A General Equilibrium Model of
Commodity Production and Financial Investment
Patrick Grüning
Christian Schlag
ZEF Bonn
February 1, 2013
Introduction
Model Setup and Equilibrium
Results
Conclusion
Motivation
Start of economic research on commodities: Keynes (1930)
Recently: commodities and their markets at the center of the
public debate
strong increase in overall size of positions held by financial
investors
simultaneous increase in commodity spot and futures prices
Research question: increase explained ...
... by fundamental economic conditions?
... by the presence of this new class of investors?
Grüning, Schlag
A General Equilibrium Model of Commodity Production and Financial Investment
1
Introduction
Model Setup and Equilibrium
Results
Conclusion
Motivation
Political debate views financial speculation as troublesome,
since it increases prices of basic resources (e.g., food) for
’mere profit’
Empirical (academic) evidence rather mixed
Practical obstacles:
short history (phenomenon can only be observed since 2004)
general data availability and quality (e.g., often hard to
distinguish financial speculators from other types of investors
and to obtain exact position sizes (Sanders, 2010))
Theory: some papers provide a rationale explaining rising
commodity prices fitting recent data well with (Liu, Qiu, Tang
(2011)) and without financial speculation (Casassus,
Collin-Dufrense, Routledge (2009))
Overall: financial speculation might or might not be a very
important source of rising commodity prices
Grüning, Schlag
A General Equilibrium Model of Commodity Production and Financial Investment
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Introduction
Model Setup and Equilibrium
Results
Conclusion
Our Model
Implication: need for a general equilibrium model which
incorporates commodity markets into the general system of
financial markets (equity, fixed income, derivatives, ...)
Elements of our paper:
integrate financial speculator in discrete-time GE production
economy
producer consumes and utilizes commodity in the production
of the final consumption good
financial speculator consumes his exogenous consumption
stream and can trade in bonds and futures with the producer
Heterogenous agents with Epstein-Zin preferences
producer: per period utility via CES aggregation over the
non-commodity and the commodity
speculator: CRRA over non-commodity good only
Grüning, Schlag
A General Equilibrium Model of Commodity Production and Financial Investment
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Introduction
Model Setup and Equilibrium
Results
Conclusion
Our Model
Model should explain ...
... under which conditions a financial speculator drives
commodity spot and/or futures prices
compare model with homogenous producers to one with producer and
financial investor
... under which conditions these price changes are
predominantly driven by fundamentals
compare impulse-response function for consumption shocks of the
different agents
Technically: solution by perturbation methods
Particular approach described in Mertens (2012)
Grüning, Schlag
A General Equilibrium Model of Commodity Production and Financial Investment
4
Introduction
Model Setup and Equilibrium
Results
Conclusion
Our Model: Results
Model output: qualitative explanation of ...
... general asset pricing moments:
risk-free rate
equity premium
equity premium volatility
... commodity-related quantities:
term structure of futures prices
term structure of futures price volatility (Samuelson effect)
futures and spot prices mean-reverting after consumption
shocks
Grüning, Schlag
A General Equilibrium Model of Commodity Production and Financial Investment
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Introduction
Model Setup and Equilibrium
Results
Conclusion
Related Literature
Johnson (2011)
similar preferences and technology; assumes representative agent in
classical ’two trees’ setting
Liu, Qiu, and Tang (2011)
multiple agents; partial equilibrium; exogenous convenience yield process
and demand
Casassus, Collin-Dufrense, Routledge (2009)
representative agent; Cobb-Douglas production function with capital and
commodity (oil)
Pirrong (2008)
stochastic volatility in economy-wide productivity; representative investor
Grüning, Schlag
A General Equilibrium Model of Commodity Production and Financial Investment
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Introduction
Model Setup and Equilibrium
Results
Conclusion
Model Setup
Agents:
producer (consumes commodity and uses it as input to
production process)
financial speculator (receives exogenous endowment stream
and trades bonds, futures with producer)
Goods:
commodity (supply Q)
capital good for production (supply K )
terminal good for producer’s consumption (result of
production)
consumption flow for speculator (supply Z )
Financial markets:
bonds
futures contracts on commodity with different maturities
Grüning, Schlag
A General Equilibrium Model of Commodity Production and Financial Investment
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Introduction
Model Setup and Equilibrium
Results
Conclusion
Model Setup
Producer’s preferences:
U1,t
1
 ψψ−1

