Fuel Efficiency Improvements: Feedback Mechanisms and Distributional Effects in the Oil Market

Abstract

We study the interactions between fuel efficiency improvements in the transport sector and the oil market, where the efficiency improvements are policy-induced in certain regions of the world. We are especially interested in feedback mechanisms of fuel efficiency such as the rebound effect, carbon leakage and the “green paradox”, but also the distributional effects for oil producers. An intertemporal numerical model of the international oil market is introduced, where OPEC-Core producers have market power. We find that the rebound effect has a noticeable effect on the transport sector, with the magnitude depending on the oil demand elasticity. In the benchmark simulations, we calculate that almost half of the energy savings may be lost to a direct rebound effect and an additional 10% to oil price adjustments. In addition, there is substantial intersectoral leakage to other sectors through lower oil prices in the regions that introduce the policy. There is a small green paradox effect in the sense that oil consumption increases initially when the fuel efficiency measures are gradually implemented. Finally, international carbon leakage will be significant if policies are not implemented in all regions; we estimate leakage rates of 35% or higher when only major consuming regions implement fuel economy policies. Non-OPEC producers will to a larger degree than OPEC producers cut back on its oil supply as a response to fuel efficiency policies due to high production costs.

Appendix: A Formal Description of the Petro2 Model

1.1 Demand Side

We have seven regions i, where both demand and production take place: OPEC, Western Europe (EU/EFTA), U.S., Rest-OECD, Russia, China and Rest of the World (on the supply side we can divide OPEC into OPEC-Core and Non-Core OPEC). Demand for final energy goods in each region is divided into six end-user sectors s: Industry, Households, Other sectors (private and public services, defense, agriculture, fishing, other), Electricity, Road and rail transport, and Domestic and international aviation and domestic shipping. In addition, there is one global sector: International shipping. Further, the model has one transformation sector: Power generation. We have six energy commodities/fuels f: Oil (aggregate of different oil products), Gas, Electricity, Coal, Biomass and Biofuels for transport (Table 4).

All variables are functions of time. However, we generally skip the time notation in the following. The functional forms and parameters are generally constant over time.

Table 4 List of regions, sectors and energy goods in the Petro2 model

List of symbols

Endogenous variables

$Q_{s,i}^{f}$ :

Demand for fuel f in sector s in region i

$Q_{s,i}$ :

Demand for energy aggregate in sector s in region i (index)

$P_{i}^{f}$ :

Producer price (node price) of fuel f in region i

$PP_{s,i}^{f}$ :

End-user price of fuel f in sector s in region i

$PI_{s,i}$ :

Price index for a fuel aggregate in sector s in region i

Exogenous variables and parameters

$GDP_{s,i}$ :

Economic activity per capita index in sector s in region i

$Pop_{i}$ :

Population index in region i

$AEEI_{s,i}$ :

Autonomous improvements in energy efficiency index in sector s in region i

$\beta _{s,i}$ :

Long-term income per capita elasticity in sector s in region i

$\alpha _{s,i}$ :

Long-term price elasticity of the fuel aggregate in sector s in region i

$\varepsilon _{s,i}$ :

Long-term elasticity of population growth in sector s in region i

$b_{s,i}$ :

Short-term income per capita elasticity in sector s in region i

$a_{s,i}$ :

Short-term price elasticity of the fuel aggregate in sector s in region i

$e_{s,i}$ :

Short-term population elasticity in sector s in region i

$\sigma _{s,i}$ :

Elasticity of substitution in sector s in region i

$\theta _{s,i}^{f}$ :

Initial budget share of fuel f in sector s in region i

$\omega _{s,i}$ :

Constant in demand function in sector s in region i

$v_{s,i}^{f}$ :

Existing taxes/subsidies on fuel f in sector s in region i

$z_{s,i}^{f}$ :

Costs of transportation, distribution and refining on fuel f in sector s in region i

$\gamma _{s,i}$ :

Lag parameter in demand function in sector s in region i

The end-user price of fuel f in sector s in region i is equal to the regional producer price of the fuel (node price) plus costs of transportation, distribution and refining in addition to existing taxes/subsidies:

$$\begin{aligned} PP_{s,i}^{f} =P_{i}^{f} +z_{s,i}^{f} +v_{s,i}^{f} \end{aligned}$$

(2)

We assume that demand for energy goods can be described through CES demand functions. Hence, we construct weighted aggregated fuel price index for each sector s and region i:

$$\begin{aligned} PI_{s,i} =\frac{\left[ {\sum \nolimits _f{\left\{ {\theta _{s,i}^{f} \left( PP_{s,i}^{f} \right) ^{(1-\sigma _{s,i} )}} \right\} }} \right] ^{1/(1-\sigma _{s,i} )}}{\left[ {\sum \nolimits _f {\left\{ {\overline{\theta }_{s,i}^{f} \left( \overline{PP}_{s,i}^{f} \right) ^{(1-\sigma _{s,i} )}} \right\} } } \right] ^{1/(1-\sigma _{s,i} )}} \end{aligned}$$

