storing power: on the importance of market structure
TRANSCRIPT
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Storing Power: on the Importance of Market Structure
David Andrés-Cerezo1 and Natalia Fabra2
1EUI, 2UC3M
January 27, 2020
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The “Duck curve”
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Introduction
Storage should play a key role in the energy transition:By making a more efficient use of existing resources (e.g. excess renewable production).By providing energy when renewables are not available.By reducing the need to invest in (polluting) back-up generation capacity.
Goals:Analyze whether the private and social incentives for investing in storage are aligned.Understand decentralized storage operation and its impact on market outcomes.Asses how this depends on market structure.
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Introduction
Storage should play a key role in the energy transition:By making a more efficient use of existing resources (e.g. excess renewable production).By providing energy when renewables are not available.By reducing the need to invest in (polluting) back-up generation capacity.
Goals:Analyze whether the private and social incentives for investing in storage are aligned.Understand decentralized storage operation and its impact on market outcomes.Asses how this depends on market structure.
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Storing technologies and market structure
Different types of storage facilities...
Figure: Pumped hydro Figure: Grid-scale batteries Figure: Electric vehicle fleet
...that imply different horizontal and vertical market structures:Competitive vs. strategic storage.Competitive vs. strategic productionDifferent ownership structures ⇒ stand-alone vs. vertically integrated storage firms.
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Storing technologies and market structure
Different types of storage facilities...
Figure: Pumped hydro Figure: Grid-scale batteries Figure: Electric vehicle fleet
...that imply different horizontal and vertical market structures:Competitive vs. strategic storage.Competitive vs. strategic productionDifferent ownership structures ⇒ stand-alone vs. vertically integrated storage firms.
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Relevant questions
1 Investment in storage capacity .Will it be socially optimal?Does it depend on market structure?
2 Productive efficiency and prices.What are the effects of strategic behavior and the ownership structure?
3 Do storage facilities confer market power? Do they mitigate market power in generation?
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Related literature
Storage and commodity speculation.McLaren (1999) Williams& Wright (1991); Mitraille, Thille (2014); etc...
Natural resource extraction.Hotelling (1931); Salant (1976); etc...
Electricity storage.Schmalensee (2019); Ambec&Crampes (2018); Karaduman (2020); Crampes&Trochet(2019); Helm&Mier (2018); García& Stacchetti (2001); etc...
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Outline
1 Model set-up
2 First Best
3 Market solutionSecond BestCompetitive storageIndependent storage monopolistVertically integrated firm
4 Welfare comparison
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Modelling set-up
DemandPrice- inelastic demand θ; consumers’ valuation v.θ is distributed according to a symmetric G (θ) in
[θ, θ
].
θ can be interpreted as demand net of renewables.Known at the production stage → Focus on seasonal variation.
GenerationExisting assets allow to produce with costs C̃(Q) increasing and convex.
StorageStorage capacity K (in MWh); Investment cost C(K) increasing and convex.qB(θ), qS(θ) : quantities bought (B) and sold (S) by the storage facility.
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Modelling set-up
DemandPrice- inelastic demand θ; consumers’ valuation v.θ is distributed according to a symmetric G (θ) in
[θ, θ
].
θ can be interpreted as demand net of renewables.Known at the production stage → Focus on seasonal variation.
GenerationExisting assets allow to produce with costs C̃(Q) increasing and convex.
StorageStorage capacity K (in MWh); Investment cost C(K) increasing and convex.qB(θ), qS(θ) : quantities bought (B) and sold (S) by the storage facility.
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Modelling set-up
DemandPrice- inelastic demand θ; consumers’ valuation v.θ is distributed according to a symmetric G (θ) in
[θ, θ
].
θ can be interpreted as demand net of renewables.Known at the production stage → Focus on seasonal variation.
GenerationExisting assets allow to produce with costs C̃(Q) increasing and convex.
StorageStorage capacity K (in MWh); Investment cost C(K) increasing and convex.qB(θ), qS(θ) : quantities bought (B) and sold (S) by the storage facility.
