Dynamic monopolistic competition with sluggish adjustment of entry and exit
aa r X i v : . [ m a t h . O C ] J un Dynamic monopolistic competition withsluggish adjustment of entry and exit
Yasuhito TanakaFaculty of Economics, Doshisha University,Kamigyo-ku, Kyoto, 602-8580, Japan.E-mail:[email protected]
Abstract
We study a steady state of a free entry oligopoly with differentiated goods, that is,a monopolistic competition, with sluggish adjustment of entry and exit of firms undergeneral demand and cost functions by a differential game approach. Mainly we showthat the number of firms at the steady state in the open-loop solution of monopolisticcompetition is smaller than that at the static equilibrium of monopolistic competition, andthat the number of firms at the steady state of the memoryless closed-loop monopolisticcompetition is larger than that at the steady state of the open-loop monopolistic competition,and may be larger than the number of firms at the static equilibrium.
Keywords: monopolistic competition; differential game; general demand function; generalcost function; open-loop; closed-loop
There are many studies of an oligopoly by differential game theory, for example, Cellini and Lambertini(2003a), Cellini and Lambertini (2003b), Cellini and Lambertini (2004), Cellini and Lambertini(2005), Cellini and Lambertini (2007), Cellini and Lambertini (2011), Fujiwara (2006), Fujiwara(2008) and Lambertini (2018). Most of these studies used a model of specific (linear or ex-ponential) demand functions and specific (quadratic or linear) cost functions. We study asteady state of a dynamic free entry oligopoly with differentiated goods, that is, a monopolisticcompetition with sluggish adjustment of entry and exit of firms under general demand andcost functions by a differential game approach. In the next section we present a model andassumptions. We consider a dynamics of the number of firms which enter into the industryaccording to the rule that the number of firms increases or decreases proportionally to the total1rofits of the firms . In Section 3 we consider an open-loop solution of a differential gameanalysis of monopolistic competition. We present both a general analysis and a linear example.In Section 4 we examine a general model of a memoryless closed-loop solution. In Section 5we consider an example with linear demand and cost functions of the memoryless closed-loopsolution. We compare open-loop and memoryless closed-loop solutions, and mainly show thefollowing results.1. The number of firms at the steady state in the open-loop solution of monopolisticcompetition is smaller than that at the static equilibrium of monopolistic competition.2. The number of firms at the steady state in the memoryless closed-loop solution ofmonopolistic competition is larger than that at the steady state of the open-loop solutionof monopolistic competition, and may be larger than the number of firms at the staticequilibrium.We also show that when the discount rate (denoted by ρ ) approaches to positive infinity, orthe speed of adjustment of the number of firms approaches to zero, the steady states of theopen-loop and the closed-loop solutions approach to the static equilibrium of monopolisticcompetition. There is a symmetric oligopoly where, at any t ∈ [ , ∞) , n firms, Firms 1, 2, . . . , n producedifferentiated goods. The firms maximize their discounted profits. Let x i ( t ) , i ∈ { , , . . . , n } ,be the outputs of the firms, p i ( t ) be the price of the good of Firm i at t .The inverse demand function for Firm i , i ∈ { , , . . . , n } , is p i ( t ) = p i ( x ( t ) , x ( t ) , . . . , x n ( t )) , i ∈ { , , . . . , n } . For simplicity we denote p i ( x ( t ) , x ( t ) , . . . , x n ( t )) , ∂ p i ( x ( t ) , x ( t ) ,..., x n ( t )) ∂ x i ( t ) , ∂ p i ( x ( t ) , x ( t ) ,..., x n ( t )) ∂ x j ( t ) , ∂ p i ( x ( t ) , x ( t ) ,..., x n ( t )) ∂ x i ( t ) , ∂ p i ( x ( t ) , x ( t ) ,..., x n ( t )) ∂ x i ( t ) ∂ x j ( t ) , j , i , by p i , ∂ p i ∂ x i ( t ) , ∂ p i ∂ x j ( t ) , ∂ p i ∂ x i ( t ) , ∂ p i ∂ x i ( t ) ∂ x j ( t ) , and so on.We assume ∂ p i ∂ x i ( t ) < , i ∈ { , , . . . , n } ,∂ p i ∂ x j ( t ) < , j , i , (cid:12)(cid:12)(cid:12)(cid:12) ∂ p i ∂ x j ( t ) (cid:12)(cid:12)(cid:12)(cid:12) < (cid:12)(cid:12)(cid:12)(cid:12) ∂ p i ∂ x i ( t ) (cid:12)(cid:12)(cid:12)(cid:12) , and ∂ p i ∂ x j ( t ) + ∂ p i ∂ x i ( t ) ∂ x j ( t ) x i ( t ) < , i ∈ { , , . . . , n } , j , i . (1) Alternatively, we can assume that the number of firms increases or decreases proportionally to the averageprofit of the firms. Essentially the same result is obtained in both cases. ∂ p i ∂ x j ( t ) + ∂ p i ∂ x i ( t ) ∂ x j ( t ) x i ( t ) = ∂ p i x i ( t ) ∂ x i ( t ) ∂ x j ( t ) . Similarly, ∂ p i ∂ x j ( t ) + ∂ p i ∂ x j ( t ) ∂ x k ( t ) x k ( t ) = ∂ p i x k ( t ) ∂ x j ( t ) ∂ x k ( t ) , j , i , k , i , j . We assume (cid:12)(cid:12)(cid:12)(cid:12) ∂ p i x i ( t ) ∂ x i ( t ) ∂ x j ( t ) (cid:12)(cid:12)(cid:12)(cid:12) ≥ (cid:12)(cid:12)(cid:12)(cid:12) ∂ p i x k ( t ) ∂ x j ( t ) ∂ x k ( t ) (cid:12)(cid:12)(cid:12)(cid:12) . Then, we obtain ∂ p i ∂ x j ( t ) + ∂ p i ∂ x j ( t ) ∂ x k ( t ) x k ( t ) < , j , i , k , i , j . (2)and (cid:12)(cid:12)(cid:12)(cid:12) ∂ p i ∂ x j ( t ) + ∂ p i ∂ x i ( t ) ∂ x j ( t ) x i ( t ) (cid:12)(cid:12)(cid:12)(cid:12) ≥ (cid:12)(cid:12)(cid:12)(cid:12) ∂ p i ∂ x j ( t ) + ∂ p i ∂ x j ( t ) ∂ x k ( t ) x k ( t ) (cid:12)(cid:12)(cid:12)(cid:12) . (3)By symmetry of the model at the steady states of open-loop and closed-loop solutions x i ( t ) = x j ( t ) = x k ( t ) .About the derivative of p i with respect to n we have ∂ p i ∂ n ( t ) = ∂ p i ∂ x j ( t ) x j ( t ) . The cost function of Firm i , i ∈ { , , . . . , n } , is c ( x i ( t )) , i ∈ { , , . . . , n } . All firms have the same cost functions. It satisfies c ′ ( x i ( t )) >
0. The instantaneous profit ofFirm i , is π i ( t ) = x i ( t ) p i ( x ( t ) , x ( t ) , . . . , x n ( t )) − c ( x i ( t )) , i ∈ { , , . . . , n } . The moving of the number of firms is governed by dn ( t ) dt = s " n Õ i = x i ( t ) p i ( x ( t ) , x ( t ) , . . . , x n ( t )) − n Õ i = c ( x i ( t )) , s > . (4)The number of firms increases or decreases proportionally to the total profit of the firms.The problem of Firm i ismax x i ( t ) ∫ ∞ e − ρ t [ x i ( t ) p i ( x ( t ) , x ( t ) , . . . , x n ( t )) − c ( x i ( t ))] dt , subject to (4). ρ > i , i ∈ { , , . . . , n } , is H i ( t ) = e − ρ t { x i ( t ) p ( x ( t ) , x ( t ) , . . . , x n ( t )) − c ( x i ( t )) + λ i ( t ) s n Õ j = x j ( t ) p j ( x ( t ) , x ( t ) , . . . , x n ( t )) − n Õ j = c ( x j ( t )) . The current value Hamiltonian function of Firm i , i ∈ { , , . . . , n } , isˆ H i ( t ) = e ρ t H ( t ) = x i ( t ) p i ( x ( t ) , x ( t ) , . . . , x n ( t )) − c ( x i ( t )) + λ i ( t ) s n Õ j = x j ( t ) p j ( x ( t ) , x ( t ) , . . . , x n ( t )) − n Õ j = c ( x j ( t )) . Let µ i ( t ) = e − ρ t λ i ( t ) , i ∈ { , , . . . , n } .