(3)

where $\overline{PP}_{s,i,0}$ denotes the (exogenous) actual price levels in the initial data year 2007. The budget shares for fuel f in the base year are given by:

$$\begin{aligned} \overline{\theta }_{s,i}^f = {{{\overline{PP} _{s,i,0}^f \cdot Q_{s,i}^f}\Bigg / {\sum \limits _{f f} \left( {\overline{PP} _{s,i}^{ff} \cdot Q_{s,i}^{ff}}\right) }}} \end{aligned}$$

(4)

where prices and quantities in (4) are measured at $t =$ 0. We allow for exogenous changes in $\theta _{s,i}^{f}$ to better model future changes in the composition of fuel consumption. So far we have only let oil as a share of total energy-use decline in the transport sector. Long-term demand for a fuel aggregate in sector sand region i is assumed to be on the following form:

$$\begin{aligned} Q_{s,i,t} =K_{s,i,t} \cdot \left( {AEEI_{s,i,t} PI_{s,i,t} } \right) ^{\alpha _{s,i} } \end{aligned}$$

(5)

where $K_{s,i,t}$ is an exogenous factor representing other variables than the price:

$$\begin{aligned} K_{s,i,t} =\omega _{s,i,t} \cdot GDP_{s,i,t}^{\beta _{s,i} } \cdot Pop_{i,t}^{\varepsilon _{s,i} } \cdot AEEI_{s,i,t}. \end{aligned}$$

Note that energy efficiency improvements beyond the reference level are modelled by reducing the AEEI-parameters. However, as efficiency improvements imply lower costs of energy services, there will be a rebound effect as long as the price elasticity is strictly negative. Following eq. (3) in Kverndokk and Rosendahl (2013), the inverse demand function for fuel $P_{f }(Q^{f})$ can be expressed as $P_{f} \left( {Q^{f}} \right) =\frac{1}{AEEI}P_{s} \left( {\frac{Q^{f}}{AEEI}} \right)$, where $P_{s}$ denotes the underlying inverse demand function for energy services.²¹ From this expression we can derive the expression in (5).

In order to take account of short- and medium-term effects, the demand functions are specified in the following partial adjustment way (here we include the time notation):

$$\begin{aligned} Q_{s,i,t}= & {} K_{s,i,t} \cdot PI_{s,i,t}^{a_{s,i} } \cdot Q_{s,i,t-1}^{\gamma _{s,i} }\nonumber \\= & {} \omega _{s,i,t} \cdot PI_{s,i,t}^{a_{s,i} } \cdot GDP_{s,i,t}^{b_{s,i} } \cdot Pop_{i,t}^{e_{s,i} } \cdot \left( {AEEI_{s,i,t} } \right) ^{1+\alpha _{s,i} }\cdot Q_{s,i,t-1}^{\gamma _{s,i} } \end{aligned}$$

(6)

where $\gamma _{s,i}$ is the lag-parameter (i.e. the effect of demand in the previous period $(0\le \gamma _{s,i} < 1)$. Then the long-term elasticities are given by: $\alpha _{s,i} =\frac{a_{s,i} }{1-\gamma _{s,i}}, \beta _{s,i} =\frac{b_{s,i} }{1-\gamma _{s,i}}$ and $\varepsilon _{s,i} =\frac{e_{s,i} }{1-\gamma _{s,i} }$. In the present model version $\gamma _{s,i} = 0$. Hence, we have no lags on the demand side and the short- and the long-term effects are equal (i.e., $\alpha _{s,i} = a_{s,i}, \beta _{s,i} = b_{s,i}, \varepsilon _{s,i} = e_{s,i})$. Then (6) is identical to (5). We normalize $Q_{s,i,0}=\textit{ 1}$ and $\textit{PI}_{s,i,0} =\textit{1}$ in the base year. Then, since GDP, Pop and AEEI all are indices equal to 1 in the base year, it must be that $\omega =\textit{ 1}$ when $\gamma _{s,i} = 0$.

Demand for fuel f in sector s in region i is a function of the demand for the fuel aggregate as well as the changes in the end-user price of the fuel aggregate relative to the end-user price of the fuel:

$$\begin{aligned} Q_{s,i}^{f} =\overline{Q}_{s,i,0}^{f} Q_{s,i} \frac{\theta _{s,i}^{f} }{\overline{\theta }_{s,i}^{f} }\left( \frac{{{PI_{s,i} }\bigg / {\overline{PI}_{s,i} }}}{{{PP_{s,i}^{f} } \bigg /{\overline{PP}_{s,i}^{f} }}}\right) ^{\sigma _{s,i} } \end{aligned}$$

(7)

where $\overline{Q}_{s,i,0}^{f}$ is the (exogenous) actual demand in the data year. The elasticities of substitution ($\sigma _{s,i} )$ can vary over sectors and regions.

1.2 Oil Supply Side

We have seven or eight oil producing regions i, depending on whether or not OPEC is split into OPEC-Core and Non-Core OPEC (this is the case in the current paper). Below we refer to OPEC as the cartel (C)—if OPEC is split into two, only OPEC-Core is assume to act as a cartel, while Non-Core OPEC is assumed to act as a competitive producer. The six Non-OPEC regions (NO) are always modelled as competitive producers.

