Inattention cost of energy

Cap-and-trade in Energy Capacity Investment when the planner is inattentive

Cea-Echenique, Feijoo and Muñoz

UAndes, PUCV

July 2023

last update 2023-07-12

Outline

Motivation

  • Decarbonization
    • Contrast Tax versus Cap-and-trade
    • Carbon pricing
  • Environment responsability as an attention problem

Literature

  • 2-stage stochastic capacity investment problems

(Ehrenmann and Smeers 2011; de Maere d ’ Aertrycke et al. 2017; Abada, de Maere d ’ Aertrycke, and Smeers 2017)

  • Re-trading: Amigo, Cea-Echenique, and Feijoo (2021)

  • Investment segmentation: Cea-Echenique and Torres-Martínez (2018)

  • Rational Inattention: Dewan and Neligh (2020)

Model

  • \bar t periods indexed by T
  • \Omega finite set of states of nature
  • P allowance long position
  • V allowance short position

Demand

In the first stage period t=0,

  • Exogenous demand D(0)\in\mathbb{R}_+.

In the second stage t\in T:=\{1,...\bar{t}\}, \omega\in\Omega

  • Probability Pr(\omega)\in[0,1].
  • Exogenous stochastic demand level D(t,\omega)\in\mathbb{R}_+^{T\times\Omega},

D=\left(D(0),(D(t,\omega))_{(t,\omega)\in T\times\Omega}\right)\in\mathbb{R}_+\times\mathbb{R}_+^{T\times\Omega}.

Producers: first stage

t=0, each producer i chooses

  • Q_i(t)\in\mathbb{R}_+ generation quantity such that their revenue is maximized price \pi^d(0)\in\mathbb{R}_+
  • \bar{Q}_i installed capacity for each producer at t=0
  • A_i\in\mathbb{R}_+ carbon allowances at a price \pi^{a}\in\mathbb{R}_+
  • x_i(t)\in\mathbb{R}_+ additional capacity with a capital expenditure of I_i\in\mathbb{R}_+ (The additional capacity becomes available after a predefined building time (in years))

Producers: second stage

In the second stage, for each pair (t,\omega), producer i maximizes its profit by choosing

  • Q_i(t,\omega)\in\mathbb{R}_+ the generation level at a price \pi^d(t,\omega)\in\mathbb{R}_+
  • TC_i(t) \in\mathbb{R}_+ technology change that adjusts the marginal cost of different technologies.
  • x_i(t,\omega)\in\mathbb{R}_+ additional capacity with an investment cost of TCR_i(t)\cdot I_i, where TCR_i(t)\in\mathbb{R}_+ is the change in the investment cost or capital expenditure over time
  • P_i(\omega)\in\mathbb{R}_+ long position in permits from other producers if they need to surpass the initial allowances allocation A_i
  • V_i(\omega)\in\mathbb{R}_+ short position in allowances
  • \pi^v(\omega)\in\mathbb{R}_+ permits retrade price

Producers

The choice set of the producer i over a time horizon of \bar{t} years is given by:

  • an amount of capacity expansion x_i:=\left(x_i(0),(x_i(t,\omega))_{(t,\omega)\in T\times\Omega}\right)\in\mathbb{R}_+\times\mathbb{R}_+^{T\times\Omega},
  • a production plan Q_i:=\left(Q_{i}(0),(Q_{i}(t,\omega)_{(t,\omega)\in T\times\Omega})\right)\in\mathbb{R}_+\times\mathbb{R}_+^{T\times\Omega},
  • allowances A_i\in\mathbb{R}_+ bought in the first period t=0,
  • allowances P_i(\omega)\in\mathbb{R}_+^{\Omega} bought in t=1 for time interval t\in[1,\bar{t}]
  • allowances V_i(\omega)\in\mathbb{R}_+^{\Omega} sold in t=1 for time interval t\in[1,\bar{t}].

Given electricity prices \pi^d:=\left(\pi^d(0),\left(\pi^d(t,\omega)\right)_{(t,\omega)\in T\times\Omega}\right)\in\mathbb{R}_+\times\mathbb{R}_+^{T\times\Omega} and parameters (a_i,b_i)_{i\in\{1,...N\}}\in(\mathbb{R}^2_+)^N, f_i(p,q)=\Big(a_i\cdot q+\frac{b_i}{2}\cdot q^{2}\Big)-p\cdot q revenue function.