µ i ( t ) is the costate variable.Assume that the outputs of all firms are equal. The free entry condition is p i ( x , x , . . . , x ) x − c ( x ) − f = . From this dndx = − p i ( x , x , . . . , x ) + ∂ p i ∂ x i ( t ) x + ( n − ) ∂ p i ∂ x j ( t ) x − c ′ ( x ) ∂ p i ∂ x j ( t ) x . Suppose that a monopolistic firm produce n substitutable goods. It determines the output ofeach good. By symmetry we assume that the outputs of all goods are equal. Let x be the outputof each good. Its profit is np i ( x , x , . . . , x ) x − nc ( x ) . The condition for profit maximization at t in the static equilibrium is n (cid:20) p i ( x , x , . . . , x ) + ∂ p i ∂ x i ( t ) x + ( n − ) ∂ p i ∂ x j ( t ) x − c ′ ( x ) (cid:21) = . If p i ( x , x , . . . , x ) + ∂ p i ∂ x i ( t ) x + ( n − ) ∂ p i ∂ x j ( t ) x − c ′ ( x ) ≥ , the output of each firm in the steady states of open-loop and closed-loop solutions should besmaller than (or equal to) the output of each good by the above monopolist. Therefore, weassume p i ( x , x , . . . , x ) + ∂ p i ∂ x i ( t ) x + ( n − ) ∂ p i ∂ x j ( t ) x − c ′ ( x ) < . Then, dndx < . (5)This holds in all cases. 4e can assume p i − c ′ ( x i ( t )) > , i ∈ { , , . . . , n } . This means that the price of the good is larger than the marginal cost of the firms.Consider a case such that each firm determines its output given the prices of the goods ofother firms. Then, the profit maximization condition for Firm i in the static oligopoly is p i + ∂ p i ∂ x i ( t ) x i ( t ) + Õ j , i ∂ p i ∂ x j ( t ) x i ( t ) dx j ( t ) dx i ( t ) − c ′ ( x i ( t )) = . (6)From the condition that p j ( x , x , . . . , x n ) is constant for each j , i , we have ∂ p j ∂ x j ( t ) dx j ( t ) dx i ( t ) + Õ k , i , j ∂ p j ∂ x k ( t ) dx k ( t ) dx i ( t ) + ∂ p j ∂ x i ( t ) = . By symmetry ∂ p j ∂ x k ( t ) = ∂ p j ∂ x i ( t ) and dx k ( t ) dx i ( t ) = dx j ( t ) dx i ( t ) . Then, dx j ( t ) dx i ( t ) = − ∂ p j ∂ x i ( t ) ∂ p j ∂ x j ( t ) + ( n − ) ∂ p j ∂ x i ( t ) . Again by symmetry ∂ p j ∂ x j ( t ) = ∂ p i ∂ x i ( t ) , ∂ p j ∂ x i ( t ) = ∂ p i ∂ x j ( t ) at the equilibrium. Thus, (6) is rewritten as p i + (cid:16) ∂ p i ∂ x i ( t ) (cid:17) + ( n − ) ∂ p i ∂ x i ( t ) ∂ p i ∂ x j ( t ) − ( n − ) (cid:16) ∂ p i ∂ x j ( t ) (cid:17) ∂ p i ∂ x i ( t ) + ( n − ) ∂ p i ∂ x j ( t ) x i ( t ) − c ′ ( x i ( t )) = p i + (cid:16) ∂ p i ∂ x i ( t ) − ∂ p i ∂ x j ( t ) (cid:17) h ∂ p i ∂ x i ( t ) + ( n − ) ∂ p i ∂ x j ( t ) i ∂ p i ∂ x i ( t ) + ( n − ) ∂ p i ∂ x j ( t ) x i ( t ) − c ′ ( x i ( t )) = . Since ∂ p i ∂ x i ( t ) < ∂ p i ∂ x j ( t ) < (cid:12)(cid:12)(cid:12) ∂ p i ∂ x i ( t ) (cid:12)(cid:12)(cid:12) > (cid:12)(cid:12)(cid:12) ∂ p i ∂ x j ( t ) (cid:12)(cid:12)(cid:12) , we have ∂ p i ∂ x i ( t ) − ∂ p i ∂ x j ( t ) < , ∂ p i ∂ x i ( t ) + ( n − ) ∂ p i ∂ x j ( t ) ∂ p i ∂ x i ( t ) + ( n − ) ∂ p i ∂ x j ( t ) > . (7)If p i + (cid:16) ∂ p i ∂ x i ( t ) − ∂ p i ∂ x j ( t ) (cid:17) h ∂ p i ∂ x i ( t ) + ( n − ) ∂ p i ∂ x j ( t ) i ∂ p i ∂ x i ( t ) + ( n − ) ∂ p i ∂ x j ( t ) x i ( t ) − c ′ ( x i ( t )) ≤ , at the steady state of open-loop and closed-loop solutions, the output of each firm is larger than(or equal to) that under the above Bertrand type behaviors of firms. Thus, we assume p i + (cid:16) ∂ p i ∂ x i ( t ) − ∂ p i ∂ x j ( t ) (cid:17) h ∂ p i ∂ x i ( t ) + ( n − ) ∂ p i ∂ x j ( t ) i ∂ p i ∂ x i ( t ) + ( n − ) ∂ p i ∂ x j ( t ) x i ( t ) − c ′ ( x i ( t )) > ,
