奢侈品公司的产品策略与定价研究 ——考虑新时代消费者特点

doi:10.3969/j.issn.0253-2778.2020.04.014

摘要/Abstract

摘要： 基于寡头市场的研究背景，将地位效用应用于奢侈品定制，研究了奢侈品市场中不同产品(标准和定制)的定价以及四种不同产品策略的比较分析. 提出了四种产品策略:①纯标准产品策略;②纯定制品产品策略;③无法自由定制的混合型产品策略;④消费者可自由选择定制程度的混合型产品策略. 针对这四种策略探索了不同产品策略的最优定价以及公司不同生产成本的最优决策,从而得出以下结论:首先,地位效用对定制产品定价的影响比标准产品更为重要;其次,基于对利润的比较分析,给出了产品策略的最优选择,而公司产品策略的选择取决于生产的灵活性.在拓展研究中,针对纯标准产品策略进行了竞争市场下的公司利润分析.

关键词: 奢侈品, 炫耀性消费, 地位效用, 定制

Abstract: Based on the research background of the oligopoly market, status utility was applied to luxury goods customization and the pricing of different products (standard and customization) in the luxury market was studied and a comparative analysis of four different product strategies was given. Four product strategies was proposed: ①product strategy with only pure-standard products, ②product strategy with customized products, ③multi product strategy that consumers can not freely choose the customization, ④multi product strategy that consumers can freely choose the customization. For these four strategies, the optimal pricing of different product strategies and the optimal decision of the company’s different production costs were explored, which led to the following conclusions: first, the effect of status utility on the pricing of customized products is more important than that of standard products; second, based on the comparative analysis of profits, the optimal choice of product strategy is given, and the choice of the company’s product strategy depends on the flexibility of production. In the expansion research, some profit analysis under the competitive market was discussed for the pure standard product strategy.

Key words: luxury, conspicuous consumption, status utility, customization

中图分类号:

F274

张佳蕾，王漫漫. 奢侈品公司的产品策略与定价研究 ——考虑新时代消费者特点[J]. 中国科学技术大学学报, 2020, 50(4): 497-515.

ZHANG Jialei, WANG Manman. Product strategy and pricing research of the luxury companies： Considering the characteristics of consumers in the new era[J]. Journal of University of Science and Technology of China, 2020, 50(4): 497-515.

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附录
A.1 纯标准产品策略（PS策略）
在临界值vs这点，购买商品的效用等于不购买商品的效用.得出
vsQ+λ(vs+1)2-ps=λvs2(A1)

当 vs=ps-λ2Q, 式(A1)成立.利润函数变为 π1=ps(1-ps-λ2Q).通过π1对ps求一阶导，得到下面最优解：

dπdps=1-2ps-λ2Q（A2）
p*s1=Q2+λ4（A3）
vs1=ps-λ2Q（A4）

联立 (A2)～(A4)，得到临界值 vs1 和最优利润π*1:

vs1=Q2-λ4Q（A5）
π*1=Q2+λ42Q（A6）

A.2 纯定制产品策略（PC策略）
和PS策略相同, 通过下式求解临界值 vc ：
vc(Q+x)+λ(vc+1)2-pc=λvc2（A7）

当vc=pc-λ2Q+x, 式（A7）成立.利润函数变为π2=pc(1-pc-λ2Q+x)-ax2-bx.π2对pc求一阶导，得到下面最优解：

p*c2=Q2+λ4+x2（A8）

vc2=Q2-λ4+x2Q+x（A9）
π*2=Q2+λ4+x22Q+x-ax2-bx（A10）

该利润函数的二阶导数也满足条件.
A.3 无法自由定制的混合产品策略（MN策略）
在这种情况下，计算两个平衡点vs3和vc3.财富水平为vs3的消费者购买标准产品和不购买任何产品来获得的效用相同;当消费者财富水平为vc3时，消费者获得的标准产品效用和定制产品效用没有区别.根据均衡条件，建立以下方程组：

vc3Q+λ(vs3+vc3)2-ps=vc3(Q+x)+λ(vc3+1)2-pc（A11）

vs3Q+λ(vs3+vc3)2-ps=λvs32（A12）

该方程关于vs3,vc3的解集为

vs3=4xps+2λps+λ2-2λpc4xQ+λ2（A13）
vc3=2λps-4Qps-2λQ+4Qpc4xQ+λ2（A14）
vc-vs=4Qpc-4Qps-4xps-2λQ-λ2+2λpc4xQ+λ2（A15）
1-vc=4xQ+2λQ+λ2-2λps+4Qps-4Qpc4xQ+λ2（A16）
π3=ps(vc-vs)+pc(1-vc)-ax2-bx（A17）