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Demand process
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Outline
1 Model set-up
2 First Best
3 Market solutionSecond BestCompetitive storageIndependent storage monopolistVertically integrated firm
4 Welfare comparison
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First Best Problem
Welfare is gross consumer surplus minus production and investment costs:
maxqB(θ),qS(θ),K
W =
∫ θ
θvθdG (θ)−
∫ θ
θC̃(θ − qS(θ) + qB(θ)
)dG (θ)− C (K)
s.t. (λ) :
∫ θ
θqB(θ)dG (θ) ≥
∫ θ
θqS(θ)dG (θ)
(µ) :
∫ θ
θqB(θ)dG (θ) ≤ K
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First Best
θ
θ θ̄
θ
q(θ), p(θ)
Figure: First Best
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First Best
θ
θ θ̄
qFBS
qFBB
θFB2
θFB2
θFB1
θFB1
MC(θ)
θ
q(θ), p(θ)
Figure: First Best
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First Best
θ
θ θ̄
qFBS
qFBB
θFB2
θFB2
θFB1
θFB1
MC(θ)
µFBC′(K) =
θ
q(θ), p(θ)
Figure: First Best
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First Best
Optimal storage management:Store when demand is low and release when demand is high.Equalization of marginal costs within storing and releasing regions.Minimization of total costs of production.
Optimal investment in storage:Marginal benefit ⇒ Marginal cost saving from storing one more unit of output.No full marginal cost equalization.
First Best
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Outline
1 Model set-up
2 First Best
3 Market solutionSecond BestCompetitive storageIndependent storage monopolistVertically integrated firm
4 Welfare comparison
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Horizontal market structure: Production
Existing assets are owned by:a dominant firm (α share), with costs C̃D(q) = q2
2α ·a competitive fringe (1 − α share) with costs C̃F(q) = q2
2(1−α) ·α ∈ (0, 1)
Fringe produces qF = (1 − α) p(θ),
Dominant firm faces an elastic residual demandD(θ) = θ − qS(θ) + qB − (1 − α) p(θ)(θ).
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Horizontal market structure: Production
Existing assets are owned by:a dominant firm (α share), with costs C̃D(q) = q2
2α ·a competitive fringe (1 − α share) with costs C̃F(q) = q2
2(1−α) ·α ∈ (0, 1)
Fringe produces qF = (1 − α) p(θ),
Dominant firm faces an elastic residual demandD(θ) = θ − qS(θ) + qB − (1 − α) p(θ)(θ).
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Independent storage: Dominant firm
Maximize profits over the residual demand:
maxp(θ)
πD = p (θ)D(θ)− [D(θ)]2
2α
Optimal prices:
p(θ) = θ − qS(θ) + qB(θ)
1 − α2
Constant mark-up equal to α.Distorted market shares:
Dominant produces α/(1 + α) < α.Fringe produces 1/(1 + α) > 1 − α
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Outline
1 Model set-up
2 First Best
3 Market solutionSecond BestCompetitive storageIndependent storage monopolistVertically integrated firm
4 Welfare comparison
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Second Best
Maximize total welfare taking production decisions as given:
maxqB(θ),qS(θ),K
W =
∫ θ
θvθdG (θ)−
∫ θ
θ
(q2D
2α +q2
F2(1 − α)
)dG (θ)− C (K)
subject to the storage constraints and taking as given that:
qD =α
1 + α
[θ − qS(θ) + qB(θ)
]qF =
11 + α
[θ − qS(θ) + qB(θ)
]
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Second Best
θ θ̄
θ
θSB1 θSB
2
θ
p(θ),MC(θ)
Figure: Second Best
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Second Best
θ θ̄
θ
θSB1 θSB
2
WMC(θ)
θ
p(θ),MC(θ)
Figure: Second Best
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Second Best
θ θ̄
θ
WMC(θ)
θSB1 θSB
2
C′(K) = µSB
θ
p(θ),MC(θ)
Figure: Second Best
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Second Best
Optimal storage management:Similar to first best.Equalization of industry (weighted) marginal costs within storing and releasing regions.Weighted marginal cost: sum of the product of each firm’s market share and marginal cost.
Optimal investment in storage:Marginal benefit ⇒ Marginal cost saving from adding one unit of storage.Market power amplifies differences in industry marginal costs.
Second Best
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Outline
1 Model set-up
2 First Best
3 Market solutionSecond BestCompetitive storageIndependent storage monopolistVertically integrated firm
4 Welfare comparison
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Problem of competitive storage firms
Perfect competition:1 Large set of small owners (e.g. electric cars).2 Take generation prices as given.3 Free entry in the market ⇒ zero-profit condition.
