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## 经济代写|产业经济学代写Industrial Economics代考|Corollary

Under assumptions 1 and 2 there is at most one Cournot equilibrium with strictly positive outputs for each firm.

At the Cournot price there is only one vector of sales quantities, the Cournot vector, that does not result in incentives to offer non-selective discounts. At any list price below the Cournot price, there is a range of firms’ market shares for which such discounts are unprofitable. For example, the unilateral output expansion incentives are negative at a point such as $X$ in figure $8.1$ and at all points between $W$ and $Y$ on the constant-industry-output hyperplane through $X$. The incentive to offer a small non-selective discount decreases as one moves along any ray away from the origin. Hence, at an industry output higher than that at $X$, the range of market shares without such incentives to discount would include some market shares that are outside the range of market shares determined by points on the line $W Y$. In this sense the common contract price $\bar{p}$ becomes more stable with respect to discounting as it falls from the Cournot price towards the competitive price.

The preceding analysis pertains to the unprofitability of discounting in the neighbourhood of the list price $\bar{p}$. Even if small discounts are unprofitable, large discounts may be profitable in the presence of significant cost asymmetries. For example, suppose that two firms’ marginal costs differ, but that each is constant, and that each firm’s sales quantity is equal to $\bar{Q} / 2$ at the common list price $\bar{p}$. It will be profitable for the low-cost firm to offer a discount that is slightly below the other’s marginal cost if the resulting profits for the low-cost firm at an output of $Q$ exceed the firm’s profits at price $\bar{p}$ and quantity $\bar{Q} / 2$. It is easy to construct numerical examples in which the point $(\bar{Q} / 2, \bar{Q} / 2)$ is to the right of the firms’ reaction functions in output space and a large discount is profitable. Such below-cost discounts are unprofitable when firms have identical cost functions.

## 经济代写|产业经济学代写Industrial Economics代考|Equilibrium

The remaining issue to be considered with respect to non-selective discounting is whether supracompetitive prices that are impervious to discounting can be established in an equilibrium of the two-stage game. A strategy for firm $i$ in the two-stage game involves public advance notification of a list price, denoted $\bar{p}i$, in the first stage and a discount price, $p{d i}(\bar{p})$, for the second stage that can depend on the prevailing list price, $\bar{p}$, at which contracts are signed. Since $\bar{p} \leq \bar{p}i$, a complete specification of the $i$ th firm’s strategy requires only that $p{d i}(\bar{p})$ be defined for $\bar{p} \leq \bar{p}i$. For example, the (two-stage) strategy $\left(\bar{p}_i, p{d i}(\bar{p})=\bar{p}\right.$ for $\left.\bar{p} \leq \bar{p}_i\right)$ would involve setting the discount price equal to the prevailing list price. We shall refer to this strategy as a ‘no-discount strategy’.

A firm will not refuse sales at the common list price if that price exceeds its marginal cost. But marginal costs will depend on the allocation of buyers at the common list price. We make the following assumption about demand allocation.

When all firms charge the same list price, buyers randomly select sellers.
Recall that there is a continuum of buyers, so that this assumption implies that firms’ market shares will be equal when the common list price exceeds firms’ marginal costs at the resulting outputs. (As the number of balls goes to infinity, the fraction of balls in each of a fixed number, $n$, of urns goes to $1 / n$.) Assumption 4 allows us to characterize equilibria for the symmetric-cost case.

# 产业经济学代考

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