NBS8337
Economics of Commodity Markets
Trends in Commodity Prices
Trends in Commodity Prices
• Classical Economists held the view that the trend in commodity prices in
relation to manufactured goods should be positive as the supply of primary
commodities would be constrained by the fixed amount of land while the
supply of manufactures would be augmented by technological progress.
• Prebisch (1950) and Singer (1950) claimed that commodity prices should
decline in relation to manufactures in the long run, labelled as the
Prebisch-Singer hypothesis (PSH).
• Deaton and Laroque (2003) set out a model showing prices of commodities
in developing countries contain no trend by linking them to the Lewis
model.
Trends in Commodity Prices
• Price of non-renewable resources would rise with the rate of interest
and that the production trajectory would be monotonically declining
until the resource is exhausted (Hotelling 1931).
• Empirical evidence is contradictory.
• Slade (1982) suggest a U-shaped time path for natural resource prices
when there is exogenous technical change and endogenous change in
the grade of metals mined.
Prebisch-Singer Hypothesis
• PSH claimed that the relative price od commodities in terms of
manufactured goods shows a downward trend.
• PSH rested their case on three stylised facts:
• Developing countries were highly specialised in the production and export of
primary commodities.
• Technical progress was concentrated mainly in industry.
• The relative price of commodities to manufactured goods had fallen steadily
since the late 19th century.
• Because of developing countries specialisation in primary
commodities, PSH claim that it is unlikely to benefit their economy.
Prebisch-Singer Hypothesis
• If technical progress in the manufacturing sector exceeds that of the
primary commodity sector then we would see the supply of
manufactures growing faster than the supply of commodities and this
would lead the relative supply of commodities to manufactures
shifting to the left.
• The relative price of primary commodities would increase
• This was the Classical view that technical progress in industrialised
countries translates into welfare gains for developing countries.
Prebisch-Singer Hypothesis
• World Market for Primary Commodities relative to Manufactures.
S’
[P(c)/P(m)]’
[P(c)/P(m)]
S
E’
E
D
Q(c)/Q(m)
Prebisch-Singer Hypothesis
• PSH pointed put that this mechanism did not work.
• Instead of commodity prices increasing, they had actually fallen
• Prebisch (1950) and Singer (1950) based their conclusion on a visual
inspection of the data : Inverse of the NBTT of UK from 1876 to 1947.
Prebisch-Singer Hypothesis
• Why commodity prices might experience declining trend
• By way of the diagram, the demand schedule could shift to the left
along with relative supply and the shift of demand would be greater
causing the new equilibrium to be at point E” (see diagram in next
slide) associated with a lower relative commodity price.
• Or alternatively, the supply schedule could shift to the right causing
the price to fall corresponding to equilibrium point E’’’ (see diagram
in next slide).
Prebisch-Singer Hypothesis
• World Market for Primary Commodities relative to Manufactures.
S’
[P(c)/P(m)]
[P(c) )/P(m)]”
[P(c )/P(m)]’’’
S
E’
[P(c)/P(m)]’
E
E”
E’’’
D
Q(c)/Q(m)
Demand Side Argument: Singer (1950)
• Demand for primary commodities have lower income elasticity.
• 1. Income growth tends to lower the relative demand for, and hence
relative price of primary commodities.
• 2. Also, technical progress in manufacturing can cause the demand
for primary commodities to grow slower than manufactures.
• Both 1 and 2 would cause the demand schedule to shift relatively
more thn that of the supply schedule.
Supply Side Argument: Prebisch (1950)
• Strong labour unions in industrialised countries cause wages in
manufacturing to ratchet upwards during booms and to be sticky
downwards during recessions.
• This causes a ratchet up of the cost of manufactures.
• Weak labour unions in developing countries fail to increase wages during
booms and cannot prevent falls during recessions.
• Thus cost of primary commodities rises by less than manufactures during
booms and falls by more during recessions creating a decline in the relative
cost of primary commodities.
• This causes rightward movement in the supply schedule.
Prebisch-Singer Hypothesis
• Policy Implications
• The PSH argues that developing countries should adopt
industrialisation.
• Diversification of commodities
• Commodity Stabilisation programmes
Deaton-Laroque Model
• Demand is given by:
𝑑𝑡 = 𝐴𝑦𝑡 − 𝐵𝑝𝑡 + 𝑘 + 𝜀𝑡𝑑
• Where 𝑑𝑡 denotes quantity demanded, 𝑦𝑡 denotes income, 𝑝𝑡
denotes price, 𝜀𝑡𝑑 is a stationary unobservable I(0) random variable.