Storage firms maximize:
maxqS(θ),qB(θ)
ΠS =
∫ θ
θp (θ)
[qS(θ)− qB(θ)
]g (θ) dθ − C(K)
subject to the storage constraints and the zero-profit condition.
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Competitive storage
θ
θ θ̄θC2θC
1
θ
q(θ), p(θ)
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Competitive storage
θ
θ θ̄θC2θC
1
p(θ)
θ
q(θ), p(θ)
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Competitive storage
θ
θ θ̄θC2θC
1
p(θ)
C(K)/K = µC
θ
q(θ), p(θ)
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Competitive storage
Optimal storage management:Storage operators exploit arbitrage opportunities.Prices (not marginal costs) equalized within storage and releasing regions.
Equilibrium investment in storage:Marginal value of storage capacity equals price differential that an extra unit of capacityallows to arbitrage.Market power in the product market amplifies arbitrage profits.
Competitive
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First Best vs. Second Best vs. Competitive
θ θ̄
θ
θ1 θ2
MC(θ)
µFB
WMC(θ)
µSB
p(θ)
µC
θ
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First Best vs. Second Best vs. Competitive
Under competitive storage with free-entry in the market, there is over-investment andover-utilization of storage ⇒ KC > KSB > KFB.
Price differential higher than marginal cost savings.
θ2 − θ11 − α2︸ ︷︷ ︸
µC
>(θ2 − θ1)(1 + α− α2)
(1 − α2)(1 + α)︸ ︷︷ ︸µSB
> θ2 − θ1︸ ︷︷ ︸µFB
Cost convexity → Higher infra-marginal profits → C(K)/K < C′(K)
KSB > KFB → Storage mitigates market power by reducing residual demand at highdemand levels.
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First Best vs. Second Best vs. Competitive
Under competitive storage with free-entry in the market, there is over-investment andover-utilization of storage ⇒ KC > KSB > KFB.
Price differential higher than marginal cost savings.
θ2 − θ11 − α2︸ ︷︷ ︸
µC
>(θ2 − θ1)(1 + α− α2)
(1 − α2)(1 + α)︸ ︷︷ ︸µSB
> θ2 − θ1︸ ︷︷ ︸µFB
Cost convexity → Higher infra-marginal profits → C(K)/K < C′(K)
KSB > KFB → Storage mitigates market power by reducing residual demand at highdemand levels.
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Outline
1 Model set-up
2 First Best
3 Market solutionSecond BestCompetitive storageIndependent storage monopolistVertically integrated firm
4 Welfare comparison
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Storage Monopolist
The storage monopolist internalizes the effects of its decisions on market prices:
maxqS(θ),qB(θ),K
ΠS =
∫ θ
θ
θ − qS(θ) + qB(θ)
1 − α2[qS(θ)− qB(θ)
]g (θ) dθ − C(K)
subject to the storage constraints.
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Storage monopolist
θ θ̄
θ
q(θ)
θM1 θM
2
θ
q(θ), p(θ)
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Storage monopolist
θ θ̄
θ
q(θ)
θM1 θM
2
p(θ)
θ
q(θ), p(θ)
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Storage monopolist
θ θ̄
θ
q(θ)
θM1 θM
2
p(θ)
MRS
MCSC′(K) = µM
θ
q(θ), p(θ)
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Storage monopolist
Optimal storage management:Less production smoothing ⇒ Storage monopolist avoids a strong price increase (decrease)when it buys (sells).
Storage monopolist equalizes marginal revenues (costs) when selling (buying).No price-equalization.
Optimal investment in storage:Marginal value of storage capacity equals the difference between MR and MC that anextra unit of capacity allows to arbitrage.Increasing in α
Monopolist
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Infra-utilization of storage capacity
θ θ̄E(θ)
q(θ)
θ
q(θ)
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Infra-utilization of storage capacity
θ θ̄E(θ)
q(θ)
KFB
KFB
qFB(θ)
θ
q(θ)
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Infra-utilization of storage capacity
θ θ̄E(θ)
q(θ)
KFB
KFB
qFB(θ)
KM
KM qM(θ)
θ
q(θ)
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Second Best & Second Best vs. Storage monopolist
Second Best: The storage monopolist under-invests → KM < KSB.1 Market power in storage ⇒ lower storage utilization.
