Optimal Dynamic Information Acquisition for Emission Control with Fixed Cost Viet Anh Nguyen, Thomas Weber ´ Ecole Polytechnique F´ ed´ erale de Lausanne EURO 2015 13 July 2015 1
Introduction and Motivation In the news 2
Introduction and Motivation In the news 3
Introduction and Motivation But how to achieve this? Goals are easy to set, but harder to meet. Different policies can be used: subsidies to green energy, emission quotas, carbon taxes. Carbon tax is an effective way to discourage carbon consumption and lower climate change risks. Optimal carbon tax rate is dependent on the CO2 concentration in the atmosphere. However, it is costly to measure this concentration. 4
Introduction and Motivation Even the funding for collecting information is in danger 5
Introduction and Motivation Even the funding for collecting information is in danger 6
Contributions Main Results We use the model by Hoel and Karp (2001), which is essentially a Linear Quadratic Gaussian model. The optimal policy is of the threshold type: when the variance is above a threshold, we acquire information. Without fixed cost, there are analytical solutions for the optimal policy (presented last year at IFORS). In this talk, we present some preliminary results for the case with fixed cost. 7
Contributions Hierarchy of the Solution Procedure Results from traditional LQG Parameter Bellman Separability Reduction Equation 1 ( ) K x P P’ P’’ K 2 ( ) y Our contributions 8
Contributions Model Setup We consider a Linear-Quadratic Gaussian setup. Evolution of the state ˜ x t x t +1 = a ˜ ˜ x t + bu t + ˜ ε t , (1) where u t is the control, a , b are scalars, ˜ ǫ t is the system noise. Lagged observation ˜ z t +1 = ˜ x t + (˜ η t / v t ) , (2) where v t is the control of the observation noise’s variance When v t = 0, the observation is pure noise. When v t = + ∞ , we have perfect information. v t can be called the precision. Assumption: ˜ ǫ t and ˜ η t are independent and normally distributed with mean 0 and variance N and M respectively. Wlog, M = 1. 9
Contributions Information Acquisition Cost The cost to acquire information consists of a fixed cost f plus a quadratic cost of the precision v . � 0 if v = 0 cost ( v ) = f + v 2 if v > 0 = f ✶ v > 0 + v 2 10
Contributions Belief Propagation - Kalman Filtering The mean and variance of the estimation satisfy av t ˆ V t x t +1 ˆ = a ˆ x t + bu t + ω t , (3) � t ˆ 1 + v 2 V t � � t ˆ v 2 V t ˆ ˆ N + a 2 V t +1 = 1 − V t , (4) t ˆ 1 + v 2 V t where ω t is an i.i.d. standard normal distribution. 11
Contributions Decision Problem After parameter reduction, the problem can be simplified to solving ∞ � β t E � � � x t + r ˆ x 2 V t + u 2 t + v 2 min (1 − r )ˆ t + s ˆ t + f ✶ { v t > 0 } � ¯ x 0 , N 0 , � ( u , v ) t =0 a v t ˆ V t s.t. ˆ x t +1 = a ˆ x t + bu t + ω t , x 0 = ¯ ˆ x 0 , � t ˆ 1 + v 2 V t � � t ˆ v 2 V t V t +1 = N + a 2 ˆ ˆ ˆ 1 − V t , V 0 = N 0 , t ˆ 1 + v 2 V t ( u t , v t ) ∈ U × V , t ∈ T , (P’) Problem (P’) has 7 parameters θ = ( a , b , s , r , f , N , β ) . 12
Contributions Bellman equation By dynamic programming principle, solving problem (P’) is equivalent to solving the Bellman equation (1 − r ) x 2 + sx + ry + u 2 + v 2 + f ✶ { v > 0 } + β E [ K (˜ x ′ , y ′ ) | x , y ] � � K ( x , y ) = min avy ˜ ω x ′ = ax + bu + s.t. ˜ , � 1 + v 2 y v 2 y � � y ′ = N + a 2 1 − y , 1 + v 2 y ( u , v ) ∈ U × V (P”) 13
Contributions Separability of K in x and y Proposition 1 Let K 1 ( x ) = Px 2 + Qx + R for all x ∈ X , where P , Q , R ∈ R . Then K ( x , y ) ≡ K 1 ( x ) + K 2 ( y ) , and the Bellman equation (P”) is separable, so (1 − r ) x 2 + sx + u 2 + β P ( ax + bu ) 2 + Q ( ax + bu ) + R � � �� K 1 ( x ) = min , u ∈U (SS) to find the best stabilizing input u ∗ , and � Pa 2 v 2 y 2 v 2 y � � � � ��� ry + v 2 + f ✶ { v > 0 } + β N + a 2 · K 2 ( y ) = min 1 + v 2 y + K 2 1 − · y 1 + v 2 y v ∈V (IA) to find the optimal informational input v ∗ , for all ( x , y ) ∈ X × R + . 14