Producers: Objective

Defining T_0:=\{0\}\cup T, the optimization problem of producer i is given by choosing

  • (x_i,Q_i, A_i,P_i,V_i)\in\mathbb{X}:=\left(\mathbb{R}_+\times\mathbb{R}_+^{T\times\Omega}\right) \times\left(\mathbb{R}_+\times\mathbb{R}_+^{T\times\Omega}\right) \times \mathbb{R}_+\times\mathbb{R}_+^{\Omega}\times\mathbb{R}_+^{\Omega}
  • given prices (\pi^d, \pi^a , \pi^v)\in\Pi:=\left(\mathbb{R}_+\times\mathbb{R}_+^{T\times\Omega}\right)\times\mathbb{R}_+\times\mathbb{R}^{\Omega}_+,
  • parameters \left((a_i,b_i),I_i, TC_i(t,\omega), TCR_i(t,\omega), CF_i,\bar{Q}_i, RP_i , \varepsilon_i\right)\in \Xi:=\mathbb{R}_+^2\times\mathbb{R}_+^7, \tau\in \mathbb{R}_+
  • and probability (Pr(\omega))_{\omega\in\Omega}\in\Delta:=\left\{\left(Pr(\omega)\right)_{\omega\in\Omega}\in[0,1]^K:\sum_{\omega\in\Omega}Pr(\omega)=1\right\} as a solution of

\min_{(x_i,Q_i,A_i,P_i,V_i)\in \mathbb{X}} f_i \big( \pi^d(0),Q_i(0)\big)+ A_i \pi^{a} + I_i x_i(0)

+ \sum_{\omega} Pr(\omega) \Bigg[ \sum_{t>0} \frac{1}{(1+R)^t} \Big[ TC_i(t)\cdot f_i \big( \pi^d(t,\omega),Q_i(t,\omega) \big)

+ TCR_i(t) \cdot I_i\cdot x_i(t,\omega) \Big] + \pi^v(\omega)\cdot \big(P_i(\omega)-V_i(\omega)\big) \Bigg]

\Big(CF_i \cdot\tau\Big) \Bigg[\bar{Q}_i + x_i(0)+\sum_{t^{\prime}<t-lag_i} x_i(t^\prime,\omega) \Bigg] - Q_i(t,\omega) \geq 0,\forall \, i,\omega, t > 0 \, (\alpha_{i,\omega,t})

(CF_i\cdot\tau)\bar{Q}_i-Q_i(0)\geq 0 \qquad \forall \quad i\quad (\kappa_i) RP_i - \bar{Q}_i - x_i(0) - \sum_{t > 0} x_i(t,\omega) \geq 0 \qquad \forall \quad i,\omega \quad (\psi_{i,\omega})

A_{i}-V_i(\omega)\geq 0 \qquad \forall \quad i,\omega \quad (\beta_{i,\omega})

A_{i} + (P_i(\omega) - V_i(\omega))-\sum_{t>0}Q_i(t, \omega)\cdot \varepsilon_{i}-Q_i(0)\varepsilon_{i} \geq 0 \qquad \forall \quad i, \omega \quad (\gamma_{i,\omega})

Auctioneer: Amigo et alii (2021)

  • \theta\in \mathbb{R}_+ available allowances at a price \pi^a in stage t=0.
  • The price is set such that it clears the CO_2 permit market among generators and the auctioneer.
  • CAP\in\mathbb{R}_+ carbon budget (maximum level of CO_2 emissions allowed in the second stage t\in T) and is drawn from a normal distribution, i.e., CAP \thicksim N(\mu, \sigma^2).

Pr(\theta \geq CAP) \leq \epsilon

  • Objective \min_{\theta} -\theta \pi^{a} + \mathcal{F}(\theta)\quad\text{s.t }\quad\phi^{-1}(\epsilon) \sigma + \mu - \theta \geq 0

Equilibrium

The data that parameterize the capacity investment model is given by the tuple (\tau,(Pr(\omega))_{\omega\in\Omega},\left(CAP,\mu,\sigma,\epsilon\right),((a_i,b_i),I_i, TC_i(t,\omega), TCR_i(t,\omega), CF_i,

\bar{Q}_i, RP_i , \varepsilon_i)_{i\in N})\in \mathbb{R}_+\times \Delta\times\mathbb{R}_+^4\times\Xi^N,

DEFINITION. An equilibrium in the capacity investment model is a vector of prices and production decisions \left((\pi^{d*},\pi^{a*},\pi^{v*}),(x_i^*,Q_i^*,A_i^*,P_i^*,V_i^*)_{i\in\{1,...,N\}},\theta^*\right)\in\Pi\times\mathbb{X}^N \times \mathbb{R}_+ such that:

(x_i^*,Q_i^*,A_i^*,P_i^*,V_i^*) minimizes the cost for each producer i\in\{1,...,N\} - \theta^* minimizes the auctioneer cost - Market clearing conditions are satisfied:

  • available allowances t=0 \sum_{i} A_{i}^* = \theta \ \ {\color{blue}{(\pi^{a*})}}
  • equilibrium in trading market t>0 \sum_{i} P_{i,\omega}^* = \sum_{i} V_{i,\omega}^* \forall \ \omega \ \ {\color{blue}{\left(\pi^{v*}(\omega)\right)}}
  • fulfillment of the demand: first stage \sum_{i} Q_i(0)^* = D(0), \ \ {\color{blue}{(\pi^{d*}(0))}}
  • fulfillment of the demand: second stage \begin{array}{ll}\sum_{i} Q_i(t,\omega)^* = D(t,\omega), &\forall \ \omega, t\\ & {\color{blue}{(\pi^{d*}(t,\omega))}}\end{array}

Rational Inattention

Following Dewan and Neligh (2020) we consider a quadratic cost function

C(P)=\begin{cases}0,&Perf\leq d\\c(Perf-d)^2,&Perf>d\end{cases},

where Perf\in[0,1] is a performance metric, c an associated marginal cost and d the performance level wiht public information.

