Does add-on presence always lead to lower baseline prices? Theory and evidence

Abstract

In many industries, firms give consumers the opportunity to add (at a price) optional goods and services to a baseline product. The aim of our paper is to clarify the effect that offering add-ons has on baseline prices. In order to do that, we develop a theoretical model of add-on pricing in competitive environments with two distinctive features. First, we discuss the choice of offering the add-on, if this entails a fixed cost. Second, we allow firms to have a varying degree of market power over the add-on. In symmetric equilibria, the presence of add-on always reduces baseline prices. In asymmetric equilibria in which only one firm offers the add-on, its presence increases the baseline price if the firm’s market power over the add-on is limited. The latter prediction of the model is confirmed by a hedonic price regression using a dataset of cruises offered worldwide, a situation in which it is possible to control for the level of add-on market power.

Appendices

Appendix A

1.1 Proof of Proposition 1

We now show how equilibrium prices are computed in each case.

Case (a)

If no firm offers A, consumers’ utility functions are given by:

$$ u_{i} \left( 0 \right) = u^{B} - p_{0}^{B} - \tau x_{i} $$

(A1)

$$ u_{i} \left( 1 \right) = u^{B} - p_{1}^{B} - \tau (1 - x_{i} ) $$

(A2)

The location of the indifferent consumer (for a given price) is obtained by solving:

$$ u^{B} - p_{0}^{B} - \tau \hat{x} = u^{B} - p_{1}^{B} - \tau \left( {1 - \hat{x}} \right) $$

(A3)

which yields:

$$ \hat{x} = \frac{1}{2} + \frac{{p_{1}^{B} - p_{0}^{B} }}{2\tau } $$

(A4)

Demand for firm 0 is then equal to $\hat{x}$, while demand for 1 is equal to $1 - \hat{x}$. Profit functions are thus as follows:

$$ \varPi_{0} = \left( {\frac{1}{2} + \frac{{p_{1}^{B} - p_{0}^{B} }}{2\tau }} \right)p_{0}^{B} $$

(A5)

$$ \varPi_{1} = \left( {\frac{1}{2} + \frac{{p_{0}^{B} - p_{1}^{B} }}{2\tau }} \right)p_{1}^{B} $$

(A6)

As the two firms are symmetric, we need to consider the first-order condition of one firm only, say firm 0:

$$ \left( {\frac{1}{2} + \frac{{p_{1}^{B} - p_{0}^{B} }}{2\tau }} \right) - \frac{1}{2t}p_{0}^{B} = 0 $$

(A7)

Imposing symmetry ($p_{1}^{B} = p_{0}^{B}$) and solving equation (A7), we obtain equilibrium prices $p_{1}^{B} = p_{0}^{B} = \tau$. Putting equilibrium prices into (A5) and (A6) yields the equilibrium profits.

Case (b)

If only firm 0 offers A, the utility functions are:

$$ u_{i} \left( 0 \right) = \left( {1 - \alpha } \right)u^{A} + u^{B} - p_{0}^{B} - \tau x_{i} $$

(A8)

$$ u_{i} \left( 1 \right) = u^{B} - p_{1}^{B} - \tau (1 - x_{i} ) $$

(A9)

The location of the indifferent consumer (for a given price) is obtained, in this case, by solving:

$$ \left( {1 - \alpha } \right)u^{A} + u^{B} - p_{0}^{B} - \tau x_{i} = u^{B} - p_{1}^{B} - \tau (1 - x_{i} ) $$

(A10)

which yields:

$$ \hat{x} = \left( {\frac{1 - \alpha }{2\tau }} \right)u^{A} + \frac{1}{2} + \frac{{p_{1}^{B} - p_{0}^{B} }}{2\tau } $$

(A11)

Profits are given by:

$$ \varPi_{0} = \left( {\frac{1 - \alpha }{2\tau }u^{A} + \frac{1}{2} + \frac{{p_{1}^{B} - p_{0}^{B} }}{2t}} \right)\left( {p_{0}^{B} + \alpha u^{A} } \right) $$

(A12)

$$ \varPi_{1} = \left( {\frac{\alpha - 1}{2\tau }u^{A} + \frac{1}{2} + \frac{{p_{0}^{B} - p_{1}^{B} }}{2\tau }} \right)p_{1}^{B} $$

(A13)

From the first-order conditions, we obtain firms’ best response functions:

$$ p_{0}^{B} = \frac{{\tau + p_{1}^{B} + \left( {1 - 2\alpha } \right)u^{A} }}{2} $$

(A14)

$$ p_{1}^{B} = \frac{{\tau + p_{0}^{B} - \left( {1 - \alpha } \right)u^{A} }}{2} $$

(A15)

Solving (A14)–(A15) yields equilibrium prices. Substituting equilibrium prices into (A12) and (A13) yields equilibrium profits.