List of symbols

Endogenous variables

$P^{o}$ :

Oil producer price (equal across regions, hence index i is not needed)

$X^{C}$ :

OPEC production (includes only OPEC-Core if OPEC is split into two)

$X_{i}^{NO}$ :

Production in Non-OPEC region i (includes Non-Core OPEC if OPEC is split into two)

$A^{C}$ :

Accumulated OPEC production

$A_{i}^{NO}$ :

Accumulated Non-OPEC production in region i

$C^{C}$ :

Total costs for OPEC

$C_{i}^{NO}$ :

Total costs for Non-OPEC in region i

$c^{C}$ :

Unit costs for OPEC

$c_{i}^{NO}$ :

Unit costs for Non-OPEC region i

$\lambda ^{C}$ :

Lagrange multiplier for OPEC

$\lambda _{i}^{NO}$ :

Lagrange multiplier for Non-OPEC region i

$\mu ^{C}$ :

Current shadow price for OPEC

$\mu _{i}^{NO}$ :

Current shadow price for Non-OPEC region i

Exogenous variables and parameters

$\varphi _{i}^{NO}$ :

Lag parameter for Non-OPEC region i

$\eta ^{C}$ :

Convexity parameter for OPEC

$\eta _{i}^{NO}$ :

Convexity parameter for Non-OPEC region i

$\tau ^{C}$ :

Rate of technological progress for OPEC

$\tau _{i}^{NO}$ :

Rate of technological progress for Non-OPEC region i

r :

Discount rate

$K_{s,i}$ :

Exogenous term representing other variables than price in the demand function in region i (i.e., $K_{s,i} =\omega _{s,i} \cdot GDP_{s,i}^{\beta _{s,i} } \cdot Pop_{i}^{\varepsilon _{i} } \cdot AIEE_{s,i}$)

Consumption of fuel (aggregate) in Eq. (6) can be written:

$$\begin{aligned} Q_{s,i} =PI_{s,i}^{\alpha _{s,i} }K_{s,i} \end{aligned}$$

(8)

where $K_{s,i}$ denotes the exogenous parts of the RHS of (6).

Global oil consumption is given by (where we use (2), (7) and (8)):

$$\begin{aligned} Q^{o}= & {} \sum \limits _s {\sum \limits _i {Q_{s,i}^{o} } } =\sum \limits _s {\sum \limits _i {\left\{ {\overline{Q}_{s,i,0}^{o} Q_{s,i} \frac{\theta _{s,i}^{f} }{\overline{\theta }_{s,i}^{f} }\left( {\frac{PI_{s,i} /\overline{PI}_{s,i} }{PP_{s,i}^{o} /\overline{PP} _{s,i}^{o} }} \right) ^{\sigma _{s,i} }} \right\} } } \nonumber \\= & {} \sum \limits _s {\sum \limits _i {\left\{ {\overline{Q}_{s,i,0}^{o} \frac{\theta _{s,i}^{f} }{\overline{\theta }_{s,i}^{f} }\left( {\frac{\overline{PP}_{s,i}^{o} }{\overline{PI}_{s,i} }} \right) ^{\sigma _{s,i} }K_{s,i} \left( {PP^{o}} \right) ^{-\sigma _{s,i} }PI_{s,i}^{\alpha _{s,i} +\sigma _{s,i} }} \right\} } } \nonumber \\= & {} \sum \limits _s {\sum \limits _i {\left\{ {\varGamma _{1,s,i} \left( {PP^{o}} \right) ^{-\sigma _{s,i} }PI_{s,i}^{\alpha _{s,i} +\sigma _{s,i} }} \right\} } } \end{aligned}$$

(9)

where $\Gamma _{1,s,i} =\overline{Q}_{s,i,0}^{o} \frac{\theta _{s,i}^{f} }{\overline{\theta }_{s,i}^{f} }\left( {\frac{\overline{PP}_{s,i,0}^{o} }{\overline{PI}_{s,i,0} }} \right) ^{\sigma _{s,i} }K_{s,i}$ include only exogenous terms.

1.3 The Optimization Problem for the Oil Producers

OPEC’s residual demand (for fixed Non-OPEC production $X^{NO})$ is:

$$\begin{aligned} X^{C}=Q^{o}-X^{NO} \end{aligned}$$

(10)

OPEC maximizes the following discounted profit over time:

$$\begin{aligned} \varPi =\,\sum \limits _t {\left\{ {\left( {1+r} \right) ^{-t}\left( {P_{t}^{o} X_{t}^{C} -C_{t}^{C} \left( {X_{t}^{C} ,A_{t}^{C} } \right) } \right) } \right\} } \end{aligned}$$

(11)

s.t. $A_{t}^{C} -A_{t-1}^{C} =X_{t}^{C}$

The cost function of OPEC in period t has the following functional form:

$$\begin{aligned} C_{t}^{C} \left( X_{t}^{C} ,A_{t}^{C} \right) =c_{t}^{C} \left( A_{t}^{C} \right) X_{t}^{C} \end{aligned}$$