We drop the chance constraint approach, by using

  • \min_{(\theta,Perf)\in[0,(1+\epsilon)CAP]\times[0,1]}-\theta\pi^aPerf+c(Perf-d)^2 profit oriented
  • \min_{\theta\in[0,CAP]}-\theta\pi^a+c\left(1-\frac{|\theta-CAP|}{CAP}-d\right)^2 welfare oriented

Results: Profit oriented

  • Calibration wrt Amigo, Cea-Echenique, and Feijoo (2021)
    • CAP: 100 MtCO_2
    • \pi^a=315.83
    • \epsilon=0.202 Andres et alii (2014) \Rightarrow c=920M

CAP=100

Perf \theta \frac{\theta}{CAP} \pi^a
0.798 120,200,000 1.20 270.13
0.8 120,200,000 1.20 270.13
0.85 113,654,321 1.14 25.75
0.9 113,568,965 1.14 93.65
0.95 118,470,479 1.18 188.86
0.99 114,916,396 1.15 298.11
1 118,332,486 1.18 317.24

Results: multiple CAPs, Perf=0.8

CAP \theta \frac{\theta}{CAP} \pi^a
100,000,000 120,200,000 1.20 270.13
200,000,000 240,400,000 1.20 60.57
300,000,000 360,600,000 1.20 119.47
400,000,000 454,798,037 1.14 95.83
500,000,000 601,000,000 1.20 79.31
600,000,000 605,279,899 1.01 66.57
700,000,000 605,279,899 0.86 56.37
800,000,000 961,600,000 1.20 152.50
900,000,000 605,279,899 0.67 42.26
1,000,000,000 605,279,899 0.61 35.98

Results: Profit oriented, allowances distributions

Results: Welfare oriented

  • Calibration wrt Amigo, Cea-Echenique, and Feijoo (2021)
    • CAP: 100 MtCO_2
    • \pi^a=315.83

\Rightarrow c=9920M

CAP (M) \theta Perf \frac{\theta}{CAP} \pi^a
100 144,944,410 0.798 1.449 232.82
200 289,888,820 0.798 1.449 141.56
300 434,833,230 0.798 1.449 103.33
400 579,777,640 0.798 1.449
500 653,754,577 0.905 1.308 37.27
600 619,939,821 0.999 1.033 40.43
700 661,258,561 0.997 0.945 36.42
800 610,913,181 0.944 0.764 31.28
900 609,127,279 0.896 0.677 31.21
1,000 698,676,376 0.909 0.699 30.00

Final Remarks

  • Respect of aggregate demand law
  • Unclear relation of inattention
  • Multiple solutions and comparative statics

References

Abada, Ibrahim, Gauthier de Maere d ’ Aertrycke, and Yves Smeers. 2017. “On the Multiplicity of Solutions in Generation Capacity Investment Models with Incomplete Markets: A Risk-Averse Stochastic Equilibrium Approach.” Mathematical Programming 165 (1): 5–69. https://doi.org/10.1007/s10107-017-1185-9.
Amigo, Pía, Sebastián Cea-Echenique, and Felipe Feijoo. 2021. “A Two Stage Cap-and-Trade Model with Allowance Re-Trading and Capacity Investment: The Case of the Chilean NDC Targets.” Energy 224 (June): 120129. https://doi.org/10.1016/j.energy.2021.120129.
Cea-Echenique, Sebastián, and Juan Pablo Torres-Martínez. 2018. “General Equilibrium with Endogenous Trading Constraints.” PLoS ONE 13 (9). https://doi.org/10.1371/journal.pone.0203814.
de Maere d ’ Aertrycke, G., A. Ehrenmann, D. Ralph, and Y. Smeers. 2017. “Risk Trading in Capacity Equilibrium Models.” Working {Paper}. University of Cambridge. https://doi.org/10.17863/CAM.17552.
Dewan, Ambuj, and Nathaniel Neligh. 2020. “Estimating Information Cost Functions in Models of Rational Inattention.” Journal of Economic Theory 187 (May): 105011. https://doi.org/10.1016/j.jet.2020.105011.
Ehrenmann, Andreas, and Yves Smeers. 2011. “Stochastic Equilibrium Models for Generation Capacity Expansion.” In Stochastic Optimization Methods in Finance and Energy: New Financial Products and Energy Market Strategies, edited by Marida Bertocchi, Giorgio Consigli, and Michael A. H. Dempster, 273–310. International Series in Operations Research & Management Science. New York, NY: Springer. https://doi.org/10.1007/978-1-4419-9586-5_13.