Firms’ Information Acquisition with Heterogeneous Consumers and Trend

Abstract

This paper analyzes firms’ location choices and information acquisition in a model of product differentiation with trend. The presence of an uncertain trend spot induces the firms not to follow the principle of maximal differentiation, unless the trend spot is expected to be near the ends of the city. Second, each firm has an incentive to acquire information about the exact trend spot. Consumer surplus is also higher when both firms are informed. Third, firms’ choice of product differentiation is excessive relative to the social optimum. Furthermore, in the social optimum, welfare is also higher when the planner is informed.

Appendix: Derivations of (38) and (39)

1.1 A.1 Derivation of (38)

$$ E \varPi_{1}^{C} = \left( {z/15,552} \right)[\alpha^{3} {-}24 \alpha^{2} E \left( t \right) + 192 \alpha E \left( {t^{2} } \right){-}512E \left( {t^{3} } \right)], $$

(37)

where α ≡ 27 + 8E(t) – 3(9 – 8var(t))^1/2.

$$ \begin{aligned} E \varPi_{1}^{B} & = z\left[ {\left( {3/8} \right) + \left( {var \left( t \right)/3} \right) + \left( {2/27} \right)\left( {var \left( t \right)} \right)^{2} } \right] \\ & = \left( {z/1944} \right)\left[ {729 + 648 var \left( t \right) + 144 \left( {var \left( t \right)} \right)^{2} } \right]. \\ \end{aligned} $$

(27)

Let Δ ₁ ≡ EΠ ^C ₁ – EΠ ^B ₁ . Δ ₁ is increasing in var(t):

$$ \begin{aligned} \partial \Delta_{1} /\partial var\left( t \right) = \left( {z/15,552} \right)\{ [3\alpha^{2} {-}48\alpha E\left( t \right) + 192\alpha \left( {\left( {E\left( t \right)} \right)^{2} + var\left( t \right)} \right)]\partial \alpha /\partial var\left( t \right) + 192\alpha \} \hfill \\ {-}\left( {z/1944} \right)\left[ {648 + 288var\left( t \right)} \right]. \hfill \\ \end{aligned} $$

(52)

$$ \partial \alpha /\partial var\left( t \right) \, = \, 12/\left( {9 \, {-} \, 8var\left( t \right)} \right)^{1/2} . $$

(53)

Substituting (53) and the value of α into (52) and organizing, we get

$$ \partial \Delta_{1} /\partial var\left( t \right) = (z/1296\sqrt {} )[1998 + \left( {128E\left( t \right){-}486{-}192var\left( t \right)} \right)\sqrt {} + 360var\left( t \right)], $$

(54)

where $\sqrt {} \equiv (9 - 8 var (t))^{1/2}$ for ease of notation.

Calculation shows that the terms in the square brackets [] on the RHS of (54) is itself increasing in var(t), so that ∂Δ ₁ /∂var(t) > ∂Δ ₁ /∂var(t) evaluated at var(t) = 0. Therefore,

$$ \begin{aligned} \partial \Delta_{1} /\partial var(t) > \partial \Delta_{1} /\partial var(t)|_{{var (t ) { = 0}}} \hfill \\ \quad = [z/(1296 \times 3)][1998 + \left( {128E\left( t \right){-}486} \right) \times 3] > 0. \hfill \\ \end{aligned} $$

(55)

Accordingly, Δ ₁ ≡ EΠ ^C ₁ – EΠ ^B ₁ takes the minimum value at var(t) = 0.

At var(t) = 0, α = 27 + 8E(t) – 3×3 = 18 + 8E(t), and

$$ \begin{aligned} E\varPi_{1}^{C} {-}E\varPi_{1}^{B} > E\varPi_{1}^{C} {-}E\varPi_{1}^{B} |_{{var (t ) { = 0}}} \hfill \\ \quad = \left( {z/1944} \right)\left[ {729 + 64\left( {E\left( t \right)} \right)^{3} {-}64E\left( {t^{3} } \right)} \right]{-}\left( {729z/1944} \right) \hfill \\ \quad = \left( {8z/243} \right)\left[ {\left( {E\left( t \right)} \right)^{3} {-}E\left( {t^{3} } \right)} \right] \hfill \\ \quad = 0\quad {\text{if the distribution of}}\;t\;{\text{has zero skewness}} .\hfill \\ \end{aligned} $$