(12)

where $c_{t}^{C}$ are the unit costs given by the following function:

$$\begin{aligned} c_{t}^{C} \left( A_{t}^{C} \right) =c_{0}^{C} \cdot e^{\eta \cdot ^{C}A_{t}^{C} -\tau ^{C} t} \end{aligned}$$

(13)

We assume that unit costs are increasing in accumulated extraction $A^{C.}$. Hence, the Lagrangian function becomes:

$$\begin{aligned} L= & {} \,\sum \limits _t {\left\{ {\left( {1+r} \right) ^{-t}\left( {P_{t}^{o} \left( {Q_{t}^{o} -X_{t}^{NO} } \right) -c_{t}^{C} (A_{t}^{C} )\cdot \left( {Q_{t}^{o} -X_{t}^{NO} } \right) } \right) } \right\} }\nonumber \\&+\sum \limits _t {\left\{ {\mu _{t}^{C} \cdot \left( {1+r} \right) ^{-t}\cdot \left( {A_{t}^{C} -A_{t-1}^{C} -\left( {Q_{t}^{o} -X_{t}^{NO} } \right) } \right) } \right\} } \end{aligned}$$

(14)

where $\mu _{t}^{C} >0$ is the current value of the shadow price of the resource at period t, and where $Q^{o}$ is a function of $P^{o}$ [see Eq. (9) above].

Before differentiating L wrt $P^{o}$, it is useful to differentiate $Q^{o}$ wrt $P^{o}$:

$$\begin{aligned} \frac{\partial Q^{o}}{\partial P^{o}}= & {} -\sum \limits _s {\sum \limits _i {\left\{ {\varGamma _{1,s,i} \sigma _{s,i} \left( {PP_{s,i}^{o} } \right) ^{-\sigma _{s,i} -1}PI_{s,i}^{\alpha _{s,i} +\sigma _{s,i} }\left( {\frac{\partial PP_{s,i}^{o} }{\partial P^{o}}} \right) } \right\} } } \nonumber \\&+\sum \limits _s {\sum \limits _i {\left\{ {\varGamma _{1,s,i} \left( {PP_{s,i}^{o} } \right) ^{-\sigma _{s,i} }\left( {\alpha _{s,i} +\sigma _{s,i} } \right) PI_{s,i}^{\alpha _{s,i} +\sigma _{s,i} -1}\left( {\frac{\partial PI_{s,i} }{\partial P^{o}}} \right) } \right\} } } \nonumber \\= & {} -\sum \limits _s {\sum \limits _i {\left\{ {\varGamma _{1,s,i} \sigma _{s,i} \left( {PP_{s,i}^{o} } \right) ^{-\sigma _{s,i} -1}PI_{s,i}^{\alpha _{s,i} +\sigma _{s,i} }} \right\} } } \nonumber \\&+\sum \limits _s {\sum \limits _i {\left\{ {\varGamma _{1,s,i} \theta _{s,i}^{0} \left( {\alpha _{s.i} +\sigma _{s,i} } \right) \left( {PP_{s,i}^{o} } \right) ^{-2\sigma _{s,i} }PI_{s,i}^{\alpha _{s,i} +\sigma _{s,i} }\left( {\sum \limits _f {\theta _{s,i}^{f} \left( {PP_{s,i}^{f} } \right) ^{1-\sigma _{s,i} }} } \right) ^{-1}} \right\} } } \nonumber \\= & {} -\sum \limits _s {\sum \limits _i {\left\{ {\overline{Q}_{s,i,0}^{o} \frac{\theta _{s,i}^{o} }{\overline{\theta }_{s,i}^{o} }\left( {\frac{\overline{PP}_{s,i}^{o} }{\overline{PI}_{s,i} }} \right) ^{\sigma _{s,i} }K_{s,i} \sigma _{s,i} \left( {PP_{s,i}^{o} } \right) ^{-\sigma _{s,i} -1}PI_{s,i}^{\alpha _{s,i} +\sigma _{s,i} }} \right\} } } \nonumber \\&+\sum \limits _s \sum \limits _i \left\{ \overline{Q}_{s,i,0}^{o} \frac{\left( {\theta _{s,i}^{o} } \right) ^{2}}{\overline{\theta }_{s,i}^{o} }\left( {\frac{\overline{PP}_{s,i}^{o} }{\overline{PI}_{s,i} }} \right) ^{\sigma _{s,i} }K_{s,i} \right. \nonumber \\&\left. \quad \times \left( {\alpha _{s,i} +\sigma _{s,i} } \right) \left( {PP_{s,i}^{o} } \right) ^{-2\sigma _{s,i} }PI_{s,i}^{\alpha _{s,i} +\sigma _{s,i} }\left( \sum \limits _f \theta _{s,i}^{f}\left( {PP_{s,i}^{f} } \right) ^{1-\sigma _{s,i} } \right) ^{-1} \right\} \nonumber \\= & {} -\sum \limits _s {\sum \limits _i {\left\{ {\overline{Q}_{s,i,0}^{o} \frac{\theta _{s,i}^{o} }{\overline{\theta }_{s,i}^{o} }\left( {\overline{PP}_{s,i}^{o} } \right) ^{\sigma _{s,i} }K_{s,i} \sigma _{s,i} \left( {PP_{s,i}^{o} } \right) ^{-\sigma _{s,i} -1}PI_{s,i}^{\alpha _{s,i} +\sigma _{s,i} }} \right\} } } \nonumber \\&+\sum \limits _s \sum \limits _i \left\{ \overline{Q}_{s,i,0}^{o} \frac{\left( {\theta _{s,i}^{o} } \right) ^{2}}{\overline{\theta }_{s,i}^{o} }\left( {\overline{PP}_{s,i}^{o} } \right) ^{\sigma _{s,i} }K_{s,i}\right. \nonumber \\&\left. \quad \times \left( {\alpha _{s,i} +\sigma _{s,i} } \right) \left( {PP_{s,i}^{o} } \right) ^{-2\sigma _{s,i} }PI_{s,i}^{\alpha _{s,i} +\sigma _{s,i} }\left( {\sum \limits _f {\theta _{s,i}^{f} \left( {PP_{s,i}^{f} } \right) ^{1-\sigma _{s,i} }} } \right) ^{-1} \right\} \end{aligned}$$

