We now turn to the case of Cournot competition in which firms choose quantities rather than prices at stage 2 after committing to quality levels at stage 1. The game played by firms is set out in 4.1 and the respective effects of LDC and developed country policies towards quality are explored in 4.2 and 4.3.
4.1 The two-stage model of firm behavior: Cournot competition.
Examining the second-stage first, we solve for pL = PL/qL and pH = PH/qH from the demand functions (5), so as to obtain the inverse demand functions:
This includes the possibility that both firms set the same qualities (i.e. qL = qH), since, setting r = 1 in (22), we obtain, P = (1- (xL + xH ))q‘ for i = L,H, as in (3). Recalling that productions are zero, for any given qualities, qL and qH , committed at stage 1, firm L sets xL to maximize its revenue, RL = PLxL, taking xH as given and firm H sets xH to maximize its revenue, RH = PHxH , taking xL as given. Thus from (22), xL and xH satisfy the first order conditions:
where the second order and stability conditions are also satisfied. Also, since 0R/0x is decreasing in xj for i,j = L,H and i g j, the outputs are strategic substitutes as is typical for Cournot competition.
From the first order conditions, we obtain output levels and then quality-adjusted prices (from (22) and (4)) at the Cournot equilibrium as follows:
From (23), it is notable that the quality-adjusted price equals the quantity sold of each product. Also, since pL and xL both fall and pH and xH both rise, greater separation in quality, as shown by an increase in r now has mixed effects on prices and output and hence the degree of competition: i.e. from (23),
Using a superscript c to distinguish functions at the Cournot equilibrium, we express the revenues
Thus, holding own qualities fixed, the revenues of the two firms respond in opposite directions with respect to an increase in r = qH/qL, with firm L’s revenue falling and firm H’s revenue rising. This implies that revenues for firm L as well as for firm H are decreased by an increase in the rivals quality: i.e.
In contrast with the Bertrand case, firm L now gains as the products become more similar, but, as before, firm H gains from a greater separation of qualities. To understand these results, first note that, for both models, an increase in r, holding xL and xH fixed, shifts up the demand curve for good H, raising the willingness of consumers to pay for the high-quality good (i.e. 0pH/0r = xL/(r)2 > 0 from (22)), but the willingness to pay for the low-quality good is unchanged (i.e. 0pL/0r = 0 from (22)). Under Cournot competition, firm H responds to this higher demand due to greater separation of products by expanding output and firm L then reacts by cutting output (xL and xH are strategic substitutes). Since quality-adjusted price and output both fall for firm L and both rise for firm H (see (24)), firm L’s revenue falls (holding qL fixed) and firm H’s revenue increases (holding qH fixed). Instead, under Bertrand competition, firm H raises price in response to an increase in r and, since pL and pH are strategic complements, firm L also raises price (see (10)) causing the revenues of both firms to increase. The idea to travell all over the world is attractive but so much money is necessary to make your dreams come true. Not to waste time just speedy cash payday loans, they will solve all your problems not only with the travelling but other as well.
Incorporating the cost of investment for general cost parameters AL and AH, the respective profits of the low and high quality firms are given by:
In this Cournot case, marginal revenue with respect to own quality is always positive: i.e. from (25)
In considering the choice of qualities, there is again a tradeoff between competition affects arising from the extent of differentiation from the rival’s product and the profitability of a particular location in quality space based on revenues and investment costs for a given quality ratio, r. Since firm L gains from a narrowing of the quality gap, this gives firm L an incentive to increase qL, which reduces r, holding qH fixed (the term -r7 ‘(r) in (29) is positive). For firm H, analogously to Bertrand competition, a greater separation of products raises revenue leading it to also want to raise qH. However, for both firm L and firm H, the profitability of an increase in quality is limited by the rising marginal cost of investment in quality. This web site www.speedy-payday-loans.com is a source of dreams realization. Everything you need is to make your mind what sum of money you desire to borrow.
Although positive profits can be made at the second-stage Cournot equilibrium when qL = qH , Lemma 3 shows (see Appendix B for the proof) that even if both firms face identical costs (i.e. if AL = AH), the first order conditions (28) imply that the quality game is asymmetric with qL < qH. and hence r > 1.
Lemma 4 (proved in Appendix B) concerns conditions under which (30) holds locally. For the case, F(q) = q2/2, commonly used in the literature, we have F'”(q) = 0 and Lemma 4(i) applies for AL/AH > 1. Letting E = qF”(q)/F'(q) represent the responsiveness (or elasticity) of the marginal cost F ‘(q) of investment with respect to quality, since F ‘(0) = 0, the assumption F “(q) > 0 (see (1)) ensures F “(q) > F ‘(q)/q and hence E > 1. If E = 1 then F “(q) = 0 and Lemma 4(i) applies. However, to allow for more general investment cost functions and for any value of AL/AH, Lemma 4(ii) requires that firm L’s marginal cost of investment increases more steeply: i.e. for qL satisfying nLL = 0, that EL = qLF”(qL)/F'(qL) > 2. For the example, F(q) = aqn, E = n-1 is a constant and Lemma 4(ii) applies if n > 3.
Firm H continues to view qL as a strategic complement to qH , but since RcLL is increased by a greater similarity of products, it follows that RcLLH < 0 and hence that firm L views qH as a strategic substitute to qL: i.e. from (29) and (31),
Thus, as shown in Figure 4, firm L’s reaction function, qL = pcL(qH), is negatively sloped in the neighborhood of equilibrium, whereas qH = pcH(qL) has a positive slope; i.e. from (28), (32) and Lemma 4,
Analogously to Proposition 1 for Bertrand competition, for a sufficiently large cost disadvantage in the LDC, we are able to prove existence and uniqueness of the pure strategy equilibrium (see Appendix B for the proof) in which the LDC firm produces qL and the developed country firm produces qH. For the remainder of the paper we assume that Proposition 9 applies and hence AL = y(1-sL) and AH = 1-sH where sL and sH represent the investment subsidy policies by LDC and developed countries respectively.
Proposition 9. Assume Cournot competition in the output market. Under conditions (1), if y > 1 is sufficiently large, there exists a unique pure strategy equilibrium in which the low quality product is produced in the LDC and the high quality product in the developed country.
4.2 LDC investment policy towards the low-quality product.
For the Cournot setting, the effects of an investment subsidy, sL, applied to firm L by the LDC on quality levels and profits are presented in Proposition 10.
Interestingly, comparing Proposition 10 with Proposition 2 for Bertrand competition, the direction of effects is the same. However, there are some critical differences behind the scenes, since, as shown in Proposition 11, the LDC has a unilateral incentive to tax the investment of its firm under Cournot competition, whereas a subsidy raises LDC welfare in the Bertrand case. For Proposition 11, we assume that LDC welfare, denoted WcL = nL(qL,qH,AL) – sLyF(qL), is locally concave at the optimal policy, scL*. As shown in Lemma 6 of Appendix B, this holds for:
This tax policy may initially seem hard to understand, since, as can be seen from Proposition 10, the LDC tax lowers the profit of firm L and at the same time, since firm H benefits from a reduction in qL, raises the profit of firm H and hence welfare in the developed country. However, LDC welfare rises because the tax revenue more than offsets the loss in firm L’s profit. Also, the fact that firm H’s profits are increased simply means that the rent-shifting aspect of the policy is entirely at the expense of consumers in the third country market. As illustrated in Figure 5, the LDC tax shifts the quality reaction function of firm L in towards the origin (shown by the dashed line) and both countries move to higher iso-welfare contours. Speedy loans! What may be easier for people to take a loan. A loan with low percantage, with no requires and no troubles. Just find the web site www.speedy-payday-loans.com it is an alternative way of having money if you feel some troubles you won’t get enough even for a month.