5t the steady states of open-loop and closed-loop solutions of dynamic oligopoly. From (7) wecan assume p i + (cid:18) ∂ p i ∂ x i ( t ) − ∂ p i ∂ x j ( t ) (cid:19) x i ( t ) − c ′ ( x i ( t )) > . (8) We seek to the general open-loop solution. The first order condition for Firm i is ∂ ˆ H i ( t ) ∂ x i ( t ) = p i + ∂ p i ∂ x i ( t ) x i ( t ) − c ′ ( x i ( t )) (9) + λ i ( t ) s " p i + ∂ p i ∂ x i ( t ) x i ( t ) − c ′ ( x i ( t )) + Õ j , i ∂ p j ∂ x i ( t ) x j ( t ) = . The second order condition is ∂ ˆ H i ( t ) ∂ x i ( t ) = ∂ p i ∂ x i ( t ) + ∂ p i ∂ x i ( t ) − c ′′ ( x i ( t )) (10) + λ i ( t ) s " ∂ p i ∂ x i ( t ) + ∂ p i ∂ x i ( t ) x i ( t ) − c ′′ ( x i ( t )) + Õ j , i ∂ p j ∂ x i ( t ) x j ( t ) < . The adjoint condition is − ∂ ˆ H i ( t ) ∂ n ( t ) = − ∂ p i ∂ x j ( t ) x i ( t ) x j ( t ) (11) − λ i ( t ) s " ∂ p j ∂ x k ( t ) x k ( t ) n Õ l = x l ( t ) + x j ( t ) p j − c ( x j ( t ))) = ∂λ i ( t ) ∂ t − ρλ i ( t ) , j , i , k , j . At the steady state we have x i ( t ) p (cid:16)Í nj = x j ( t ) (cid:17) − c ( x i ( t )) = ∂λ i ( t ) ∂ t = i ∈{ , , . . . , n } . By symmetry, all x i ( t ) ’s and all λ i ( t ) ’s are equal, and ∂ p i ∂ x j ( t ) = ∂ p j ∂ x i ( t ) = ∂ p j ∂ x k ( t ) , j , i , k , j ,∂ p j ∂ x i ( t ) = ∂ p i ∂ x j ( t ) , j , i . Denote the steady state values of x i ( t ) , λ i ( t ) and n ( t ) by x ∗ , λ ∗ and n ∗ . (9), (10) and (11) arereduced to p i + ∂ p i ∂ x i ( t ) x ∗ − c ′ ( x ∗ ) + λ ∗ s (cid:18) p i + ∂ p i ∂ x i ( t ) + ( n ∗ − ) ∂ p i ∂ x j ( t ) x ∗ − c ′ ( x ∗ ) (cid:19) = , (12)6 ∂ p i ∂ x i ( t ) + ∂ p i ∂ x i ( t ) x ∗ − c ′′ ( x ∗ ) + λ ∗ s (cid:18) ∂ p i ∂ x i ( t ) + ∂ p i ∂ x i ( t ) + ( n ∗ − ) ∂ p i ∂ x j ( t ) x ∗ − c ′′ ( x ∗ ) (cid:19) < , and − ∂ p i ∂ x j ( t ) ( x ∗ ) − λ ∗ n ∗ s ∂ p i ∂ x j ( t ) ( x ∗ ) = − ρλ ∗ . Therefore, λ ∗ s = s ∂ p i ∂ x j ( t ) ( x ∗ ) ρ − n ∗ s ∂ p i ∂ x j ( t ) ( x ∗ ) < . (13)From (12) and (13) p i + ∂ p i ∂ x i ( t ) x ∗ − c ′ ( x ∗ ) (14) + s ∂ p i ∂ x j ( t ) ( x ∗ ) ρ − n ∗ s ∂ p i ∂ x j ( t ) ( x ∗ ) (cid:18) p i + ∂ p i ∂ x i ( t ) x ∗ + ( n ∗ − ) ∂ p i ∂ x j ( t ) x ∗ − c ′ ( x ∗ ) (cid:19) = . Let ˜ x and ˜ n be the equilibrium output of each firm and the number of firms in the staticmonopolistic competition. Then, p i + ∂ p i ∂ x i ( t ) ˜ x − c ′ ( ˜ x ) = . Suppose that x = ˜ x for each firm and n = ˜ n . The left-hand side of (14) is ( n − ) s ∂ p i ∂ x j ( t ) x ρ − ns ∂ p i ∂ x j ( t ) x ∂ p i ∂ x j ( t ) x . This is positive. Thus, under the assumption that the second order condition is satisfied, theoutput of each firm in the open-loop solution is larger than that at the static equilibrium, thatis, x ∗ > ˜ x .From (5) n ∗ < ˜ n . We obtain the following result. Proposition 1.
The number of firms at the steady state in the open-loop solution of monopolisticcompetition is smaller than that at the static equilibrium of monopolistic competition.
Note that when ρ → + ∞ or s →
0, the steady state of open-loop solution approaches to thestatic equilibrium.. 7 .2 A linear example
Suppose that the inverse demand function for Firm i is p i ( t ) = a − x i ( t ) − b n Õ j , i x j ( t ) . a is a positive constant, and 0 < b <
