown parameters
to be estimated.
The estimating equations for the net investment in quasi-fixed factors follow
from taking the derivative of value function J with respect to ¨ , x, and u and
substituting into Eq. Ž8.:

˙xU s rM Ž a¨ q g 9¨ w q B¨ q Hy q f¨ u . q Ž rI y M . x q rMe¨ .

Ž 21 .

Note that if the error terms are serially correlated, they must move at the rate of
adjustment M.
The study by Epstein and Denny Ž1983. transforms the non-linear structural
form given by Eq. Ž21. into a linear reduced-form model. To derive the reduced
form for estimation, we express Eq. Ž21. as follows:

˙xU s Ž rI y M .Ž x y x . ,

Ž 22 .

where:
x s yr Ž I y M . y1 rM w a¨ q g 9¨ w q B¨ q Hy q f¨ u q e¨ x .

Ž 23.

Substituting Eq. Ž23. into Eq. Ž22. using the discrete approximation Ž x y xy1 . for
xU and Ž xy1 y x . for Ž x y x ., we obtain the following partial adjustment equations
for the quasi-fixed factors:
x s rM w a¨ q g 9¨ w q B¨ q Hy q f¨ u x q wŽ 1 q r . I y M x xy 1 q rMe¨ .

Ž 24 .

The terms in the first set of brackets on the right-hand side of Eq. Ž24. constitute
the target or long-run equilibrium level of stocks. These targets depend upon crude
oil cost shocks, rental values, final product sales, and weather shocks represented
by u. Given that the adjustment coefficients M are identified, we can estimate the
model in reduced form and recover the structural parameters ŽEpstein and Denny,
1983..
The demands for the variable inputs are obtained by taking the derivative of J
and Jx and with respect to w and substituting into Eq. Ž9.:
z s r Ž aw q g 9w w q g¨ ¨ q gx x q gy y q fw u . y gx Ž x y xy1 . q rew .

Ž 25 .

The second to last term on the right of Eq. Ž25. represents the extent of
disequilibrium in short-run input demands. Eqs. Ž24. and Ž25. constitute the
refinery inventory and input demand module. The input demand and inventory
functions are linear in their parameters. In addition, we assume that the following
specific form for H:

T.J. Considine, E. Heo r Energy Economics 22 (2000) 527]547

Hs

h11
h21
h31
0
0
0
0
h81

h12
h22
0
h42
0
0
0
h82

h13
h23
0
0
h53
0
0
h83

h14
h24
0
0
0
h64
0
h84

h15
h25
0
0
0
0
h75
h85

537

Ž 26.

.

Thus, refinery production levels for all products affect refinery capacity, crude oil,
and work-in-process inventories, while product inventories only depend on their
own production level.
Given these assumptions, the estimating equations for the refinery inventory and
demand module are as follows:
2

xit s r a¨)i q

5

Ý g¨)i j wjt q b)ii ¨it q Ý hi)j yjt q fi h uh t q fic uct
js1

js1

q Ž 1 q r y mi i . xi ty1 q

eit)

Ž 27.

,

for i s 1, 2, 8, which are the demand for crude oil and unfinished inventories, and
crude distillation capacity;
2

xit s r a¨)i q

Ý g¨)i j wjt q b)i i ¨i t q h)i iy2 yiy2 t q fi h uh t q fic uct
js1

q Ž 1 q r y mii . xity1 q e)it ,

Ž 28.

for i s 3, . . . ,7, which constitute the demands for product storage; and
2

zit s r aw i q

8

5

Ý g¨ i j wjt q Ý gx i j xjt q Ý gy i j yjt q fz i h uh t q fz ic uct
js1

js1

js1

8

y

Ý gx i j Ž xjt y xjty1 . q ew)t ,

Ž 29 .

js1

for i s 1, 2, which are the equations for consumption of crude oil and propane.
U
Note that the investment equations are in reduced form, so that hi j s mii hi j . All
the quasi-fixed factors, x, depend on input prices w, own-user cost ¨ , production
levels y, weather u, and an own-lag term weighted by the adjustment coefficients
m. The demands for variable inputs are unaffected by user costs for quasi-fixed
factors but change with quasi-fixed factors. They also depend on the input prices,
production levels for all products, and weather.
The supply of storage model assumes prices are stochastic state variables, which

