Optimal Dynamic Information Acquisition for Emission Control with - - PowerPoint PPT Presentation

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

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Introduction and Motivation

In the news

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Introduction and Motivation

In the news

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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.

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Introduction and Motivation

Even the funding for collecting information is in danger

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Introduction and Motivation

Even the funding for collecting information is in danger

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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.

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Contributions

Hierarchy of the Solution Procedure

P P’ P’’

1( )

K x

2( )

K y

Parameter Reduction Bellman Equation Separability Results from traditional LQG Our contributions

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Contributions

Model Setup

We consider a Linear-Quadratic Gaussian setup. Evolution of the state ˜ xt ˜ xt+1 = a˜ xt + but + ˜ εt, (1) where ut is the control, a, b are scalars, ˜ ǫt is the system noise. Lagged observation ˜ zt+1 = ˜ xt + (˜ ηt/vt), (2) where vt is the control of the observation noise’s variance When vt = 0, the observation is pure noise. When vt = +∞, we have perfect information. vt can be called the precision. Assumption: ˜ ǫt and ˜ ηt are independent and normally distributed with mean 0 and variance N and M respectively. Wlog, M = 1.

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Contributions

Information Acquisition Cost

The cost to acquire information consists of a fixed cost f plus a quadratic cost of the precision v. cost(v) =

• if v = 0

f + v2 if v > 0 = f ✶v>0 + v2

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Contributions

Belief Propagation - Kalman Filtering

The mean and variance of the estimation satisfy ˆ xt+1 = aˆ xt + but + avt ˆ Vt

• 1 + v2

t ˆ

Vt ωt, (3) ˆ Vt+1 = N + a2

• 1 −

v2

t ˆ

Vt 1 + v2

t ˆ

Vt

• ˆ

Vt, (4) where ωt is an i.i.d. standard normal distribution.

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Contributions

Decision Problem

After parameter reduction, the problem can be simplified to solving min

(u,v) ∞

• t=0

βt E

• (1 − r)ˆ

x2

t + sˆ

xt + r ˆ Vt + u2

t + v2 t + f ✶{vt>0}

• ¯

x0, N0

• ,

s.t. ˆ xt+1 = aˆ xt + but + a vt ˆ Vt

• 1 + v2

t ˆ

Vt ωt, ˆ x0 = ¯ x0, ˆ Vt+1 = N + a2

• 1 −

v2

t ˆ

Vt 1 + v2

t ˆ

Vt

• ˆ

Vt, ˆ V0 = N0, (ut, vt) ∈ U × V, t ∈ T , (P’) Problem (P’) has 7 parameters θ = (a, b, s, r, f , N, β).

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Contributions

Bellman equation

By dynamic programming principle, solving problem (P’) is equivalent to solving the Bellman equation

K(x, y) = min

• (1 − r)x2 + sx + ry + u2 + v 2 + f ✶{v>0} + β E [K(˜

x′, y ′)| x, y]

• s.t.

˜ x′ = ax + bu + avy ˜ ω

• 1 + v 2y

, y ′ = N + a2

• 1 −

v 2y 1 + v 2y

• y,

(u, v) ∈ U × V (P”)

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Contributions

Separability of K in x and y

Proposition 1 Let K1(x) = Px2 + Qx + R for all x ∈ X, where P, Q, R ∈ R. Then K(x, y) ≡ K1(x) + K2(y), and the Bellman equation (P”) is separable, so K1(x) = min

u∈U

• (1 − r)x2 + sx + u2 + β
• P(ax + bu)2 + Q(ax + bu) + R
• ,

(SS) to find the best stabilizing input u∗, and

K2(y) = min

v∈V

• ry + v 2 + f ✶{v>0} + β

Pa2v 2y 2 1 + v 2y + K2

• N + a2 ·
• 1 −

v 2y 1 + v 2y

• · y
• (IA)

to find the optimal informational input v ∗, for all (x, y) ∈ X × R+.

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Contributions

Separability of K in x and y

The value of P, Q, R in K1(x) = Px2 + Qx + R can be found similarly to the traditional LQG. The subproblem of K2(y) can be written as

K2(y) =ry − 1 y + βP(N + a2y)+ min

y ′∈[N,N+a2y]

• f ✶{y ′<N+a2y} +

a2 y ′ − N − βPy ′ + βK2 (y ′)

• ,

Instead of optimizing over the control v, we are optimizing over the future variance y′ through the bijective mapping from v to y′ y′ = N + a2y − v2y2 1 + v2y

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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.

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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∗.

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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.

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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

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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∗.

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Contributions

Sample of variance trajectory

Figure 4: Sample of variance trajectory when f = 3

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Contributions

Sample of variance trajectory

Figure 5: Sample of variance trajectory when f = 8

Higher fixed cost, lower the frequency of acquiring information.

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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.

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Conclusions

Thank you!

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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.

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