Product Differentiation in a Vertical Structure

A: Bertrand Competition Under Free Entry in the Input Sector

For the inverse market demand function P i = 1 − α q i − g ∑ j = 1 i ≠ j n q j , where α = 1 + s ( n − 1 ) ( 1 − g ) , the corresponding demand function is q i = α − g − p i ( α + g ( n − 2 ) ) + g ∑ i ≠ j p j ( α − g ) ( α + g ( n − 1 ) ) .

Given the input price w, the ith final goods producer maximises (p _i  − w)q _i with respect to price. The equilibrium price and output can be found as p i b ∗ = − ( 1 + ( n − 1 ) s ) ( 1 + w ) + g ( 1 + 2 w − n w + ( n − 1 ) ( 1 + w ) s ) ( − 2 + 3 g − g n − 2 ( 1 − g ) ( n − 1 ) s ) and q i b ∗ = ( 1 − 2 g + g n + ( 1 − g ) ( n − 1 ) s ) ( 1 − w ) ( 2 − 3 g + g n + 2 ( 1 − g ) ( n − 1 ) s ) ( 1 + g ( n − 1 ) ( 1 − s ) − s + n s ) , i = 1, 2, …, n, respectively.

The total input demand is n q i b ∗ = I = n ( 1 − 2 g + g n + ( 1 − g ) ( n − 1 ) s ) ( 1 − w ) ( 2 − 3 g + g n + 2 ( 1 − g ) ( n − 1 ) s ) ( 1 + g ( n − 1 ) ( 1 − s ) − s + n s ) , which gives w b = n − ( 2 − g ) I − g n I − 2 s I ( 1 − g ) ( n − 1 ) + g 2 ( n − 1 ) I ( 1 − 2 g + g n + ( 1 − g ) ( n − 1 ) s ) n .

If m input suppliers entered the market, the kth input supplier determines its output to maximise Max I k ( w b I k ) , where I = ( I k + ∑ z = 1 k ≠ z m I z ) . The equilibrium input production is I k b ∗ = n ( 1 + n − 1 s + g − 2 + n + s − n s ) 1 + m − 2 + 3 g − g n − 2 s 1 − g n − 1 − 1 − g n − 1 1 − s + s − n s . Hence, for a given number of input suppliers, m, the equilibrium input price is w b ∗ ( m ) = 1 1 + m , which is independent of g. This is similar to the result under Cournot competition in the final goods market and as mentioned in section 2.1, it is due to the reason provided in Dhillon and Petrakis (2002).

Given the input price, w b ∗ ( m ) = 1 1 + m , the equilibrium profit of the kth input supplier who entered the market is π I k b = n ( 1 + ( n − 1 ) s + g ( − 2 + n + s − n s ) ) ( 1 + m ) 2 ( − 2 + 3 g − g n − 2 s ( 1 − g ) ( n − 1 ) ) ( − 1 − g ( n − 1 ) ( 1 − s ) + s − n s ) − K . Hence, the free entry equilibrium number of input suppliers is given by π I k b = n ( 1 + ( n − 1 ) s + g ( − 2 + n + s − n s ) ) ( 1 + m ) 2 ( − 2 + 3 g − g n − 2 s ( 1 − g ) ( n − 1 ) ) ( − 1 − g ( n − 1 ) ( 1 − s ) + s − n s ) − K = 0 or

Assume K < 1 8 so that at least one input supplier enters the market for any values of n and g.

The equilibrium profit of the ith final goods producer is π i b ∗ ( m b ∗ ) = ( K ( 1 − g ) ( 1 + ( n − 1 ) s ) ( 2 − 3 g + g n + 2 ( 1 − g ) ( n − 1 ) s ) n ( 2 − 3 g + g n + 2 s ( 1 − g ) ( n − 1 ) ) 2 ) ( − 1 + n ( 1 + ( n − 1 ) s + g ( − 2 + n + s − n s ) ) K n ( 1 − 2 g + g n + ( 1 − g ) ( n − 1 ) s ) ( − 2 + 3 g − g n − 2 ( 1 − g ) ( n − 1 ) s ) ( − 1 − g ( n − 1 ) ( 1 − s ) + s − n s ) ) 2 .