T.J. Considine, E. Heo r Energy Economics 22 (2000) 527]547

538

requires that its corresponding value and cost functions have at least non-zero
third]order derivatives. To conform to the non-zero third-order derivative requirement, we assume a square-root quadratic approximation for the trader’s cost
function. Using Eq. Ž14., we estimate the following expression for spot prices for
refinery products:
2

pit s gii q

Ý

lji

js1

q ti

ž /
tt

yi t

wjt

ž /

1r2

yit

5

q

Ý gi j
j/i

yjt

ž /

1r2

yit

5

Ý

q

di j

js1

xjt

ž /

1r2

yit

q ki

ž /
kt

1r2

yit

1r2

Ž 30 .

,

for i s 1,...,4, where g, l, d, k and t are unknown parameters, and li j s lji ,
di j s dji given symmetry. In other words, spot prices depend upon input prices,
production, own product inventory, capacity, and a time trend for technological
change.
Using Eq. Ž17., the returns-to-storage equation is as follows, using the difference
between the 1-month futures price and the spot price for the expected price
change:
2

pfit

y Ž 1 q r . pit s bii q

Ý mji
js1

q hi

ž /
kt

xit

wjt

ž /

1r2

xit

1r2

q ri

5

q

Ý dji
js1

ž /
tt

xit

yjt

ž /
xit

1r2

5

q

Ý bi j
j/i

xjt

ž /

1r2

xit

1r2

,

Ž 31 .

for i s 1,...,4, where pitf is the nearby futures price. In the model simulations
reported below, we solve this expression for the futures prices. The spot price and
the arbitrage relations for the refinery products, are affected by input prices,
inventory and production levels of all products, capacity k, and time trend t. In
addition, the arbitrage relations depend on the opportunity cost of capital. As in
Considine and Larson Ž1998. and Heo Ž1998., this study also hypothesizes that
futures and spot prices affect current inventory levels.
The estimated form of the net import equations are obtained by applying Eq.
Ž19. to a simple quadratic function for the net import expenditure function for
each product:
nit s aei q hi i Ž pritr pit . q ki ,iq2 xiq2,t q ki8 x8 t q li i sit q ri h uh t q ric uct q eit ,
Ž 32.
for i s 1, . . . ,5. This formulation essentially implies strong separability between
various net product import demands. One-time lags of independent variables are
added to each estimation equation to treat autocorrelation.
In addition to the above three modules, an import price module is added to

T.J. Considine, E. Heo r Energy Economics 22 (2000) 527]547

539

facilitate this model’s forecasting ability. This study adopts four simple autoregressive models for Rotterdam product spot prices for the import price module, which
were originally proposed by the Energy Information Administration Ž1993.:
prit s spri q spi pit q sci wt q sli pri ,ty1 ,

Ž 33 .

for i s 1, . . . ,4, where pr, p and w are the vectors of Rotterdam spot prices, New
York spot prices, and crude oil price, respectively.
User costs for inventories are defined as the sum of the financial opportunity
cost of capital, storage cost and the difference between futures and spot prices:
f

¨ it s rpit q st y pit y pit ,

Ž 34 .

for i s 1,...,7. Again, for model simulation, estimates of spot prices provided by Eq.
Ž30. and futures prices given by Eq. Ž31. would make user costs an endogenous
variable in the model. A similar rental price formulation is used for crude
distillation capacity, except storage costs are replaced by depreciation.
The entire refined petroleum market model consists of 27 behavioral equations:
the seven inventory and the capacity equations wEqs. Ž27. and Ž28.x, and the two
variable input demand functions wEq. Ž29.x for the refinery inventory and
demand module;
the four spot price equations wEq. Ž30.x and the four arbitrage relations wEq.
Ž31.x;
the five net product import equations wEq. Ž32.x for the net import module; and
the four imported oil price equations wEq. Ž33.x for the import price module.

v

v

v
v

In addition, there are the market clearing identities defined in Eq. Ž18. and the
user cost definitions given by Eq. Ž34..

