The choice between product and logistic innovation in a spatial model with income distribution

Abstract

We consider the strategic choice between product and logistic innovation in a spatial framework where consumers are distributed across a linear city and have different incomes depending on their location. Equilibria depend on the relative efficiency of product and logistic innovation and on the level of income inequality. In a single-plant duopoly, multiple equilibria where firms differentiate their strategy may arise. In particular, we show that, if income inequality is sufficiently low, the firm located close to poor residents might choose to increase quality (product innovation) whereas the rival might choose to reduce transportation costs (logistic innovation). We also consider a multiplant monopoly framework and a joint-venture set-up, finding the conditions for different innovation equilibria to arise. Finally, we find that a social planner should mimic the behaviour of the multi-plant monopolist.

Appendix

Proof of Proposition 1

Following Lederer and Hurter (1986), we introduce the following sharing rule: if the utility of a consumer is the same when he buys from firm A and when he buys from firm B, he chooses the firm with the higher competitive advantage. In our framework, firm J is said to have a competitive advantage over firm $- J$ at location x if $\gamma (1 - i_{J} )f(x) - (t - \varepsilon i_{J} )T_{J} (x) \ge \gamma (1 - i_{ - J} )f(x) - (t - \varepsilon i_{ - J} )T_{ - J} (x)$ holds. This assumption is standard in spatial models, and allows avoiding the technicality of ε-equilibria.¹⁷ Recall that the utility of a customer located at x and buying from firm J$( - J)$ is $v + \gamma (1 - i_{J} )f(x) - p_{J} (x)$$(v + \gamma (1 - i_{ - J} )f(x) - p_{ - J} (x))$. Suppose that x is such that $(t - \varepsilon i_{ - J} )T_{ - J} (x) + \gamma f(x)(i_{ - J} - i_{J} ) \ge (t - \varepsilon i_{J} )T_{J} (x)$. Consider firm J. Firm J serves customer x if $p_{J} (x) \le \gamma f(x)(i_{ - J} - i_{J} ) + p_{ - J} (x)$. First, we show that $p_{ - J} (x) > (t - \varepsilon i_{ - J} )T_{ - J} (x)$ cannot be an equilibrium. When $p_{ - J} (x) > (t - \varepsilon i_{ - J} )T_{ - J} (x)$, the best-reply of firm A consists in setting $p_{J} (x) = \gamma f(x)(i_{ - J} - i_{J} ) + p_{ - J} (x)$: customer x buys from firm J, which obtains a non-negative mark-up, as $(t - \varepsilon i_{ - J} )T_{ - J} (x) + \gamma f(x)(i_{ - J} - i_{J} ) \ge (t - \varepsilon i_{J} )T_{J} (x)$ by assumption. Firm $- J$ has the incentive to undercut firm J by setting a price equal to: $p_{ - J} '(x) = p_{ - J} (x) - \eta$, where $\eta$ is a positive and small number. Since $p_{ - J} (x)$ is higher than $(t - \varepsilon i_{ - J} )T_{ - J} (x)$ by hypothesis, $p_{ - J} '(x)$ is higher than the transportation costs sustained by firm $- J$ (i.e. the mark-up of firm $- J$ on customer x is positive). Therefore, $p_{ - J} (x) > (t - \varepsilon i_{ - J} )T_{ - J} (x)$ cannot be an equilibrium, because firm $- J$ would obtain higher profits by setting $p_{ - J} '(x)$. We now show that $p_{ - J} (x) = (t - \varepsilon i_{ - J} )T_{ - J} (x)$ is an equilibrium. The best-reply of firm J is still $p_{J} (x) = \gamma f(x)(i_{ - J} - i_{J} ) + p_{ - J} (x)$. With such a price, firm $- J$ obtains zero profits from customer x, which buys from firm J, but it has no incentive to change the price, because by increasing the price it would continue to obtain zero profits, and setting a price lower than zero would entail a loss (the mark-up would be negative). It follows that the couple $p_{J} (x) = \gamma f(x)(i_{ - J} - i_{J} ) + (t - \varepsilon i_{ - J} )T_{ - J} (x)$ and $p_{ - J} (x) = (t - \varepsilon i_{ - J} )T_{ - J} (x)$ represents the (unique) price equilibrium. □

1.1 Relevant thresholds for Proposition 2

$$ \begin{aligned} & \gamma_{1} \equiv \frac{{t(2 + m) - \sqrt {t[t(2 + m)^{2} - 2\varepsilon (1 + m)^{2} ]} }}{{(1 + m)^{2} }};\quad \gamma_{2} \equiv \frac{{(t - \varepsilon )(m - 2) + \sqrt {(t - \varepsilon )[t(2 - m)^{2} - \varepsilon (2 - m^{2} )]} }}{{(1 - m)^{2} }} \\ & \gamma_{3} \equiv \frac{{t(2 - m) - \sqrt {t[t(2 - m)^{2} - 2\varepsilon (1 - m)^{2} ]} }}{{(1 - m)^{2} }};\quad \gamma_{4} \equiv \frac{{(\varepsilon - t)(m + 2) + \sqrt {(t - \varepsilon )[t(2 + m)^{2} - \varepsilon (2 - m^{2} )]} }}{{(1 + m)^{2} }} \\ & m* \equiv 1 - \frac{{2[t - \sqrt {t(t - \varepsilon )]} }}{\varepsilon }. \\ \end{aligned} $$

1.2 Relevant thresholds for Proposition 5

$$ \begin{aligned} \gamma_{1}^{CS} & \equiv \frac{{2\varepsilon (1 + m) - t(2 + m) + \sqrt {t[t(2 + m)^{2} - 2\varepsilon (1 - m^{2} )]} }}{{(1 + m)^{2} }} \\ \gamma_{2}^{CS} & \equiv \frac{{t(2 - m) - m\varepsilon - \sqrt {(t - \varepsilon )[t(2 - m)^{2} - \varepsilon (2 - 4m + 3m^{2} )]} }}{{(1 - m)^{2} }} \\ \gamma_{3}^{CS} & \equiv \frac{3\varepsilon }{2};\quad m^{CS} * \equiv \frac{1}{3} - \frac{{2[t - \sqrt {t(t - \varepsilon )]} }}{3\varepsilon }. \\ \end{aligned} $$