Hotelling Under Pressure Soren Anderson (Michigan State) Ryan Kellogg (Michigan) Stephen Salant (Maryland) October 2015
Hotelling has conceptually underpinned most of the resource extraction literature since 1931 Classic analogy is to a cake eating problem in which extractors can produce at any rate Leads to Hotelling Rule(s): Individual firm: Extract when price minus marginal cost of extraction is highest in present value Market equilibrium: Price minus marginal cost of extraction should rise at the rate of interest Empirical literature has focused on testing whether prices rise at rate of interest 2
Goal: Reformulate Hotelling in a way that is empirically compatible with oil extraction We focus on key observables: oil production, oil prices, drilling, and drilling costs. Empirical results using Texas data: 1. Oil production from existing wells does not respond to oil price variation Instead, there is a steady decline 2. Oil drilling responds strongly to price signals Rate of drilling and rental prices for drilling rigs 3
Goal: Reformulate Hotelling in a way that is empirically compatible with oil extraction We build a new Hotelling model: Flow capacity (reservoir pressure) from existing wells declines with remaining reserves (Darcy s Law) Effectively zero marginal production cost Firms invest in capacity by drilling new wells; drilling costs are convex Hotelling Rule for drilling rather than for production (a keg tapping problem) 4
The model leads to a series of predictions that align with industry stylized facts Local predictions: behavior of price-taking producers Flow constraint binds given observed price history Drilling activity and costs respond to oil prices & tech shocks Long-run production peaks (can be reversed with shocks) Global predictions: endogenous prices Flow constraint binds under weak sufficient conditions Long-run production peaks (can be reversed with shocks) Trajectories of prices, production, and drilling following shocks Positive demand shock: prices jump up and then decline 5
Crude oil spot prices and January futures curves Backwardation after positive shocks due to rapid growth among emerging Asian markets (Kilian 2009; Kilian and Hicks 2013) Iraq / Kuwait Contango after negative shock due to Asian financial crisis, OPEC (Kilian 2009) 9
Plot of spot price and rate of futures price increase over 12 months 10
Production from existing wells in Texas declines asymptotically toward zero ( 10% per year) 11
Drilling of new wells increases with oil prices 13
Drilling rig rental prices increase with oil prices 14
Empirical facts are rationalized by the following industry cost structure: 1. Max production is physically constrained and declines toward zero with volume of reserves ( exponential decay) Consistent with petroleum geology and engineering (Darcy s Law) 2. Marginal production cost is very low below the constraint Will explain (along with #1) the zero response to price shocks Not true for surface mining with high costs (e.g., Canada oil sands) 3. Fixed operating costs Explains shut-in of marginal wells during 1998-1999 4. Upward-sloping supply curve for drilling rig rentals Explains why drilling and rig dayrates increase with oil prices 15
We recast Hotelling as drilling problem: Drill wells that are flow-constrained due to falling pressure INTEREST RATE OIL FLOW RATE OF DRILLING UTILITY FROM OIL (increasing, concave) U = oil price DRILLING COST (increasing, convex) D = d = rig rental rate State variables: R(t): stock of undrilled wells K(t): production capacity Assume zero marginal extraction costs for consistency with TX data Assume zero fixed operating costs for tractability 18
Oil flow and resource constraints SHADOW PRICES OIL FLOW CAPACITY CONSTRAINT 19
Oil flow and resource constraints SHADOW PRICES OIL FLOW CAPACITY CONSTRAINT MAX FLOW FROM NEW WELL CAPACITY DECAY RATE EACH WELL HAS RESERVES OF X / λ K(t) = λ [TOTAL RESERVES IN DRILLED WELLS] LOWER BOUND ON θ(t) IS THE STREAM OF FUTURE PRICES, DISCOUNTED AT r + λ 20
Oil flow and resource constraints SHADOW PRICES OIL FLOW CAPACITY CONSTRAINT MAX FLOW FROM NEW WELL CAPACITY DECAY RATE RESOURCE CONSTRAINT (finite amount of wells) 21
Flow constraint binds when today s price is higher than value of deferred production Consider tradeoff of producing a marginal barrel: Earn revenue of U (F(t)) (i.e., oil price) But lose λ units of capacity valued at θ(t) Production is constrained iff U (F(t)) > λθ(t) 22
Flow constraint binds when today s price is higher than value of deferred production Consider tradeoff of producing a marginal barrel: Earn revenue of U (F(t)) (i.e., oil price) But lose λ units of capacity valued at θ(t) Production is constrained iff U (F(t)) > λθ(t) θ(t) is a function of anticipated future prices Price anticipated to rise slower than r forever: P(t) > λθ(t) guaranteed What happens if price is anticipated to rise faster than r and then level off? 23
