Technical Appendix to Accompany “Capacity Payment Mechanisms and Investment Incentives in Restructured Electricity Markets” by David P. Brown First, more detailed derivations of several proofs are provided. Then, extensions of the basic model are considered. Proof of Lemma 3: Using (1) – (4), and Lemma 1 and 2, firm i’s expected profit function: E[π −
PI
k− −r
Z
k−
Z
−ck − g J (·)dθE dθC +
] = 0 1
Z
k− −r
C
(P
E
− J
E
1
=
(P
− γ)k − dGE (θE ) +
Z
k−r
k
Z
−ck g J (·)dθE dθC + 1
Z
+
(P 0
Z
k−r K−r
Z
K
1
(P k−r 1
Z
Z
K k
K−r 1
Z
C
− γ)k − + P k − − ck − g J (·)dθE dθC E
C
Z
1Z 1
0
0
ck − g J (·)dθE dθC
0
C −
P k dGC (θC ) − ck − ;
Z
k−r
E
C
− γ)
Z
K
θE − ck g J (·)dθE dθC − γ) 2 K−r Z k C1 C P (θ + r) − ck g J (·)dθE dθC 2 k−r 0 E
θE C1 C +P (θ + r) − ck g J (·)dθE dθC 2 2
− γ)k + P
C1
J
2 E
(θC + r) − ck g J (·)dθE dθC Z
C
0
Z
1
Z
(P K−r K 1Z K
E
K
(P K−r
1
+
k
E
− γ)
θE C + P k − ck g J (·)dθE dθC 2
C
− γ)k + P k − ck g J (·)dθE dθC
Z 1Z 1 θE J E E C g (·)dθ dθ + (P − γ)k g J (·)dθE dθC (P − γ) 2 0 k 0 K Z 1 Z 1 Z K−r Z 1 C C1 C P k g J (·)dθE dθC + P (θ + r) g J (·)dθE dθC − ck 2 K−r 0 k−r 0 Z K Z 1 θE E E E E (P − γ) dG (θ ) + (P − γ)k dGE (θE ) 2 k K Z K−r Z 1 C1 C C (θ + r) dGC (θC ) + P P k dGC (θC ) − ck; and 2 k−r K−r
Z
+
E
k−
P k g (·)dθ dθ −
P k − ck g (·)dθ dθ +
+
=
E
k
Z
+
+
E
(P Z
1
C − J
K K−r
+
=
1
(P 0 k Z − γ)k − ck g J (·)dθE dθC +
0 k−r
Z
k− −r
0
Z
1
Z
k−
E[π =P I ] =
Z
k−
E
1
− γ)k g (·)dθ dθ +
k−
0
1
Z
C
− γ)k − − ck − g J (·)dθE dθC (P
0
= Z
Z
k− −r
1Z 1
Z
E
(P k−
P k − − ck − g J (·)dθE dθC +
+
1
Z
0
0 k−
Z
k− −r
Z
E
1
E[π
+P I
Z
k− −r
k−
Z
+ J
E
C
k− −r
Z
K
Z
−ck g (·)dθ dθ +
] = 0
Z
k− −r
1
Z
(P 0 K−r
Z
k− −r K−r
Z
K k−
C
K
(P k− −r K−r
k− 1
Z
+
(P k− −r K 1 Z k−
Z
K−r 1
K
k− 1
Z
+
(P K−r K 1Z K
Z =
0
+
1
K−r K
Z =
E
E
C
− γ)[θE − k − ] + P k + − ck + g J (·)dθE dθC C
− γ)k + + P k + − ck + g J (·)dθE dθC
E
(P k− K−r
Z +
E
−
J
E
1Z 1
Z
C
E
− γ)[θ − k ] g (·)dθ dθ + (P − γ)k + g J (·)dθE dθC k− 0 K Z 1 Z K−r Z 1 C + J C E C P k g (·)dθ dθ + P [θC + r − k − ] g J (·)dθE dθC − ck + (P
Z
C
− γ)k + + P [θC + r − k − ] − ck + g J (·)dθE dθC
C
(P K−r 1
E
C
− γ)[θE − k − ] + P [θC + r − k − ] − ck + g J (·)dθE dθC
0
Z
+ Z
E
P k + − ck + g J (·)dθE dθC
+ Z
− γ)k + − ck + g J (·)dθE dθC
0
+ Z
− γ)[θE − k − ] − ck + g J (·)dθE dθC
P [θC + r − k − ] − ck + g J (·)dθE dθC
+ Z
E
E
k−
0
0
+ Z
(P
k− −r
0 E
− γ)[θE − k − ] dGE (θE ) +
0 1
Z
(P
E
− γ)k + dGE (θE )
K C
P [θC + r − k − ] dGC (θC ) +
k− −r
Z
1
C
P k + dGC (θC ) − ck + .
K−r
Proof of Lemma 5: Using (1) - (4), Lemmas 1 and 2, and that all electricity and capacity auction costs are passed down to consumers, expected consumer surplus is: Z k− −r Z k− Z k− −r Z K E E[CS P I ] = (v − γ)θE g J (·)dθE dθC + (v − P )θE g J (·)dθE dθC 0
Z
0 k− −r
0 1
Z
0
Z
K−r
Z
K K
Z
K−r
k− −r E
Z
k− k−
C
Z
1
E
C
(v − P )K − P (θC + r) g J (·)dθE dθC
K
2
C
(v − γ)θE − P (θC + r) g J (·)dθE dθC
0
k−
+ k− −r
K−r
(v − P )θE − P (θC + r) g J (·)dθE dθC
+ k− −r
E
(v − P )K g J (·)dθE dθC +
+
Z
1
Z
k−
Z
C
E
J
E
1
Z
C
Z
K
K−r 1
Z
0 E
C
k−
K−r 1
Z
E
(v − P )θE − P K g J (·)dθE dθC
(v − γ)θ − P K g (·)dθ dθ +
+
C
(v − P )K − P K g J (·)dθE dθC
+ K−r K 1 Z k−
Z
1Z K
Z
E
(v − P )θE g J (·)dθE dθC 0 0 0 Z 1Z 1 Z K−r Z 1 E C + (v − P )K g J (·)dθE dθC − P (θC + r) g J (·)dθE dθC (v − γ)θE g J (·)dθE dθC +
=
k−
0
k− −r
K 1
Z
1
Z
C
P K g J (·)dθE dθC
− K−r k−
0
0
Z
E
E
K
Z
E
E
(v − γ)θ dG (θ ) +
E
E
Z
E
(v − P )θ dG (θ ) + 0 k− Z K−r Z 1 C C C C C − P (θ + r) dG (θ ) − P K dGC (θC ). =
k− −r
1
E
(v − P )K dGE (θE )
K
(54)
K−r
Proof of Lemma 7: Using (1) - (4), and Lemmas 1 and 6, firm i’s expected profit function is: E[π
−P E
θb1C
Z
k−
Z
0
θb1C
pc+ ∗ k −
1
Z
(P
+ θb1C Z 1
1
Z
(P θb2C
E
E
1
(P k− 1 Z 1
+ θb2C
(P
E[π
θb1C
0
Z
C
P k − − ck − g J (·)dθE dθC
0
C
− γ)k − + P k − − ck − g J (·)dθE dθC
E
− J
E
θb2C
Z
C
− γ)k g (·)dθ dθ +
1
Z
pc+ ∗ k − g J (·)dθE dθC
0
C
P k − g J (·)dθE dθC − ck −
E
−
E
Z
E
− γ)k dG (θ ) +
k J
E
C
Z
−ck g (·)dθ dθ + Z
θb2C
(P
E
θb1C
Z
pc+ ∗ k − dGC (θC )
K
(P 0
1
+ 0
k−
− γ)k − + pc+ ∗ k − − ck − g J (·)dθE dθC
θb1C
0 θb1C
Z
θb2C
θb1C
] = Z
1
Z
C
− ck g (·)dθ dθ +
k−
Z
− γ)k − − ck − g J (·)dθE dθC
0
1
=
=P E
E
k−
1Z
=
Z
E
k−
+
Z
(P k−
0
θb2C
0
− J
1
Z
0 k−
Z
+
Z
θb1C
Z
C
0 θb2C
Z
E
−ck g (·)dθ dθ +
] = Z
− J
E
Z
3
C
P k − dGC (θC ) − ck − ;
θb2C
θE − ck g J (·)dθE dθC 2
θb2C
k
θb1C
K
1
+
− γ)
k
− γ)k − ck g J (·)dθE dθC +
Z
Z 0
1e pc+ ∗ D(·) − ck g J (·)dθE dθC 2
Z
θb2C
Z
K
(P
+ θb1C
Z
+
θb2C
E
− γ)
k
Z
1
θE 1e − ck g J (·)dθE dθC + pc+ ∗ D(·) 2 2
