Bertrand Meets Ford: Benefits and Losses

Abstract

The paper carries out the detailed comparison of two types of imperfect competition in a general equilibrium model. The price-taking Bertrand competition assumes the myopic income-taking behavior of firms, another type of behavior, price competition under a Ford effect, implies that the firms’ strategic choice takes into account their impact to consumers’ income. Our findings suggest that firms under the Ford effect gather more market power (measured by Lerner index), than “myopic” firms, which is agreed with the folk wisdom “Knowledge is power.” Another folk wisdom implies that increasing of the firms’ market power leads to diminishing in consumers’ well-being (measured by indirect utility.) We show that in general this is not true. We also obtain the sufficient conditions on the representative consumer preference providing the “intuitive” behavior of the indirect utility and show that this condition satisfy the classes of utility functions, which are commonly used as examples (e.g., CES, CARA and HARA.)

Appendix

15.1.1 Proof of Theorem 15.1

Combining Zero-profit condition (15.17) $m=\frac {f}{L}n=\varphi n$ with formula for symmetric equilibrium demand x = n ⁻¹ − φ ⇔ n = (x + φ)⁻¹ we can rewrite the equilibrium mark-up equation for income-taking firms (15.20) as follows

$$\displaystyle \begin{aligned} \frac{\varphi}{x+\varphi}=\varphi+(1-\varphi)r_{u}(x). \end{aligned}$$

Solving this equation with respect to x we obtain the symmetric equilibrium consumers’ demand x(φ), parametrized by φ = f∕L, which cannot be represented in closed form for general utility u(x), however, the inverse function φ(x) has the closed-form solution

$$\displaystyle \begin{aligned} \varphi=\frac{1-x}{2}-\sqrt{\left(\frac{1-x}{2}\right)^{2}-\frac{xr_{u}(x)}{1-r_{u}(x)}}. {} \end{aligned} $$

(15.25)

It was mentioned above that graph of indirect utility V (n) is bell-shaped and equilibrium masses of firms satisfy $n^*\leq \bar {n}\leq \bar {n}^F$ if and only if $V'(\bar {n})\leq 0$. Calculating the first derivative V ′(n) = u(n ⁻¹ − φ) − n ⁻¹ ⋅ u′(n ⁻¹ − φ) and substituting both n = (x + φ)⁻¹ and (15.25) we obtain that

$$\displaystyle \begin{aligned} n^*\leq \bar{n}\leq \bar{n}^F\iff u(x)\leq\left(\frac{1+x}{2}-\sqrt{\left(\frac{1-x}{2}\right)^{2}-\frac{xr_{u}(x)}{1-r_{u}(x)}}\right)u'(x), \end{aligned}$$

at $x=\bar {x}$—the equilibrium consumers demand in case of income-taking firms. The direct calculation shows that this inequality is equivalent to

$$\displaystyle \begin{aligned} \varDelta_{u}(x)\leq(1-r_{u}(x))\frac{1-x}{2}\left[1-\sqrt{1-\frac{4x r_u(x)}{(1-r_{u}(x))(1-x)^{2}}}\right].{} \end{aligned} $$

(15.26)

We shall prove that this inequality holds for all sufficiently small x > 0, provided that Δ′(0) < r _u(0). To do this, consider the following function

$$\displaystyle \begin{aligned} A_{u}(x)=\frac{x\cdot r_{u}(x)}{1-x}, \end{aligned}$$

which satisfies $A_{u}(0)=0=\varDelta _{u}(0), \ \varDelta _{u}^{\prime }(0)<A^{\prime }_{u}(0)=r_{u}(0)$. This implies that inequality Δ _u(x) ≤ A _u(x) holds for all sufficiently small x > 0.

Applying the obvious inequality $\sqrt {1-z}\leq 1-z/2$ to

$$\displaystyle \begin{aligned} z=\frac{4x r_u(x)}{(1-r_{u}(x))(1-x)^{2}}, \end{aligned}$$

we obtain that the right-hand side of inequality (15.26)

$$\displaystyle \begin{aligned} (1-r_{u}(x))\frac{1-x}{2}\left[1-\sqrt{1-\frac{4x r_u(x)}{(1-r_{u}(x))(1-x)^{2}}}\right]\geq A_{u}(x)\geq \varDelta_{u}(x) \end{aligned}$$

for all sufficiently small x > 0, which completes the proof of statement (a).

Applying the similar considerations to Eq. (15.19), which determines the equilibrium markup under a Ford effect, we obtain the following formula for inverse function φ(x)

$$\displaystyle \begin{aligned} \varphi=\frac{1-r_{u}(x)-x}{2}-\sqrt{\left(\frac{1-r_{u}(x)-x}{2}\right)^{2}-xr_{u}(x)} \end{aligned}$$

Using the similar considerations, we obtain that

$$\displaystyle \begin{aligned} \bar{n}^{F}\leq n^{*}\iff u(x)\geq\left(\frac{1-r_{u}(x)+x}{2}-\sqrt{\left(\frac{1-r_{u}(x)-x}{2}\right)^{2}-xr_{u}(x)}\right)u'(x) \end{aligned}$$

at $x=\bar {x}^{F}$—the equilibrium demand under Bertrand competition with Ford effect. The direct calculation shows that the last inequality is equivalent to

$$\displaystyle \begin{aligned} \varDelta_{u}(x)\geq\frac{1-r_{u}(x)-x}{2}\left[1-\sqrt{1-\frac{4xr_{u}(x)}{\left(1-r_{u}(x)-x\right)^{2}}}\right].{} \end{aligned} $$

(15.27)

Now assume

$$\displaystyle \begin{aligned} \delta_{u}>\frac{r_{u}(0)}{1-r_{u}(0)}, \end{aligned}$$

which implies that

$$\displaystyle \begin{aligned} \alpha\equiv\frac{r_{u}(0)+(1-r_{u}(0))\delta_{u}}{2r_{u}(0)}>1. \end{aligned}$$

Let

$$\displaystyle \begin{aligned} B_{u}(x)\equiv \frac{\alpha xr_{u}(x)}{1-r_{u}(x)-x}, \end{aligned}$$

it is obvious that Δ _u(0) = B _u(0) = 0, and

$$\displaystyle \begin{aligned} B^{\prime}_{u}(0)=\frac{\alpha r_{u}(0)}{1-r_{u}(0)}=\frac{r_{u}(0)+(1-r_{u}(0))\delta_{u}}{2(1-r_{u}(0))}<\delta_{u}=\varDelta^{\prime}_u(0), \end{aligned}$$

which implies that inequality Δ _u(x) ≥ B _u(x) holds for all sufficiently small x.

On the other hand, the inequality $\sqrt {1-z}\geq 1-\alpha z/2$ obviously holds for any given α > 1 and $z\in \left [0,\frac {4(\alpha -1)}{\alpha ^{2}}\right ]$. Applying this inequality to

$$\displaystyle \begin{aligned} z=\frac{4xr_{u}(x)}{\left(1-r_{u}(x)-x\right)^{2}}, \alpha=\frac{r_{u}(0)+(1-r_{u}(0))\delta_{u}}{2r_{u}(0)}, \end{aligned}$$

we obtain that the right-hand side of (15.27) satisfies

$$\displaystyle \begin{aligned} \frac{1-r_{u}(x)-x}{2}\left[1-\sqrt{1-\frac{4xr_{u}(x)}{\left(1-r_{u}(x)-x\right)^{2}}}\right]\leq B_{u}(x){} \end{aligned} $$

(15.28)

for all sufficiently small x > 0, because x → 0 implies z → 0. This completes the proof of Theorem 15.1.