4. Empirical analysis
We estimate each of the four modules separately to avoid convergence problems
that may occur when estimating large simultaneous equation systems. Our estimator is the Generalized Method of Moments ŽGMM. developed by Hansen Ž1982..3
The lags on the moving average components of the errors are from the optimal
bandwidths formulas developed by Newey and West Ž1994.. All variables in the
model are stationary. Using the methods developed by Hylleberg et al. Ž1990., we
also found no seasonal unit roots.
The instrumental variables used in this study are seasonally unadjusted. They
include the M1 measure of the money supply, housing starts, manufacturing labor
hours, consumer price index net of energy, the S & P 500 index, the industrial
3

We use the GMM procedure in DOSrWIN TSP 4.3 by TSP International for the estimation.

T.J. Considine, E. Heo r Energy Economics 22 (2000) 527]547

540

Table 2
Summary fit statistics and tests of overidentifying restrictions
Dependent variables

Inventory and demand module
Price module
Net import module
Import price module

x 2 -Statistics
Test

Critical Ž95%.

P-Value

553.26
439.35
319.65
260.75

555.24
519.44
320.03
298.61

0.06
0.82
0.05
0.48

production index, long-term bond rates, yields on federal funds, and deviations of
heating and cooling degree days from their 30-year means. Lagged values of
refinery productions, net imports, NYMEX and Rotterdam spot prices, inventories
of petroleum products, and lagged world petroleum production are also included.
In addition, four OPEC regime shift variables are included for the crude oil price
collapse from February to November 1986, the invasion of Kuwait by Iraq on
August 1990, and the ‘Desert Shield’ and ‘Desert Storm’ from September to
December 1990 and from January to February 1991, respectively. A set of monthly
dummy variables is also included to account for fixed seasonal effects.
Estimation uses monthly data from January 1985 to December 1994. The input
price vector includes propane prices and the refiner acquisition price for crude oil,
which includes domestic and imported supplies. We use spot and 1-month futures
prices for gasoline and heating fuel from the New York Mercantile Exchange
ŽNYMEX. for current and expected prices. Import product prices are Rotterdam
spot prices. Jet fuel prices equal an average of gasoline, distillate, and residual
prices and residual fuel futures prices equal crude oil futures for prices due to data
unavailability. Inventories are month-end values. Consequently, we used month-end
prices for crude oil and products. All other data are monthly averages from the
Department of Energy’s Energy Information Administration ŽEIA. database.
We found that the model was much more stable in out-of-sample simulations
when we included monthly dummies in the input and product inventory equations.
Thus, the inventory equations contain fixed seasonal effects captured by the
monthly dummy variables and stochastic weather effects represented by deviations
of heating and cooling degree-days from their 30-year means.
First, we performed tests for the overidentifying restrictions of the model to test
the maintained restrictions, such as dynamic optimization, and the quadratic
approximation of the cost function.4 The P values of the test are larger than 0.05
for all modules, indicating that we cannot reject the four modules Žsee Table 2..
The estimated residuals have no unit root at the 5% significance level in augmented Dickey]Fuller tests, indicating an absence of serious dynamic misspecification.
Values of the estimated parameters are listed in Table A1 through Table A4 of
4

See Davison and MacKinnon Ž1993, pp. 232]237 and 614]620., for more details of the test.

T.J. Considine, E. Heo r Energy Economics 22 (2000) 527]547

541

Table 3
Short-run marginal cost elasticities Žasymptotic t-statistics .
Explanatory
variables

Input prices
Crude oil
Propane
Production le¨ els
Gasoline
Distillate
Residual
Jet fuel
Other
In¨ entory le¨ els
Gasoline
Distillate
Residual
Jet fuel
Other
Capacity
Time

Short-run marginal costs C y
Gasoline

Distillate
fuel

Residual
fuel

Jet
fuel

0.66
Ž10.87.
y0.13
Žy1.04.

0.68
Ž13.54.
y0.02
Žy0.23.

0.85
Ž18.04.
y0.17
Žy1.70.

0.72
Ž15.03.
y0.08
Žy0.83.

0.15
Ž0.51.
y0.06
Žy0.54.
0.05
Ž1.72.
0.02
Ž0.64.
y0.21
Žy0.85.

y0.15
Žy0.54.
0.78
Ž4.59.
y0.02
Žy0.54.
0.11
Ž2.44.
y0.53
Žy2.20.

0.43
Ž1.72.
y0.07
Žy0.54.
0.15
Ž2.28.
0.08
Ž1.53.
y0.23
Žy1.06.