1e − γ)k + pc+ ∗ D(·) − ck g J (·)dθE dθC 2 K Z θbC Z k Z θbC Z K 3 3 θE C1 C E C1 C P (P − γ) (θ + r) − ck g J (·)dθE dθC + +P (θ + r) C C 2 2 2 b b θ2 0 θ2 k Z θbC Z K 3 θE E C1 C ck g J (·)dθE dθC + (P − γ) +P (θ + r) − ck g J (·)dθE dθC C 2 2 b θ2 k Z θbC Z 1 3 E C1 C (θ + r) − ck g J (·)dθE dθC (P − γ)k + P C 2 b K θ2 Z 1Z k Z 1Z K θE C E C J E C (P − γ) P k − ck g (·)dθ dθ + + P k − ck g J (·)dθE dθC 2 θb3C 0 θb3C k Z 1Z 1 E C (P − γ)k + P k − ck g J (·)dθE dθC (P
E
θb1C
+ − + + +
θb3C
K
Z θbC Z 1 Z 1Z 1 2 θE J E C J E C = (P − γ) g (·)dθ dθ + k g (·)dθ dθ + pc+ ∗ × C 2 b 0 k 0 K θ1 0 ! Z Z Z C Z 1 b θ 1 1 e 3 D(·) 1 C g J (·)dθE dθC + P (θC + r) g J (·)dθE dθC + k g J (·)dθE dθC − ck C C 2 2 b b θ2 0 θ3 0 Z K E Z θbC Z 1 2 θ 1e E pc+ ∗ D(·) = (P − γ) dGE (θE ) + k dGE (θE ) + dGC (θC ) C 2 2 b k θ K ! 1 Z θbC Z 1 3 1 C k dGC (θC ) − ck; and + P (θC + r) dGC (θC ) + C C 2 b b θ2 θ3
E[π
+P E
Z
θb1C
Z
k− + J
Z
θb1C
Z
1
K θb2C Z k−
+ θb1C
Z
θb2C
E
θb2C
θb3C
Z
− γ)[θE − k − ] − ck + g J (·)dθE dθC
k−
E
1
E
∗
− + J E C e − γ)[θE − k − ] + pC + [D(·) − k ] − ck g (·)dθ dθ ∗
− + J E C e − γ)k + + pC + [D(·) − k ] − ck g (·)dθ dθ
K
Z
+ θb2C
E
− γ)k + − ck + g J (·)dθE dθC
k−
(P Z
(P
∗
K
+ θb1C
K
− + J E C e pC + [D(·) − k ] − ck g (·)dθ dθ
(P Z
Z
0
Z
+ θb1C
θb1C
Z 0
(P Z
C
0
+ 0
E
−ck g (·)dθ dθ +
] = 0
1Z K
Z
E
k−
C
P [θC + r − k − ] − ck + g J (·)dθE dθC
0
4
θb3C
Z
K
Z
E
(P
+ θb2C θb3C
Z
1
Z
(P
+ θb2C
E
k−
Z
(P
+ θb3C Z 1
C
+ J
1
(P
E
(P
E
− γ)[θE − k − ] + (P
C
− c)k + g J (·)dθE dθC
k−
C
− γ)k + + P k + − ck + g J (·)dθE dθC
Z
1Z K
− γ)
θb2C
1
Z
+ θb1C Z 1 θb3C
−
E
J
E
1Z 1
Z
C
[θ − k ] g (·)dθ dθ + 0
Z
θb3C
+ J
E
k g (·)dθ dθ
k−
e − k − ] g J (·)dθE dθC + pc+ ∗ [D(·)
0 1
Z
θb2C
K 1
Z
C
C
P [θC + r − k − ] g J (·)dθE dθC
0
C
P k + g J (·)dθE dθC − ck +
+ 0
Z
K E
−
E
Z
E
C
Z
1
θb3C
Z k dG (θ ) + +
[θ − k ] dG (θ ) +
− γ) k−
+ P
K
Z
− c)k g (·)dθ dθ +
0
= (P
1
K
E
E
Z
C
0
Z
θb3C
Z
E
θb3C
+ = (P
C
− γ)k + + P [θC + r − k − ] − ck + g J (·)dθE dθC
K
1
Z
C
− γ)[θE − k − ] + P [θC + r − k − ] − ck + g J (·)dθE dθC
k−
E
E
θb1C
K −
C
C
C
+
[θ + r − k ] dG (θ ) +
e − k − ] dGC (θC ) pc+ ∗ [D(·)
!
1
Z
θb2C
C
C
k dG (θ )
− ck + .
θb3C
θb2C
Proof of Lemma 9: Using (1) - (4), Lemmas 1 and 6, and that all electricity and capacity auction costs are passed down to consumers, expected consumer surplus is: Z θbC Z k− Z θbC Z K 1 1 E PE E J E C E[CS ] = (v − γ)θ g (·)dθ dθ + (v − P )θE g J (·)dθE dθC 0
Z
0 θb1C
Z
0 1
0
Z
E
(v − P )K g J (·)dθE dθC +
+ Z
K
+ θb1C
Z
θb2C
k−
Z
1
+ θb1C
Z
θb3C
θb2C θb3C
θb2C θb3C
E
e g J (·)dθE dθC (v − P )θE − pc+ ∗ D(·)
K
Z
k−
C
(v − γ)θE − P (θC + r) g J (·)dθE dθC
0
Z
K
E
C
(v − P )θE − P (θC + r) g J (·)dθE dθC
k−
Z
1
+ θb2C
e g J (·)dθE dθC (v − γ)θE − pc+ ∗ D(·)
0
E
+ Z
k− k−
Z
e g J (·)dθE dθC (v − P )K − pc+ ∗ D(·)
+ Z
θb2C
θb1C
K θb2C
Z
E
C
(v − P )K − P (θC + r) g J (·)dθE dθC
K
5
1
Z
k−
Z
C
E
J
E
Z
C
1
Z
K
θe3C 1
Z
θe3C
0
Z
1
E
C
E
J
E
1Z K
Z
C
(v − γ)θ g (·)dθ dθ +
= 0
1
0
−
θb2C
Z
1
C
P (θC + r) g J (·)dθE dθC −
0
k−
Z
E
E
K
Z
E
0 θb2C
Z −
θb1C
Z
∗ C C e pC + D(·) dG (θ ) −
1
Z
∗ J E C e pC + D(·) g (·)dθ dθ
0
Z
1
Z
θb3C
0
E
E
1
C
P K g J (·)dθE dθC E
E
Z
1
(v − P )θ dG (θ ) +
(v − γ)θ dG (θ ) +
=
θb2C
θb1C
K θb3C
Z
E
(v − P )K g J (·)dθE dθC −
+
Z
E
(v − P )θE g J (·)dθE dθC
k−
0
0 1Z
Z
k−
K
1 Z k−
Z
C
(v − P )K − P K g J (·)dθE dθC
+ θe3C
E
(v − P )θE − P K g J (·)dθE dθC
(v − γ)θ − P K g (·)dθ dθ +
+
k− b θ3C
C
P (θC + r) dGC (θC ) −
Z
K 1
E
(v − P )K dGE (θE ) C
P K dGC (θC ).
(55)
θb3C
θb2C
Continuation Equilibria Extension If θE ∈ (k + , K] or θC > k − − r there exists the potential for multiple equilibria to arise where firm i is the marginal bidder, while firm j bids sufficiently low to ensure that undercutting is unprofitable ∀ i, j = 1, 2 with i 6= j. In the basic model under both CPMs, it was assumed that the large firm is the marginal bidder. In this section, I relax this assumption by considering all continuation equilibria. CPM with Price-Inelastic Demand The bidding behavior in the energy and capacity auctions are analogous to that specified in Lemmas 1 and 2. For the demand regions θE ∈ (k + , K] and θC ∈ (k + − r, K − r] firm i is the marginal bidder with probability ρji in auction ∀ i = 1, 2 where ρj1 + ρj2 = 1 ∀j ∈ {E, C}. Conclusion 1. Firm i’s expected profit ( under CPM-PI: PI πi− if ki = k − ≤ kh = k + , and πiP I (ki , kh ) = PI if ki = k + > kh = k − πi+ where PI πi−
Z
k+
=
(P k− 1
Z +
(P K 1
Z +
E
E
−
E
E
Z
− γ)k dG (θ ) + Z − γ)k − dGE (θE ) +
K
(P
E
E + E − − γ) ρE dGE (θE ) i (θ − k ) + (1 − ρi )k
C
k+ K−r
P
k− −r C
P k − dGC (θC ) − ck − , and
K−r
6
C + C − ρC dGC (θC ) i (θ + r − k ) + (1 − ρi )k
(56)
PI πi+
Z
k+
=
(P k− 1
−
E
E
K
Z
E
− γ)[θ − k ] dG (θ ) +
(P
E
k+
Z +
E
E
(P
− γ)k + dGE (θE ) +
k+ −r
C
P (θC + r − k − ) dGC (θC )
k− −r
K K−r
Z +
Z
E − E + dGE (θE ) − γ) ρE i (θ − k ) + (1 − ρi )k
P
C
k+ −r
C ρC i (θ
−
+ r − k ) + (1 −
+ ρC i )k
C
Z
C
1
C
P k + dGC (θC ) − ck + .
