On hierarchical competition in oligopoly

Abstract

In this paper, we consider a hierarchical oligopoly model, in which firms compete on quantities of an homogeneous product. We provide a proof and an interpretation that under the three necessary and sufficient conditions of linear aggregate demand, constant and identical marginal costs, the strategy of leaders at any stage depends neither on the number of leaders who play after nor on the number of remaining stages. So, all firms behave as Cournotian oligopolists on the residual demand. We show that these three assumptions are not only sufficient but also necessary. Any departure from any of these assumptions rules out this property.

Appendices

Appendix A: Proof of Lemma 1

The proof is by backward induction and structured in three steps.

Step 1 property 1 is true for cohort $t=T-1$ (with $T>1$).

The inverse demand function faced by firms is defined by:

$$\begin{aligned} p(X) = a - bX \quad \text{ with} X=\sum \limits _{\tau =1}^{T} X_\tau , \end{aligned}$$

(H1)

where $X_\tau = \sum _{i=1}^{n_\tau } x_\tau ^i\ge 0$ is the aggregate production of cohort $\tau$. For any quantity of output $X_{T-1}$ produced by cohort $T-1$, the resulting residual demand faced by followers of cohort $T$ is:

$$\begin{aligned} p\left(\sum \limits _{\tau =1}^{T}X_\tau \right) = \bar{a}_{T-1} - bX_{T}\quad \text{ with} \bar{a}_{t}\equiv a-b\sum \limits _{\tau =1}^{t}X_\tau , \end{aligned}$$

(38)

where $\bar{a}_{T-1}$ is considered as given by followers. Geometrically (see Fig. 1), followers must select a couple $(X,p)$ on the segment $[D,A]$.

When acting symmetrically, the associated marginal revenue of cohort-$T$ firms (dashed line on Fig. 1) is defined by:⁹

$$\begin{aligned} R_m(X_T) = \bar{a}_{T-1} - b\frac{1+n_{T}}{n_{T}}X_{T}. \end{aligned}$$

(39)

Considering the following derivatives:

$$\begin{aligned} \frac{\partial R_m}{\partial X_T}(X_T)&= \frac{DF}{CF} = -b\frac{1+n_{T}}{ n_{T}}\end{aligned}$$

(40)

$$\begin{aligned} \frac{\partial p}{\partial X_T}(X_T)&= \frac{DF}{AF} = -b, \end{aligned}$$

(41)

it comes that:

$$\begin{aligned} CF=\frac{n_{T}}{1+n_{T}}AF, \quad \text{ or} \text{ equivalently} AC=\frac{1}{1+n_{T}}AF. \end{aligned}$$

(42)

Finally, applying Thales’ theorem to triangles $ABC$ and $ADF$ leads to:

$$\begin{aligned} BC=\frac{1}{1+n_{T}}DF=EF. \end{aligned}$$

(43)

Actually, $EF$ is the markup of a leader after the entrance of the last cohort, while $DF$ is the markup of a leader before the entrance of cohort $T$. Equation (43) can be rewritten as:

$$\begin{aligned} p\left(\sum \limits _{\tau =1}^{T}X_\tau \right) - c = \frac{1}{1+n_{T}}\left[ p\left(\sum \limits _{\tau =1}^{T-1}X_\tau \right) - c\right]. \end{aligned}$$

(44)

Step 2 assume property (1) is true for any cohort $t=T-h$ ($1\le h \le T-2$) then it is true for cohort $T-h-1$.

If property (1) holds for cohort $T-h$ then:

$$\begin{aligned} \left[p\left(\sum \limits _{\tau =1}^{T}X_\tau \right)-c\right]x^i_{T-h} = \gamma _{T-h} \left[p\left(\sum \limits _{\tau =1}^{T-h}X_\tau \right)-c\right] x^i_{T-h}, \end{aligned}$$

(45)

with $\gamma _{T-h}\equiv \prod _{\tau = T-h+1}^{T} \frac{1}{1+n_\tau }$.

Thus, maximizing firm $i$’s profit is tantamount to maximize the myopic profit defined as follows:

$$\begin{aligned} \max _{x^i_{T-h}} \left[p\left(\sum \limits _{\tau =1}^{T-h}X_\tau \right)-c \right]x^i_{T-h}. \end{aligned}$$

(46)

When firms of cohort-$(T-h)$ act symmetrically, the myopic marginal revenue (dashed line) is defined by:¹⁰

$$\begin{aligned} \tilde{R}_m(X_{T-h}) = \bar{a}_{T-h-1} - b \frac{1+n_{T-h}}{n_{T-h}}X_{T-h}. \end{aligned}$$

(47)

In the same way as in step 1, it can be shown that:

$$\begin{aligned} p\left(\sum \limits _{\tau =1}^{T-h}X_\tau \right) - c = \frac{1}{1+n_{T-h}} \left[p\left(\sum \limits _{\tau =1}^{T-h-1}X_\tau \right) - c\right]. \end{aligned}$$

(48)

By assumption, the following property is satisfied:

$$\begin{aligned} p\left(\sum \limits _{\tau =1}^{T}X_\tau \right)-c = \gamma _{T-h} \left[ p\left(\sum \limits _{\tau =1}^{T-h}X_\tau \right)-c\right]. \end{aligned}$$

(49)

We deduce from the two previous equations that:

$$\begin{aligned} p\left(\sum \limits _{\tau =1}^{T}X_\tau \right)-c&= \frac{\gamma _{T-h}}{ 1+n_{T-h}} \left[p\left(\sum \limits _{\tau =1}^{T-h-1}X_\tau \right)-c\right]\end{aligned}$$

(50)

$$\begin{aligned}&= \gamma _{T-h-1} \left[p\left(\sum \limits _{\tau =1}^{T-h-1}X_\tau \right)-c \right] \end{aligned}$$

(51)

Step 3 from steps 1 and 2 we conclude by backward induction that Property 1 is true for any cohort $t$ (with $1\le t\le T-1$).

Appendix B: Proof of Lemma 2

Applying Thales’ theorem to triangles ABC and ADF leads to:

$$\begin{aligned} CF=\frac{n_{t}}{1+n_{t}}AF. \end{aligned}$$

(52)

Actually, $CF$ is the optimal output produced by cohort $t$, that is $X_t$, while $AF$ is the maximal quantities cohort $t$ can produce to generate non-negative profit (equal to the difference between the perfect competition equilibrium supply and the output already produced by the previous cohorts). The property above can be rewritten as:

$$\begin{aligned} X_t =\frac{n_t}{1+n_t}(X_t + AC), \quad \text{ or} \text{ equivalently} AC =\frac{X_t}{n_t} = x_t. \end{aligned}$$

(53)

Notice that $AC$ is also the maximal quantities cohort $t+1$ can produce to generate non-negative profit. Then, equality (52) applied to cohorts $t$ and $t+1$ becomes:

$$\begin{aligned} X_{t+1} =\frac{n_{t+1}}{1+n_{t+1}}AC, \quad \text{ leading} \text{ to} \frac{X_{t+1}}{n_{t+1}} = x_{t+1} = \frac{1}{1+n_{t+1}}x_t. \end{aligned}$$

(54)

By backward induction, it turns out that:

$$\begin{aligned} x_t = \eta _{1,t} x_1,\quad \text{ where} \eta _{1,t}\equiv \prod \limits _{\tau = 2}^{t} \frac{1}{1+n_\tau }. \end{aligned}$$

(55)