First Best: The storage monopolist under-invests (over-invest) if α < α̂ (α < α̂), withα ∈ (0, 1) → KM ⋛ KSB.
1 Market power in storage ⇒ lower storage utilization.2 Market power in generation ⇒ arbitrage profits higher than at the first best.
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Second Best & Second Best vs. Storage monopolist
Second Best: The storage monopolist under-invests → KM < KSB.1 Market power in storage ⇒ lower storage utilization.
First Best: The storage monopolist under-invests (over-invest) if α < α̂ (α < α̂), withα ∈ (0, 1) → KM ⋛ KSB.
1 Market power in storage ⇒ lower storage utilization.2 Market power in generation ⇒ arbitrage profits higher than at the first best.
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Outline
1 Model set-up
2 First Best
3 Market solutionSecond BestCompetitive storageIndependent storage monopolistVertically integrated firm
4 Welfare comparison
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Vertically Integrated Firm
Dominant firm vertically integrated with storage monopolist.
maxp(θ),qB(θ),qS(θ)
πS =
∫ θ̄
θ
[p(θ)D (p; θ)− [D (p; θ)− qS(θ) + qB(θ)]2
2α
]g (θ) dθ,
subject to the storage constraints.
Higher residual demand (firm controls storage) → D (p; θ) = θ − (1 − α)p(θ).
Storage facilities ⇒ Help the dominant producer smooth its production over time.
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Vertically integrated firm
θ θ̄
θ
q(θ)
θI1 θI
2
θ
q(θ), p(θ)
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Vertically integrated firm
θ θ̄
θ
q(θ)
θI1 θI
2
MCD(θ)
θ
q(θ), p(θ)
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Vertically integrated firm
θ θ̄
θ
q(θ)
θI1 θI
2
MCD(θ)
p(θ)
θ
q(θ), p(θ)
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Vertically integrated firm
θ θ̄
θ
q(θ)
θI1 θI
2
MCD(θ)
p(θ)
C′(K) = µI
θ
q(θ), p(θ)
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Vertically integrated firm
Optimal storage management:Vertically integrated firm uses storage to smooth own production.Under-utilization of given storage capacity with respect to first best.
Optimal investment in storage:Marginal value of storage capacity equals own marginal cost savings.Investment decreases in α.
Integrated
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First Best vs. Vertically integrated firm
In a market with a vertically integrated dominant firm, there is under-investment instorage, KI < KFB < KSB.
In contrast to previous cases, KI is decreasing in α.Efficiency gains from higher α dominate larger arbitrage opportunities
The under-investment problem is aggravated with respect to the case of an independentmonopolist, KI < KM
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Outline
1 Model set-up
2 First Best
3 Market solutionSecond BestCompetitive storageIndependent storage monopolistVertically integrated firm
4 Welfare comparison
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Consumer’s surplus
Consumer’s surplus only depends on the price profile (i.e. weighted average price)
CS = vθ −∫ θ̄
θp(θ)θg(θ)dθ.
Market power in generation increases the price level.Market power in storage increases the variance of the market price.
The ranking of consumer surplus across market structures is
CSFB > CSC ≥ CSSB > CSM > CSI > CSNS.
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Consumer’s surplus
Consumer’s surplus only depends on the price profile (i.e. weighted average price)
CS = vθ −∫ θ̄
θp(θ)θg(θ)dθ.
Market power in generation increases the price level.Market power in storage increases the variance of the market price.The ranking of consumer surplus across market structures is
CSFB > CSC ≥ CSSB > CSM > CSI > CSNS.
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Price profile: no capacity restrictions
pC(θ)E(θ)1−α2
E(θ)
E(θ) pFB(θ)
θ
pNS(θ)
pM(θ)
pI(θ)
θ̄
θ
p(θ)
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Total welfare
Total welfare is just a function of the total costs of production.Market power creates static & dynamic productive inefficiencies:
Generation (static) ⇒ Distorted market shares.Storage (dynamic) ⇒ Lower storage usage, production not flatenned.Aggravated with vertical integration ⇒ Fringe absorbs demand variations.
The ranking of total welfare across market structures is
TWFB > TWSB > TWC > TWM > TWI > TWNS.