dG (θ ) + K−r
Proof. Using (1) – (4), Lemmas 1 and 2, a full derivation of firm i’s expected profit function is analogous to that in the Proof of Lemma 3. Conclusion 2 details the firms’ investment incentives in the current setting. Conclusion 2. The equilibrium capacity levels (k − , k + ) exists and satisfy: (P
E
− γ)[1 − GE (K)] + (1 − ρE i )(P
C
E
− γ)[GE (K) − GE (k + ) − k − g E (k + )]
C
C C + − C + +P [1 − GC (K)] + (1 − ρC i )P [G (K) − G (k ) − k g (k )] = c, and
(P
E
(57)
− γ)[1 − GE (K) + GE (k + ) − GE (k − ) − k − g E (k − )]
+ (1 − ρE i )(P
E
C
− γ)[GE (K) − GE (k + )] + P [1 − GC (K) + g C (k + ) C
C C + − GC (k − ) − k − g C (k − )]+ (1 − ρC i )P [G (K) − G (k )] = c.
(58)
Proof. Using Theorem 1 in Fabra et al. (2011), an asymmetric PSNE in capacity choices (and no symmetric NE) exist and entail the solution to the firms’ first-order conditions if (i) the game is PI
submodular; (ii) limk+ ↓k
dπi+ dk+
PI
> limk− ↑k
dπi− dk−
PI
PI
dπi+ (0,0) dπi− (1,1) > 0 and < 0. + dk dk− +P I −P I = limk− ↑k πi such that the limk+ ↓k πi E
; and (iii)
Using (56), it is straightforward to show that
function is continuous in the capacity choice. Further, using (56), because P PI
profit
> γ:
PI
dπi+ dπi− E E C C C lim − lim = ρE i (P − γ)kg (k) + ρi P Kg (K) > 0. + − + − k ↓k dk k ↑k dk
(59)
(59) implies that there is a kink in the profit function at the symmetric capacity limit. This implies that the best-reply functions do not cross the diagonal, i.e., there is a discontinuous jump in the best-reply functions. Thus, no symmetric equilibrium in pure strategies exists. In addition, PI
using (56), it is straightforward to show that E −(1 − ρE i )(P
dπi+ (0,0) dk+
C C − γ)g E (1) − (1 − ρC i )P g (1) − c
=P
E
−γ+P
C
PI
− c > 0 and
dπi− (1,1) dk−
=
θj g j (θj )
< 0. Lastly, if ∀ j ∈ {E, C} is increasing, then the game is submodular (for more details see the proof of Propositon 3 in Fabra et al. (2011) and Proposition 4 in de Frutos and Fabra (2011)). This implies that the asymmetric equilibrium exists and satisfied the firms’ first-order conditions. Condition (57) and (58) reflect the both the large and small firms’ profit maximizing capacity decision, respectively. Conclusion 3 details the expected aggregate welfare in the current environment. 7
Conclusion 3. Aggregate expected welfare: # "Z − Z 1 Z K k E E (v − P )K dGE (θE ) (v − P )θE dGE (θE ) + E[W P I ] = α (v − γ)θE dGE (θE ) + − K k 0 Z Z 1 K E E E E E E E (P − γ)K dG (θ ) − cK + (1 − α) (P − γ)θ dG (θ) + k−
Z
K
K−r
− (2α − 1)
C
C
C
C
Z
1
P (θ + r) dG (θ ) + k− −r
P K dG (θ) ) . C
C
C
(60)
K−r
Proof. Using (1) - (4), Lemmas 1 and 2, and that all electricity and capacity auction costs are passed down to consumers, expected consumer surplus is analogous to that in (16). Using (2), (16), and (56), (60) follows directly. C
Under CPM-PI, the regulator chooses the capacity demand parameters (P , r) to maximize expected welfare anticipating how the firms will respond in their subsequent capacity investment decisions and the bidding behavior in the energy and capacity auctions. The optimal capacity payment parameters under CPM-PI follow analogously to those detailed in Lemma 5. CPM with Price-Elastic Demand Next, under CPM-PE I relax the assumption that the large firm is the marginal bidder if θE ∈ (k + , K] or if θC > k − − r with certainty. For a subset of the demand realizations, there exists multiple PSNE where either the small or large firm can be the marginal bidder. Consider the continuation equilibria where if there are multiple equilibria in a demand region, firm i is the marginal bidder with probability ρji ∀ i = 1, 2 in auction j ∈ {E, C}. The bidding behavior in the electricity procurement auction is analogous to that specified in Lemma 1. However, by allowing the small firm to be the marginal bidder, the capacity auction bidding behavior differs from that specified in Lemma 6. The following conclusions will play an important role in characterizing the bidding behavior in the capacity auction under CPM-PE. Conclusion 4. Suppose Assumption 1 holds. Then, pc+ ∗ > pc− ∗ where i h i c e c C ∀ i, j ∈ {−, +} with i 6= j. D(p , θ ; ·) − k pcj ∗ = max p c p
Proof. Using Assumption 1 and (61): " C # " C # b P b P c ∗ C − c ∗ C + p+ = +θ +r−k > p− = +θ +r−k ⇔ k + − k − > 0. 2 b 2 b
(61)
(62)
Conclusion 5. Suppose Assumption 1 holds. Then, there exists a θb2C ∈ (0, 1) such that: e c+ ∗ , θC ; ·) − k − ] pc− ∗ k + R pc+ ∗ [D(p
8
as
θC R θb2C .
(63)
Proof. Using Assumption 1 and Conclusion 4: e c+ ∗ , θC ; ·) − k − ] M (θC ) = pc− ∗ k + − pc+ ∗ [D(p " C " C # #" C # P b P 1 P C + + C − C − = +θ +r−k k − +θ +r−k − (θ + r − k ) 2 b 2 b b
(64)
Using (64), because k + > k − : " C # C 1 P P C − C − 0 C + +θ +r−k − +θ +r−k = k + − k − + θC + r > 0. M (θ ) = k + 2 b b This implies that M (θC ) is monotonically increasing in θC . Therefore, there exists a θb2C where b M (θ2C ) = 0. This implies that M (θC ) R 0 as θC R θb2C . C Conclusion 5 reveals that the large firm will only remain as the infra-marginal bidder (bC + < b− ) if θC ≥ θb2C . Otherwise, it is profitable for the large firm to unilaterally deviate to set the market 0 c ∗ c ∗ clearing price bC + = p+ > p− . Conclusion 6. Suppose Assumption 1 holds. There exists θb1C < θb2C < θb3C < θb4C where: e c = 0, θbC ; ·) = k − ; θb1C : D(p 1
(65)
e c ∗ , θbC ; ·) − k − ]; θb2C : pc− ∗ (θb2C )k + = pc+ ∗ (θb2C )[D(p + 2
(66)
C θb3C : pc+ ∗ (θb3C ) = P ; and
(67)
C θb4C : pc+ ∗ (θb4C ) = P .
(68) C
Proof. Using Assumption 1, Conclusion 4, (1) and (21), θb1C = k − − r − Pb . Using (64): b − b − k − k+ k+ − 0 = k − k + k + < 0. M (θb1C ) = 2 2 From Conclusion 5, because M 0 (θC ) > 0, M (θbC ) < 0, and M (θbC ) = 0, then θbC < θbC . 1
2
1
2
Using Assumption 1, Conclusion 4, (1): C
P θb3C = k − − r + b
(69)
C
P θb4C = k + − r + b
Using (69) and (70), it is straightforward to show that θb3C < θb4C .
(70)
Conclusion 7. Suppose Assumption 1 holds. If the large firm is the marginal bidder bidding C c ∗ bC + = min{P , p+ }, the small firm has no incentive to unilaterally deviate from bidding sufficiently low to ensure that its entire capacity is procured.
9
Proof. Suppose Assumption 1 holds and there is a bid profile in the capacity auction where C c ∗ C C0 c ∗ c bC + = min{P , p+ } > b− . Suppose the small firm unilaterally deviates to b− = min{max{p− , b+ + 0
C
C
c }}, P } for some > 0. From Conclusion 4, pc+ ∗ > pc− ∗ such that bC − = min{b+ + , P }. As → 0, the change in the small firm’s payoff from the capacity auction is: 0
0
C + C − e C C ∆π− = bC − [D(b− , θ ; ·) − k ] − b+ k .