1. Also suppose that the cost function of Firm i , i ∈ { , , . . . , n } , is c ( x i ( t )) = cx i ( t ) + f , c > . f > dn ( t ) dt = s " a − x i ( t ) − b n Õ j , i x j ( t ) ! n Õ i = x i ( t ) − c n Õ i = x i ( t ) − n ( t ) f , s > . The current value Hamiltonian function isˆ H i ( t ) = x i ( t ) a − x i ( t ) − b n Õ j , i x j ( t ) ! − cx i ( t ) − f + λ i ( t ) s © « a − x j ( t ) − b n Õ k , j x k ( t ) ª®¬ n Õ j = x j ( t ) − c n Õ j = x j ( t ) − n ( t ) f . The first order and the second order conditions for Firm i , i ∈ { , , . . . , n } , are ∂ ˆ H i ( t ) ∂ x i ( t ) = ( + λ i ( t ) s ) a − x i ( t ) − b n Õ i , i x j ( t ) − c ! − λ i ( t ) sb Õ j , i x j ( t ) = , and ∂ ˆ H i ( t ) ∂ x i ( t ) = − ( + λ i ( t ) s ) < . The adjoint condition for Firm i , i ∈ { , , . . . , n } , is − ∂ ˆ H i ( t ) ∂ n ( t ) = bx i ( t ) x j ( t ) − λ i ( t ) s © « a − x j ( t ) − b Õ k , j x k ( t ) ª®¬ x j ( t ) . − cx j ( t ) − f − bx k ( t ) n Õ l = x l ( t ) = ∂λ i ( t ) ∂ t − ρλ i ( t ) . At the steady state we have ( a − Í nj = x j ( t )) x i ( t ) − cx i ( t ) − f = ∂λ i ( t ) ∂ t = i ∈ { , , . . . , n } . By symmetry, all x i ( t ) ’s and all λ i ( t ) ’s are equal. Denote the steady statevalues of x i ( t ) , λ i ( t ) and n ( t ) by x ∗ , λ ∗ and n ∗ . Then, the above adjoint condition is reduced to b ( x ∗ ) + λ ∗ n ∗ sb ( x ∗ ) = − ρλ ∗ . (cid:2)(cid:3)(cid:4)(cid:1)(cid:5)(cid:1)(cid:5)(cid:3)(cid:4)(cid:1)(cid:6)(cid:1)(cid:6)(cid:3)(cid:4)(cid:1)(cid:7)(cid:1)(cid:7)(cid:3)(cid:4)(cid:1)(cid:4) (cid:1)(cid:8) (cid:1)(cid:2) (cid:1)(cid:5) (cid:1)(cid:6) (cid:1)(cid:7) (cid:1)(cid:4) (cid:1)(cid:9) (cid:1) (cid:1) ρ n ∗ Figure 1: The numbers of firms in open-loop and ρ From this λ ∗ = − b ( x ∗ ) ρ + n ∗ sb ( x ∗ ) < , or λ ∗ s = − bs ( x ∗ ) ρ + n ∗ sb ( x ∗ ) < . Since ρ >
0, when s → λ ∗ s →
0. Similarly, when ρ → + ∞ we have λ ∗ s → ( + λ ∗ s ) ( a − x ∗ − ( n ∗ − ) bx ∗ − c ) − λ ∗ s ( n ∗ − ) bx ∗ (15) = ( a − x ∗ − ( n ∗ − ) bx ∗ − c ) − b s ( x ∗ ) ρ + n ∗ bs ( x ∗ ) ( a − ∗ x ∗ − ( n ∗ − ) bx ∗ − c ) = . On the other hand, the free entry condition at the steady state is ( a − x ∗ − ( n ∗ − ) bx ∗ ) x ∗ − c ( x ∗ ) − f = . (16)Solving (15) and (16) we get the steady state values of x ∗ and n ∗ . We give graphical represen-tations in Figure 1 assuming a = , f = , c = , s = , b = and in Figure 2 assuming a = , f = , c = , ρ = , b = .When s → ρ → + ∞ , (15) is further reduced to a − x ∗ − ( n ∗ − ) bx ∗ − c = . This is equivalent to the static equilibrium condition.9 (cid:2)(cid:3)(cid:4)(cid:1)(cid:5)(cid:1)(cid:5)(cid:3)(cid:4)(cid:1)(cid:6)(cid:1)(cid:6)(cid:3)(cid:4)(cid:1)(cid:7)(cid:1)(cid:7)(cid:3)(cid:4)(cid:1)(cid:4) (cid:1)(cid:8) (cid:1)(cid:8)(cid:3)(cid:2) (cid:1)(cid:8)(cid:3)(cid:5) (cid:1)(cid:8)(cid:3)(cid:6) (cid:1)(cid:8)(cid:3)(cid:7) (cid:1)(cid:8)(cid:3)(cid:4) (cid:1)(cid:8)(cid:3)(cid:9) (cid:1) (cid:1) sn ∗ Figure 2: The numbers of firms in open-loop and s We seek to a memoryless closed-loop solution. The first order condition and the second ordercondition are the same as those in the open-loop solution as follows. ∂ ˆ H i ( t ) ∂ x i ( t ) = p i + ∂ p i ∂ x i ( t ) x i ( t ) − c ′ ( x i ( t )) (17) + λ i ( t ) s " p i + ∂ p i ∂ x i ( t ) x i ( t ) − c ′ ( x i ( t )) + Õ j , i ∂ p j ∂ x i ( t ) x j ( t ) = , and ∂ ˆ H i ( t ) ∂ x i ( t ) = ∂ p i ∂ x i ( t ) + ∂ p i ∂ x i ( t ) − c ′′ ( x i ( t )) + λ i ( t ) s " ∂ p i ∂ x i ( t ) + ∂ p i ∂ x i ( t ) x i ( t ) − c ′′ ( x i ( t )) + Õ j , i ∂ p j ∂ x i ( t ) x j ( t ) < . The