Lesiones cutáneas como factores pronósticos

Quantity competition, in which rms take demand and forward contracts as given

In the third stage, each firm’s prohibitive price and forward contracted amount is given.

Thus, the profit of each firm can be stated as:

Π_i = (a− xi− xj)

x_i− ˜f_i

− (c − Ii) x_i (4.20)

Remark: In the context of advertising, the profit function should rather look likeΠi = (a+

Ii−xi−xj)(xi− ˜fi)−cxi. To ensure comparability with the long-term investment decision, the profit function used above is taken. This can be done without loss of generality, since both profit functions are equivalent.

The best quantity response of a firm due to the quantity set by the competitor is found by maximizing the profit function with respect to the quantityxi.

∂Π_i The quantities set by each firm in Nash-equilibrium is given byx^∗_i,j. The quantities depend on both firms’ forward contracted amount and both firms’ investment.

xi = 1

With the equilibrium quantitiesx^∗_i,x^∗_j the spot market pricep^∗_sm in the third stage can be

Investment decision, in which rms take forward contracts as given

In the second stage firms decide about their investments, knowing each firm’s forward contracted amount and anticipating the quantity decision each firm makes in the third stage. Thus, the profit functions can be reduced to a relationship of forward contracts, amount of investment, marginal costs and the prohibitive price and look as follows:

Πi = (p^∗_sm− c + Iⁱ) x^∗_i − Ii² The best investment response of each firm in the second stage due to the investment of the competitor is given by: Each firm’s investment depends positively on the prohibitive pricea and on its own for-ward traded amount ˜fi. Each firm’s investment depends negatively on the competitor’s forward traded amount ˜fi, the competitor’s investmentIj and on marginal costsc.

Remark: Again, for the cost-cutting interpretation positive marginal costs after the investment have to be ensured. When interpreting the investment decision in the second stage as advertising, this is not necessary. Therefore and to ensure comparability with the results of section 4.3.2, this is not explicitly modeled in the presented model.

The investment chosen by each firm in Nash-equilibrium is given byI_i^∗, I_j^∗ and found in the intersection of the best investment reaction functions.

I_i = 1

Each firm’s investment depends positively on the prohibitive pricea and its own forward traded amount ˜fi. It depends negatively on the competitor’s forward traded amount ˜fj. With the equilibrium of investments the quantities of each firms can easily be determined as: Each firm’s quantityxi depends positively on the prohibitive pricea and positively on its own forward traded amount ˜f_i. It depends negatively on the competitor’s forward traded amount ˜fj and marginal costsc.

The equilibrium price in the second stage is easily determined either by inserting these quantities into the inverse linear demand function or by inserting the second-stage forward traded amount into the equilibrium spot market price (equation 4.23).

p^∗ = a− x^∗i − x^∗j

The price in the second stage depends positively on the prohibitive pricea and the marginal costsc. It depends negatively on each firm’s forward traded amount ˜fi, ˜fj.

Decision on forward contracts, under anticipation of investment and spot market competition

In the first stage firms decide about the amount of forward contracts they supply on the market. In doing so they perfectly anticipate the consequences on both firms’ investments as well as on the quantity supplied on the spot market. Thus, the profit can be reduced to a function solely dependent on each firm’s amount contracted forward as well as the fundamental market conditions, which are given by marginal costs and the prohibitive price. The profit function is given by the contribution surplus multiplied by the amount sold to the market less the investment costs resulting from both firms’ positions on the forward market. The optimal amount of forward contracts of each firm in the second stage due to the forward contracted amount of the rival is given by:

∂Πi

The Nash-equilibrium forward traded amount is found in the intersection of both firms’

forward contract best response function.

f˜i = 1

The subgame-perfect quantity is found by inserting the subgame-perfect forward traded amount ˜fi

∗ (equation 4.31) into the subgame-perfect quantity on the 2 stage (equation 4.27):

The subgame-perfect investment is found by inserting the subgame-perfect forward traded amount ˜fi

∗ (equation 4.31) into the subgame-perfect investment on stage 2 (equation 4.26):

The subgame-perfect price is found by inserting the subgame-perfect forward traded amount ˜f_i^∗ (equation 4.31) into the inverse demand function:

p^∗ = a− xi− xj

The subgame-perfect profit is found by inserting the subgame-perfect price (equation 4.34), investment (equation 4.33) and quantity (equation 4.32) into the profit function (equation 4.29):

The consumer surplusσ can be easily determined using the subgame-perfect price (equa-tion 4.34) each firms’ quantity (equa(equa-tion 4.32):

σ = 1

The social welfareω can be determined by summing up the consumer surplus and each firms profit:

Note: For each subgame-perfect outcome of the decision structure forward trading, in-vestment and quantity competition the letters ˜F , I, Q are added. This will be helpful to compare the market results with the results for other structures of decision. For example, pF ,I,Q˜ means the price that emerges, when (as described in this section) first forward trad-ing, then the investment, and afterwards quantity competition takes place.

xF ,I,Q˜ = 18

Table 4.1 gives as a benchmark the market outcome for the decision structure, first in-vestment and then quantity competition, as well as for the decision structure first forward trading and then quantity competition (Allaz and Villa, 1993). The derivation of all re-sults for the case of investment and then quantity competition is shown by equation B.1 to equation B.7 in the appendix. The derivation of all results for the case of forward trading and then quantity competition is shown by equation B.8 to equation B.14 in the appendix.

Table 4.2 gives the market outcome for the decision structure, first forward trading,