1− 1
ψ1  1

h
i
1
1−γ
1
1−
1−γ1
= (1 − β1 ) [v (C1,t , Lt )] ψ1 + β1 Et U1,t+1


CES-type aggregation
terminal good consumption C1,t
commodity good consumption Lt
ρ
v (C1,t , Lt ) = φC1,t
+ (1 − φ)Lρt
ρ1
Characteristics:
risk aversion: γ1 = 12
subjective discount factor: β1 = 0.999
EIS: ψ1 = 0.065
φ = 0.3
ρ = −12.5
Grüning, Schlag
A General Equilibrium Model of Commodity Production and Financial Investment
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Introduction
Model Setup and Equilibrium
Results
Conclusion
Model Setup
CES production function for terminal good: F (N, K )
N: amount of commodity good, K : amount of second factor (like capital);
ν = 0.1
Capital:
Kt = e µ·t+kt
with
kt = ϕkt−1 + εk,t
Exogenous endowment for speculator:
Zt = e µ·t+zt
zt = µc (1 − ϕ) + ϕzt−1 + εz,t
with
Total commodity good supply:
Qt = e µ·t+qt
with qt = ϕqt−1 + εq,t
Market clearing for commodity good:
!
Qt = Lt + Nt
∀t
commodity good used for either consumption (L) or for production (N)
Grüning, Schlag
A General Equilibrium Model of Commodity Production and Financial Investment
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Introduction
Model Setup and Equilibrium
Results
Conclusion
Model Setup
Budget constraint:
C1,t + B1,t+1 = F (Kt , Nt ) + B1,t Rf ,t−1 + φ1,t (St − Ft−1 )
C1 : terminal good consumption
B1,t : amount held in bonds from t − 1 to t
φ1 : (one-period) futures contracts held from t − 1 to t
Rf : gross risk-free rate
S: commodity spot price
F : commodity futures price
New futures contract always has value zero!
Grüning, Schlag
A General Equilibrium Model of Commodity Production and Financial Investment
10
Introduction
Model Setup and Equilibrium
Results
Conclusion
Model Setup: Financial Speculator
Preferences:
U2,t
2