(71)
C C C0 e C 0 , θC ; ·) = θC + r. Hence, and D(b Using Assumption 1 and (1), if bC + = P , then b− = P − C ≤ 0. Similarly, if bC = pc ∗ , then bC 0 = pc ∗ − . As → 0, for any for any θC ≤ K − r, ∆π− + + − + C ≤ 0. θC ≤ K − r the reduction in output dominates the price benefit such that ∆π−
The following conclusion characterizes the equilibrium bidding behavior in the capacity auction when the multiplicity of PSNE are taken into account. Conclusion 8. Equilibrium bidding behavior in the capacity auction with price-elastic demand: 1. If θC ≤ θbC , then the unique PSNE entails both firms bidding at zero and earning a payoff of 1
zero. 2. If θb1C < θC ≤ θb2C , then the PSNE entails the large firm bidding at pc+ ∗ and serving residual demand D(pc+ ∗ , θC ; ·) − k − , while the small firm procures its entire capacity by bidding sufficiently low such that undercutting is unprofitable. 3. If θbC < θC ≤ θbC , then there are multiple PSNE where bC = pc ∗ , while the firm j procures 2
3
i
i
its entire capacity by bidding sufficiently low such that undercutting is unprofitable ∀ i, j ∈ {−, +} with i 6= j. C c∗ 4. If θb3C < θC ≤ θb4C , then there are multiple PSNE where bC i = min{P , pi }, while the firm j procures its entire capacity by bidding sufficiently low such that undercutting is unprofitable ∀ i, j ∈ {−, +} with i 6= j. C 5. If θbC < θC ≤ K − r, then there are multiple PSNE where bC = P , while the firm j 4
i
procures its entire capacity by bidding sufficiently low such that undercutting is unprofitable ∀ i, j ∈ {−, +} with i 6= j. C
6. If θC > K − r, then there are two PSNE where a firm i bids at P , firm h bids sufficiently low such that undercutting is unprofitable, and both firms procure their entire capacities with i, h = 1, 2 and i 6= h. Proof. For θC ∈ [0, θb1C ] or θC ∈ [K − r, 1], the findings follow directly from Proposition 1 in c ∗ Fabra et al. (2006). Suppose θC ∈ (θb1C , θb2C ] and bC + = p+ , while the small firm bids sufficiently low such that a unilateral deviation by the marginal bidder is unprofitable. From Conclusion 7, the inframarginal bidder has no incentive to unilaterally deviate to become the marginal bidder. Further, from Conclusion 5 the small firm will never be the marginal bidder in this demand region as the large firm always has an incentive to unilaterally deviate to set the equilibrium price. C c∗ Suppose θC ∈ [θb2C , K − r] and bC i = min{P , pi }, while the firm j procures its entire capacity by bidding sufficiently low such that undercutting is unprofitable. Thus, the marginal bidder has no incentive to unilaterally deviate to undercut the inframarginal bidder. From Conclusion 5, if i = − and j = +, because θC > θb2C the large firm has no incentive to unilaterally deviate to become 10
the marginal bidder. From Conclusion 7, if i = + and j = −, the small firm has no incentive to unilaterally deviate to become the marginal bidder. For the demand regions θE ∈ (k + , K] and θC ∈ (θb2C , K − r] firm i is the marginal bidder with probability ρji in auction j ∀ i = 1, 2 where ρj1 + ρj2 = 1 ∀j ∈ {E, C}. Conclusion 9. Firm i’s expected profit ( under CPM-PE: PE πi− if ki = k − ≤ kh = k + , and πiP E (ki , kh ) = P E πi+ if ki = k + > kh = k −
(72)
where PE πi−
Z
k+
(P
= k− 1
Z +
(P
E
E
−
E
Z
E
− γ)k − dGE (θE ) +
θb3C
θb2C θb4C
+ θb3C
Z
P θb4C 1
C
C C − ρi (θ + r − k + ) + (1 − ρC dGC (θC ) i )k
C
P k − dGC (θC ) − ck − , and
+ K−r k+
Z =
(P k− 1
E
− γ)[θE − k − ] dGE (θE ) +
(P
E
− γ)k + dGE (θE ) +
Z
+ θb2C
Z
θb4C
+ θb3C
Z
θb2C
θb1C
K θb3C
Z
K
(P k+
Z
Z
pc+ ∗ k − dGC (θC )
C
K−r
+
E + E − − γ) ρE dGE (θE ) i (θ − k ) + (1 − ρi )k
− c ∗ C e + C C P (1 − ρC i )k + p− ρi [D(·) − k ] dG (θ )
Z
PE
E
− c ∗ C e + C C pc+ ∗ (1 − ρC i )k + p− ρi [D(·) − k ] dG (θ )
+
πi+
k+ θb2C
θb1C
+ Z
(P
Z
K
Z
K
− γ)k dG (θ ) +
E
E − E + − γ) ρE dGE (θE ) i (θ − k ) + (1 − ρi )k
e − k − ] dGC (θC ) pc+ ∗ [D(·)
− c ∗ C − C C e pc+ ∗ ρC i [D(·) − k ]p− (1 − ρi )k dG (θ ) C
C − c ∗ C + C C P ρC i [θ + r − k ]p− (1 − ρi )k dG (θ )
K−r
P
+ θb4C 1
Z +
C
C C + ρi (θ + r − k − ) + (1 − ρC dGC (θC ) i )k
C
P k + dGC (θC ) − ck + .
K−r
Proof. Using (1) – (4) and Lemma 1 and Conclusion 8, a full derivation of firm i’s expected profit function is analogous to that in the Proof of Lemma 7. Conclusion 10 details the firms’ investment incentives in the current setting. 11
Conclusion 10. The equilibrium capacity levels (k − , k + ) exists and satisfy: E
E
C
E E + − E + C − γ)[1 − GE (K)] + (1 − ρE i )(P − γ)[G (K) − G (k ) − k g (k )] + P [1 − G (K)] "Z bC # Z K−r c ∗ θ4 dp C + (1 − ρC pc− ∗ + −+ k + dGC (θC ) + P dGC (θC ) = c, and (73) i ) dk θb4C θb2C
(P
E
E
E E + − γ)[1 − GE (K) + GE (k + ) − GE (k − ) − k − g E (k − )] + (1 − ρE i )(P − γ)[G (K) − G (k )] Z θbC 3 dpc+ ∗ − C C C C bC C c ∗ )P G (K − r) − G ( θ ) + (1 − ρ ) k dGC (θC ) p + + P [1 − GC (K)] + (1 − ρC i 3 i + − C dk b θ2 h i C b dθ2 C c ∗ bC − e c ∗ , θbC ; ·)] = c. + ρ p+ (θ2 )k − pc− ∗ (θb2C )[D(p (74) − 2 dk − i
(P
Proof. Using Theorem 1 in Fabra et al. (2011), an asymmetric PSNE in capacity choices (and no symmetric NE) exist and entail the solution to the firms’ first-order conditions if (i) the game is dπ +
PI
dπ −
PI
dπ +
PI
PI
(0,0)
submodular; (ii) limk+ ↓k dki + > limk− ↑k dki − ; and (iii) i dk+ > 0 and Using (61), (64) - (68), the following conditions follow directly:
dπi− (1,1) dk−
< 0.
lim pc− ∗ = lim pc+ ∗
(75)
lim lim θb2C = lim θb1C
(76)
lim θb3C = lim θb4C
(77)
k− ↑k
k+ ↓k
k− ↑k k+ ↓k
k− ↑k
k− ↑k
k+ ↓k
PE
PE
Using (72) and (75) – (77), it is straightforward to show that limk+ ↓k πi+ = limk− ↑k πi− such that the profit function is continuous in the capacity choice. Further, using (61), (64) - (68), E ∗ bC bC (75) – (77), that pC + (θ1 ) = 0, M (θ2 ) = 0 defined in (64), and (72), because P > γ: PI
PI
dπi+ dπi− E E lim lim − = ρE i (P − γ)kg (k) > 0. + − + − dk k ↓k k ↑k dk
(78)
(78) implies that there is a kink in the profit function at the symmetric capacity limit. This implies that the best-reply functions do not cross the diagonal, i.e., there is a discontinuous jump in the best-reply functions. Thus, no symmetric equilibrium in pure strategies exists. In addition, PI E C ∗ dπ + (0,0) = P −γ +P −c > 0 using (72) and that pC (θbC ) = 0, it is straightforward to show that i + PI
dπi− (1,1) dk−
+ E
1
dk
γ)g E (1)
θj g j (θj )
and = −(P − − c < 0. Lastly, if ∀ j ∈ {E, C} is increasing, then the game is submodular (for more details see the proof of Propositon 3 in Fabra et al. (2011) and Proposition 4 in de Frutos and Fabra (2011)). This implies that the asymmetric equilibrium exists and satisfied the firms’ first-order conditions. Condition (73) and (74) reflect the both the large and small firms’ profit maximizing capacity decision, respectively.