• We assume 𝐴 > 0 so that demand exclusive of price movements is
increasing with world income.
Deaton-Laroque Model
• The supply process is a simple version of the Lewis Model
𝑠𝑡 = 𝑠𝑡−1 + 𝐷 𝑝𝑡 − 𝑝∗ + 𝜀𝑡𝑠
• Where 𝑠𝑡 denotes quantity supplied, 𝑝∗ denotes the marginal cost of
production on marginal land or marginal cost of extraction of mineral,
𝑝𝑡 denotes price, 𝜀𝑡𝑠 is a stationary unobservable I(0) random variable
which is the supply shock.
• We assume 𝐷 > 0 so that supply increases when price is above
marginal cost and vice versa.
Deaton-Laroque Model
• Assuming no inventories and equating supply and demand:
𝐴𝑦𝑡 − 𝐵𝑝𝑡 + 𝑘 + 𝜀𝑡𝑑 = 𝑠𝑡−1 + 𝐷 𝑝𝑡 − 𝑝∗ + 𝜀𝑡𝑠
𝐵 + 𝐷 𝑝𝑡 = 𝐴𝑦𝑡 + 𝑘 − 𝑠𝑡−1 + 𝐷𝑝∗ + 𝜀𝑡𝑑 − 𝜀𝑡𝑠
𝑝𝑡 = 𝐵 + 𝐷
−1
𝐴𝑦𝑡 + 𝑘 − 𝑠𝑡−1 + 𝐷𝑝∗ + 𝜀𝑡𝑑 − 𝜀𝑡𝑠
(A.1)
• Substituting this result in the supply process:
𝑠𝑡 = 𝑠𝑡−1 + 𝐷𝑝𝑡 − 𝐷𝑝∗ + 𝜀𝑡𝑠
𝑠𝑡 = 𝑠𝑡−1 + 𝐷 𝐵 + 𝐷
−1
𝐴𝑦𝑡 + 𝑘 − 𝑠𝑡−1 + 𝐷𝑝∗ + 𝜀𝑡𝑑 − 𝜀𝑡𝑠 − 𝐷𝑝∗ + 𝜀𝑡𝑠
Deaton-Laroque Model
• After some rearranging:
𝑠𝑡 = 𝐵 + 𝐷 −1 𝐵𝑠𝑡−1 + 𝐴𝐷𝑦𝑡 + 𝐷𝑘 − 𝐷𝐵𝑝∗ + 𝐷𝜀𝑡𝑑 + 𝐵𝜀𝑡𝑠
Δ𝑠𝑡 = 𝐵 + 𝐷
−1
𝐷(𝐴𝑦𝑡−1 − 𝑠𝑡−1 ) + 𝐴𝐷Δ𝑦𝑡 + 𝐷𝑘 − 𝐷𝐵𝑝∗ + 𝐷𝜀𝑡𝑑 + 𝐵𝜀𝑡𝑠
Deaton-Laroque Model
• From the demand and supply processes we can write:
Δ𝑑𝑡 = 𝐴Δ𝑦𝑡 − 𝐵Δ𝑝𝑡 + Δ𝜀𝑡𝑑
Δ𝑠𝑡 = 𝐷 𝑝𝑡 − 𝑝∗ + 𝜀𝑡𝑠
• Equating the two:
𝐴Δ𝑦𝑡 − 𝐵Δ𝑝𝑡 + Δ𝜀𝑡𝑑 = 𝐷 𝑝𝑡 − 𝑝∗ + 𝜀𝑡𝑠
• or, 𝐴Δ𝑦𝑡 + Δ𝜀𝑡𝑑 − 𝜀𝑡𝑠 = 𝐷𝑝𝑡 − 𝐷𝑝∗ + 𝐵𝑝𝑡 − 𝐵𝑝𝑡−1
• Adding and subtracting 𝐵𝑝∗ on the LHS of the eqn.
