# 经济代写|产业经济学代写Industrial Economics代考|ECON4438

#### Doug I. Jones

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

We use the traditional model of Cournot oligopoly with homogeneous goods. Demand is given by $p(X)$, where $p$ is price, $X$ is industry output, and $p^{\prime}(X)<0$; we write for the absolute value of the elasticity of demand, $\varepsilon(X) \equiv-p(X) / X p^{\prime}(X)$. The number of firms, $n$, is exogenous (although of course it changes with a merger), reflecting some important barriers to entry ${ }^8$ We denote firm i’s cost function by $c^i\left(x_i\right)$, where $x_i$ is firm $i$ ‘s output. For notational ease, we write $c^i \equiv c^i\left(x_i\right)$ for firm $i$ ‘s total cost and $c_x^i \equiv c_x^i\left(x_i\right)$ for firm $i$ ‘s marginal cost. Importantly, we permit the firms to differ in efficiency.

In the Cournot equilibrium, each firm $i$ picks its output $x_i$ to maximize its profits, given its rivals’ outputs. Writing $y_i \equiv \sum_{j \neq i} x_j=X-x_i$ for aggregate output of all firms other than firm $i$, firm $i$ ‘s profits are $\pi^i\left(x_i, y_i\right) \equiv p\left(x_i+y_i\right) x_i-c^i\left(x_i\right)$. Firm $i$ ‘s first-order condition, $\partial \pi^i / \partial x_i=0$, is
$$p(X)+x_i p^{\prime}(X)-c_x^i\left(x_i\right)=0, \quad i=1, \ldots, n$$
A Cournot equilibrium is a vector $\left(x_1, \ldots, x_n\right)$ such that equation (2) holds for all $n$ firms. We denote firm i’s market share by $s_i \equiv x_i / X$.

Comparing two firms $i$ and $j$, the Cournot equilibrium conditions, (2), tell us that $x_i>x_j$ if and only if $c_x^i<c_x^j$. In equilibrium, larger firms have lower marginal costs. In any Cournot equilibrium in which different firms produce different quantities, marginal costs differ across firms, so that costs are not minimized given the aggregate output level, and consequently aggregate output, $X$, is not in general a sufficient statistic for welfare.

Throughout the chapter, we make two weak assumptions on the Cournot equilibrium; we require both to hold throughout a relevant range, as will become clear below. First, we assume that each firm’s reaction curve slopes downward. Equivalently, an increase in rivals’ output, $y_i$, lowers firm $i$ ‘s marginal revenue
$$p^{\prime}(X)+x_i p^{\prime \prime}(X)<0, \quad i=1, \ldots, n .$$
Inequality (3) is a very weak assumption and is standard in Cournot analysis; see Dixit (1986) and Shapiro (1989). It holds if the industry demand curve satisfies $p^{\prime}(X)+X p^{\prime \prime}(X)<0 . .^9$

## 经济代写|产业经济学代写Industrial Economics代考|Price effects of horizontal mergers

We model a merger as a complete combination of the assets and of the control of the merging firms, whom we call the ‘insiders’. After the merger, a new Cournot equilibrium is established between the merged entity $M$ and the non-participant firms, whom we call the ‘outsiders’. In this section, we examine the effect of a merger on aggregate output, $X$. This is, of course, the central question if merger analysis is concerned only with consumer welfare (ignoring consumers’ ownership of profits). As we shall see in section 3 , it is also an essential component of an analysis of overall economic welfare, $W$.

Mergers differ enormously in the extent to which productive assets can usefully be recombined, and in the extent to which output decisions can usefully, or anticompetitively, be coordinated. At one extreme, consider a production technology in which all firms have constant and equal marginal costs, and the merged entity has the same costs. In this special case, mergers are purely anticompetitive: there is no other motive. For a slightly rosier view, recall that by equation (2), firms’ marginal costs typically differ in Cournot equilibrium, so that a merger may offer an opportunity to rationalize production – that is, without changing total output, to shift output to the facility with lower marginal cost. A still sunnier view is that mergers may create synergies. For example, two firms that own complementary patents may combine and produce much more efficiently than either could alone (without a licensing agreement).

For a theory of horizontal mergers to be useful for policy purposes, it should be general enough to allow for all these possibilities, which can be captured in assumptions about the relationship between the merged entity’s cost function, $c^M(\cdot)$, and those of the insiders. Our theory is very general in this regard, as we make no a priori assumptions on $c^M($.$) beyond those implied by conditions (3) and (4). Our first$ proposition gives a necessary and sufficient condition on $c^M(\cdot)$ for equilibrium output to fall with the merger.

# 产业经济学代考

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

$$p^{\prime}(X)+x_i p^{\prime \prime}(X)<0, \quad i=1, \ldots, n .$$

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