1)the dominant firm choose k, capacity needed for a fixed input production

2)the fringe firms(small firm) enter determining n, number of firms

3)the dominant firm chooses q, its output

4)the small firms choose x, its output

then we may define

Players={A dominant firm, many small firm}

Strategy of the dominant={k,n,q,x}

Strategy of the small firm={x}

Backward induction

At 4), A Fringe(small) firm

max P(q + nx)x-c(x)

f.o.c P(q + nx)=c'(x)

At 3), The dominant firm

max P[(q + nx) + r]q-kC(q/k)

s.t P(q + nx)=c'(x)

At 2), number of small firms are determined by Zero profit condition

P(q + nx)x-c(x)-R(k+n) ==> c'(x)x-c(x)=R(k+n)

where R is a fixed input cost for the final good and R is a increasing function of k and n. A dominant firm may affect zero profit condition by increasing or decreasing its k.

At 1), the dominant firm chooses (k,n)

-max [P(q + nx) + r]q-kC(q/k)-R(k+n)(k-k0) where k0 is the capacity that the dominant firms has initially.

The above is rough description of EQ in this game.

Lemma 1; x is decreasing in k and n. Each output of the fringe firm decreases as capacity and number of firms increase

Using P(q + nx)=c'(x) and c'(x)x-c(x)=R(k+n) which are the EQ in At 4) and At2),

we may rewrite 1st stage problem into [c'(x) + r]q(x,n)-kC(y)-[c'(x)x-c(x)](k-k0)

then optimal k* satifying II=[c'(x) + r]q(x,n)-kC(y)-[c'(x)x-c(x)](k-k0)

f.o.c dII/dk=0

k* is optimal capacity that the firm has. K is industry capacity as a whole

we also define welfare function as W and find optimal dW/dk

Note:

The capacity of the dominant firm can be regarded as the degree of the vertical integration.

The more k dominant firm has, the higher prices will be realized in both intermediate and final goods. In addition, the amount of k that the dominant firm has, will affect the condition of small firms entry so the number of firms is a function of k.

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Published by seandaddy

Ph.D
Department of Economics
Bloomsburg University of Pennsylvania
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