𝐴Δ𝑦𝑡 + Δ𝜀𝑡𝑑 − 𝜀𝑡𝑠 = 𝐷𝑝𝑡 − 𝐷𝑝∗ + 𝐵𝑝𝑡 − 𝐵𝑝𝑡−1 + 𝐵𝑝∗ − 𝐵𝑝∗
Deaton-Laroque Model
• Rearranging:
𝐴Δ𝑦𝑡 + Δ𝜀𝑡𝑑 − 𝜀𝑡𝑠 = (𝐵 + 𝐷)𝑝𝑡 − 𝐵𝑝𝑡−1 + 𝐵𝑝∗ − (𝐵 + 𝐷)𝑝∗
𝐵𝑝𝑡−1 − 𝐵𝑝∗ + 𝐴Δ𝑦𝑡 + Δ𝜀𝑡𝑑 − 𝜀𝑡𝑠 = (𝐵 + 𝐷)𝑝𝑡 − (𝐵 + 𝐷)𝑝∗
𝐵 + 𝐷 (𝑝𝑡 −𝑝∗ ) = 𝐵(𝑝𝑡−1 −𝑝∗ ) + 𝐴Δ𝑦𝑡 + Δ𝜀𝑡𝑑 − 𝜀𝑡𝑠
• The reduced form price process:
(𝑝𝑡 −𝑝∗ ) = 𝐵 + 𝐷
−1
𝐵(𝑝𝑡−1 −𝑝∗ ) + 𝐴Δ𝑦𝑡 + Δ𝜀𝑡𝑑 − 𝜀𝑡𝑠
(1)
Deaton-Laroque Model
• The result (equation 1) shows that in the short run, price will respond
to fluctuations in demand and supply. If prices deviate from their
steady-state inter-temporal equilibrium, given by 𝐵 + 𝐷 −1 𝐴Δ𝑦𝑡
where Δ𝑦𝑡 is the mean growth rate of income, then prices will show
signs of adjustment.
• In other words, (empirically) if we were to conduct unit root tests on
the price series, we would expect to obtain a stationary process with
linear adjustment.
Price Dynamics of Natural Resources: Slade
• The relative price considered: ratio of extractive-industry price index
to an overall price index.
• 𝑄(𝑡):
• 𝑔(𝑡):
the output of metal in the extractive industry at time 𝑡.
the grade of ore mined at time 𝑡. [grade is ordered by
increasing extractions costs].
• 𝐵(𝑄):
the willingness to pay for 𝑄.
• 𝐶 𝑄, 𝑔, 𝑡 : the total extraction cost.
• 𝑓(𝑔):
the density of the metal for grade 𝑔.
• 𝜌:
the social discount rate.
Price Dynamics of Natural Resources: Slade
• Objective: to choose a time path for extraction rates that will
maximise the discounted stream of current and future benefits minus
costs.
• The maximisation problem is:
𝑚𝑎𝑥
• 𝑔
∞ −𝜌𝑡
𝑒
0
𝐵 𝑄 − 𝐶(𝑄, 𝑔, 𝑡) 𝑑𝑡
s.t.
𝑄 𝑡 = 𝑔 𝑡 𝑓(𝑔 𝑡 )
Price Dynamics of Natural Resources: Slade
• The optimal control problem can be solved by introducing the costate variable 𝜆(𝑡)
• The Hamiltonian is set up as:
• 𝐻 = 𝑒 −𝜌𝑡 𝐵 𝑄 − 𝐶(𝑄, 𝑔, 𝑡) − 𝜆𝑔
• 𝐻 = 𝑒 −𝜌𝑡 𝐵 𝑔𝑓 − 𝐶(𝑔𝑓, 𝑔, 𝑡) − 𝜆𝑔
• To choose the control variable:
• 𝐻𝑔 = 𝑒 −𝜌𝑡 𝐵′ 𝑓 − 𝐶 ′ 𝑓 − 𝜆 = 0
• Or,
𝐵′ 𝑓
=
𝜆
𝑒
′𝑓
+
𝐶
−𝜌𝑡
[imposing the constraint]
Price Dynamics of Natural Resources: Slade
• Or, 𝐵′ =
• Or, 𝑃 𝑄
𝜆𝑒 𝜌𝑡
+ 𝐶′
𝑓
𝜆𝑒 𝜌𝑡
=
+
𝑓
𝐶′
(equation B.1)
• where 𝑃 𝑄 is the inverse demand function.