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Total welfare
Total welfare is just a function of the total costs of production.Market power creates static & dynamic productive inefficiencies:
Generation (static) ⇒ Distorted market shares.Storage (dynamic) ⇒ Lower storage usage, production not flatenned.Aggravated with vertical integration ⇒ Fringe absorbs demand variations.
The ranking of total welfare across market structures is
TWFB > TWSB > TWC > TWM > TWI > TWNS.
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Conclusions
The market does not provide adequate investment incentives in storage capacity.Market power in generation leads to over-investment.Market power in storage to under-investment.
Vertical integration between storage and generation yields the most inefficient outcome.Texas regulator: utilities are not permitted to use storage.
Storage reduces the ability to exercise market power in generation, conditional on beingindependently owned.
Storage capacity auctions.Solve investment problem, although inefficient storage operation.
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Ways forward
Introduce other cases (e.g. load-owned storage).
Introduce stochastic demand and/or production.
Empirical simulation for the Spanish market.
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First Best (cont)
Optimal storage management:For given K, storage decisions are
qFBB (θ) = max
{θFB
1 − θ, 0}
and qFBS (θ) = max
{θ − θFB
2 , 0}
whereθFB
1 = E [θ]− µ
2 ≤ θFB2 = E [θ] +
µ
2 ,
and µ = µFB(K) is the unique solution to∫ θFB1 (µ)
θ
[θFB
1 (µ)− θ]
g(θ)dθ = K.
Optimal investment in storage:∂W∂K = 0 ⇒ µ
(KFB
)= C′
(KFB
)⇒ θFB
2 − θFB1 = C′
(KFB
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Second Best (cont)
Optimal storage management:For given K, storage decisions are
qSBB (θ) = max
{θSB
1 − θ, 0}
and qSBS (θ) = max
{θ − θSB
2 , 0}
whereθSB
1 = E [θ]− µ
2(1 − α2)(1 + α)
1 + α− α2 ≤ θSB2 = E [θ] +
µ
2(1 − α2)(1 + α)
1 + α− α2 ,
and µ = µFB(K) is the unique solution to∫ θSB1 (µ)
θ
[θSB
1 (µ)− θ]
g(θ)dθ = K.
Optimal investment in storage: Back
∂W∂K = 0 ⇒ µ
(KSB
)= C′
(KSB
)⇒ (θSB
2 − θSB1 )
1 + α− α2
(1 − α2)(1 + α)= C′
(KSB
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Competitive storage (cont)
Optimal storage management:For given K, the equilibrium storage decisions are
qCB(θ) = max
{θC
1 − θ, 0}
and qCS(θ) = max
{θ − θC
2 , 0}
whereθC
1 = E [θ]−µ(1 − α2)
2 ≤ θC2 = E [θ] +
µ(1 − α2)
2 ,
with µ = µC(K) implicitly defined by:∫ θC1 (µ)
θ
[θC
1 (µ)− θ]
g(θ)dθ = K.
Investment in storage:
µC(K) = (θC2 − θC
1 )/(1 − α2) = C (K) /K < C′(K).
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Storage Monopolist (cont)
Optimal storage management:For given K, the equilibrium storage decisions are
qMB (θ) = max {(θ1 − θ) /2, 0} and qM
S (θ) = max {(θ − θ2) /2, 0} ,
whereθM
1 = E [θ]− µ(1 − α2)/2 ≤ θM2 = E [θ] + µ(1 − α2)/2,
with µ = µM(K) is the unique solution to∫ θM
1 (µ)θ
θM1 (µ)−θ
2 g(θ)dθ = KOptimal investment in storage:
C′(K) = µM(K) = (θM2 − θM
1 )/(1 − α2)
Back
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Vertically Integrated Firm (cont)
Optimal storage management:For given K, the equilibrium storage decisions are
qIB(θ) = max
{(θI
1 − θ)/2, 0
}and qI
S(θ) = max{(
θ − θI2
)/2, 0
},
whereθI
1 = E [θ]− µ(1 + α)/2 ≤ θI2 = E [θ] + µ(1 + α)/2,
with µ = µI(K) is the unique solution to∫ θI1(µ)
θ
θI1 (µ)− θ
2 g(θ)dθ = K.
Optimal investment in storage:
C′(K) = µI(K) = (θI2 − θI
1)/ (1 + α) .
BackD.Andrés-N.Fabra Storing Power: on the Importance of Market Structure January 27, 2020