Competing Firms under Price Uncertainty Ferenc Szidarovszky ReliaSoft, 1450 S. Eastside Loop, Tucson, Arizona 85710, USA Phone: +1 (520) 901-­‐9072, E-­‐mail: [email protected] Melissa Lynn Rosario Raytheon, 1151 E. Hermans Road, Tucson, Arizona 85756, USA Phone: +1 (520) 647-­‐6118, E-­‐mail: [email protected] Abstract The optimum behavior of competing firms is examined under price uncertainty. The firms want to maximize their expected profits and in order to reduce risk, they also want to minimize the variances of the profits. This two-­‐objective problem is realized by optimizing the certainty equivalents. In addition, the firms also want to limit the probabilities of very low profits, which requirement is formulated as chance-­‐constraints. Two methods are suggested to find the equilibrium output levels. Based on the Kuhn-­‐Tucker conditions as feasible solutions of a system of equalities and inequalities have to be identified. The second method requires to solve a single-­‐objective nonlinear programming problem. 1 Introduction The theory of competing firms is one of the most frequently discussed topics in mathematical economics. The oligopoly theory dates back to the mid nineteenth century and since then many researchers devoted their efforts to this subject. A comprehensive summary of the results up to the mid 70’s can be found in Okuguchi (1976), and their multi-­‐product generalizations with case studies are discussed in Okuguchi and Szidarovszky (1999). The most recent developments with special focus on nonlinear models are reported in Bischi et al. (2010). In most models in the literature the uncertainty in the market price and production costs are ignored by assuming complete information about these economic factors. In this paper we will examine 𝑛-­‐firm oligopolies under price uncertainty where the resulted uncertainty of the profit is modeled by its certainty equivalent (Sargent, 1979). In order to avoid very low profits, chance constraints are added to the optimum problems of the firms. The equilibrium can be found by solving for the feasible solutions of a system of nonlinear equalities and inequalities or solving a single-­‐
objective nonlinear programming problem. 2 Mathematical Model Consider 𝑛 competing firms producing the same type of product and sell it in a common market. Let 𝑥! denote the output of firm 𝑘 with marginal cost 𝑐! and assumed price function 𝑃! 𝑠 = 𝐴! − 𝐵! 𝑠, where 𝑠 = !!!! 𝑥! is the total industry output. It is also assumed that firm 𝑘 knows that 𝑃! 𝑠 is only an estimate of the price function and the “true” price function is 𝐴! − 𝐵! 𝑠 + 𝛼! , where 𝛼! is a random variable with known cumulative distribution 𝐹! , expectation µμ! and variance 𝜎!! . The profit of firm 𝑘 is given as 𝜋! = 𝑥! (𝐴! − 𝐵! 𝑠 + 𝛼! − 𝑐! ) , (1) which is random and depends on the output levels of the competitors. It is assumed that the firms want to maximize their expected profits, and minimize the variances of the profits in order to decrease uncertainty. They also want to avoid very low profits by requiring that certain chance-­‐constraints are satisfied. The expected profit of firm 𝑘 is 𝐸 𝜋! = 𝑥! (𝐴! − 𝐵! 𝑠 + 𝜇! − 𝑐! ) (2) and the variance of the profit equals 𝑉𝑎𝑟 𝜋! = 𝑥!! 𝜎!! . (3) The certainty equivalent for firm 𝑘 is therefore 𝐶𝐸! = 𝑥! 𝐴! − 𝐵! 𝑠 + 𝜇! − 𝑐! − ɤ! 𝑥!! 𝜎!! , (4) where ɤ! > 0 shows the risk acceptance level of the firm. If ɤ! = 0, then risk is not considered at all, and as ɤ! → ∞ only risk is considered. The firms also want to limit the probability of very low actual profits by requiring that 𝑃 𝜋! < 𝜀! ≤ 𝑝! , (5) where 𝜀! is the lower bound of the profit, which can be violated with probability at most 𝑝! . Hence 𝑝! is a small probability level. Notice that condition (5) can be rewritten as 𝑝! ≥ 𝑃 𝑥! 𝐴! − 𝐵! 𝑠 + 𝛼! − 𝑐! < 𝜀! = 𝑃 𝛼! <
𝜀!
