Question 1

PartA

Demand for electricity is given by \(p(Q)=A-Q\), where: \[Q=Q_r+\sum_{n=1}^{i=3}K_i\]

Our residual demand function is:\[Q_r=\sum_{n=1}^{i=3}q_i\]

Therefore \(p(Q)=A-\sum_{n=1}^{i=3}q_i-\sum_{n+1}^{i=3}K_i\).

Our profit maximisation function with respect to \(q_1\) can be written as: \[\max \Pi_i=(A-q_1-2q_j-\sum_{n=1}^{i=3}K_i-c)q_i\]

First Order Condition: \(A-2q_i-2q_j-\sum_{n=1}^{i=3}K_i-c=0\)

Second Order condition: \(-2\).

Since firms are symmetrical, \(q_i=q_j\).

\[\therefore q_i=\frac{A-\sum_{n=1}^{i=3}K_i-c}{4}\] \[p(Q)=A-\sum_{n=1}^{i=3}q_i-\sum_{n=1}^{i=3}K_i\]

\[p(Q)=A-\frac{3(A-\sum_{n=1}^{i=3}K_i-c)}{4}-\sum_{n=1}^{i=3}K_i\] \[p(Q)=\frac{A-\sum_{n=1}^{i=3}K_i+3c}{4}\]

Figure 1

Here is an example of a cournot equilibrium with two identical firms:

Part B

\[Q_1=q_1+K_1\]  

\[Q_1=\frac{A+3K_1-K_2-K_3-c}{4}\]

Therefore the marginal effect of \(K_1\) on Firm 1's production quantity is: \[\frac{dQ_1}{dK_1}=-\frac{3}{4}.\]

Given that firm's are symmetrical, the marginal effect of \(K_1\) on Firm 2's production quantity is: \[\frac{dQ_2}{dK_1}=-\frac{1}{4}.\]

Result

We see that adopting a forward contract has a positive effect on that firm's total production levels, while it decreases the amount produced by other competing firms in the market.

Question 2

• In period 2 of the game, firms compete in a spot market identical to the market we solved for in question 1.
• Due to arbritrage, the forward contract price F in period 1 must be identical to the spot price in period 2.

Therefore the spot price for period 1 is given by: \[F=\frac{A-\sum_{n=1}^{i=3}K_i+3c}{4}\]

As follows, the firm's first period profit function is given by:

\[\Pi_i=(p(Q)-c)q_i+(F-c)K_i\]

In the first period, there are no existing forward contracts. Then \(Q=\sum_{n=1}^{i=3}q_i\),

Our profit maximisation problem with respect to \(K_i\) is given by:

\[max \Pi_i=(A-\sum_{n=1}^{i=3}q_i-c)q_i+(\frac{A-\sum_{n=1}^{i=3}K_i+3c}{4}-c)K_i\]

FOC: \(\frac{A-2K_i-2K_j-c}{4}=0\)

SOC:\(-2\)

Since firms are symmetrical, \(K_i=K_j\) and \(K_i=\frac{A-c}{4}\).

\[\therefore F=\frac{A-\sum_{n=1}^{i=3}K_i+3c}{4}=\frac{A-\frac{3(A-C)}{4}+3c}{4}\] \[F=\frac{A+15c}{8}\]

When we plug \(K_i\) into \(q_i\),

\[q_i=\frac{A-C}{8}\] \[Q_i=\frac{3(A-c)}{4}\]

Question 3

• In both papers, the authors try to show whether firms have strategic incentives for forward contracts in energy markets.
• They seek to identify whether firms sell future contracts for hedging reasons or for strategic reasons.

• Brandt et al. (2008), on the other hand, focus only on strategic incentives in quantity competition and supply function competition.
• Mainly, they try to figure out how forward contracts help to reduce prices in spot markets by increasing quantity produced.

• Results of the paper show that the introduction of forward markets lead to a decline in prices and a rise in quantity produced.
• This situation mitigates the market power of firms and leads to more competition in the market.

• As indicated in the van Eiken and Morraga-Gonzales paper, every firm has an incentive to enter the forward market, as aggregate production is higher than in the absence of a forward contract.

Figure 2

Empirical data on electricity markets in Europe: