Equilibrium commodity prices with irreversible investment and non-linear technologies

Abstract

We model oil price dynamics in a general equilibrium production economy with two goods: a consumption good and oil. Production of the consumption good requires drawing from oil reserves at a fixed rate. Investment necessary to replenish oil reserves is costly and irreversible. We solve for the optimal consumption, production and oil reserves policy for a representative agent. We analyze the equilibrium price of oil, as well as the term structure of oil futures prices. Because investment in oil reserves is irreversible and costly, the optimal investment in new oil reserves is periodic and lumpy. Investment occurs when the crude oil is relatively scarce in the economy. This generates an equilibrium oil price process that has distinct behavior across two regions (characterized by the abundance/scarcity of oil). We undertake three empirical tests suggested by our model. First, we estimate key parameters using SMM to match moments of oil price futures as well as other macroeconomic properties of the data. Second, we estimate an affine regime switching model of the oil price, which captures the main features of our equilibrium model and preserves the tractability of reduced-form models. Lastly, we compare the risk premium in short-maturity oil futures implied by our model to the data.

Introduction

Oil prices, and energy costs in general, play an important role in the economy. Hamilton, 2003, Hamilton documents that nine out of ten of the U.S. recessions since World War II were preceded by a spike up in oil prices. The “oil-shocks” of the 1970’s are the most dramatic example. More interesting, Hamilton (2003) documents that the connection between oil and economic activity is not simple. Oil expenditures account for roughly 4% of US GDP. However, the 7.8% reduction in world oil production in 1973 is associated with a large 3.2% drop in real US GDP. In contrast, the 8.8% drop in production that occurred at the time of the 1991 Persian Gulf War lead only a 0.1% drop on GDP. Many papers have studied the role of oil in real-business-cycle (RBC) models, which turns out to be a difficult task.² In this paper we focus on a related question. We investigate the properties of the price of oil (and oil futures) in a general equilibrium model where oil is an important input to the production of the consumption good, and where oil production is characterized by significant adjustment costs and irreversibility.

Oil prices display a number of interesting characteristics. First, there is tremendous variation in the spot price of oil. Since 1947 the nominal spot price has fluctuated between $10 and $140 per barrel. This large volatility is not just present in the spot price, but also in futures prices. Although the volatility is smaller for longer-horizon delivery contracts (the “Samuelson Effect”), even three-year contracts exhibit substantial volatility. More generally, there is substantial variation in the slope of the futures curve. Futures curves tend to be decreasing (i.e., in“backwardation”) most of the time (63% in our sample). But at other times futures curves are increasing (i.e., in “contango”) or display a single hump. This fact explains why many reduced-form commodity derivative models require at least two factors to fit the futures curve. Typically, these models consider a second factor, the so-called “convenience yield”, to capture the variation in the slope of the futures curve. This also implies that the spot price is not a sufficient statistic to characterize the shape of the futures price curve. While oil and futures prices display a number of salient stylized facts, the evidence about whether oil-price risk commands a risk premium is unclear. Gorton and Rouwenhorst (2006) document that there is no risk-premium to holding commodity risk while Erb and Harvey (2006) come to the opposite conclusion. Specifically, the evidence is inconclusive as to whether the futures price of oil is, on average, equal to the future spot price. Understanding the theoretical connection between futures prices and the price of risk will shed some light on this ambiguous evidence.

To understand the properties of the oil price we consider a two-good economy within a general equilibrium model. Such a model is important to connect oil to the fundamentals and to generate an endogenous stochastic discount factor useful to track down the convenience yield and the risk premium in oil spot and futures prices. The consumption good is used for consumption and investment. Production of this consumption good is uncertain and requires as inputs both consumption good and a second good, oil (which could more generally be seen as energy). Oil used in production flows at a fixed rate from existing oil reserves which are also uncertain. New oil can be added to reserves when needed, but this requires an irreversible investment of the consumption good with a fixed-cost component. We first solve for the optimal consumption, investment, and oil reserves policy of a representative agent, and then derive the equilibrium price of oil as well as the term structure of oil futures prices.