𝜀!
− 𝐴! + 𝐵! 𝑠 + 𝑐! = 𝐹!
− 𝐴! + 𝐵! 𝑠 + 𝑐! 𝑥!
𝑥!
or !!
!!
− 𝐴! + 𝐵!
!
!!! 𝑥!
+ 𝑐! ≤ 𝐹!!! 𝑝! , (6) where 𝐹!!! is the inverse of 𝐹! . It is also natural to require that 0 ≤ 𝑥! ≤ 𝐿! , where 𝐿! is the capacity limit of firm 𝑘 . Hence firm 𝑘 is solving the following optimization problem: maximize subject to 𝑥! 𝐴! − 𝐵!
!
!!! 𝑥!
+ 𝜇! − 𝑐! − ɤ! 𝑥!! 𝜎!! 𝑥! ≥ 0 𝐿! − 𝑥! ≥ 0 𝐹!!! 𝑝! −
!!
!!
!
!!! 𝑥!
+ 𝐴! − 𝐵!
(7) − 𝑐! ≥ 0. The objective function and all constraints are concave in 𝑥! , so the Kuhn-­‐Tucker conditions are sufficient and necessary: 𝑢! , 𝑣! , 𝑤! ≥ 0 𝑥! ≥ 0 𝐿! − 𝑥! ≥ 0 𝐹!!!
𝐴! − 2𝐵! 𝑥! − 𝐵!
𝑝!
!!! 𝑥!
𝜀!
−
+ 𝐴! − 𝐵!
𝑥!
!
𝑥! − 𝑐! ≥ 0 !!!
+ 𝜇! − 𝑐! − 2ɤ! 𝑥! 𝜎!! + 𝑢! − 𝑣! + 𝑤!
!!
!!!
− 𝐵! = 0 (8) 𝑢! 𝑥! = 0 𝑣! 𝐿! − 𝑥! = 0 𝑤!
𝐹!!!
𝑝!
𝜀!
−
+ 𝐴! − 𝐵!
𝑥!
!
𝑥! − 𝑐! = 0. !!!
The equilibrium of this situation is an output level for all firms such that every form is in optimum. Since the Kuhn-­‐Tucker conditions are sufficient and necessary, condition (8) suggests a computer method to find the equilibrium. Create the system of equalities and inequalities by requiring all conditions (8) of all firms, then the set of feasible solutions and the set of all equilibria are the same. In this way the computation of the equilibria are reduced to finding feasible solutions of a large set of inequalities and equalities with unknowns 𝑥! , 𝑢! , 𝑣! , 𝑤! for 𝑘 = 1,2, … , 𝑛. Hence the problem is 4𝑛-­‐dimensional. It is easy to see that we can rewrite the problem as a 4𝑛-­‐dimensional single-­‐objective optimization problem: minimize subject to !
!!!
𝑢! 𝑥! + 𝑣! 𝐿! − 𝑥! + 𝑤! 𝐹!!! 𝑝! −
!!
!!
+ 𝐴! − 𝐵!
!
!!! 𝑥!
− 𝑐! 𝐹!!! 𝑝! −
𝐴! − 2𝐵! 𝑥! − 𝐵!
!!! 𝑥!
!!
!!
𝑢! , 𝑣! , 𝑤! ≥ 0
𝑥! ≥ 0
𝐿 ! − 𝑥! ≥ 0
+ 𝐴! − 𝐵!
!
!!! 𝑥!
− 𝑐! ≥ 0
+ 𝜇! − 𝑐! − 2ɤ! 𝑥! 𝜎!! + 𝑢! − 𝑣! + 𝑤!