0.13
Ž0.64.
0.24
Ž2.44.
0.04
Ž1.53.
0.08
Ž2.05.
y0.32
Žy1.58.

y0.90
Žy4.12.
y0.30
Žy2.56.
y0.07
Žy1.17.
y0.13
Žy4.18.
0.04
Ž0.20.
0.80
Ž2.01.
0.08
Ž1.11.

y0.27
Žy1.31.
y0.32
Žy2.77.
0.18
Ž2.37.
y0.01
Žy0.27.
0.17
Ž0.83.
y0.64
Žy1.62.
0.08
Ž1.19.

y0.11
Žy0.57.
y0.10
Žy1.01.
y0.02
Žy0.24.
y0.01
Žy0.21.
0.16
Ž0.91.
y1.07
Žy2.99.
0.12
Ž2.00.

y0.45
Žy2.60.
y0.26
Žy2.62.
0.02
Ž0.41.
y0.02
Žy0.55.
0.13
Ž0.76.
y0.32
Žy0.96.
0.10
Ž1.64.

Appendix A. Among the seven user cost terms in the inventory equations, those of
gasoline, distillate, jet fuel, and other petroleum products stocks are negative,
showing correct signs consistent with the theory of storage. Estimated rates of
adjustment mii for inventories are rapid. For gasoline, 29% of the total adjustment
of stocks occurs within 1 month. The adjustment rate for residual fuel is even
quicker with more than half of the adjustment occurring within 1 month. These
rates suggest that, on average, it may take less than 6 months for the petroleum
stocks to adjust to equilibrium levels.
The weather shock variables show that refiners reduce stocks of distillate and
residual fuels and modestly increase demand for crude oil with colder weather, but
they rapidly increase stocks of unfinished oil with hotter weather, signaling the

542

T.J. Considine, E. Heo r Energy Economics 22 (2000) 527]547

Table 4
Scaled derivatives for short-run benefits of holding inventory Žasymptotic t-statistics .
Explanatory
variables

Input prices
Crude oil
Propane
Production le¨ els
Gasoline
Distillate
Residual
Jet fuel
Other
In¨ entory le¨ els
Gasoline
Distillate
Residual
Jet fuel
Other
Capacity
Time

Short-run benefits of holding inventory C x
Gasoline

Distillate
fuel

Residual
fuel

Jet
fuel

y0.45
Žy2.29.
y0.71
Žy1.78.

0.25
Ž1.59.
0.63
Ž2.00.

y0.10
Žy2.87.
y0.06
Žy0.83.

y0.09
Žy1.02.
y0.06
Žy0.35.

y0.82
Žy4.12.
y0.10
Žy1.31.
y0.01
Žy0.57.
y0.08
Žy2.60.
0.50
Ž0.62.

y0.43
Žy2.56.
y0.19
Žy2.77.
y0.02
Žy1.01.
y0.07
Žy2.62.
1.12
Ž1.70.

y0.28
Žy1.17.
0.28
Ž2.37.
y0.01
Žy0.24.
y0.01
Žy0.41.
0.23
Ž1.25.

y0.47
Žy4.18.
y0.01
Žy0.27.
0.00
Žy0.21.
y0.01
Žy0.55.
0.62
Ž1.99.

1.62
Ž1.76.
0.57
Ž2.06.
y0.03
Žy0.89.
0.19
Ž2.46.
y0.74
Žy1.10.
y0.08
Ž0.06.
0.15
Ž0.70.

0.88
Ž2.06.
0.86
Ž2.98.
0.03
Ž1.24.
0.19
Ž4.29.
0.91
Ž1.56.
y4.31
Žy4.81.
0.14
Ž0.84.

y0.14
Žy0.89.
0.09
Ž1.24.
0.16
Ž1.44.
0.03
Ž0.61.
y0.29
Žy2.05.
0.23
Žy0.78.
y0.15
Žy3.37.