• To obtain the path of the state variable:
•
𝜕𝜆
𝜕𝑡
= 𝐻𝑔 = 𝑒 −𝜌𝑡 𝐵′ 𝑔𝑓 ′ − 𝐶 ′ 𝑔𝑓 ′ − 𝐶𝑔
(equation B.2)
Price Dynamics of Natural Resources: Slade
• Differentiating B.1 w.r.t. time 𝑡:
′
• 𝑃=𝐶 +
𝑓 𝜆𝑒 𝜌𝑡 𝜌+𝑒 𝜌𝑡 𝜆 −𝑓′𝑔𝜆𝑒 𝜌𝑡
𝑓2
• Rearranging the expression:
′
•𝑃=𝐶 +
𝜆𝑒 𝜌𝑡 𝜌+𝑒 𝜌𝑡 𝜆
𝑓
−
𝑓′𝑔𝜆𝑒 𝜌𝑡
𝑓2
equation B.3
Price Dynamics of Natural Resources: Slade
𝜕𝜆
(note:
𝜕𝑡
• Substituting B.2 in B.3 we get:
•𝑃=
𝐶′
′
+
•𝑃=𝐶 +
𝜆𝑒 𝜌𝑡 𝜌+𝐵′ 𝑔𝑓′ −𝐶 ′ 𝑔𝑓′ −𝐶𝑔
𝑓
𝜆𝑒 𝜌𝑡 𝜌+𝑃𝑔𝑓′ −𝐶 ′ 𝑔𝑓′ −𝐶𝑔
𝑓
−
−
𝑓′𝑔𝜆𝑒 𝜌𝑡
𝑓2
𝑓′𝑔𝜆𝑒 𝜌𝑡
𝑓2
= 𝜆)
Price Dynamics of Natural Resources: Slade
• Now, rearranging the terms:
′
•𝑃=𝐶 +
𝑃𝑔𝑓′
𝑓
−
𝜆𝑒 𝜌𝑡 𝜌
+
𝑓
𝑓
𝐶𝑔
−
𝐶 ′ 𝑔𝑓′
𝑓
+
𝑓′𝑔𝜆𝑒 𝜌𝑡
𝑓2
• The expression in square brackets can be shown to be equal to
• Therefore; 𝑃 = 𝐶 ′ −
𝐶𝑔
𝑓
+
𝜆𝑒 𝜌𝑡 𝜌
𝑓
equation B.4
𝑃 𝑔𝑓′
𝑓
Price Dynamics of Natural Resources: Slade
• Slade makes the assumption about marginal cost (i.e. 𝐶 ′ ) that it is an
additive function such that:
• 𝐶 = ℎ 𝑔 + 𝑘(𝑡) 𝑄
• Therefore we have:
𝐶 ′ = ℎ 𝑔 + 𝑘(𝑡)
𝐶 ′ = ℎ′𝑔 + 𝑘
𝐶𝑔 = ℎ′ 𝑄
Substituting these expressions in B.4
𝑃 = ℎ′𝑔 + 𝑘 −
ℎ′ 𝑄
𝑓
+
𝜆𝑒 𝜌𝑡 𝜌
𝑓
Price Dynamics of Natural Resources: Slade
• Finally, some more rearranging:
• 𝑃 = ℎ′𝑔 + 𝑘 −
• 𝑃 = (ℎ′ 𝑔 −
ℎ′ 𝑄
𝑓
ℎ′ 𝑄
𝑓
)
𝜆𝑒 𝜌𝑡 𝜌
+
𝑓
𝜆𝑒 𝜌𝑡 𝜌
+𝑘+
𝑓
• or,
•
•
•
𝑄
′
𝑃 = ℎ (𝑔 − ) + 𝑘
𝑓
𝜆𝑒 𝜌𝑡 𝜌
𝑃=𝑘+
𝑓
𝑄
since 𝑔 −
= 0;
𝑓
+
𝜆𝑒 𝜌𝑡 𝜌
𝑓
equation B.5
constraint of optimisation
Price Dynamics of Natural Resources: Slade
• Define 𝜆 = 𝜆𝑒 𝜌𝑡 𝑓:
rental rate or the marginal value of the
resource in the ground.
• Going back to equation B.1:
𝑃 𝑄 =
𝜆𝑒 𝜌𝑡
𝑓
+ 𝐶′
• We can write: 𝑃 = 𝐶 ′ + 𝜆
• So that price equals marginal extract cost plus rent.
• Going back to equation B.5:
• We can write ∶ 𝑃 = 𝑘 + 𝜌𝜆
• So that the rate of change of marginal cost due to changes in
technology plus the discount rate times rent.
Reading
• 1. Cuddington, J.T., Ludema, R., Jayasuriya, S. 2002. The PrebischSinger Redux. Office of Economics Working Paper. U.S. International
Trade Commission. No. 2002-06-A.
• 2. Deaton, A., and Laroque, G., 2003. A model of commodity prices
after Sir Arthur Lewis. Journal of Development Economics, 71(2),
pp.289-310.
• 3. Slade, M. 1982. Trends in Natural Resource Commodity Prices: An
Analysis of the Time Domain. Journal of Environmental Economics and
Management. 9, pp. 122 – 137.