!!
!!!
(9) − 𝐵! = 0
for 𝑘 = 1,2, … , 𝑛 . Since there are professional software packages for solving nonlinear optimization problems, the equilibria can be obtained easily. The optimum value of the objective function shows the existence of equilibrium. If it is positive, then there is no solution which satisfies all conditions of (8), so no equilibrium exists. If the optimum value is zero, then any optimal solution provides equilibrium. Because of the first four conditions of (8), the objective function of (9) has to be always nonnegative. The dimension of problem (9) can be reduced by 𝑛 in the following way. By solving the last condition for 𝑢! , and replacing 𝑢! with the resulted expression in the objective function and requiring that the expression is nonnegative, we can reduce the number of unknowns by 𝑛, and also the last condition can be eliminated. 3 Numerical Example Consider a duopoly, 𝑛 = 2, with 𝐴! = 𝐴! = 10, 𝐵! = 𝐵! = 1, 𝐿! = 𝐿! = 10, 𝑐! = 𝑐! = 2, so the duopoly is symmetric. Assume in addition that both ∝! and ∝! are normally distributed with 𝜇! = 𝜇! =
0 and 𝜎!! = 𝜎!! = 2. So 𝐹! ∝ = 𝐹! ∝ = Φ
∝
!
, where Φ is the standard normal distribution function, and therefore 𝐹!!! 𝑧 = 𝐹!!! 𝑧 = 2Φ !! 𝑧 . It is also assumed that the lower bounds for the profits of both firms are 𝜀! = 𝜀! = 2 with probability levels 𝑝! = 𝑝! = 1%, and the risk acceptance factors are ɤ! = ɤ! = 0.5. Then problem (9) has the special form: minimize !
!!!
𝑢! 𝑥! + 𝑣! 10 − 𝑥! + 𝑤!
subject to 2Φ !! 0.01 −
!
!!
+ 10 − 𝑥! + 𝑥! − 2 𝑢! , 𝑣! , 𝑤! ≥ 0
𝑥! ≥ 0
10 − 𝑥! ≥ 0
2Φ !! 0.01 −
!!
!!
𝑘 = 1,2 + 10 − 𝑥! + 𝑥! − 2 ≥ 0
10 − 𝑥! − 𝑥! + 𝑥! + 0 − 2 − 2𝑥! + 𝑢! − 𝑣! + 𝑤!
with unknowns 𝑥! , 𝑥! , 𝑢! , 𝑢! , 𝑣! , 𝑣! , 𝑤! , 𝑤! . !
!!!
−1 =0
4 Conclusion A mathematical model was formulated and solved to find optimal production levels of competing firms under price uncertainty. The firms want to maximize expected profit and in order to minimize risk they also want to minimize the variances of their profits. Each firm also requires that the probability that the profit falls below a certain lower bound is very small. The expectation and variance of each profit are combined in the certainty equivalent which is then optimized. Two methods were suggested to find the equilibrium. In the first case the Kuhn-­‐Tucker conditions for all firms present a system of equalities and inequalities with 4𝑛 unknowns, and every feasible solution gives equilibrium. In the second method a single-­‐variable nonlinear optimum problem is constructed and should be solved. The zero optimal value of the objective function indicates the existence of equilibrium. 5 References Bischi, G-­‐I., C. Chiarella, M. Kopel and F. Szidarovszky (2010) Nonlinear Oligopolies: Stability and Bifurcations. Springer-­‐Verlag, Heidelberg/New York. Okuguchi, K. (1976) Expectations and Stability in Oligopoly Models. Springer-­‐Verlag, Berlin. Okuguchi, K. and F. Szidarovszky (1999) The Theory of Oligopoly with Multi-­‐product Firms. Springer-­‐
Verlag, Berlin (2nd edition). Sargent, T.J. (1979) Macroeconomic Theory. Academic Press, New York.