(15)

where we used $\frac{\partial PP_{s,i}^{o} }{\partial P^{o}}=1$ (cf. Eq. 2) and

$$\begin{aligned} \frac{\partial PI_{s,i} }{\partial P^{o}}= & {} \frac{\partial }{\partial P^{o}}\left\{ {{{\left[ {\sum \limits _f {\theta _{s,i}^f{{\left( {PP_{s,i}^f} \right) }^{1 - {\sigma _{s,i}}}}} } \right] }^{\frac{1}{{(1 - {\sigma _{s,i}})}}}}}\Bigg / {{{\left[ {\sum \limits _f {\overline{\theta }_{s,i}^f{{\left( {\overline{PP} _{s,i}^f} \right) }^{1 - {\sigma _{s,i}}}}} } \right] }^{\frac{1}{{(1 - {\sigma _{s,i}})}}}}}\right\} \\= & {} \frac{1}{\varGamma _{2} }\cdot \frac{\partial }{\partial P^{o}}\left[ {\sum \limits _f {\theta _{s,i}^{f} \left( {PP_{s,i}^{f} } \right) ^{1-\sigma _{s,i} }} } \right] ^{\frac{1}{(1-\sigma _{s,i} )}} \\= & {} \frac{1}{\varGamma _{2} }\frac{1}{(1-\sigma _{s,i} )}\left[ {\sum \limits _f {\theta _{s,i}^{f} \left( {PP_{s,i}^{f} } \right) ^{(1-\sigma _{s,i} )}} } \right] ^{\frac{1}{(1-\sigma _{s,i} )}-1}\cdot \theta _{s,i}^{o} \cdot \left( {1-\sigma _{s,i} } \right) \cdot \left( {PP_{s,i}^{o} } \right) ^{-\sigma _{s,i} } \\= & {} \frac{1}{\varGamma _{2} }\theta _{s,i}^{o} \cdot \left( {PP_{s,i}^{o} } \right) ^{-\sigma _{s,i} }\cdot \left[ {\sum \limits _f {\theta _{s,i}^{f} \left( {PP_{s,i}^{f} } \right) ^{(1-\sigma _{s,i} )}} } \right] ^{\frac{1}{(1-\sigma _{s,i} )}-1}\\= & {} \theta _{s,i}^{o} \cdot \left( {PP_{s,i}^{o} } \right) ^{-\sigma _{s,i} }\cdot \frac{\left[ {\sum \nolimits _f {\theta _{s,i}^{f} \left( {PP_{s,i}^{f} } \right) ^{(1-\sigma _{s,i} )}} } \right] ^{\frac{1}{(1-\sigma _{s,i} )}-1}}{\left[ {\sum \nolimits _f {\overline{\theta }_{s,i}^{f} \left( {\overline{PP}_{s,i}^{f} } \right) ^{1-\sigma _{s,i} }} } \right] ^{\frac{1}{(1-\sigma _{s,i} )}}} \\= & {} \theta _{s,i}^{o} \cdot \left( {PP_{s,i}^{o}} \right) ^{-\sigma _{s,i} }PI_{s,i} \left( {\sum \limits _f {\theta _{s,i}^{f} \left( {PP_{s,i}^{f} } \right) ^{(1-\sigma _{s,i} )}}} \right) ^{-1},\\ \hbox {where}&\varGamma _{2} =\left[ {\sum \limits _f {\overline{\theta }_{s,i}^{f} \left( {\overline{PP} _{s,i}^{f} } \right) ^{1-\sigma _{s,i} }} } \right] ^{\frac{1}{(1-\sigma _{s,i} )}}. \end{aligned}$$