Why produce at constraint even when future price is higher in present value terms? Expected price path PP 0 Flow rate on the constraint, declines at rate λ t 24
Why produce at constraint even when future price is higher in present value terms? PP 0 ee rrrr Expected price path Price less than PP 0 in present value PP 0 Price exceeds PP 0 in present value Flow rate on the constraint, declines at rate λ t 25
Why produce at constraint even when future price is higher in present value terms? PP 0 ee rrrr Expected price path Standard Hotelling intuition: profitably reallocate some production PP 0 Flow rate on the constraint, declines at rate λ t 26
Why produce at constraint even when future price is higher in present value terms? PP 0 ee rrrr Expected price path PP 0 Standard Hotelling intuition: profitably reallocate some production But this is infeasible! Only a fraction of deferred production is available immediately Flow rate on the constraint, declines at rate λ t 27
Why produce at constraint even when future price is higher in present value terms? PP 0 ee rrrr Expected price path Deferred production is instead recovered gradually over a long period PP 0 Length of time overlaps with the period for which price has lower present value Flow rate on the constraint, declines at rate λ t 28
Was producing at the constraint optimal during 1990 2007 (in particular, 1998 1999)? YES! Front month price λθ(t) No change forecast E[Price] = futures Baseline assumptions: interest rate r = 10%, production decline rate λ = 10%. Futures prices follow actual data out to 60 months and then are assumed to plateau 31
Drilling incentives: A modified Hotelling rule If drilling is positive (a>0), then marginal discounted revenue of drilling minus marginal cost of drilling must rise at interest rate. Shadow value of oil flow capacity at t Marginal discounted revenue from drilling 1 more well at t Marginal cost of drilling 1 more well at t Shadow cost at t of losing 1 untapped well Key insight: firms solve a drilling timing problem, not an extraction timing problem 36
Fixed oil price: Constant revenues, but rental price on drilling rigs adjusts to clear market If expected oil price is fixed at PP, then θθ = PP/(rr + λλ), and our modified Hotelling rule is given by: Exogenous oil price Discounted revenue stream (constant) Value at t of each untapped well (rises at interest rate) Cost at t to rent a rig (must fall) Implies declining rig rental rates and therefore falling drilling rates 39
Drilling starts high and declines, leading to a classic peaked production path in region Drilling rate (wells/yr) Oil flow (million bbl/yr) Parameters: Prices: Low = $20/bbl, High = $40/bbl 100 wells to drill 0.5 million bbl / well Drilling MC = 1 + 5a (multiply by ee.03tt with technical change) r = λ = 0.1 40
We observe production peaks in real world TX LA AK 41
Shale oil boom led to rising drilling & production
Endogenous oil price: drilling & production paths starting from K 0 = 0 Drilling rate (wells / yr), Oil flow (million bbl/yr) 5.0 4.0 Drilling falls monotonically, while production rises, peaks, and then declines toward zero 3.0 2.0 1.0 Drilling rate Oil flow rate Parameters: 100 wells to drill 0.5 million bbl / well Drilling MC = 1 + 5a r = λ = 0.1 Demand: P = 200 200F 0.0 0 10 20 30 40 50 60 70 80 90 100 Time (years) 45
Endogenous oil price: drilling revenues, costs, & profits starting from K 0 = 0 $million per well 50 40 Marginal discounted revenue from drilling (θx) 30 20 10 0 Marginal profit per well increases at r until drilling stops Marginal cost of drilling 0 10 20 30 40 50 60 70 80 90 100 Time (years) 46
Simulated response of drilling to demand shocks (one negative shock, two positive shocks) 2.4 Drilling Rate (wells/yr) 2.0 1.6 1.2 0.8 20 25 30 35 40 Time (years) Original demand: P = 200 200F Intercept jumps to 180 at t = 25 to 200 at t = 30 to 220 at t = 35 Drilling MC = 1 + 5a Drilling rate (and drilling cost) responds immediately to demand shocks 48
Simulated response of production and oil price to demand shocks Flow Rate (million bbl/yr) 0.9 0.9 0.8 0.8 Oil flow does not jump on impact; gradual response from new well drilling Oil price ($/bbl) 0.7 20 25 30 35 40 70 55 40 25 10 20 25 30 35 40 Price jumps on impact Contango after negative shock Backwardation after positive shock Time (years) 49
Observed patterns of price expectations following large shocks align with model s predictions Backwardation after positive shocks due to rapid growth among emerging Asian markets (Kilian 2009; Kilian and Hicks 2013) Iraq / Kuwait Contango after negative shock due to Asian financial crisis, OPEC (Kilian 2009) 50