12
Assume firm 1 is the small firm. Define ρj1 to be the probability that firm 1 is the marginal bidder in auction j ∈ {E, C}. Conclusion 11 details the expected aggregate welfare in the current environment. Conclusion 11. Aggregate expected welfare under CPM-PE: "Z − # Z K Z 1 k E E E PE E E E E E E E E[W ] = α (v − γ)θ dG (θ ) + (v − P )θ dG (θ ) + (v − P )K dG (θ ) 0 Z k− K Z 1 K E E (P − γ)K dGE (θE ) − cK + (1 − α) (P − γ)θE dGE (θ)E + k−
Z − (2α − 1)
K θb2C
θb1C
e c ∗ , θC ; ·) dGC (θC ) pc+ ∗ D(p +
c ∗e c ∗ C C C +ρC 1 p− D(p− , θ ; ·) dG (θ )
Z
K−r
+
Z
θb4C
+ θb3C
Z
θb3C
+ θb2C
c ∗e c ∗ C (1 − ρC 1 )p+ D(p+ , θ ; ·)
C
C C C c ∗e c ∗ C C ρC 1 p− D(p− , θ ; ·) + (1 − ρ1 )P (θ + r) dG (θ )
P (θ + r) dG (θ ) + P K[1 − G (K − r)] . C
C
C
C
C
C
(79)
θb4C
Proof. Using (1) - (4), Lemma 1 and Conclusion 8, and that all electricity and capacity auction costs are passed down to consumers, expected consumer surplus equals:49
E[CS
PE
Z
k− E
E
Z
E
(v − γ)θ dG (θ ) +
]=
E
E
E
Z
E
Z −
θb2C
θb1C
C
C
Z
K−r
+ r) dG (θ ) +
E
(v − P )K dGE (θE )
K
e c ∗ , θC ; ·) dGC (θC ) pc+ ∗ D(p +
c ∗e c ∗ C C C +ρC 1 p− D(p− , θ ; ·) dG (θ )
C C ρC 1 )P (θ
1
(v − P )θ dG (θ ) + k−
0
+(1 −
K
Z
θb3C
+ θb2C
Z
θb4C
+ θb3C
c ∗e c ∗ C (1 − ρC 1 )p+ D(p+ , θ ; ·)
c ∗e c ∗ C ρC 1 p− D(p− , θ ; ·)
P (θ + r) dG (θ ) + P K[1 − G (K − r)] . C
C
C
C
C
C
(80)
θb4C
(79) follows directly from (72) and (80). C
Under CPM-PE, the regulator chooses the capacity demand parameters (P , r, b) to maximize expected welfare anticipating how the firms will respond in their subsequent capacity investment decisions and the bidding behavior in the energy and capacity auctions. Using (79), the characterC ization the optimal parameters (P , r) are analogous to those specified in Lemma 9. However, to show that the findings of the basic model are robust to an environment with multiple equilibria, it will be important to provide conditions under which b∗ ∈ (0, ∞). Further characterization is difficult for general demand distributions. Therefore, attention is focused on a setting in which capacity and electricity demand is sufficiently high. Suppose P (θE ≤ 49
A full characterization of the consumer surplus in this setting is analogous to (24).
13
k + ) = 0 and P (θC ≤ θb2C ) = 0. This condition ensures that we focus on demand realizations in which no single firm can supply electricity or capacity demand. Because electricity demand reflects the maximum peak demand (and capacity demand reflects a forecast of this maximum peak demand), this assumption is not restrictive in practice. Conclusion 12. Suppose Assumption 1 holds and P (θE ≤ k + ) = P (θC ≤ max{θb2C , k − − r}) = 0. If α > 21 , P (k − − r < θC ≤ θb4C ) is sufficiently large, and P (θ ≤ K − r) = 1, then b∗ ∈ (0, ∞). Proof. Suppose Assumption 1 holds, α >
1 2
and P (θE ≤ k + ) = P (θC ≤ θb2C ) = 0. Using (79):
i dK h dE[W P E ] E C = α (v − P )[1 − GE (K)] − P [1 − GC (K − r)] db db i dK h E C +(1 − α) (P − γ)[1 − GE (K)] + P [1 − GC (K − r)] − c db ! Z θbC e c ∗ , θC ; ·) 3 dpc− ∗ D(p − ρC −(2α − 1) 1 db θb2C ! ! Z θbC c ∗ , θ C ; ·) c ∗ D(p c ∗ , θ C ; ·) c ∗ D(p e e 4 dp dp + + − − C C C C C C +(1 − ρ1 ) dG (θ ) + ρ1 dG (θ ) . db db θb3C
(81)
where using (28), (82) simplifies to: i dK h dE[W P E ] E C = α (v − P )[1 − GE (K)] − P [1 − GC (K − r)] db db h i dK E C +(1 − α) (P − γ)[1 − GE (K)] + P [1 − GC (K − r)] − c db ! 2 Z θbC C 4 1 b + dk + 1 P 1 C + + (θ + r − k )k − k − + (θC + r − k + )2 dGC (θC ) −(2α − 1) ρC 1 C 2 2 db 4 b 4 b θ2 ! Z θbC C 2 − 3 1 b dk 1 P 1 C C − − − − 2 C C C +(1 − ρ1 ) (θ + r − k )k − k − + (θ + r − k ) dG (θ ) . (82) 2 db 4 b 4 θb2C 2 −
+
It will be necessary to determine the signs of dkdb and dkdb . Using (73) and (74) and that P (θE ≤ k + ) = P (θC ≤ θb2C ) = 0 by assumption, comparative statics yields the following system: " dk− # A B C × dkdb+ = (83) D E F db where: A = −ρE 1 (P
E E
C
C C C bC − C bC − γ)g E (K) − ρC 1 P g (K − r) − (1 − ρ1 )[G (θ3 )b + k g (θ3 )]; C
E C C B = −ρE 1 (P − γ)g (K) − ρ1 P g (K − r);
(84) (85)
14
C = −(1 −
D=
ρC 1)
"Z
θb3C
θb2C
E −(1 − ρE 1 )(P
# C 1 C P − C bC − − C C (θ + r − k )k dG (θ ) + k g (θ3 ) ; 2 2b
− γ)g
E
C C (K) − (1 − ρC 1 )P g (K
(86) (87)
− r);
E C C E C C C bC C bC C bC E = −(1 − ρE 1 )(P − γ)g (K) − (1 − ρ1 )P g (K − r) − ρ1 [bG (θ4 ) + g (θ4 ) + bG (θ4 )]; (88) "Z bC # C θ4 P 1 F = −ρC (θC + r − k + )k + dGC (θC ) + g C (θb4C )k + . 1 2b θb2C 2 (89) Under the maintained assumptions, A, B, C, D, and E are negative. Using Cramer’s Rule: dk − dk + AF − CD CE − BF = = (90) db AE − BD db AE − BD
where using (84) - (87): ih i h E E C C C C C bC C C bC − γ)g (K) + ρ P g (K − r) 2bρ G ( θ ) + ρ g ( θ )+ AE − BD = ρE (P 1 1 4 1 4 1 +(1 −
ρC 1)
i h E C bC C bC − E (1 − ρE G (θ3 )b + g (θ3 )k 1 )(P − γ)g (K)
C C C C bC C C bC g (K − r) + 2bρ G ( θ ) + ρ g ( θ ) > 0. +(1 − ρC )P 1 4 1 4 1
(91)
(90) and (91) implies: dk − s = CE − BF. db
dk + s = AF − CD db
(92) −
−
Recall, A, B, C, D, and E are negative. If F is sufficiently positive, then dkdb < 0 and dkdb ?0. F is sufficiently negative when P (θb2C ≤ θC ≤ θb4C ) is sufficiently large such that the (negative) integral term in (89) is sufficiently large such that F > 0.50 Further, using (84) – (89), (91) when F is sufficiently positive: dK dk + dk − AF − CD + CE − BF s = + = = (A − B)F − C(D − E) db db db AE − BD = −(1 − ρC )F [GC (θbC )k − ] − ρC C[2bGC (θbC ) + g C (θbC )] < 0. 1
3
1
4
PE
4
(93)
PE
] ] > 0 and limb→∞ dE[W < 0 such that b∗ ∈ (0, ∞). It will now be shown that limb→0 dE[W db db Using (61), (64) - (68), and (75) – (77), as b → 0 it is straightforward to show that θb1C , pc− ∗ , and pc+ ∗ → 0 and θb2C , θb3C , and θb4C → ∞. Therefore, suppose P (θC ≤ K − r) = 1 and P (θb2C ≤ θC ≤ θb4C ) is sufficiently large, using (82) because dK db < 0:
dE[W P E ] dK = −(1 − α)c > 0. b→0 db db lim
(94)
Using (61), (64) - (68), and (75) – (77), as b → ∞ it is straightforward to show that θb1C = 50
This condition reflects demand realizations that result in a market clearing price on the sloped portion of the demand curve.
