Assumptions and General Remarks

Collusive behavior of firms can occur, if and only if, there is no incentive for any firm to deviate from the collusive agreement unilaterally. If the net present value of profits gained by collusion is greater or equal than the net present value of profits gained by ending collusion, no incentive for any firm to break the collusive agreement unilaterally exists. The highest profit that can be earned in each period is the monopoly profit, which is shared equally by both (symmetric) firms.

It is assumed that firms face a linear demand function (D =a−p) and bear constant marginal costs c. The exact outcome of prices, quantities and profits is stochastic and depends on the difference between the reservation priceaand marginal costsc. I do not distinguish between demand and supply shocks. The difference between the reservation price and marginal costs (γ = a−c) will be denoted "spread" in the analysis. The first

two moments of this “spread” are given byE[γ]andV[γ]. In order to give comparative static results, the König-Huygens theorem is used later in this paper to decompose the expectation of the squared “spread” into its variance and its squared expectation (E[γ²] = V[γ] +E[γ]²).

Whenever I use monopoly prices, quantities and profits for the argumentation, I refer to monopoly prices, quantities and profits for a given realization of the stochastic difference between reservation price and marginal costs. As shown by Liski and Montero (2006, p.

226) assuming a linear demand function is possible without loss of generality. I denote the price, quantity and profit associated with the one-period monopoly solution byp^M = ^a+c₂ , q^M = ^a−c₂ andΠ^M = (p^M −c)q^M = ^(a−c)₄ ².

The spot and the forward market are connected as follows: During the first period, both firms simultaneously choose the amount of forward contracts they want to trade (for-ward market period). During the second period, contracts are settled and firms choose the amount they additionally want to sell on the spot market (spot market period). The forward market opens in the even periods (t = 0,2, . . .) and the spot market in the odd periods (t = 1,3, . . .). To maintain comparability with pure-spot market games the per period discount factor is given by√

δ. Alternatively the spot market opens for a marginal unit of time right after the forward market and the discount factor is given by δ. The important fact is that discounting only takes place between two spot markets, two forward markets or the forward market int and the spot market in t+ 1. Hence, no discounting takes place between consecutive forward and spot markets. The structure of trading ini-tially on the forward market and settling contracts afterwards as well as meeting residual demand on the spot market is infinitely repeated. One can think of firms deciding around Christmas each year about forward contracts to be delivered in the following year.

Firms compete in prices and sell a homogenous product, which seems a valid assump-tion especially for electricity markets. Whenever firm isets a price lower than its com-petitorj firmimeets the whole spot market demand. However, this spot market demand is decreased by the total amount of forward contracts that have been sold before by firm i and by firm j (F˜ = ˜fi+ ˜fj). Thus, the demand to serve and the spot market profit can be

stated as:

D^SM_i =

a−F˜−pi

Π^SM_i = (pi−c)

a−F˜−pi

(3.1)

When prices are equal, firms split the market equally. Consider the following trigger strategy to ensure collusive behavior: In the first forward market round (period 0), firm i sells an amount of contracts that are settled right in the next period,f˜_i^0,1, and sells no contracts that are settled in subsequent periods, f˜_i^0,l = 0 for all l > 1, wherel denotes the period of delivery. Hence firms only sell forward contracts that will be settled in the following spot market, since no forward contracts with delivery in t > 1 are sold (f˜_i^0,l = 0 ∀ l >1). In this following spot market period firmisets the monopoly price (p^t_i =p^M) if and only if in every period preceding period t both firms have set monopoly prices in the spot market and have contracted the collusive amount f˜_i^0,1 = ˜f_j^0,1 = ˜f in the forward market one period ahead. Whenever firmjdeviates from this agreement, firm isets a price equal to marginal costs in the spot market and sells any arbitrary amount of forward contracts forever. This can be seen as the grim trigger strategy for games, where firms are allowed to trade on a spot as well as on a forward market. It corresponds to the grim trigger strategy analyzed by Friedman (1971), when firms were solely allowed to trade on a spot market.

Liski and Montero (2006) do not allow forward contracts exceeding monopoly quantity in their model of forward trading and collusion in a deterministic market structure. How-ever, in a volatile market, firms do not know in any forward market period the demand and cost structure they will face in the following spot market period. Hence, firms might have traded forward more than the quantity they can sell with monopoly prices on spot market. This may happen e.g. for a relatively small realization of the difference of reser-vation price and marginal costs. Therefore the critical discount factor will be derived for the forward traded amount being less than monopoly quantity as well as for the forward traded amount being larger than monopoly quantity.