 ψψ−1
1
ψ2  2

h
i 1−
1− ψ1
1−γ2
1−γ2
= (1 − β2 )C2,t 2 + β2 Et U2,t+1


Speculator’s preferences only over the non-commodity
consumption good
Budget constraint:
C2,t + B2,t+1 = Zt + B2,t Rf ,t−1 + φ2,t (St − Ft−1 )
Characteristics:
risk aversion: γ2 = 8
subjective discount factor: β2 = 0.997
EIS: ψ2 = 0.1
Grüning, Schlag
A General Equilibrium Model of Commodity Production and Financial Investment
11
Introduction
Model Setup and Equilibrium
Results
Conclusion
Model Setup: Comparison of Agents
Financial investor is
less risk averse
has higher IES
is more impatient
’Typical’ characteristics for financial investor
Grüning, Schlag
A General Equilibrium Model of Commodity Production and Financial Investment
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Introduction
Model Setup and Equilibrium
Results
Conclusion
Equilibrium
Individual constrained optimization problems are solved
Objective: maximize lifetime utility
Choice variables:
consumption
allocation of the commodity good for production
bond and futures holdings
Grüning, Schlag
A General Equilibrium Model of Commodity Production and Financial Investment
13
Introduction
Model Setup and Equilibrium
Results
Conclusion
Asset Prices
Commodity spot price St : marginal product
St = FN (Kt , Nt )
Futures price Ft,t+1 ≡ Ft : expected spot price under Q
Ft = EtQ [St+1 ]
Long-maturity futures prices (term structure):
Ft,t+n = EtQ [St+n ]
Backwardation (contango):
Ft,t+n+1 − Ft,t+n
< (>)0
Ft,t+n
Convenience yield
δt = 1 −
Grüning, Schlag
Ft
St Rf ,t−1
A General Equilibrium Model of Commodity Production and Financial Investment
14
Introduction
Model Setup and Equilibrium
Results
Conclusion
Asset Prices
risk-free rate
futures return
spot return
convenience yield
equity market
First Moments
2 Agent Model 1 Agent
2.61%
0.95%
3.76%
4.36%
3.92%
Model
2.39%
1.39%
1.57%
2.44%
5.17%
Data
≈ 1.50%
≈ 4% (oil)
≈ 3.5% (oil)
≈ 3% (oil)
≈ 5%
Numbers are annualized. Equity premium and stock returns are levered, i.e.,
adjusted for the debt-equity ratio.
Grüning, Schlag
A General Equilibrium Model of Commodity Production and Financial Investment
15
Introduction
Model Setup and Equilibrium
Results
Conclusion
Asset Prices
risk-free rate
futures return
spot return
convenience yield
equity market
Volatilities
2 Agent Model 1 Agent Model
3.23%
3.47%
12.38%
16.20%
13.84%
18.36%
7.70%
7.14%
28.94%
31.15%
Data
≈ 1%
≈ 25% (oil)
≈ 20% (oil)
≈ 5% (oil)
≈ 20%
Numbers are annualized. Equity premium and stock returns are levered, i.e.,
adjusted for the debt-equity ratio.
Grüning, Schlag
A General Equilibrium Model of Commodity Production and Financial Investment
16
Introduction
Model Setup and Equilibrium
Results
Conclusion
Futures Term Structure
Futures prices:
monotonically decreasing with respect to maturity
(backwardation)
slope from 1 to 4 quarters in the model (−2%) slightly flatter
than in the data (−5% for oil)
Futures price volatilities
monotonically decreasing with respect to maturity (Samuelson
effect)
slope from 1 to 4 quarters:
−13% in the model, −33% in the data
Grüning, Schlag
A General Equilibrium Model of Commodity Production and Financial Investment
17
Introduction
Model Setup and Equilibrium
Results
Conclusion
Portfolio Holdings
Portfolio Holdings
E [B1 ]
0.73
−0.24
E [φ1 ]
σB1
147.53%
σ φ1
24.27%
AC1 (B1 )
0.98
0.93
AC1 (φ1 )
Producer is ...
... bonds long
... futures short
uses financial markets for insurance purposes
Grüning, Schlag
A General Equilibrium Model of Commodity Production and Financial Investment
18
Introduction
Model Setup and Equilibrium
Results
Conclusion
Impulse-Response Functions
Figure: Futures and Spot Price after a Commodity Shock in the Representative Agent Model
Grüning, Schlag
A General Equilibrium Model of Commodity Production and Financial Investment
19
Introduction
Model Setup and Equilibrium
Results
Conclusion
Impulse-Response Functions
Figure: Futures and Spot Price after a Commodity Shock for the Producer
in Multiple Agent Model
Grüning, Schlag
A General Equilibrium Model of Commodity Production and Financial Investment
20
Introduction
Model Setup and Equilibrium
Results
Conclusion
Conclusion
Model with two heterogenous agents:
producer
speculator
differ with respect to their endowment streams and their
preferences
Model can qualitatively explain
downward sloping term structure of futures prices
Samuelson effect
Grüning, Schlag
A General Equilibrium Model of Commodity Production and Financial Investment
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