15
C k − − r, pc− ∗ = pc+ ∗ = P , θb3C = k − − r, and θb4C = k + − r. Therefore, suppose P (θE ≤ k + ) = P (θC ≤ + k − − r) = 0 and P (θb2C ≤ θC ≤ θb4C ) is sufficiently large, using (82), because dkdb < 0 and dK db < 0:
dK (2α − 1) dE[W P E ] = −(1 − α)c − lim b→∞ db db 4 + (θC + r − k + )2 −
Z
θb4C
2(θC + r − k + )k +
k− −r
dk + ∞ dGC (θC ) = −∞ < 0. db
(95)
(94) and (95) implies that b∗ ∈ (0, ∞). Conclusion 12 provides plausible conditions where the optimal slope parameter is interior. Capacity demand must be distributed such that the probability that the equilibrium capacity auction price is on the sloped portion of the demand curve is sufficiently high.51 Proposition 5 follows directly from this result. Therefore, the core results of the model are robust to the consideration of multiple equilibria in the capacity and electricity auction. The CPM-PE outperforms the CPM-PI in terms of expected welfare.
Installed Generation Capacity Extension The basic model focused on a setting in which there are two entrants choosing their capacity limits. In practice, in addition to the potential entrants, there is often a set of heterogeneous installed generation units. In this section, I consider an environment where the new entrants are peak-load generation technologies, while there is a set of installed generation units. Assume there are two incumbents with installed generation units {1, 2} with installed capacities k1 and k2 , respectively. Further, there are two potential entrants {3, 4} who will choose their capacity limits ki ∀ i = 3, 4 at a cost c > 0 per-unit of capacity. It is without loss of generality to assume that the two incumbents have a constant marginal electricity generation costs γ1 < γ2 up to their capacity limits. Further, the entrants’ have constant marginal electricity generation costs γE > γ2 up to their limits. Pcapacity j Define k(j) = k to be the capacity of the first j generation units ∀ j = 1, 2, 3, 4. Define i=1 i P4 K = i=1 ki to be the aggregate capacity level. Define k − = min{k3 , k4 } and k + = max{k3 , k4 } to be the minimum and maximum entrant’s capacity, respectively. The timing and structure of the game is analogous to that in the basic model where the incumbents’ capacities are taken as fixed. Analogous to the entrants’ bids in the basic model, the incumbents submit a bid into both the energy and capacity auctions. Conclusion 13. Equilibrium bidding behavior in the electricity auction: 1. If θE ≤ k(1), then the unique PSNE entails firm 1 supplying all electricity demand at marginal cost γ2 . 2. If k(1) < θE ≤ k(2), then there are multiple PSNE where firm i bids at γE and supplies residual demand θE − kj , while firm j procures its entire capacity by bidding sufficiently low such that undercutting is unprofitable ∀ i, j = 1, 2 with i 6= j. 51
This assumption is supported empirically. In practice, the capacity auction price has always been on the sloped portion of the demand curve.
16
3. If k(2) < θE ≤ k(2)+k − , then there is a unique PSNE where the entrants bid at γE , while the incumbents procure their entire capacities by bidding sufficiently low such that undercutting is unprofitable. 4. If k(2) + k − < θE ≤ k(2) + k + , then there is a unique PSNE where the large entrant bids E at P and supplies residual demand θE − k(2) − k − , while the all other firms procure their entire capacities by bidding sufficiently low such that undercutting is unprofitable. E
5. If k(2) + k + < θE ≤ K, then there are multiple PSNE where a bidder i bids at P and supplies residual demand, while the all other firms j procure their entire capacities by bidding sufficiently low such that undercutting is unprofitable ∀ i, j = 1, 2, 3, 4 with i 6= j. E
6. If θE > K, then there are multiple PSNE where a bidder i bids at P and supplies its entire capacity, while the all other firms j procure their entire capacities by bidding sufficiently low such that undercutting is unprofitable ∀ i, j = 1, 2, 3, 4 with i 6= j. Proof : Follows directly from Proposition 1 in Fabra et al. (2006). If k(1) < θE ≤ k(2) or k(2)+k + < θE ≤ 1, there are multiple PSNE. I focus on the environment where: (i) if k(1) < θE ≤ k(2), the least efficient incumbent is the marginal bidder and (ii) if k(2) + k + < θE ≤ 1, the large entrant is the marginal bidder. The Continuation Equilibrium Extension reveals that the core findings of the analysis are robust to considering continuation equilibrium in which the small firm can set the market clearing price. Further, the findings are robust to the environment where the most efficient incumbent sets the market clearing price in the region k(1) < θE ≤ k(2). CPM with Price-Inelastic Demand Next I consider the bidding behavior in the capacity auction with price-inelastic demand. Recall, at the capacity auction stage, capacity investment decisions are taken as given such that the cost of capacity investments are sunk. Conclusion 14. The equilibrium bidding behavior in the capacity auction: 1. If θC ≤ k(2) + k − − r, then the unique PSNE where all bidders bid at zero. 2. If k(2) + k − − r < θC ≤ k(2) + k + − r, then there is a unique PSNE where the large entrant C bids at P and supplies residual demand θC + r − k(2) − k − , while the all other firms procure their entire capacities by bidding sufficiently low such that undercutting is unprofitable . C
3. If k(2) + k + − r < θC ≤ K − r, then there are multiple PSNE where a bidder i bids at P and supplies residual demand, while the all other firms j procure their entire capacities by bidding sufficiently low such that undercutting is unprofitable ∀ i, j = 1, 2, 3, 4 with i 6= j. E
4. If θC > K − r, then there are multiple PSNE where a bidder i bids at P and supplies its entire capacity, while the all other firms j procure their entire capacities by bidding sufficiently low such that undercutting is unprofitable ∀ i, j = 1, 2, 3, 4 with i 6= j. Proof : Follows directly from Proposition 1 in Fabra et al. (2006). If θC > k(2) + k + − r, there are multiple equilibria. I focus on the environment where the large firm sets the market clearing price. The Continuation Equilibrium Extension reveals that the core findings of the analysis are robust to considering continuation equilibrium in which the small firm can set the market clearing price. 17
Conclusion 15. The firm’s profit functions are defined as follows:52 E[π1P I ]
k(1)
Z
E
E
E
Z
E
Z
k(1) 1
k(2)+k− −r
k(2)
k(2)+k−
(96)
Z (γE −γ2 )[θ −k(1)] dG (θ )+ E
k(2)+k−
Z (γE −γ2 )k2 dG (θ )+ E
E
1
1
E
k(2)+k−
k(2)
k(1)
Z
E
(P −γ1 )k1 dGE (θE )
P k1 dGC (θC );
E
=
1
C
+
Z
Z
E
(γE −γ1 )k1 dG (θ )+
(γ2 −γ1 )θ dG (θ )+
= 0
E[π2P I ]
k(2)+k−
E
(P −γ2 )k2 dGE (θE )
C
+ k(2)+k− −r
P k2 dGC (θC );
(97)
−P I πi πi= P I πiP I (ki , kh ) = π+P I i
if ki = k − < kh = k + , if ki = kh = k, and if ki = k + > kh = k −
(98)
where PI πi−
πi= P I
πi+
PI
Z
1
=
E
(P k(2)+k− K
−
E
Z
E
− γE )k dG (θ ) +
1
C
P k − dGC (θC ) − ck − ,
k(2)+k− −r
Z 1 1 E E E E (P − γE )k dGE (θE ) = (P − γE ) (θ − k(2)) dG (θ ) + 2 K k(2)+k Z K−r Z 1 1 C C + (θC + r − k(2)) dGC (θC ) + P P k dGC (θC ) − ck, and 2 k(2)+k−r K−r Z K Z 1 E E E − E E = (P − γE )[θ − k − k(2)] dG (θ ) + (P − γE )k + dGE (θE ) Z
E
k(2)+k− K−r
Z
C
P [θC + r − k − − k(2)] dGC (θC ) +
+ k(2)+k− −r
Z
K 1
C
P k + dGC (θC ) − ck + .