Case (c)

The solution of case (c) is symmetric to case (b).

Case (d)

If both firms offer A, consumers’ utility functions are given by:

$$ u_{i} \left( 0 \right) = \left( {1 - \alpha } \right)u^{A} + u^{B} - p_{0}^{B} - \tau x_{i} $$

(A16)

$$ u_{i} \left( 1 \right) = \left( {1 - \alpha } \right)u^{A} + u^{B} - p_{1}^{B} - \tau (1 - x_{i} ) $$

(A17)

The location of the indifferent consumer (for a given price) is obtained by solving:

$$ \left( {1 - \alpha } \right)u^{A} + u^{B} - p_{0}^{B} - \tau \hat{x} = \left( {1 - \alpha } \right)u^{A} + u^{B} - p_{1}^{B} - \tau \left( {1 - \hat{x}} \right) $$

(A18)

which yields:

$$ \hat{x} = \frac{1}{2} + \frac{{p_{1}^{B} - p_{0}^{B} }}{2\tau } $$

(A19)

Demand for firm 0 is then equal to $\hat{x}$, while demand for 1 is equal to $1 - \hat{x}$. Profit functions are thus as follows:

$$ \varPi_{0} = \left( {\frac{1}{2} + \frac{{p_{1}^{B} - p_{0}^{B} }}{2\tau }} \right)\left( {p_{0}^{B} + \alpha u^{A} } \right) $$

(A20)

$$ \varPi_{1} = \left( {\frac{1}{2} + \frac{{p_{0}^{B} - p_{1}^{B} }}{2\tau }} \right)\left( {p_{1}^{B} + \alpha u^{A} } \right) $$

(A21)

As the two firms are symmetric, we need to consider the first-order condition of one firm only, say firm 0:

$$ \left( {\frac{1}{2} + \frac{{p_{1}^{B} - p_{0}^{B} }}{2\tau }} \right) - \frac{1}{2\tau }\left( {p_{0}^{B} + \alpha u^{A} } \right) = 0 $$

(A22)

Imposing symmetry ($p_{1}^{B} = p_{0}^{B}$) and solving equation (A22), we obtain equilibrium prices $p_{1}^{B} = p_{0}^{B} = \tau - \alpha u^{A}$. Putting equilibrium prices into (A20) and (A21) yields the equilibrium profits.

1.2 Proof of Proposition 3

The proof is almost immediate. If $k < \frac{1}{2}\tau - \left( {\tau - \frac{{u^{A} }}{3}} \right)\left( {\frac{1}{2} - \frac{{u^{A} }}{6\tau }} \right) \equiv \frac{{u^{A} }}{3} - \frac{{\left( {u^{A} } \right)^{2} }}{18\tau }$, OffA is the best response to OffA. If $k > \left( {\tau - \frac{{u^{A} }}{3}} \right)\left( {\frac{1}{2} - \frac{{u^{A} }}{6\tau }} \right) - \frac{\tau }{2} \equiv \frac{{u^{A} }}{3} - \frac{{\left( {u^{A} } \right)^{2} }}{18\tau }$, NotOffA is the best response to NotOffA. If $\frac{1}{2}\uptau - \left( {\uptau - \frac{{u^{A} }}{3}} \right)\left( {\frac{1}{2} - \frac{{u^{A} }}{{6\uptau}}} \right) < k < \left( {\uptau + \frac{{u^{A} }}{3}} \right)\left( {\frac{1}{2} + \frac{{u^{A} }}{{6\uptau}}} \right) - \frac{1}{2}\uptau$, OffA is the best response to NotOffA and vice versa, which leads to asymmetric equilibria. In asymmetric equilibrium, offering the add-on entails higher profit if $k < \left( {\uptau + \frac{{u^{A} }}{3}} \right)\left( {\frac{1}{2} + \frac{{u^{A} }}{{6\uptau}}} \right) - \left( {\tau - \frac{{u^{A} }}{3}} \right)\left( {\frac{1}{2} - \frac{{u^{A} }}{{6\uptau}}} \right)$. This inequality is always satisfied in the asymmetric equilibria region since $\left( {\uptau + \frac{{u^{A} }}{3}} \right)\left( {\frac{1}{2} + \frac{{u^{A} }}{{6\uptau}}} \right) - \frac{1}{2}\uptau \equiv \frac{{u^{A} }}{3} - \frac{{\left( {u^{A} } \right)^{2} }}{{18\uptau}} < \left( {\uptau + \frac{{u^{A} }}{3}} \right)\left( {\frac{1}{2} + \frac{{u^{A} }}{{6\uptau}}} \right) - \left( {\uptau - \frac{{u^{A} }}{3}} \right)\left( {\frac{1}{2} - \frac{{u^{A} }}{{6\uptau}}} \right) \equiv \frac{{2u^{A} }}{3}$.

Appendix B

See Table 4.

Table 4 Descriptive statistics