该种产品策略下的利润函数为 π3=ps(vc-vs)+pc(1-vc)-ax2-bx. 联立 (A15)～(A17)，得出公司关于ps和pc的利润函数如下：

π3=ps4Qpc-4Qps-4xps-2λQ-λ2+2λpc4xQ+λ2+pc4xQ+2λQ+λ2-2λps+4Qps-4Qpc4xQ+λ2-ax2-bx（A18）


式(A18)是ps和pc的二元函数.通过对ps和pc分别取偏导数，得到以下公式：

π3ps=8Qpc-8Qps-8xps-2λQ-λ24xQ+λ2=0（A19）

π3pc=8Qps-8Qpc+4xQ+2λQ+λ24xQ+λ2=0（A20）

联立 (A19) 和 (A20)，有
p*s3=Q2（A21）
p*c3=Q2+λ4+x2+λ28Q（A22）
将（A21）和（A22）代入（A18），得

π*3=Q4+λ4+x4+λ216Q-ax2-bx（A23）

命题3.3证明 MN与PS策略的利润公式分别为

π*3=Q4+λ4+x4+λ216Q-ax2-bx（A24）
π*1=Q4+λQ4+λ216（A25）

为了讨论产品策略的选择，不妨假设π*3>π*1,则有
ax2+bx-x4-(λ216Q+λ4-λQ4-λ216)<0（A26）

令G=λ216Q+λ4-λQ4-λ216,改写为G=λ216Q-λ216+λ4-λQ4.当Q>1时,λ216Q-λ216<0,λ4-λQ4<0，故G<0；同理可得：当Q<1时，G>0.

当Q>1时，有G>0,有ax2+bx-x4-G<0,解该一元二次不等式：
当G<0,解得0因为当Q>1时，总有G>0,当G>(b-14)24a时，该不等式无解，故π*3>π*1无解，得到命题3.3③；
当Q>1,G<(b-14)24a,两个解14-b± (b-1/4)2+4aG2a均小于0，由于解代表定制程度必须大于0，故在定义域内无解，得到命题3.3④；
当Q>1,G<(b-14)24a,b<1/4时，求得定义域内两解为14-b± (b-1/4)2+4aG2a,可得到不等式的解集为14-b- (b-1/4)2+4aG2a
A.4 可自由选择定制程度的混合产品策略（MF策略）

π4=∫kx+vsvsQ2+λ4+M(vi-vs)kdvi+(1-kx-vs)Q2+λ4+Mx-(ax2+bx)（A27）


化简为
π4=Q4+λ4+λ216Q+Mx2+Mxλ4Q-kMx22-ax2-bx（A28）

式（A28）对x求一阶导，得到以下结果：
dπ4dx=M2+Mλ4Q-kMx-2ax-b=0（A29）

x*4=2QM-λM-4Qb8Qa+4QMk（A30）

命题3.6证明 x*4=2QM-λM-4Qb8Qa+4QMk,x*4对M求导得到
dx*4M=(2Q-λ)(8Qa+4QMk)-4Qk(2QM-λM-4Qb)(8Qa+4QMk)2（A31）

化简为

dx*4M=16Q2a-8λQa+16Q2bk(8Qa+4QMk)2=8Qa2vs3+16Q2bk(8Qa+4QMk)2>0(0
得证.

命题4.1证明改写需求函数公式：

D1=c+142+λ2-p1-(c+1)c4（A33）

令m=2+λ2-p1,化简后得：D1=14cm+m-(c+1)2c4，为了求得需求函数随价格差的变化关系，对需求函数求一阶导，令一阶导等于0，求得c=± 4+48m-46，根据函数导数判断单调性，得到命题4.1.

命题4.2证明公司1的利润函数如下：

π1=2cp1+λ2cp1-cp21-cp1(c+1)c4+2p1+λ2p1-p21-(c+1)c4p14（A34）

同样为了求得公司1利润与价格差c的关系，将利润函数对c求导：
π1c=142p1+λ2p1-p21-3c2p1+4cp1+p14（A35）

令π1c=0,即
3c2p1+4cp1+4p21-7p1-2λp1=0（A36）
令n=4p21-7p1-2λp1;将式（A36）改写为3c2p1+4cp1+n=0.求得
c=-4p1± 16p21-12p1n6p1（A37）
通过对根号内的数值正负性加以考虑，进行分类讨论：
①当p1<2512+λ2时，16p21-12p1n>0,可正常求解得到命题4.2①和4.2②；
②当p1>2512+λ2时，16p21-12p1n<0,π1c<0,得到命题4.2③.

()
()

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