K−r
∀ i, h = 3, 4 with i 6= h. Proof : Using the bidding behavior in Conclusions 13 and 14, the characterizations of the firms’ profit functions follow analogously to those in Lemmas 3 and 7. The introduction of the installed generation units does not change the strategic nature of the entrants’ capacity investment decision. That is, there is a discontinuity at symmetric capacities eliminating the potential for a symmetric equilibrium. Conclusion 16 details the firms’ investment incentives in any equilibrium. Conclusion 16. The equilibrium capacity levels (k − , k + ) always exists and satisfy:: (P 52
E
C
− γ)[1 − GE (K)] + P [1 − GC (K − r)] = c, and
(99)
It is assumed that if the entrants have symmetric capacities, they are the marginal bidder with probability
18
1 . 2
Z
1
(P
E
k(2)+k−
Z
K
(P
=
E
k(2)+k− Z K−r
− γE )k − dGE (θE ) +
Z
1
−
E
C
P k − dGC (θC )
k(2)+k− −r E
E
1
Z
− γE )[θ − k − k(2)] dG (θ ) + C
−
C
C
Z
C
K 1
k(2)+k− −r
− γE )k + dGE (θE )
C
P k + dGC (θC ).
P [θ + r − k − k(2)] dG (θ ) +
+
E
(P
(100)
K−r +P I
] Proof : From Lemma 1, the equilibrium capacity levels (k − , k + ) satisfy: dE[π = 0 and π + dk+ PI ck + = π − + ck − . Using (98), (99) and (100) follow directly from these conditions.
Next, in order to understand the regulator’s incentives to choose P terize the aggregate welfare function.
C
PI
+
and r, we need to charac-
Conclusion 17. Aggregate expected welfare under CPM-PI: E[W P I ] = α
k(1)
Z
Z (v−γ2 )θE dGE (θE )+
0
Z
1
+
E
K
Z
k(1)
Z (γ2 −γ1 )θ dG (θ )+
k(2)+k− E
E
+
K
Z
k(2) 1
k(2)
E
γE θE −γ1 k1 −γ2 [θE −k(1)]dGE (θE )
k(1)
(γE −γ1 )k1 +(γE −γ2 )k2 dG (θ )+ Z
E
0
+
E
k(2)+k−
P θE −γ1 k1 −γ2 k2 −γE [θE −K(2)] dGE (θE )
Z P K −γ1 k1 −γ2 k2 −γE (k +k ) dG (θ )−c(k +k ) +(1−2α) E
−
+
E
−
E
K−r
C
+
P θC dGC (θC )
k(2)+k− −r
K
Z
E
(v−P )θE dGE (θE )
k(2)+k−
E
E
K
Z (v−γE )θE ) dGE (θE )+
k(1)
Z (v−P )K dG (θ ) +(1−α) E
k(2)+k−
1
P KdG (θ ) . C
+
C
C
K−r
(101) Proof : Using (1) - (4), (96) – (98), Conclusions 13 and 14, and that all electricity and capacity auction costs are passed down to consumers, expected consumer surplus is: E[CS P I ] =
Z
K(1)
(v−γ2 )θE dGE (θE )+
0
Z
1
Z
k(2)+k−
k
Z
E
E
E
"Z
K−r
C
C
C
C
Z
K
1
P (θ + r) dG (θ ) + k(2)+k− −r
E
(v−P )θE dGE (θE )
k(2)+k−
k(1)
(v − P )K dG (θ ) −
+
(v−γE )θE dGE (θE )+
# C
C
C
P K dG (θ ) .
(102)
K−r
(101) follows directly from (2), (96) – (98) and (102). C
Using (101), the characterization of the optimal capacity demand parameters (P , r) under CPM-PI are analogous to those specified in Lemma 5.
19
CPM with Price-Elastic Demand Next I consider the bidding behavior in the capacity auction with price-elastic demand. The electricity auction bidding behavior with installed generation capacity is analogous to that specified in Conclusion 13. Recall, at the capacity auction stage, capacity investment decisions are taken as given such that the cost of capacity investments are sunk. Throughout the analysis, suppose Assumption 1 holds. Conclusion 18. The equilibrium bidding behavior in the capacity auction:53 1. If θC ≤ θb1C , then the unique PSNE where all bidders bid at zero. 2. If θb1C < θC ≤ θb2C , then there is a unique PSNE where the large entrant bids at pc+ ∗ and e C ∗ , ·) − k(2) − k − , while the all other firms procure their entire supplies residual demand D(p + capacities by bidding sufficiently low such that undercutting is unprofitable. C 3. If θbC < θC ≤ θbC , then a PSNE entails the large entrant bidding at P and supplying residual 2
3
demand θC + r − k(2) − k − , while the all other firms procure their entire capacities by bidding sufficiently low such that undercutting is unprofitable. E 4. If θC > θb3C , then there are multiple PSNE where a bidder i bids at P and supplies its entire capacity, while the all other firms j procure their entire capacities by bidding sufficiently low such that undercutting is unprofitable ∀ i, j = 1, 2, 3, 4 with i 6= j.
where ∗ pC +
:
∗ C e C D(p + , θ ; ·)
−
− k − k(2) +
∗ dd(·) pC + c
e
dp
=0 ⇒
∗ pC +
" C # b P C − = + θ + r − k − k(2) ; 2 b (103) C
e c = 0, θb1C ; ·) = k − + k(2) ⇒ θb1C = k(2) + k − − r − P ; θb1C : D(p b
(104)
C
P C ⇒ θb2C = k(2) + k − − r + θb2C : pc+ ∗ (θb2C ) = P ; b
(105)
e c = P C , θbC ; ·) = K ⇒ θbC = K − r; θb3C : D(p 3 3 (106) Proof : Follows directly from Proposition 1 in Fabra et al. (2006). Using the electricity and capacity auction bidding behavior, Conclusion 19 characterizes the firms’ profit functions. Conclusion 19. The firm’s profit functions are defined as follows:54 E[π1P E ]
Z
k(1) E
E
E
Z
k(2)+k−
(γ2 −γ1 )θ dG (θ )+
= 0
E
E
(γE −γ1 )k1 dG (θ )+ k(1)
Z
1
k(2)+k−
E
(P −γ1 )k1 dGE (θE )
53 As show in the Continuation Equilibria extension, there exists multiple equilibria for a subset of the region C bC b (θ1 , θ3 ]. However, I focus on the environment where the large firm is the marginal bidder in this section. An analysis analogous to that in the Continuation Equilibria section could be carried out for the current setting to reveal that the results are robust to this assumption. 54 It is assumed that if the entrants have symmetric capacities, they are the marginal bidder with probability 12 .
20
θb2C
Z +
θb1C
E[π2P E ]
Z
∗
C C pC + k1 dG (θ ) +
1
Z
θb2C
C
P k1 dGC (θC );
k(2) E
E
E
k(2)+k−
Z
E
k(2)
k(1) θb2C
+ θb1C
Z
E
(γE −γ2 )k2 dG (θ )+
(γE −γ2 )[θ −k(1)] dG (θ )+
= Z
(107)
∗ C C pC + k2 dG (θ )
Z
1
+ θb2C
1
k(2)+k−
E
(P −γ2 )k2 dGE (θE )
C
P k2 dGC (θC );
−P E πi PE πi= P E πi (ki , kh ) = π+P E i
(108) if ki = k − < kh = k + , if ki = kh = k, and if ki = k + > kh = k −
(109)
where πi−
PE
Z
1
=
(P
E
k(2)+k−
πi= P E
− γE )k − dGE (θE ) +
Z
θb2C
θb1C
∗
− C C pC + k dG (θ ) +
Z
1
C
P k − dGC (θC ) − ck − ,
θb2C
Z 1 1 E E E E (P − γE )k dGE (θE ) = (P − γE ) (θ − k(2)) dG (θ ) + 2 K k(2)+k Z θbC Z C b h i θ3 2 ∗1 C1 C e − k(2) dGC (θC ) + + pC D(·) P (θ + r − k(2)) dGC (θC ) + C C 2 2 b b θ1 θ2 Z 1 C + P k dGC (θC ) − ck, and Z
K
E
θb3C
πi+
PE
Z
K
=
(P
E
k(2)+k−
Z
θb2C
+ θb1C 1
Z +
− γE )[θE − k − − k(2)] dGE (θE ) +
Z
1
(P
E
− γE )k + dGE (θE )
K
Z i h − C C C∗ e p+ D(·) − −k k(2) dG (θ ) +
θb3C
C
P (θC + r − k − − k(2)) dGC (θC )
θb2C
C
P k dGC (θC ) − ck + .