(56)

cf. Let μ be the mean and σ the standard deviation of the probability distribution t.

$$ \begin{aligned} {\text{The formula expressing skewness }} & = E\{ [(t{-}\mu )/\sigma ]^{3} \} \\ & = \{ E\left( {t^{3} } \right){-}3\mu [E\left( {t^{2} } \right){-}\mu E\left( t \right)]{-}\mu^{3} \} /\sigma^{3} \\ & = [E\left( {t^{3} } \right){-}3\mu \sigma^{2} {-}\mu^{3} ]/\sigma^{3} \\ \end{aligned} $$

Skewness evaluated at σ ² = 0 equals [E(t ³) –μ ³]/σ ³.

1.2 A.2 Derivation of (39)

$$ E\varPi_{2}^{C} = \left( {z/15,552} \right)[\alpha \beta^{2} + 8\beta (2\alpha {-}\beta )E\left( t \right) + 64(\alpha {-}2\beta )E\left( {t^{2} } \right){-}512E\left( {t^{3} } \right)], $$

(37)

where α ≡ 27 + 8E(t) – 3(9 – 8var(t))^1/2, and β ≡ 9 – 8E(t) + 3(9 – 8var(t))^1/2.

$$ \varPi_{2}^{A} = 3z/8 = 729z/1944. $$

(19)

$$ \partial \beta /\partial var\left( t \right) = {-}12/\left( {9{-}8var\left( t \right)} \right)^{1/2} = {-}\partial \alpha /\partial var\left( t \right). $$

(57)

Let Δ ₂ ≡ EΠ ^C ₂ – Π ^A ₂ . Δ ₂ is decreasing in var(t):

$$ \begin{aligned} \partial \Delta_{2} /\partial var\left( t \right) = & \left( {z/15,552} \right)[(\beta^{2} + 16\beta E\left( t \right) + 64E\left( {t^{2} } \right))\partial \alpha /\partial var\left( t \right) + 64(\alpha {-}2\beta ) \\ & + (2\alpha \beta + 16\alpha E\left( t \right){-}16\beta E\left( t \right){-}128E\left( {t^{2} } \right))\partial \beta /\partial var\left( t \right)]. \\ \end{aligned} $$

(58)

Substituting (53), (57), and the values of α and β into (58) and organizing, we get

$$ \begin{aligned} \partial \Delta_{2} /\partial var\left( t \right) \, & = (z/432\sqrt {} )\{ {-}207 + 8E\left( t \right) + \left[ {\left( {128/3} \right)E\left( t \right){-}5} \right]\sqrt {} + 120var\left( t \right)\} \\ & < (z/432\sqrt {} )[{-}207 + 6 + \left( {32{-}5} \right)\sqrt {} + 120var\left( t \right)] \\ & < (z/(432\sqrt {} )\left[ {{-}201 + 81 + 120var\left( t \right)} \right] \\ & < 0 \, \left( {{\text{cf}}. \, \left( {33} \right)} \right). \\ \end{aligned} $$

(59)

Accordingly, Δ ₂ ≡ EΠ ^C ₂ – EΠ ^B ₁ takes the maximum value at var(t) = 0.

At var(t) = 0, α = 18 + 8E(t), and

$$ \begin{aligned} E\varPi_{2}^{C} {-}\varPi_{2}^{A} & < E\varPi_{2}^{C} {-}\varPi_{2}^{A} |_{{var (t ) { = 0}}} \\ & = \left( {z/1944} \right)\left[ {729 + 64\left( {E\left( t \right)} \right)^{3} {-}64E\left( {t^{3} } \right)} \right]{-}\left( {729z/1944} \right) \\ & = \left( {8z/243} \right)\left[ {\left( {E\left( t \right)} \right)^{3} {-}E\left( {t^{3} } \right)} \right] \\ & = 0\quad {\text{if the distribution of}}\;t\;{\text{has zero skewness}} .\\ \end{aligned} $$

(60)

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