To ease computation the following formulation of (15) is used in GAMS:

$$\begin{aligned} \frac{\partial Q^{o}}{\partial P^{o}}&=-\sum \limits _s {\sum \limits _i {\left\{ {\sigma _{s,i} \overline{Q}_{s,i,0}^{o} \frac{\theta _{s,i}^{o} }{\overline{\theta }_{s,i}^{o} }\left( {\overline{PP}_{s,i}^{o} } \right) ^{\sigma _{s,i} }K_{s,i} PI_{s,i}^{\alpha _{s,i} +\sigma _{s,i} }\left( {PP_{s,i}^{o} } \right) ^{-\sigma _{s,i} -1}} \right\} } } \nonumber \\&\quad +\sum \limits _s \sum \limits _i \left\{ \left( {\alpha _{s,i} +\sigma _{s,i} } \right) \overline{Q}_{s,i,0}^{o} \frac{\left( {\theta _{s,i}^{o} } \right) ^{2}}{\overline{\theta }_{s,i}^{o} }\left( {\overline{PP}_{s,i}^{o} } \right) ^{\sigma _{s,i} }K_{s,i} PI_{s,i}^{\alpha _{s,i} +\sigma _{s,i} -1}\left( {PP_{s,i}^{o} } \right) ^{-\sigma _{s,i} }\right. \nonumber \\&\left. \quad \times \left( {PP_{s,i}^{o} } \right) ^{-\sigma _{s,i} }PI_{s,i} \frac{1}{\sum \nolimits _f {\theta _{s,i}^{f}\left( {PP_{s,i}^{f} } \right) ^{1-\sigma _{s,i} }} } \right\} \nonumber \\&=-\sum \limits _s {\sum \limits _i {\left\{ {\sigma _{s,i} \overline{Q} _{s,i,0}^{o} \frac{\theta _{s,i}^{o} }{\overline{\theta }_{s,i}^{o} }\left( {\overline{PP}_{s,i}^{o} } \right) ^{\sigma _{s,i} }K_{s,i} PI_{s,i} ^{\alpha _{s,i} +\sigma _{s,i} }\left( {PP_{s,i}^{o} } \right) ^{-\sigma _{s,i} -1}} \right\} } } \nonumber \\&\quad +\sum \limits _s \sum \limits _i \left\{ \left( {\alpha _{s.i} +\sigma _{s,i} } \right) \overline{Q}_{s,i,0}^{o} \frac{\left( {\theta _{s,i}^{o} } \right) ^{2}}{\overline{\theta }_{s,i}^{o} }\left( {\overline{PP}_{s,i}^{o} } \right) ^{\sigma _{s,i} }K_{s,i} PI_{s,i}^{\alpha _{s,i} +\sigma _{s,i} }\left( {PP_{s,i}^{o} } \right) ^{-2\sigma _{s,i} }\right. \nonumber \\&\left. \quad \times \frac{1}{\sum \nolimits _f {\theta _{s,i}^{f} \left( {PP_{s,i}^{f} } \right) ^{1-\sigma _{s,i} }} } \right\} \nonumber \\&=-\sum \limits _s {\sum \limits _i {\left\{ {\sigma _{s,i} \overline{Q} _{s,i,0}^{o} \frac{\theta _{s,i}^{o} }{\overline{\theta }_{s,i}^{o} }\left( {\overline{PP}_{s,i}^{o} } \right) ^{\sigma _{s,i} }K_{s,i} PI_{s,i} ^{\alpha _{s,i} +\sigma _{s,i} }\left( {PP_{s,i}^{o} } \right) ^{-\sigma _{s,i} -1}} \right\} } } \nonumber \\&\quad +\sum \limits _s \sum \limits _i \left\{ \left( {\alpha _{s,i} +\sigma _{s,i} } \right) \overline{Q}_{s,i,0}^{o} \frac{\left( {\theta _{s,i}^{o} } \right) ^{2}}{\overline{\theta }_{s,i}^{o} }\left( {\overline{PP}_{s,i}^{o} } \right) ^{\sigma _{s,i} }K_{s,i} PI_{s,i}^{\alpha _{s,i} +2\sigma _{s,i} -1}\left( {PP_{s,i}^{o} } \right) ^{-2\sigma _{s,i} }\right. \nonumber \\&\left. \quad \times \frac{1}{\sum \nolimits _f {\theta _{s,i}^{f} \left( {\overline{PP}_{s,i}^{f} } \right) ^{1-\sigma _{s,i} }} } \right\} \end{aligned}$$

(15*)

We now differentiate L wrt $P^{o}$:

$$\begin{aligned} ({1+r})^{t}\frac{\partial L}{\partial P_{t}^{o} }=\,\left( {Q_{t}^{o} -X_{t}^{NO} } \right) +\left( {P_{t}^{o} -c_{t}^{C} \left( {A_{t}^{C} } \right) -\mu _{t}^{C} } \right) \frac{\partial Q_{t}^{o} }{\partial P_{t}^{o} }=0 \end{aligned}$$

(16)

where we can insert for ${\partial Q_{t}^{o} } \big /{\partial P_{t}^{o} }$ from Eq. (15) above.

