National and store brand advertising strategies

Abstract

As the propensity of premium store brands (SBs) increases, retailers must consider different ways to drive sales besides promotional strategies. With this in mind, we consider a national brand (NB) and a (premium) SB co-existing in a market. Each brand has to decide the amount to invest in advertising its product and the prices to charge its customers, which can be determined separately or in unison. When either advertising expenditures or pricing decisions are set, each brand must keep in mind that the advertising efforts and revenue may spillover between the two brands, customers who intend to purchase the NB may end up purchasing the SB and vice versa. We derive an analytical model of the situations described and characterize equilibrium advertising decisions. We find that the characteristics of a premium SB may depend on which marketing/promoting instrument (advertising or pricing) is the primary method for driving demand; and in some situations an NB may be better off to not advertise at all and instead let the premium SB carry out all of the advertising.

Appendices

Appendix A

List of mathematical symbols and notation

SB::

Store Brand

NB::

National Brand

p _S ::

the retail price of the SB good

p _N ::

the retail price of the NB good

A _S ::

the market size of the SB, also referred to as the advertising effort exerted by the SB

A _N ::

the market size of the NB, also referred to as the advertising effort exerted by the NB

::

the base market size for the NB good

β::

the fraction of the NB sales that are retained by the SB, retailer, we assume β∈(0, 1)

α _S ::

the advertising spillover from the NB to the SB, we assume α _S ∈(0, 1)

α _N ::

the advertising spillover from the SB to the NB, we assume α _N ∈(−1, 1); α _N <0 models a competitive advertising environment and α _N >0 models a complementary advertising environment

a _S ::

the negative effect of p _S on demand for the SB good

a _N ::

the negative effect of p _N on demand for the NB good

b _S ::

the positive effect of p _N on demand for the SB good

b _N ::

the positive effect of p _S on demand for the NB good

c _S ::

the cost of advertising effort exerted by the SB

c _N ::

the cost of advertising effort exerted by the NB

π _S (A _S , p _S |A _N , p _N )::

the profit function of the SB

π _N (A _S , p _N )::

the profit function of the NB

Appendix B

Mathematical appendix

In this appendix we provide the details of the results presented in Section 3.

B.1. Advertising only case

Its FOC leads to the profit maximizer A _S ^*:

In (B2) A _S ^* is an interior point solution, including non-negativity constraint on A _S we have:

Note that we know the interior point is a maximum as π _S (A _S |A _N ) is concave, (d²π _S (A _S |A _N ))/dA _S ²=−2c _S ⩽0.

Using (B3), we may rewrite π _N (A _N ) as:

We now find the FOC for π _N (A _N ):

First note that π _N (A _N ) is concave, d²π _N (A _N )/(dA _N ²)=−2c _N ⩽0. For A _N ^* to be non-negative, it implies Substituting the interior point value of A _N ^* into the condition of interior A _S ^* in (B3), we have:

Using the notation defined in Section 3.2, we let and φ=(1−β)α _S (1−α _N α _S )(c _S )/(c _N )−βα _N . For each advertising decision, A _S ^* and A _N ^*, we have four possible scenarios: both solutions are interior, only one of them is interior, or none of them are interior. They are defined with respect to the relationship between p _N and and independently between (p _S )/(p _N ) and φ.

Note that the second case of (C1) follows from A _S =0 meaning the profit of the NB is Using FOC we have:

Note that in this case d²π _N (A _N )/(dA _N ²)=−1<0 meaning that π _N (A _N ) is concave.

Similarly for A _S ^* we have:

Note that the third case of (C2) follows from A _N =0 meaning the profit of the SB is Using FOC we have:

Just as with the other cases, d²π _S (A _S )/dA _S ²=−2c _S ⩽0 meaning that π _S (A _S ) is concave.

B.2. Pricing only case

We now explore the FOC of the SB profit function assuming advertising decisions are fixed.