0.77
Ž2.46.
0.51
Ž4.29.
0.03
Ž0.61.
0.14
Ž3.36.
y0.04
Žy0.16.
y1.41
Žy2.50.
y0.03
Žy0.37.

preparation for higher gasoline demand as summer approaches. Inventories and
variable input demands all rise with higher production.
In the price module, spot prices increase with higher crude oil prices and
own-production levels Žsee Table 3.. All four short-run price equations conform to
the conventional notion of an upward sloping supply curve. In addition, several of
the cross-elasticities of marginal cost with respect to production are significant,
indicating joint production interactions. Spot prices also decline with higher ‘own’
inventory levels } significantly for gasoline and distillate fuels.
The convenience yield elasticities appear in Table 4. Since Cx , the partial
derivative of cost with respect to inventories, or the benefits of holding inventories
can be negative, we define these elasticities by multiplying the derivatives by the

T.J. Considine, E. Heo r Energy Economics 22 (2000) 527]547

543

level of the variable in the partial equilibrium differentiation. The scaled derivatives of the estimated Cx function with respect to own inventory levels are positive
and significant, consistent with the theory of storage. Negative scaled derivatives
with respect to own-production levels indicate that the Cx functions shift up Ždown.
with lower Žhigher. production levels. This finding suggests that, ceteris paribus,
marginal convenience yields increase when production declines.
Ratios of Rotterdam and NYMEX spot prices play a limited role in net imports.
Own-product shipment levels have positive effects on most net imports, indicating
that refiners import more as product demands increase. Capacity and weather
variables have very little effect on petroleum product net imports. The import price
equations show significant positive correlation between Rotterdam and NYMEX
spot prices.

5. Simulation analysis
The simulation model consists of 41 equations, which we solve simultaneously. In
addition to the 27 behavioral equations, we include five refinery product production identities based on the material balances for out-of-sample forecasting. The
simulation model also includes an identity for gross product worth, which is a
weighted average of product prices, and seven identities that define user costs for
quasi-fixed factors. Finally, the model includes an equation for net refinery
processing gains by simply taking the difference between total refining inputs and
outputs.
We first performed a static simulation of the model Žsee Table 5.. Inventories
and production forecasts show very low root mean squared errors ŽRMSEs., less
than 5% on average. RMSEs of the product spot and futures prices are relatively
higher than those of inventory equations averaging slightly higher than 10%. The
model shows very good fits for refinery production and inventories; all with lower
than 10% RMSEs, except for those of distillate fuel. Finally, the model does not
display any explosive tendencies during all simulations even with different sets of
prices5, further illustrating the model’s stability. These goodness-of-fit simulations
imply that the model would perform well in evaluating future policy implications.
The next test of the model involves measuring the impacts associated with
demand and supply shocks. Our demand shock involves a colder than average
winter over a 3-month period represented by a 10% increase in heating degree-days.
The model assumes final product sales are exogenous. To estimate the impacts of
weather on product sales, we estimate five simple product demand equations that

5

Month-average prices, such as average spot prices and resale prices, when compared to the results
using the month-end prices, give better fits in the in-sample simulation and worse fits in the out-of
sample simulations for the equations of marginal costs and arbitrage relations. For other equations,
both prices give similar results in both simulations.

544

T.J. Considine, E. Heo r Energy Economics 22 (2000) 527]547

Table 5
Root mean squared errors for in-sample static and out-of-sample dynamic simulation solutions
In-sample

Out-of-sample

In¨ entories
Crude oil
Unfinished oil
Gasoline
Distillate fuel
Residual fuel
Jet fuel
Other
Refinery capacity

0.02
0.03
0.05
0.01
0.05
0.04
0.04
0.01

0.09
0.10
0.09
0.21
0.09
0.06
0.09
0.01

Input demands
Crude oil
Other

0.04
0.15

0.04
0.25

Production
Gasoline
Distillate fuel
Residual fuel
Jet fuel
Other

0.04
0.01
0.08
0.05
0.06

0.03
0.14
0.08
0.04
0.06

Spot prices
Gasoline
Distillate fuel
Residual fuel
Jet fuel

0.13
0.13
0.09
0.10

0.13
0.20
0.11
0.13

Futures prices
Gasoline
Distillate fuel
Residual fuel
Jet fuel

0.13
0.13
0.09
0.09

0.09
0.18
0.11
0.10

Import prices
Gasoline
Distillate fuel
Residual fuel
Jet fuel

0.11
0.10
0.12
0.08

0.29
0.14
0.09
0.09

are functions of relative prices, output or income, and heating and cooling
degree-days. Considine Ž1998. reports a more complete analysis of short-run energy
demand.
The only two refined petroleum product sales responsive to weather are distillate
and residual fuel oil, which increase 2.7 and 4.5%, respectively, in response to the
10% higher heating degree days Žsee Table 6.. Heating oil, which comprises