θb3C
∀ i, h = 3, 4 with i 6= h. Proof : Using the bidding behavior in Conclusions 13 and 18, the characterizations of the firms’ profit functions follow analogously to those in Lemmas 3 and 7. The introduction of the installed generation units does not change the strategic nature of the entrants’ capacity investment decision. That is, there is a discontinuity at symmetric capacities eliminating the potential for a symmetric equilibrium. Conclusion 20 details the firms’ investment incentives in any equilibrium. Conclusion 20. The equilibrium capacity levels (k − , k + ) always exists and satisfy:: (P
E
C
− γ)[1 − GE (K)] + P [1 − GC (K − r)] = c, and
21
(110)
Z
1
(P
E
k(2)+k−
Z
K
(P
=
E
k(2)+k−
Z
θb2C
+ θb1C Z 1
+
−
E
Z
E
− γE )k dG (θ ) +
θb2C
θb1C
∗ − C C pC + k dG (θ )
− γE )[θE − k − − k(2)] dGE (θE ) +
Z
1
(P
Z
1
C
P k − dGC (θC )
+
E
θb2C
− γE )k + dGE (θE )
K
Z h i ∗ e − −k − k(2) dGC (θC ) + pC D(·) +
θb3C
C
P (θC + r − k − − k(2)) dGC (θC )
θb2C
C
P k dGC (θC ).
(111)
θb3C +P E
] = 0 and π + Proof : From Lemma 1, the equilibrium capacity levels (k − , k + ) satisfy: dE[π dk+ PE ck + = π − + ck − . Using (109), (110) and (111) follow directly from these conditions.
PE
+
C
Next, in order to understand the regulator’s incentives to choose P , b, and r, we need to characterize the aggregate welfare function. Conclusion 21. Aggregate expected welfare under CPM-PE: E[W P E ] = α
Z
k(1)
Z (v−γ2 )θE dGE (θE )+
0
Z
1
+
k(1)
Z (v−P )K dG (θ ) +(1−α) E
E
E
Z (γ2 −γ1 )θ dG (θ )+ E
k(2)+k− E
+
K
Z
E
k(2) 1
E
k(2)
E
γE θE −γ1 k1 −γ2 [θE −k(1)]dGE (θE )
k(1)
(γE −γ1 )k1 +(γE −γ2 )k2 dG (θ )+ Z
E
k(2)+k−
P θE −γ1 k1 −γ2 k2 −γE [θE −K(2)] dGE (θE )
Z P K −γ1 k1 −γ2 k2 −γE (k +k ) dG (θ )−c(k +k ) +(1−2α) E
−
+
E
−
E
+
θb3C
+
θb2C
θb1C
K
Z
C
C
C
Z
C
1
P (θ +r) dG (θ )+ θb2C
E
(v−P )θE dGE (θE )
k(2)+k−
0
+
K
Z (v−γE )θE dGE (θE )+
k(1)
K
Z
k(2)+k−
∗ C C e C∗ pC + D(p+ , ·) dG (θ )
P K dG (θ ) . C
C
C
θb3C
(112) Proof : Using (1) - (4), (107) – (109), Conclusions 13 and 18, and that all electricity and capacity auction costs are passed down to consumers, expected consumer surplus is: E[CS P E ] =
Z
K(1)
(v−γ2 )θE dGE (θE )+
Z
0 1
Z
E
E
E
Z
1
Z
θb2C
θb1C
k
+
(v−γE )θE dGE (θE )+
Z
k(1)
(v − P )K dG (θ ) −
+
k(2)+k−
∗ C C e C∗ pC + D(p+ , ·) dG (θ )
C C C P K dG (θ ) .
Z
θb3C
+
K
E
(v−P )θE dGE (θE )
k(2)+k−
C
P (θC + r) dGC (θC )
θb2C
(113)
θb3C
22
(112) follows directly from (2), (107) – (109) and (113). C
Using (112), the characterization of the optimal capacity demand parameters (P , r) under CPM-PE are analogous to those specified in Lemma 9. However, to show that the findings of the basic model are robust to an environment with installed generation capacity, it will be important to provide conditions underwhich b∗ ∈ (0, ∞). Conclusion 22. Suppose Assumption 1 holds. Then, if α = 21 , b∗ ∈ (0, ∞]. If α > 21 and P ( θb1C ≤ θC ≤ k(2) + k − − r ) < P ( k(2) + k − − r < θC ≤ θb2C ) b∗ is interior. Otherwise, b∗ = ∞. Proof : Using (112): " # Z θbC C ∗ e 2 dp D(·) dE[W P E ] dk − E + = (2α−1) (P − γ)[k(2) + k − ]g E (k(2) + k − ) − dGC (θC ) = 0. C db db db b θ1 (114) Suppose Assumption 1 holds. If α = 12 , then (114) holds for any b∗ ∈ (0, ∞]. Suppose α > 12 . It will be shown that there is an interior solution if P ( θb1C ≤ θC ≤ k(2) + k − − r ) < P ( k(2) + k − − r < θC ≤ θb2C ). Otherwise, expected welfare is monotonically increasing in b and hence, b∗ = ∞. It will be shown that if P ( θb1C ≤ θC ≤ k(2) + k − − r ) < P ( k(2) + k − − r < θC ≤ θb2C ), then dE[W P E ] dE[W P E ] lim > 0 and lim < 0 for some eb < ∞ such that there does not exist a concern b→0 db db b→e b solution. Suppose α > 12 , using (114): Z θbC C ∗ e − 2 dp dE[W P E ] s E + D(·) − E − dk − dGC (θC ). = (P − γ)[k(2) + k ]g (k(2) + k ) db db db θb1C (115) Using (27) – (29), in the current environment with installed generation capacity (115) simplifies: b dE[W P E ] s dk − E [k(2)+k − ] (P − γ)g E (k(2) + k − ) + [GC (θb2C ) − GC (θb1C )] = db db 2 1 + 4
Z
θb2C
θb1C
C
P b
!2 −2(θC +r−k(2)−k − )[k(2)+k − ]− θC + r − −k(2) − k −
2
dGC (θC ).
(116) needs to be identified. Using (110) it is straightforward to show that To analyze (116), dK dk− dk+ c ∗ bC db = 0 such that db = − db . Using (103) - (106) and that p+ (θ1 ) = 0, implicitly differentiating 55 (111) yields: dk− db
Z θbC 2 dk − E C (P − γ)[2(1 − GE (K)) + GE (K) − GE (k(2) + k − ) − k − g E (k(2) + k − )] + P dGC (θC ) C db b θ1 Z
θb2C
+ θb1C 55
3 b(θC + r − [k − + k(2)]) dGC (θC ) + 2
Z
θb2C
A full derivation is analogous to that detailed in (30) - (33).
23
1
C
2P dGC (θC )
Z
θb2C
+ θb1C
dk − ⇒ = −A db
1 C (θ + r − k − k(2))[k(2) + k − ] + 2
"Z
θb2C
θb1C
P
C
− pc+ ∗ b
1 C (θ + r − k − − k(2))[k(2) + k − ] + 2
P
!
C
pc+ ∗ dGC (θC ) = 0 b ! # − pc+ ∗ pc+ ∗ dGC (θC ) b b (117)
E
− γ)[2(1 − GE (K)) + GE (K) − GE (k(2) + k − ) − k − g E (k(2) + k − )] + R1 C b(θC + r − 23 [k − + k(2)]) dGC (θC ) + θbC 2P dGC (θC ).
where A = (P
R θb2C θb1C
P
C
+
2
Notice, θC + r − k(2) − k − Q 0 as θC Q k(2) + k − − r. Further, using (104) and (105), C C bC θb2 − (k(2) + k − − r) = k(2) + k − − r − θb1C = Pb such that θc 1 and θ2 are equidistant from k(2) + k − − r. This implies that if P (θb1C ≤ θC ≤ k(2) + k − − r) < P (k(2) + k − − r < θC ≤ θb2C ), R θbC − then bC2 θC + r − −k(2) − k − dGC (θC ) > 0. Further, this implies that dkdb ≤ 0. θ1
PE
] > 0 in the current setting follows analogously from (35). From The result that limb→0 dE[W db R θbC (116), if P (θb1C ≤ θC ≤ k(2 + k − − r) < P (k(2) + k − − r < θC ≤ θb2C ), then bC2 θC + r − k(2) − θ C 2 1 − k − dGC (θC ) > 0 and dkdb ≤ 0 such that the sole positive term in (116) is Pb [GC (θb2C )−GC (θb1C )].
dE[W P E ] Thus, for a large (finite) value of b (eb) this term is sufficiently small such that lim < 0. db b→e b This implies that b∗ ∈ (0, ∞). Propositions 5 and 6 follow directly from this result. Therefore, the core results of the model are robust to the addition of installed generation capacity. The addition of elastic demand reduces the payments to both the installed and new entrants in the capacity auction during demand periods that solely reflect a rent transfer from producers to consumers. This strengthens the findings that CPM-PE outperforms CPM-PI under plausible conditions.
24