Conclusion: Oil extraction is a dynamic drilling investment problem with constrained flow Oil production involves drilling wells with flow constrained by geology and reservoir pressure Research should focus on optimal drilling, not extraction Implications: Standard Hotelling results typically do not hold But Hotelling-like intuition applies to drilling wells Basic, data-driven assumptions lead to predictions that align closely with real-world observations Long run behavior Responses to shocks 55
Next steps: enrich the supply side of the model Drilling companies invest in new rigs if the rental rate is expected to be high Results in more elastic medium / long run drilling supply Reserves are heterogeneous Localized rig markets imply that high-cost reserves may be drilled simultaneously with low-cost reserves Stochastic demand Model phenomena such as: Response to new discoveries / technology shocks Rig substitution between oil and gas 56
BONUS SLIDES 57
Decline during 1998-1999 price collapse driven entirely by shut-ins of marginal wells 58
Multi-well leases show similar response 59
Sole-operated and common-pool fields show similar response 60
Production quotas are not binding 61
Newer wells show similar responses 62
High-production wells show similar response 63
Above-ground storage rises with futures prices 64
Necessary conditions 65
When flow constraint binds, we can re-write our modified Hotelling rule in terms of primitives If constraint is binding (F=K) and drilling is positive (a>0), then we can derive a modified Hotelling rule relating price to drilling costs: Marginal utility of oil flow (price) Amortized per-barrel marginal cost of drilling Per-barrel cost of drilling now versus delaying Shadow value of oil in wells Convexity in drilling cost function implies an added opportunity cost to drilling: if the cost of drilling is falling, it s more costly to drill now If dd = 0 then Hotelling s rule re-emerges (with some important caveats) 66
Hotelling s original path can be optimal, but only if ALL of the following conditions hold: Oil flow decay rate sufficiently high relative to interest rate (also depends on concavity of utility) Initial flow constraint not too high (else, period of zero drilling and production set to constraint) Unbounded marginal utility (b/c oil flows forever) Constant marginal drilling costs (b/c convex costs lead to gradual ramping up periods) 67
Endogenous price and no scarcity: Our model is closely related to macro investment model (q-theory) REINTERPRETATION AS A MACRO MODEL U (F) downward sloping demand for output F F = K capital stock (i.e., linear production fcn.) λ depreciation rate for existing capital a rate of investment in new capital d(a) marginal investment cost (d > 0) As in macro model, drilling and capacity dynamics can be captured using a phase diagram 68
Building the phase diagram: locus of points. such that a = 0 a(t) Above the locus, a(t) is increasing Below the locus, a(t) is decreasing locus given by points where aa = 0 K(t) 69
Building the. phase diagram: locus of points such that K = 0 a(t) Above the locus, K(t) is increasing KK = 0 Below the locus, K(t) is decreasing K(t) 70
The phase diagram can be used to study the steady state and transition dynamics a(t) This region cannot be entered in eqbm F = K guaranteed in this region May have F < K if K very high F = K guaranteed in this region K(t) 71
There is a unique optimal path ( stable arm ) leading to the steady state a(t) KK = 0 For a sufficiently small K 0, production will always be constrained on the optimal path aa = 0 K(t) 72
Now consider positive demand shock: this shifts. the a = 0 locus outward a(t) KK = 0 aa = 0 K(t) 73
Positive demand shock causes drilling to jump up immediately, followed by production buildup a(t) KK = 0 Production buildup leads to price backwardation, matching our futures market data (reverse for negative shock) aa = 0 K(t) 74
Finite number of wells: Returns to drilling must rise at the rate of interest; no steady state > 0 a(t) KK = 0 With scarcity, the aa = 0 locus must shift inward over time aa tt > 0 = 0 aa tt = 0 = 0 K(t) 75
Starting from K 0 = 0, drilling falls over time. Production rises then falls, eventually to zero. a(t) KK = 0 Extraction path yields peak in production over time Impact of demand shocks similar to no scarcity case aa tt > 0 = 0 aa tt = 0 = 0 K(t) 76
A weak condition is sufficient for production always to be constrained starting from K 0 = 0 a(t) KK = 0 Sufficient condition: demand elasticity decreases in magnitude as F increases Satisfied, e.g. by any inverse demand P(F) = α - βf δ aa tt > 0 = 0 aa tt = 0 = 0 K(t) 77
For a sufficiently large K 0, production initially may be unconstrained a(t) KK = 0 Unconstrained production could result from a very large (out of sample) negative demand shock aa tt > 0 = 0 aa tt = 0 = 0 K(t) 78