If we rearrange Eq. (16) we get:

$$\begin{aligned} P_{t}^{o} =c_{t}^{C} \left( {A_{t}^{C} } \right) +\mu _{t}^{C} -\frac{\partial P_{t}^{o} }{\partial Q_{t}^{o} }\left( {Q_{t}^{o} -X_{t}^{NO} } \right) \end{aligned}$$

(17)

where the last term on the right hand side is the cartel rent.

Next, we differentiate wrt $A^{C}$:

$$\begin{aligned} \left( {1+r} \right) ^{t}\frac{\partial L}{\partial A_{t}^{C} }=\,-\frac{\partial c_{t}^{C} \left( {A_{t}^{C} } \right) }{\partial A_{t}^{C} }\left( {Q_{t}^{C} -X_{t}^{NO} } \right) +\mu _{t}^{C} -\left( {1+r} \right) ^{-1}\mu _{t+1}^{C} =0 \end{aligned}$$

(18)

or:

$$\begin{aligned} \eta ^{C}c_{t}^{C} \left( {A_{t}^{C} } \right) \left( {Q_{t}^{C} -X_{t}^{NO} } \right) -\mu _{t}^{C} +\left( {1+r} \right) ^{-1}\mu _{t+1}^{C} =0 \end{aligned}$$

(19)

Whereas the discounted shadow price decreases over time, the running shadow price $\mu$ can both decrease and increase over time. When the cartel stops producing, the shadow price reaches zero.

Let us turn to the competitive fringe’s optimization problem. The cost function of Non-OPEC regions in period t has the following functional form:

$$\begin{aligned} C_{i,t}^{NO} \left( X_{i,t}^{NO} ,A_{i,t}^{NO} \right)= & {} c_{i,t}^{NO} \left( A_{i,t}^{NO} \right) e^{\phi _{i}^{NO} \left( {\frac{X_{i,t}^{NO} }{X_{i,t-1}^{NO} }-1} \right) }X_{i,t}^{NO} \end{aligned}$$

(20)

$$\begin{aligned} c_{i,t}^{NO} \left( A_{i,t}^{NO} \right)= & {} c_{i,0}^{NO} \cdot e^{\eta _{i}^{NO} A_{i,t}^{NO} -\tau _{i}^{NO} t} \end{aligned}$$

(21)

We here assume that there are no adjustment costs for Non-OPEC production, reflected by the parameter $\phi _{i}^{NO} = 0$.

However, we assume the following cost function:

$$\begin{aligned} C_{i,t}^{NO} \left( X_{i,t}^{NO} ,A_{i,t}^{NO} \right) =\kappa _{A,t} c_{i,t}^{NO} \left( A_{i,t}^{NO} \right) \left( {X_{i,t}^{NO} } \right) ^{\kappa _{B,t} } \end{aligned}$$

(20b)

where $\kappa _{A,t}$ and $\kappa _{B,t}$ are exogenous parameters. In the initial years, $\kappa _{B,t}$ >1 to reflect increasing marginal costs also within a period. $\kappa _{B,t}$ is then gradually reduced to one over time. $\kappa _{A,t}$ is calibrated so that marginal costs at $X_{i,0}^{NO}$are the same as with $\kappa _{A,t} = \kappa _{B,t} = 1$, i.e., $\kappa _{A,t} =\left( {\kappa _{B,t} } \right) ^{-1}\left( {X_{i,t}^{NO} } \right) ^{1-\kappa _{B,t} }$

The optimization problem can be written:

$$\begin{aligned} \text{ Max }\,\sum \limits _t {\left\{ {(1+r)^{-t}\left( {X_{i,t}^{NO} P_{t} -C_{i,t}^{NO} \left( X_{i,t}^{NO} ,A_{i,t}^{NO} \right) } \right) } \right\} } \end{aligned}$$

(22)

with $A_{i,t}^{NO} -A_{i,t-1}^{NO} =X_{i,t}^{NO}$.

The Lagrangian function becomes:

$$\begin{aligned} L= & {} \sum \limits _t {\left\{ {(1+r)^{-t}\left( {X_{i,t}^{NO} P_{t} -C_{i,t}^{NO} \left( X_{i,t}^{NO} ,A_{i,t}^{NO}\right) } \right) } \right\} }\nonumber \\&+\sum \limits _t {\left\{ {\mu _{i,t}^{NO} \cdot (1+r)^{-t}\cdot \left( A_{i,t}^{NO} -A_{i,t-1}^{NO} -X_{i,t}^{NO} \right) } \right\} } \end{aligned}$$

(23)

where $\mu _{i}^{NO} =-(1+r)\lambda _{i}^{NO} >0$ is the current value of the shadow price on the resource constraint, and $\lambda _{i}^{NO}$ is the present value of the shadow price (the Lagrange multiplier).

