Players=(m upstream firms, n downstream firms, government)

m upstream firms (produce a homogeneous input at a constant) marginal cost k

the input is then sold to n downstream firms

Upstream |
produce input |
marginal cost k |
input price is c |

Downstream |
produce output using input |
only input price is its marginal cost c |

-Timing of events-

First, the upstream firms choose quantities which results in a market clearing input price, c

Second, the downstream firms choose quantities taking the input price as given

Backward Induction

From utility maximization problem, firm has inverse demand function as Eq.(2)

then EQ quantity is (4)

We derived EQ quantity in Downstream market.

Let’s move on the upstream. Let Q denote total factor demand, i.e. Q=nq*. Thus inverse demand function for the upstream market is given

Upstream firm maximizes profits

where . under symmetric assumption

EQ market quantity and input price is derived

Pulling together Q** and c(Q) gives

we have (Q**,c**), market EQ of input price and quantity.

Given this upstream EQ price, the downstream SPNE price and quantity are given by:

(,)

Then we get the welfare,

2.2 Post Merger EQ

– After merging, intermediate inputs are supplied to the downstream firm at the cost of k.

– the merging downstream firm will be more aggressive as it is operating at a low marginal cost, k while other downstream firms spend more cost at c. thus k=<c

– Net welfare depend on the original market structure and the degree of horizontal product differentiation

Then country’s welfare function is defined as follows:

Proposition 1:

Vertical integration increases welfare if the number of downstream firms, n, is smaller or equal to the number of upstream firms, m.

Welfare depends on the level of product differentiation

4. The antitrust Market Definition

Suppose that a subset of firms form a coalition. An Intervention or antitrust policy controls and affects this level of ;

that is, the authority intervene the mark-up of firms in coalition, not controlling the number of firms directly.

: the price charged by firms in coalition (upstream + downstream).

In case of coalition, as we did previous, we derived EQ quantity, price and number of firms clearing market. ()

Some comparative statics results are as follows:

1) The number of firms in coalition increases in the value of . Larger mark-up, more firms in coalition.

2) The degree of horizontal product differentiation and the size of antitrust policy have negative relationship.

If you know about this, you may refer to 13 page in the paper.

Note:

After merging, k=<c, which means intermediate good can be supplied cheaper than other firms.

The upstream firm merged by the downstream cannot charge the price over the marginal cost.

Antitrust policy is not directly controlling the number of firms but the ratio of mark-up. However, by doing so, the number of firms can be controlled.