Optimal consumption and investment are characterized by the ratio of the stock of oil to the stock of the consumption good (capital), which is the key state variable in the economy. The irreversibility of investment creates an inaction region that makes investment periodic.³ In this region, investment is postponed until crude oil is scarce enough in the economy. The fixed-cost component in the investment makes the addition of new oil reserves lumpy.⁴ Finally, the fixed flow rate of oil in production is crucial to generate a substantial degree of backwardation. These three key assumptions in the oil sector – irreversible investment, a fixed-cost component, and a fixed flow rate of oil — generate significant variation in the state variable that carries over to the equilibrium oil and futures price dynamics.

We define the price of oil as the marginal value of an additional unit of oil reserves. Given this characterization, the central feature of our model is that the lumpy, periodic investment in new oil implies that the spot price of oil is a humped-shaped function of the state variable. Indeed, the equilibrium oil price reaches a maximum price above the marginal cost of adding new oil stocks. This non-monotonic response of the oil price is due exclusively to the fixed-cost component of the investment in new oil stocks. When oil is plentiful (oil/capital ratio is high), investment in new oil reserves is a long way off. As oil is used in production the ratio of oil-to-capital decreases and the price of oil increases. In contrast, when oil is scarce (oil-to-capital ratio is low), we are near the investment boundary. In this region, as oil is used and the oil-to-capital ratio declines, the price of oil falls. Despite the fact the current quantity of oil is smaller, the marginal value of a unit of oil is smaller since the near-term investment will increase the stock of oil. The result is that the dynamic properties of the oil price (and hence futures prices) are notably different in the abundance and scarcity regions. In effect, the price of oil is governed by a process with two regimes.

Our model is similar to partial equilibrium commodity models that focus on the role of commodity storage.⁵ These models typically assume an inverse (net) demand function that maps the quantity of the commodity and a shock into a price. The fact that inventories must be non-negative produces an asymmetry in the spot price behavior at zero inventory (stock-out). This generates interesting dynamic properties for the commodity spot price and the related futures prices, in particular, backwardation arises from a fear of stock-out in the near future. Storage models are similar to our general equilibrium model in that they produce a two-regime process for the spot price. Our general equilibrium model adds to the storage approach along a few dimensions. First the zero-inventory or stock-out event that is so central to inventory models is infrequently observed in oil markets. For example, the oil-futures curve is downward sloping 63% of the time. It is hard to see zero (or even near-zero) inventory levels anywhere near this frequency. Of course, this fact by itself may not be troubling as inventories are difficult to measure and the storage technology in these models does not capture the complexities of inventory. More troubling, however, is the difficulty inventory models have explaining longer-horizon dynamics. In order to explain frequent backwardation, models must be calibrated so inventory levels hit zero frequently. As a result, the inventory state variable has little impact over price levels beyond the short horizon of inventory. Routledge et al. (2000), for example, normalize oil-futures prices so that the twelve-month futures price has zero variance. Lastly, due to computational limitations, this literature typically assume risk-neutrality and hence do not have implications for commodity risk premia.

Our paper is also related to other equilibrium models that study the behavior of commodity prices. Carlson et al. (2007) propose an equilibrium model of natural resources and study the effect of adjustment costs in the forward price dynamics. However, in contrast to our paper, they assume risk-neutrality, an exogenous demand function for commodity, and (the main friction in their model) that commodity is exhaustible, whereas in our paper commodity is essentially present in the ground in infinite supply but is costly to extract. Also, Kogan et al. (2009) identify a new pattern of futures volatility term structure that is inconsistent with standard storage models. This new pattern can be explained within their industry equilibrium model that exhibits investment constraints and irreversibility. Similar to our scarcity and abundance regions, they have investment and no-investment regions with different spot price dynamics. Unlike our model, they take the demand side and risk premia as exogenous and focus mainly on the implications for the volatility curve. Casassus et al. (2013) extend a simplified version of the model proposed here to a multi-commodity framework tailored to study the long-term correlation between futures returns. Recently, Ready (2016) and Hitzemann (2016) extend the production-based asset pricing framework with long-run productivity risk of Croce (2014) to study the relation between oil consumption, oil prices and macroeconomic fluctuations. As opposed to our model, Ready (2016) builds a partial equilibrium model with exogenous oil supply, while Hitzemann (2016) assumes that oil enters directly in the utility function of the representative household and not as an input for the production technology, like in our model.⁶