First note that d²π _S (p _S |p _N )/(dp _S ²)=−2a _S which means π _S (p _S |p _N ) is concave in p _S . Note that all of the terms in the right hand side of (B7) are positive, so the interior solution p _S ^*(p _N ) is always feasible.

We now write π _N (p _N ), given p _S ^*(p _N ):

Note that d²π _N (p _N )/(dp _N ²)=2(1−β)((b _N /2a _S ))(b _S +βb _N )−a _N , in order to ensure concavity we must have a _N ⩾(b _N /2a _S )(b _S +βb _N ). The numerator of p _N ^* is always positive, only the denominator may be negative in the case a _N <(b _N /2a _S )(b _S +βb _N ), in order to ensure p _N ^* is non-negative we have add the condition that a _N >(b _N /2a _S )(b _S +βb _N ) and to avoid division by zero, which is stronger than the concavity condition. As discussed in Section 3.3, we do not consider the case of p _N ^*⩽0 even when α _N is negative, as this would imply an SB may advertise an NB out of existence.

B.3. General case

For the SB’s best response to the NB’s actions A _N and p _N , we solve for FOCs, , to find A _S ^*(A _N , p _N ) and p _S ^*(A _N , p _N ):

⇒

⇒

⇒

Given the optimal values of p _S ^*(A _N , p _N ) and A _S ^*(A _N , p _N ) we rewrite π _N (A _N , p _N ) as:

In order to ensure the FOCs, , minimize π _N (A _N , p _N ), we have to ensure ∇²π _N (A _N , p _N ) is negative semidefinite, that is, |∇²π _N (A _N , p _N )|⩽0. Formally this means that:

Given this condition, we consider :

⇒

where Δ=(1−β)(1−α _N α _S )+4c _N (2a _S βα _N ²+2βα _N b _N +b _S α _N +2βc _S b _N ²+2b _S c _S b _N −a _N (4a _S c _S −1))/((4a _S c _S −1)(1−α _N α _S )). Note that in an interior solution, however, as and A _N ⩾0 an interior solution is not feasible and only a corner solution exists. A corner solution is the only option for A _N ∈[0, ∞). As an interior stationary point exists and by construction π _N (A _N , p _N ) is concave, we know that π _N (A _N , p _N ) is increasing for all values of and decreasing for all values , and thus is the feasible point that maximizes π _N (A _N , p _N |A _S ^*, p _S ^*).

Appendix C

Sensitivity analysis on β

The optimal decisions, pricing and advertising, determined in Section 3 may depend on the wholesale price that we model using an exogenously set parameter β. In this section we examine how these decisions change with β.

From Section 3.2 we know that when p _S and p _N are fixed, the optimal advertising decisions are:

and

We take the derivative of each decision with respect to β to determine how each will change with the wholesale price:

and

From the Equations (C3) and (C4) we note that the NB will decrease their advertising effort with β, that is, lower wholesale price will lead to lower advertising effort. Conversely, as wholesale price decreases, β increases, the SB will exert larger effort in the case advertising campaigns are complementary, α _N is positive. However, in the case advertising campaigns are competitive, the effect of β on the SB advertising decision may depend on the relative magnitude between α _N and α _S among other parameters, that is, whether the direct competing effect (α _N /2)(p _N /c _S ) is stronger than indirect spillover effect α _S ((1−α _N α _S ))/(2)(p _N /c _N ).

In the pricing only case we have:

and

Looking at the partial derivatives we have:

and

From Equation (C5) we note that p _N is always increasing in β. The same relationship holds for p _S as p _N (β)⩾0 by condition (8), and thus p _S is also increasing in β.

We finally consider the general case:

and

Taking the partial derivatives of the equilibrium decisions with respect to β we have:

and

From (C7) we note that p _N is decreasing in β for the general hand. Things are not so obvious for the SB, A _S and p _S may increase or decrease with β depending on the relationships between the terms. If the first term of each of the partials is greater than the second then, the optimal decisions decrease with β, otherwise they both increase with β.