Contributions Separability of K in x and y The value of P , Q , R in K 1 ( x ) = Px 2 + Qx + R can be found similarly to the traditional LQG. The subproblem of K 2 ( y ) can be written as K 2 ( y ) = ry − 1 y + β P ( N + a 2 y )+ a 2 � � y ′ − N − β Py ′ + β K 2 ( y ′ ) min f ✶ { y ′ < N + a 2 y } + , y ′ ∈ [ N , N + a 2 y ] Instead of optimizing over the control v , we are optimizing over the future variance y ′ through the bijective mapping from v to y ′ v 2 y 2 y ′ = N + a 2 y − 1 + v 2 y 15
Contributions Difficulty The fixed cost makes the one-period cost function become discontinuous. Because of the convex cost, the threshold policy may not be optimal: Starting from a very high variance, the decision maker can choose to reduce the variance gradually instead of making one big decrease. 16
Contributions Results Proposition 2 The threshold policy is optimal. The proof is based on the fact that when there is no fixed cost, the decision maker’s optimal policy is a threshold policy. When there is fixed cost, the threshold policy is also optimal to avoid fixed costs entailed by multiple jumps. As a consequence, we concentrate on finding the optimal policy ( y ∗ , α ∗ ). When the variance is above y ∗ , the decision maker acquire information so that the variance goes down to α ∗ . 17
Contributions Results Analytical results exist only under continuous time. For discrete time, we can do a search over the whole space of the possible policy Figure 1: An example of the search over the whole policy space. 18
Contributions A better zoom The optimal policy lies strictly above the 45-degree line, implying α ∗ < y ∗ . Figure 2: Zoom into the region of interest 19
Contributions Results Due to discrete time, there might be multiple optimal threshold y ∗ , however, α ∗ is unique. Figure 3: Relationship between the magnitude of fixed cost f and α ∗ Higher fixed cost, lower α ∗ . 20
Contributions Sample of variance trajectory Figure 4: Sample of variance trajectory when f = 3 21
Contributions Sample of variance trajectory Figure 5: Sample of variance trajectory when f = 8 Higher fixed cost, lower the frequency of acquiring information. 22
Conclusions Conclusions Extension of LQG problem with dynamic information acquisition. The accuracy of the information acquisition is endogenous. The threshold policy is optimal under the case with fixed cost. A search over the entire policy space can be done to find the optimal policy. Work in progress: multi-dimensional. 23
Conclusions Thank you! 24
References Athans, M. (1972) On the determination of optimal costly measurement strategies for linear stochastic systems. Automatica 8(4):397–412. Hoel, M., Karp, L. (2001) Taxes and quotas for a stock pollutant with multiplicative uncertainty. Journal of Public Economics 82(1):91–114. Hoel, M., Karp, L. (2002) Taxes versus quotas for a stock pollutant. Resource and Energy Economics 24(4):367–384. Karp, L. and Zhang, J. (2012) Taxes versus quantities for a stock pollutant with endogenous abatement costs and asymmetric information. Economic Theory 49(2):371–409. Lindset, S., Lund, A.C. and Matsen, E. (2009) Optimal information acquisition for a linear quadratic control problem, European Journal of Operational Research 199(2):435–441. Stokey, N. L. (2008) The Economics of Inaction: Stochastic Control Models with Fixed Costs. Princeton University Press. Veldkamp, L. (2011) Information choice in macroeconomics and finance. Princeton University Press. 1
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