Differentiating π i b ∗ ( m b ∗ ) with respect to g and evaluating it at g = 0, we get ∂ π i b ∗ ( m b ∗ ) ∂ g | g = 0 = ( n − 1 8 n 2 ( 1 + ( n − 1 ) s ) 4 ) ( n 2 ( 1 + ( n − 1 ) s ) − 2 K n ( 1 + ( n − 1 ) s ) 3 ) ( − n ( 1 − s ) 2 ( 1 + ( n − 1 ) s ) + K n ( 1 + ( n − 1 ) s ) 3 ) , which is positive for K > 2 n ( 1 − s ) 2 1 + ( n − 1 ) s ≡ K ∗ . Hence, g = 0 is not the preferred differentiation for K ∈ ( K ∗ , 1 8 ) . Since we are looking at the change in profit with respect to g and evaluating it at g = 0, it is intuitive that the critical conditions are the same under Cournot and Bertrand competition since the type of competition does not matter at g = 0.

We also get ∂ π i b ∗ ( m b ∗ ) ∂ g | g = 1 = − [ n − n 2 + n ( n − 1 ) K ] 2 ( 1 + ( n − 1 ) s ) n 3 ( n − 1 ) 3 < 0 , implying that g = 1 is not the preferred differentiation.

Hence, if K ∈ ( K ∗ , 1 8 ) , neither g = 0 nor g = 1 is the preferred differentiation, implying firms prefer moderate differentiation in this situation.

B: Bertrand Competition Under Monopolist Input Supplier with Economies of Scale

Given the demand function, the ith final goods producer maximises ( p i − w ) q i with respect to price. The equilibrium price and output can be found as p i b i ∗ = − ( 1 + ( n − 1 ) s ) ( 1 + w ) + g ( 1 + 2 w − n w + ( n − 1 ) ( 1 + w ) s ) ( − 2 + 3 g − g n − 2 ( 1 − g ) ( n − 1 ) s ) and q i b i ∗ = ( 1 − 2 g + g n + ( 1 − g ) ( n − 1 ) s ) ( 1 − w ) ( 2 − 3 g + g n + 2 ( 1 − g ) ( n − 1 ) s ) ( 1 + g ( n − 1 ) ( 1 − s ) − s + n s ) , i = 1, 2, …, n, respectively.

The total input demand is n q i b i ∗ = I = n ( 1 − 2 g + g n + ( 1 − g ) ( n − 1 ) s ) ( 1 − w ) ( 2 − 3 g + g n + 2 ( 1 − g ) ( n − 1 ) s ) ( 1 + g ( n − 1 ) ( 1 − s ) − s + n s ) , which gives

The monopolist input supplier determines its output by maximising Max I [ w b i I − ( c I − d I 2 ) ] . The equilibrium input production is

We get c − 2 d I b i ∗ > 0 for d < c ( 2 − 3 g + g n + 2 s ( 1 − g ) ( n − 1 ) ) ( 1 + g ( n − 1 ) ( 1 − s ) + s ( n − 1 ) ) n ( 1 − 2 g + g n + ( 1 − g ) ( n − 1 ) s ) ≡ d b i ‾ , which is assumed to hold.

Inserting (B2) into (B1) for I gives the equilibrium input price, w b i ∗ . Using p i b i ∗ , q i b i ∗ and w b i ∗ , we get the equilibrium profit of the ith final goods producer as

Differentiating π i b i ∗ with respect to g and evaluating it at g = 0, we get ∂ π i b i ∗ ∂ g | g = 0 = ( 1 − c ) 2 ( n − 1 ) ( 2 − s ( 4 − 2 s − n ( 2 − d − 2 s ) ) ) 4 ( 2 − d n + 2 ( n − 1 ) s ) 3 ( − n ( 1 − s ) 2 ( 1 + ( n − 1 ) s ) + K n ( 1 + ( n − 1 ) s ) 3 ) , which is positive for d > 2 ( 1 − s ) ( 1 + ( n − 1 ) s ) n s ≡ d ∗ where d ∗ > ( < ) d b i ‾ . Hence, g = 0 is not the preferred differentiation for d ∈ ( d ∗ , d b i ‾ ) . Since we are looking at the change in profit with respect to g and evaluating it at g = 0, it is intuitive that the critical conditions are the same under Cournot and Bertrand competition since the type of competition does not matter at g = 0.

We also get ∂ π i b i ∗ ∂ g | g = 1 = − ( 1 − c ) 2 ( 1 + s ( n − 1 ) ) 4 n ( n − 1 ) ( d − 1 ) 2 < 0 , implying that g = 1 is not the preferred differentiation. Since the final goods producers get zero profits under Bertrand competition if the products are homogeneous, they always prefer some differentiation in this situation.

Hence, if d ∈ ( d ∗ , d b i ‾ ) , neither g = 0 nor g = 1 is the preferred differentiation, implying firms prefer moderate differentiation in this situation.