T.J. Considine, E. Heo r Energy Economics 22 (2000) 527]547

545

Table 6
Three-month impacts from a winter sales shock
Thousand barrels per day
Gasoline
Demand
Supplies from:
Production
Net imports
Inventory change

Distillate

Residual

Jet fuel

Other

y7.75

93.40

43.52

y2.99

y4.80

y26.24
9.18
9.30

47.98
18.76
26.67

y0.72
27.01
17.23

y31.06
10.69
17.38

y51.87
y8.13
55.19

1.67
1.33
1.16
2.71

y0.42
y0.42
y0.31
4.53

0.92
0.79
0.68
y0.19

NA
NA
NA
y0.12

Percent changes
Prices
Spot
Futures
Import
Demand

1.62
1.58
1.27
y0.10

roughly 50% of distillate sales, is sensitive to home heating requirements in the
northeastern United States. The single largest market for residual fuel oil is
electric power generation, which is sensitive to heating demands and air conditioning loads. The other three products decline negligibly. Hence, we focus our
discussion on distillate and residual fuel oil markets.
The main purpose of this simulation is to determine how these sales shocks are
supplied. Do refiners supply these unanticipated demands from current production,
from inventory supplies, or from imports of refined products? Of the 93 thousand
barrels per day Žtbd. increase in distillate sales, more than half is from higher
production Žsee Table 6.. Additional supplies come from a combination of higher
net imports Ž19 tbd. and an inventory drawdown of nearly 27 tbd. These results
suggest considerable flexibility in current production to meet sales shocks. Nevertheless, supplies flowing from inventory and net imports are important. Petroleum
refiners seem to be employing a flexible strategy for dealing with distillate fuel oil
sales shocks drawing from each of three major sources of supply.
Another interesting finding is that the cold weather shock induces a price
backwardation, defined when spot prices rise above prices for future delivery.
Notice in Table 6 that the percentage increase in spot prices for distillate is greater
than the percentage increase in distillate futures prices. This finding is important
because the study by Williams and Wright Ž1991. argues that commodity price
backwardations arise from stockouts when producer inventories disappear. Our
simulations illustrate that price backwardations arise because producer returns to
stock holding rise with sales.
The simulation results for residual fuel also tell an interesting story. Nearly
all-additional supplies come from increases in net imports and inventory depletion
Žsee Table 6.. In fact, imports rise more than the drawdown from inventories,

546

T.J. Considine, E. Heo r Energy Economics 22 (2000) 527]547

Table 7
Three-month impacts from a crude oil price shock
Thousand barrels per day

Supplies from:
Production
Net imports
Inventory change

Gasoline

Distillate

Residual

Jet fuel

Other

7.29
9.92
y17.21

y10.70
6.55
4.14

1.92
0.60
y2.52

y0.78
2.18
y1.39

2.89
3.58
y0.76

5.35
5.59
3.66

8.24
8.27
6.78

5.92
5.96
4.45

Percent changes
Prices
Spot
Futures
Import

4.87
4.72
3.90

NA
NA
NA

suggesting that residual imports may serve as a buffer to meet sales shocks. US
residual fuel oil markets are unique because several large refineries in the
Caribbean region can supply residual fuel on relatively short notice given their
proximity to US markets.
We finally simulate the market impacts of a 10% increase in crude oil prices
during the winter. As expected, product prices increase. By construction, product
sales are unchanged because we do not include the sales functions in the model
simulation. Net imports for all products increase Žsee Table 7. because we
implicitly assume foreign producers do not face the same increase in crude oil
prices. Refinery inputs and stocks of crude oil Žnot shown in Table 7. decline.
Consequently, producers draw down raw material inventories when input prices
rise, consistent with cost smoothing.
With the exception of distillate fuel oil, supplies from product inventories
decline Žsee Table 7.. There are two opposing forces affecting product inventory
levels } production levels and user costs. Our econometric estimates indicate that
stocks rise with higher production for all products and fall with increased user costs
for all products except residual fuel oil. User costs, which are endogenous in the
model, increase for all inventories except distillate in this simulation. Distillate
stocks decline because imports increase, which lowers required production and
inventory levels. These results suggest that producers may employ different strategies to meet sales and cost shocks, depending upon access to imports, transport
costs, and other logistical factors.

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