_{1}

By making use of the optimal stopping theory, we construct a multi-stage stochastic Cournot model to examine the effect of increase in uncertainty and number of entrants on the amount and timing of strategic cost reduction investment. It is revealed that firms should enlarge and postpone the investment if 1) the market is more uncertain, or 2) there exist more firms in the market.

Studies on oligopolistic markets under uncertainty, which progressed by paying attention to information sharing with rivals (Basar and Ho (1974) [

The present paper attempts to push forward these studies by focusing on Cournot competition in a stochastically fluctuating market. More precisely, we construct a multi-stage stochastic Cournot model based on the optimal stopping theory to examine the effect of increase in uncertainty and number of entrants on the amount and timing of the strategic cost reduction investment.

Structure of this paper is as follows. Section 2 lays out a stochastic dynamic Cournot model and derives the conditions for the optimal amounts of the output and the investment. Section 3 derives the condition for the optimal timing of the investment. Based on these analyses, Section 4 reveals the effect of increase in uncertainty and number of entrants on the optimal amount and timing of the investment. Concluding remarks are made in Section 5.

It is revealed that firms should enlarge and postpone the investment if 1) the market is more uncertain, or 2) there exist more firms in the market.

Let us consider a stochastic oligopolistic market where time passes continuously with importance of the future diminishing with discount rate ρ. There exist n identical firms that are planning to enter the market to produce homogeneous goods in a Cournot fashion, engaging in two stage game, i.e., each firm determines amount and timing of investment in the first stage, and then, in the second stage, amount of output in each period after the investment. Let us assume that the ith firm’s investment K_{i} reduces its marginal cost c_{i}, which we assume to be constant over time once the investment was conducted. In the following, in order to simplify the analysis, we

specify the relationship between the ith firm’s marginal cost and investment as

Letting x_{i}(t) and p(t) denote the ith firm’s output and the unit price in period t, respectively, we assume p(t) is related to the total output in period t,

where a is a positive constant that expresses the choke price, while b is a positive variable that expresses the size of the market, which we assume to fluctuate stochastically according as the following geometric Brownian motion:

with initial value b_{0}, where s is a positive constant that expresses volatility in a sense that larger s means more uncertain expansion of the market, while dz is Wiener process that expresses the random movement.

Following the standard procedure of the backward induction, to begin with, let us determine the optimal output of each firm in the second stage. Since the ith firm’s profit in period t after the investment, π_{i}(t), is described as

we have the ith firm’s output and profit in the Cournot equilibrium in period t, respectively, as

from the ith firm’s first order condition for profit maximization

Since we have the following equations from Equation (3)

we can express the stochastic process of the ith firm’s profit, by making use of Ito’s lemma, as

with initial value π_{0}, where μ = s^{2} and σ = −s.

If we let

obtained as _{i}, the optimal amount of the investment is determined to equalize the marginal values of _{i}. Thus, we have the condition

which reduces to

by assuming symmetric solution, i.e.,

In this section, we derive the condition for the ith firm’s optimal timing of the investment in the first stage.

As a preliminary, let us derive the expected value of one unit of profit at _{0}) denote this value, the general solution to G(π_{0}) is expressed as

where γ_{1} < 0 and γ_{2} > 0 are solutions to the characteristic equation^{2} and σ = −s into it. Since G(π_{0}) satisfies G(∞) = 0 and_{2}, we obtain

where

Thus, we can express the ith firm’s object function V_{i} to maximize in period 0 as

By substituting the ith firm’s profit in the symmetric equilibrium

from whose derivative with respect to

which reduces to

by assuming symmetric solution, i.e., _{i} with respect to

Now we are ready to determine the optimal amount and timing of the investment. By solving Equation (5) (optimal condition for the amount of the investment) and Equation (11) (optimal condition for the timing of the in-

vestment) with respect to K and b, and making use of

If we assume for example that a = 10 and ρ = 1.2, graphs of Equation (12) and Equation (13) are depicted as in

Since K^{*} is a decreasing function as in ^{*} is an increasing function with respect to γ in a relevant range as in

in s increases K^{*} and reduces b^{*}. Since we have ^{*} is

equivalent with postponement of the investment. Since we can also see from Equation (12) and Equation (13) that graph of K^{*} shifts upward while graph of b^{*} shifts downward in accordance with an increase in n as in

Proposition: Firms should enlarge and postpone the investment if 1) the market is more uncertain, or 2) there exist more firms in the market.

By constructing a theoretical model that combines Cournot competition with stochastic motion, the present paper attempts to examine the effect of an increase in uncertainty and number of entrants on the amount and timing of the strategic cost reduction investment.

It is revealed that firms should enlarge and postpone the investment if 1) the market is more uncertain, or 2) there exist more firms in the market. We can see that the investment should be enlarged if the value of waiting rises in accordance with increase in uncertainty and number of entrants.

It is necessary to examine the robustness of the results by assuming more general demand function and cost function. It is also necessary to relax the assumption of geometric Brownian motion, as well as to give an empirical testing of this model. We take up such analysis next, hoping to contribute to the advancement of the theory of the Cournot competition in a stochastically fluctuating market.

The author is grateful for the reviewers’ valuable comments that improved the manuscript.

YasunoriFujita, (2016) Optimal Amount and Timing of Investment in a Stochastic Dynamic Cournot Competition. Theoretical Economics Letters,06,1-6. doi: 10.4236/tel.2016.61001