adjoint condition is different from that in the open-loop solution. It is written as − ∂ ˆ H i ( t ) ∂ n ( t ) − Õ j , i ∂ ˆ H i ( t ) ∂ x j ( t ) ∂ x i ( t ) ∂ n ( t ) = ∂λ i ( t ) ∂ t − ρλ i ( t ) . (18)The term in (18) − Õ j , i ∂ ˆ H i ( t ) ∂ x j ( t ) ∂ x i ( t ) ∂ n ( t ) i and the current level of the state variable. We have Õ j , i ∂ ˆ H i ( t ) ∂ x j ( t ) = ( n − ) x i ( t ) ∂ p i ∂ x j ( t ) + λ i ( t ) s Õ j , i p j + ∂ p j ∂ x j ( t ) x j ( t ) + n Õ k , j ∂ p k ∂ x j ( t ) x k ( t ) − c ′ ( x j ( t )) , j , i , k , j . From (17) ∂ x i ( t ) ∂ n ( t ) = − ∆ (cid:26) ( + λ i ( t ) s ) (cid:18) ∂ p i ∂ x j ( t ) + ∂ p i ∂ x i ( t ) ∂ x j ( t ) x i ( t ) (cid:19) x j ( t ) (19) + λ i ( t ) s ∂ p i ∂ x j ( t ) x j ( t ) + Õ j , i ∂ p j ∂ x i ( t ) ∂ x k ( t ) x j ( t ) x k ( t ) ! ) , j , i , k , i , j , where ∆ = ∂ p i ∂ x i ( t ) + ∂ p i ∂ x i ( t ) − c ′′ ( x i ( t )) (20) + λ i ( t ) s " ∂ p i ∂ x i ( t ) + ∂ p i ∂ x i ( t ) x i ( t ) − c ′′ ( x i ( t )) + Õ j , i ∂ p j ∂ x i ( t ) x j ( t ) < . At the steady state we have p (cid:0)Í nk = x k ( t ) (cid:1) x i ( t ) − c ( x i ( t )) = ∂λ i ( t ) ∂ t = i ∈{ , , . . . , n } . By symmetry, all x i ( t ) ’s and all λ i ( t ) ’s are equal. Denote the steady state valuesof x i ( t ) , λ i ( t ) and n ( t ) by x ∗∗ , λ ∗∗ and n ∗∗ . Then, using ∂ p j ∂ x i ( t ) = ∂ p i ∂ x j ( t ) , (17) and (18) are reducedto p i + ∂ p i ∂ x i ( t ) x ∗∗ − c ′ ( x ∗∗ ) (21) + λ ∗∗ s (cid:20) p i + ∂ p i ∂ x i ( t ) x ∗∗ + ( n ∗∗ − ) ∂ p i ∂ x j ( t ) x ∗∗ − c ′ ( x ∗∗ ) (cid:21) = , and − ∂ p i ∂ x j ( t ) ( x ∗∗ ) − λ ∗∗ n ∗∗ s ∂ p i ∂ x j ( t ) ( x ∗∗ ) − (cid:20) ( n ∗∗ − ) ∂ p i ∂ x j ( t ) x ∗∗ (22) + λ ∗∗ s ( n ∗∗ − ) (cid:18) p i + ∂ p i ∂ x i ( t ) x ∗∗ + ( n ∗∗ − ) ∂ p i ∂ x j ( t ) x ∗∗ − c ′ ( x ∗∗ ) (cid:19) (cid:21) ∂ x i ( t ) ∂ n ( t ) = − ρλ ∗∗ . From (21) λ ∗∗ s = − p i + ∂ p i ∂ x i ( t ) x ∗∗ − c ′ ( x ∗∗ ) p i + ∂ p i ∂ x i ( t ) x ∗∗ + ( n ∗∗ − ) ∂ p i ∂ x j ( t ) x ∗∗ − c ′ ( x ∗∗ ) , (23)11nd 1 + λ ∗∗ s = ( n ∗∗ − ) ∂ p i ∂ x j ( t ) x ∗∗ p i + ∂ p i ∂ x i ( t ) x ∗∗ + ( n ∗∗ − ) ∂ p i ∂ x j ( t ) x ∗∗ − c ′ ( x ∗∗ ) . (24)Then, (22) is rewritten as − ∂ p i ∂ x j ( t ) ( x ∗∗ ) − λ ∗∗ n ∗∗ s ∂ p i ∂ x j ( t ) ( x ∗∗ ) + ( n ∗∗ − ) (cid:20) p i + (cid:18) ∂ p i ∂ x i ( t ) − ∂ p i ∂ x j ( t ) (cid:19) x ∗∗ − c ′ ( x ∗∗ ) (cid:21) ∂ x i ( t ) ∂ n ( t ) = − ρλ ∗∗ . This means λ ∗∗ s = ∂ p i ∂ x j ( t ) s ( x ∗∗ ) − ( n ∗∗ − ) s h p i + (cid:16) ∂ p i ∂ x i ( t ) − ∂ p i ∂ x j ( t ) (cid:17) x ∗∗ − c ′ ( x ∗∗ ) i ∂ x i ( t ) ∂ n ( t ) ρ − n ∗∗ s ∂ p i ∂ x j ( t ) ( x ∗∗ ) . (25)By (8), p i + (cid:18) ∂ p i ∂ x i ( t ) − ∂ p i ∂ x j ( t ) (cid:19) x ∗∗ − c ′ ( x ∗∗ ) > . From (25) and (21) we get ∂ ˆ H i ∂ x i ( t ) = p i + ∂ p i ∂ x i ( t ) x ∗∗ − c ′ ( x ∗∗ ) (26) + ∂ p i ∂ x j ( t ) s ( x ∗∗ ) − ( n ∗∗ − ) s h p i + (cid:16) ∂ p i ∂ x i ( t ) − ∂ p i ∂ x j ( t ) (cid:17) x ∗∗ − c ′ ( x ∗∗ ) i ∂ x i ( t ) ∂ n ( t ) ρ − n ∗∗ s ∂ p i ∂ x j ( t ) ( x ∗∗ ) (cid:20) p i + ∂ p i ∂ x i ( t ) x ∗∗ + ( n ∗∗ − ) ∂ p i ∂ x j ( t ) x ∗∗ − c ′ ( x ∗∗ ) (cid:21) = . Compare (26) and (14). Suppose that x = x ∗∗ , n = n ∗∗ and (26) is satisfied, the left-hand sideof (14) is ( n ∗∗ − ) s h p i + (cid:16) ∂ p i ∂ x i ( t ) − ∂ p i ∂ x j ( t ) (cid:17) x ∗∗ − c ′ ( x ∗∗ ) i ∂ x i ( t ) ∂ n ( t ) ρ − n ∗∗ s ∂ p i ∂ x j ( t ) ( x ∗∗ ) (cid:20) p i + ∂ p i ∂ x i ( t ) x ∗∗ (27) + ( n ∗∗ − ) ∂ p i ∂ x j ( t ) x ∗∗ − c ′ ( x ∗∗ ) (cid:21) .