The first order condition wrt.$X_{i,t}^{NO}$ is:

$$\begin{aligned} P_{t}= & {} \frac{C_{i,t}^{NO} \left( X_{i,t}^{NO} ,A_{i,t}^{NO} \right) }{X_{i,t}^{NO} }+\frac{\varphi _{i}^{NO} }{X_{i,t-1}^{NO} }C_{i,t}^{NO} \left( X_{i,t}^{NO} ,A_{i,t}^{NO} \right) \nonumber \\&-(1+r)^{-1}\frac{\varphi _{i}^{NO} X_{i,t+1}^{NO} }{(X_{i,t}^{NO} )^{2}}C_{i,t}^{NO} \left( X_{i,t+1}^{NO} ,A_{i,t+1}^{NO} \right) +\mu _{i,t}^{NO} \end{aligned}$$

(24)

Note that if $\phi _{\mathrm {i}}^{\mathrm {NO}} = 0$, (24) simplifies to:

$$\begin{aligned} P_{t} =c_{i,t}^{NO} \left( A_{i,t}^{NO} \right) +\mu _{i,t}^{NO} \end{aligned}$$

(25)

The first term on the right-hand-side in Eqs. (24) and (25) is the average unit cost. The second term in (24) accounts for the rising short-term unit costs. Together, the two first terms are the marginal production costs in the short term (for an exogenous $X_{t-1}^{NO}$). The third term is negative, taking into account the positive effect on future cost reductions of increasing current production. The last term in (24) and (25) is the scarcity effect; the alternative cost of producing one unit more today as it increases future costs due to scarcity.

Alternatively, using (20b) we get the following first order condition:

$$\begin{aligned} P_{t} =\kappa _{B,t} \frac{C_{i,t}^{NO} \left( X_{i,t}^{NO} ,A_{i,t}^{NO}\right) }{X_{i,t}^{NO} }+\mu _{i,t}^{NO} \end{aligned}$$

(24b)

When we differentiate L wrt $A_{i,t}^{NO}$ we get the following condition for changes in the Lagrange multiplier (identical to the corresponding condition for OPEC above):

$$\begin{aligned} \eta _{i}^{NO} \cdot C_{i,t}^{NO} \left( X_{i,t}^{NO} ,A_{i,t}^{NO} \right) -\mu _{i,t}^{NO} +\left( {1+r} \right) ^{-1}\mu _{i,t+1}^{NO} =0 \end{aligned}$$

(26)

Relationships between price and substitution elasticities

In the model described above, $\alpha$ is the direct price elasticity of the energy aggregate, while $\sigma$ is the substitution elasticity between energy goods in the energy aggregate. The direct price elasticity of oil follows implicitly from $\alpha$ and $\sigma$, as well as the value shares $\theta$ and prices of the energy goods. Since we may want to specify the direct price elasticity of oil instead of the elasticity of the energy aggregate, it is useful to derive the exact relationship between these two, and specifically derive a reduced form expression for the latter as a function of the former (and other necessary parameters/variables).

Let $\xi _{s,i}^{o} =\frac{\partial Q_{s,i}^{o}}{\partial PP_{s,i} ^{o}}\frac{PP_{s,i}^{o}}{Q_{s,i}^{o}}$ denote the direct price elasticity of oil in sector sand region i. From (15) and (9) we have $\Big ($note that $\frac{\partial Q_{s,i}^{o}}{\partial PP_{s,i}^{o}}=\frac{\partial Q_{s,i} ^{o}}{\partial P^{o}}\Big )$ ²²:

$$\begin{aligned} \xi _{s,i}^{o} =\frac{\partial Q_{s,i}^{o}}{\partial PP_{s,i} ^{o}}\frac{PP_{s,i}^{o}}{Q_{s,i}^{o}}=-\sigma _{s,i} +\left( {\alpha _{s.i} +\sigma _{s,i} } \right) \theta _{s,i}^{o} \frac{\left( {PP_{s,i}^{o}} \right) ^{1-\sigma _{s,i} }}{\sum \nolimits _f {\theta _{s,i}^{f} \left( {PP_{s,i}^{f} } \right) ^{1-\sigma _{s,i} }}} \end{aligned}$$

This can alternatively be expressed as:

$$\begin{aligned} \alpha _{s,i} =\left( {\xi _{s,i}^{o} +\sigma _{s,i} } \right) \;\frac{\sum \nolimits _f {\theta _{s,i}^{f} \left( {\overline{PP} _{s,i}^{f} } \right) ^{1-\sigma _{s,i} }} }{\theta _{s,i}^{o} \left( {\overline{PP}_{s,i}^{o} } \right) ^{1-\sigma _{s,i} }}-\sigma _{s,i} \end{aligned}$$

(27)

In the special case where all end-user prices in a sector/region are equal across energy goods, we get:

$$\begin{aligned} \alpha _{s,i} =\frac{\xi _{s,i}^{o} +\sigma _{s,i} }{\theta _{s,i}^{o} }-\sigma _{s,i} \end{aligned}$$

(27*)

In the calibration of the model, we use (27) to derive estimates of $\alpha$, given estimates of the RHS parameters and base-year levels of the variables.

1.4 Oil Demand Share by Sector and Region

See Table 5.

Table 5 Oil demand share by sector and region in 2007 (%)