To understand why a switch from Bertrand to Cournot competition causes LDC policy to switch from an investment subsidy to an investment tax, we again appeal to the correspondence of the model with a Stackelberg leader-follower model in which firm L is the leader and there is no government intervention. Since firm H raises qH in response to an increase in qL under both Bertrand and Cournot competition, the fact that the revenues of firm L are increased by an increase in qH under Bertrand competition (i.e. RLH > 0) and are reduced by an increase in qH under Cournot competition (i.e. RcLH < 0), means that a Stackelberg leader would increase qL above its Nash equilibrium value in the Bertrand case and reduce qL in the Cournot case. If both firms play Nash in quality, the same outcome is achieved by an investment subsidy in the LDC under Bertrand competition and an investment tax under Cournot competition.
Fundamentally, by taking qH as given at the Nash quality equilibrium, firm L overestimates the effect of an increase in qL in making the products more similar under both forms of competition. However, since firm L gains from a greater separation of products in the Bertrand case and from a greater similarity of products in the Cournot case, it sets qL too low in the Bertrand case and too high in the Cournot case for maximum profit. To correct for this, the LDC policy moves firm L (and hence firm H) up the quality ladder under Bertrand competition and down the quality ladder under Cournot competition.
4.3 Developed country policy towards the high-quality product
For the Cournot setting, Proposition 12 concerns the effects of an investment subsidy, sH, applied to firm H by the developed country. An increase in sH increases quality qH, but in contrast to the Bertrand setting, firm L’s reaction function has a negative slope and qL falls. As might be expected, Firm H enjoys higher profits, but firm L’s profits are reduced.
Next, Proposition 13 shows that a shift form Bertrand to Cournot competition gives country H an incentive to subsidize rather than tax the investment of its firm. As illustrated in Figure 6, under Cournot competition, the subsidy shifts up the quality reaction function of firm H (shown as the dashed line), moving the equilibrium from point Nc to point Sc, which, as before, corresponds to the Stackelberg leader-follower point with firm H as the leader and scH = 0. Since firm H gains from a reduction in qL under both forms of competition (i.e. RcHL < 0 and RcHL < 0), in each case the policy is aimed at reducing qL. For Cournot competition, it follows from the negative slope of firm L’s reaction function that a subsidy will raise qH and hence lower qL. By contrast, in the Bertrand case, since firm L raises qL in response to an increase in qH, the relevant policy is an investment tax. For Proposition 13, we assume that the welfare function, WcH , for country H is locally concave at the optimum subsidy, scH*. From Lemma 7 (see Appendix B for the proof), this holds under the same conditions, EL > 2 and a(q) > 0, used to ensure local concavity of LDC welfare.
Finally, as in the Bertrand case, the jointly optimal investment policy corrects for the cross effects of the quality chosen by each firm on its rival’s profit. Since Firm H gains from the widening of the quality gap due to a reduction in qL and firm L gains from the narrowing of the quality gap due to a reduction in qH, joint profit maximization involves a move by both firms down the quality ladder. Consequently, as shown in Proposition 14, the policy requires that each country tax investment, with the tax given by country H. Relative to the Nash policy equilibrium, the joint choice of policies increases the investment tax in the LDC and results in a switch from a subsidy to a tax in the developed country. Since an increase in product differentiation has mixed effects on price competition (pH rises and pL falls), in contrast to the Bertrand case, there is no clear relationship between the size of the quality gap and the ability to raise prices at the expense of third country consumers. Rather the jointly optimal policy is directed at finding the optimal location in quality space taking into account revenues and the increasingly high investment costs as quality is increased.
Proposition 14. Under Cournot competition, the joint welfare of the two producing countries is maximized by an investment tax in both countries with scL J < scL* < 0 and scHJ < 0 < scH*.
Figure 4 Quality reaction functions: Cournot competition
Figure 5 The LDC’s optimal tax: Cournot competition
Figure 6. The developed country’s optimal subsidy: Cournot competition
Appendix B: Cournot Competition
Appendix B: Cournot Competition (continue)
Appendix B: Cournot Competition (continue 2)
Appendix B: Cournot Competition (continue 3)
Appendix B: Cournot Competition (continue 4)
Appendix B: Cournot Competition (continue 5)