To illustrate the quantitative features and advantages of our general equilibrium model, we undertake three (related) empirical exercises. First we estimate our model using Simulated Method of Moments (SMM) estimation proposed by Lee and Ingram (1991) and Duffie and Singleton (1993). We estimate the parameters of our model to match the sample average term-structure of the oil futures price and the sample volatility of the term-structure along with some aggregate macroeconomic moments. With (arguably) sensible parameters, we match the term-structure of futures prices and volatilities reasonably well. The model generates backwardation (negative slope in the futures curve) on average. The model also captures the level and shape of the term-structure of volatilities pretty well. In theory, the dynamic properties of the oil price are different in the scarcity and abundance regions. This is only relevant empirically if the (endogenous) state variable has a distribution that accesses these regions frequently enough to matter. Under our estimated model parameters, this is indeed the case. The economy is in scarcity of oil 14% of the time and in abundance 86% of the time.

One of the disadvantages of a general equilibrium model is that it is challenging to compute. In practice, commodity pricing is often implemented using an exogenously specified process of the spot price and a convenience yield (a second factor that captures the dynamics of the futures curve slope). In this setting, pricing futures contracts and other derivatives is straightforward (e.g., Gibson, Schwartz, 1990, Schwartz, 1997 and Casassus and Collin-Dufresne, 2005). This approach is very effective since it is tractable enough to price complex commodity-contingent claims, but often suffers from the lack of theoretical justification for the underlying dynamics chosen for the state variables.

Our second empirical exercise is motivated by trying to link both approaches. In the model, the equilibrium process for the oil price differs across the scarcity and abundance regions. However, it turns out that within those regions the spot price behavior is simple to characterize. In particular, within a regime, a linear approximation to the drift and volatility functions is a reasonable approximation of our equilibrium price. We use quasi-maximum likelihood technique of Hamilton (1989) to estimate a two-regime model with crude oil price data. The resulting parameter estimates on the drift and volatility are in line with the calibrated model. Moreover, since the stochastic process is, up to the two-regimes, affine, we expect the reduced-form approximation of our model to be useful for derivative pricing. In addition, we estimate the smoothed inference about whether the state of the economy is in the scarcity or the abundance regions. This is a helpful diagnostic on the model since we can confirm that the data are consistent with a two-regime process that is economically meaningful (both regimes are visited frequently).

One of the advantages of a model in general equilibrium is that we can investigate the risk premium properties of the oil commodity. Interestingly, in our model the risk premium on oil is state-dependent. In the abundance region, a long position in oil (e.g., an oil futures) is risky and commands a risk premium. In the scarcity region, a long position in oil is a hedge and the risk premium is negative. Our third empirical exercise is to investigate the conditional risk premium in oil-futures data. In regressing oil price return on the market return we find that the beta is significantly negative in the estimated scarcity regime and positive (though not statistically significant) in the other regime. The difference across regimes is consistent with our model. More importantly, variation in the commodity risk premium makes unconditional tests of the risk premium more difficult to interpret (e.g., Gorton and Rouwenhorst, 2006 and Erb and Harvey, 2006).

Our paper proceeds with the model developed next in Section 2. The equilibrium properties are characterized in Section 3 (with most of the proofs contained in the appendix). Section 4 contains the empirical implementations of our model where we estimate with SMM, characterize a two-regime affine model, and explore the properties of the commodity risk premium. Finally, Section 5 concludes.