12t the steady state from (19) ∂ x i ( t ) ∂ n ( t ) = Γ (cid:26) −( n ∗∗ − ) ∂ p i ∂ x j ( t ) x ∗∗ (cid:18) ∂ p i ∂ x j ( t ) + ∂ p i ∂ x i ( t ) ∂ x j ( t ) x ∗∗ (cid:19) x ∗∗ (28) + (cid:18) p i + ∂ p i ∂ x i ( t ) x ∗∗ − c ′ ( x ∗∗ ) (cid:19) (cid:20) ∂ p i ∂ x j ( t ) + ( n ∗∗ − ) ∂ p i ∂ x j ( t ) ∂ x k ( t ) x ∗∗ (cid:21) x ∗∗ (cid:27) , = Γ (cid:26) ( n ∗∗ − )( p i − c ′ ( x ∗∗ )) (cid:18) ∂ p i ∂ x j ( t ) + ∂ p i ∂ x j ( t ) ∂ x k ( t ) x ∗∗ (cid:19) x ∗∗ −( n ∗∗ − ) ∂ p i ∂ x j ( t ) (cid:20) (cid:18) ∂ p i ∂ x j ( t ) + ∂ p i ∂ x i ( t ) ∂ x j ( t ) x ∗∗ (cid:19) − (cid:18) ∂ p i ∂ x j ( t ) + ∂ p i ∂ x i ( t ) ∂ x j ( t ) x ∗∗ (cid:19) (cid:21) x ∗∗ −( n ∗∗ − ) ∂ p i ∂ x j ( t ) (cid:18) p i + ∂ p i ∂ x i ( t ) x ∗∗ − c ′ ( x ∗∗ ) (cid:19) x ∗∗ (cid:27) , where Γ = ∆ (cid:18) p i + ∂ p i ∂ x i ( t ) x ∗∗ + ( n ∗∗ − ) ∂ p i ∂ x j ( t ) x ∗∗ − c ′ ( x ∗∗ ) (cid:19) . We have ∆ < p i + ∂ p i ∂ x i ( t ) x ∗∗ + ( n ∗∗ − ) ∂ p i ∂ x j ( t ) x ∗∗ − c ′ ( x ∗∗ ) < p i − c ′ ( x ∗∗ ) >
0, and from (1),(2) and (3), ∂ p i ∂ x j ( t ) + ∂ p i ∂ x i ( t ) ∂ x j ( t ) x i ( t ) < ,∂ p i ∂ x j ( t ) + ∂ p i ∂ x j ( t ) ∂ x k ( t ) x i ( t ) < , and (cid:12)(cid:12)(cid:12)(cid:12) ∂ p i ∂ x j ( t ) + ∂ p i ∂ x i ( t ) ∂ x j ( t ) x i ( t ) (cid:12)(cid:12)(cid:12)(cid:12) ≥ (cid:12)(cid:12)(cid:12)(cid:12) ∂ p i ∂ x j ( t ) + ∂ p i ∂ x j ( t ) ∂ x k ( t ) x i ( t ) (cid:12)(cid:12)(cid:12)(cid:12) . Suppose ∂ x i ( t ) ∂ n ( t ) ≥
0. From (25), λ ∗ s ≤
0. By (21), we have p i + ∂ p i ∂ x i ( t ) x ∗∗ − c ′ ( x ∗∗ ) ≤
0. From(28) this means ∂ x i ( t ) ∂ n ( t ) <
0. It is a contradiction. Thus, we have ∂ x i ( t ) ∂ n ( t ) <
0, and then (27) ispositive (because p i + ∂ p i ∂ x i ( t ) x ∗∗ + ( n ∗∗ − ) ∂ p i ∂ x j ( t ) x ∗∗ − c ′ ( x ∗∗ ) < x ∗∗ < x ∗ and n ∗∗ > n ∗ . We have shown the following result. Proposition 2.
The number of firms at the steady state in the memoryless closed-loop solutionof monopolistic competition is larger than that in the open-loop solution of monopolisticcompetition. If ∂ p i ∂ x j ( t ) s ( x ∗∗ ) − ( n ∗∗ − ) s h p i + (cid:16) ∂ p i ∂ x i ( t ) − ∂ p i ∂ x j ( t ) (cid:17) x ∗∗ − c ′ ( x ∗∗ ) i ∂ x i ( t ) ∂ n ( t ) > x ∗∗ < ˜ x and the number of firms at the steady state in the closed-loop solution is larger than that at thestatic equilibrium of free entry oligopoly.Also note that from (26) we find that when s → ρ → + ∞ , the steady state of theclosed-loop solution approaches to the static equilibrium.13 (cid:2)(cid:1)(cid:3)(cid:1)(cid:4)(cid:1)(cid:5)(cid:1)(cid:6)(cid:1)(cid:7)(cid:1)(cid:8) (cid:1)(cid:9) (cid:1)(cid:10) (cid:1)(cid:2) (cid:1)(cid:3) (cid:1)(cid:4) (cid:1)(cid:5) (cid:1)(cid:6) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) ρ n n ∗∗ n ∗ Figure 3: The numbers of firms in open-loop, closed-loop and ρ Similarly to the example in the open-loop case, we assume that the inverse demand function is p i ( t ) = a − x i ( t ) − b n Õ j , i x j ( t ) , a > , and the cost function of Firm i , i ∈ { , , . . . , n } , is c ( x i ( t )) = cx i ( t ) + f , c > , f > . The moving of the number of firms is governed by dn ( t ) dt = s " a − x i ( t ) − b n Õ j , i x j ( t ) ! n Õ i = x i ( t ) − c n Õ i = x i ( t ) − n ( t ) f , s > . From (23), (24) and (20), at the steady state we have λ ∗∗ s = − a − x ∗∗ − ( n ∗∗ − ) bx ∗∗ − ca − x ∗∗ − ( n ∗∗ − ) bx ∗∗ − c , + λ ∗∗ s = − ( n ∗∗ − ) bx ∗∗ a − x ∗∗ − ( n ∗∗ − ) bx ∗∗ − c , ∆ = ( n ∗∗ − ) x ∗∗ a − x ∗∗ − ( n ∗∗ − ) bx ∗∗ − c , and ∆ [ a − x ∗∗ − ( n ∗∗ − ) bx ∗∗ − c )] = ( n ∗∗ − ) x ∗∗ . (cid:2)(cid:1)(cid:3)(cid:1)(cid:4)(cid:1)(cid:5)(cid:1)(cid:6)(cid:1)(cid:7)(cid:1)(cid:8) (cid:1)(cid:9) (cid:1)(cid:9)(cid:10)(cid:11) (cid:1)(cid:9)(cid:10)(cid:2) (cid:1)(cid:9)(cid:10)(cid:3) (cid:1)(cid:9)(cid:10)(cid:4) (cid:1)(cid:9)(cid:10)(cid:5) (cid:1)(cid:9)(cid:10)(cid:6) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) sn n ∗∗ n ∗ Figure 4: The numbers of firms in open-loop, closed-loop and s Therefore, ∂ x i ( t ) ∂ n ( t ) = − ( n ∗∗ − )( a − x ∗∗ − ( n ∗∗ − ) bx ∗∗ − c ) b − ( n ∗∗ − )( a − x ∗∗ − ( n ∗∗ − ) bx ∗∗ − c ) b ( n ∗∗ − ) = − [ a + ( n ∗∗ − ) x ∗∗ − ( n ∗∗ − ) bx ∗∗ − c ] b ( n ∗∗ − ) , (26) is reduced to ( a − x ∗∗ − ( n ∗∗ − ) bx ∗∗ − c ) (29) − sb ( x ∗∗ ) − s ( a − n ∗∗ bx ∗∗ − c ) [ a + ( n ∗∗ − ) x ∗∗ −( n ∗∗ − ) bx ∗∗ − c ] b ρ + n ∗∗ bs ( x ∗∗ ) ( a − n ∗∗ x ∗∗ − c ) = . On the other hand, the free entry condition at the steady state is the same as that in the open-loopcase as follows, ( a − n ∗∗ − x ∗∗ ) x ∗∗ − c ( x ∗∗ ) − f = . (30)Solving (29) and (30) we get the steady state values of x ∗∗ and n ∗∗ . We give graphicalrepresentations in Figure 3 assuming a = , f = , c = , s = , b = and in Figure 4assuming a = , f = , c = , ρ = , b = . In these figures we depict the relations betweenthe number of firms at the steady states of open-loop and closed-loop solutions and the valueof s or ρ . In this paper we analyze a dynamic free entry oligopoly with differentiated goods, that is, amonopolistic competition by differential game approach.15 cknowledgment
This work was supported by Japan Society for the Promotion of Science KAKENHI GrantNumber 18K01594.
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