simulating competitive pricing

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Simulating Competitive Pricing

Charles Higgins, PhDDept Finance/AIMS née CIS

Loyola Marymount University

1 LMU Dr.Los Angeles, CA 90045-8385

310 338 7344

[email protected]

Draft 4 August 25, 2011

Introduction

In a capitalist society, a firm seeks to maximize profits determined by:

1) Π = (P - V)●Q – F

where Π is profits, P is a firm’s price, V is variable costs, Q is quantity sold,

and F is fixed costs. It is argued that the pricing of goods and services is a

function of costs if there is competition and that Q will decrease when P

increases and vice versa. Further, if competitive prices change, a new

equilibrium will be reached in most cases.

Theoretic economic examinations argue that eventually the firm will

converge toward marginal revenue equaling marginal costs or:

2) ΔR/ΔQ = ΔC/ΔQ

where R is revenue and C is costs. Theory is generally configured in a

calculus (and often with calculus) leading to MC equaling V and then P

converging toward V.

Simulation

Instead of a theoretic approach, one could simulate individualcustomer behaviors and the price reaction behaviors of firms. Other

approaches have been investigated (Amstutz [1970] and Csik [1996].

Let there be a market of N firms and M customers and a stated

variable cost of V for each firm. An initial price P for each firm is set and

then for each iterative market cycle, each customer randomly selects any

where from 1 to N firms to choose the lowest price. The selected starting

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firm for each customer is also randomized. After each market cycle, the

firm with the lowest profits (and the most in need of a change in price) resets

its price toward the mean price of that cycle. It is often necessary to

introduce a random variation in order to perturb the system into motion

which was initially set as 1 so that a random component was added

anywhere from +.5 to –.5. Dampening of the random variation allows a later

finer gradation of subsequent price changes and is 99.9 percent of the

previous variation.

In each run, varying the number of firms, initial prices, variable costs,

and the number firms searched resulted in a convergence of firm prices

toward variable costs. The exceptions were when the customers did not

search beyond only one firm (which produced the expected result of a

steadily rising price) and when the firms had different variable costs (not

present here—for another day). Specifically, in the simulation where there

were three firms, a customer could search one, two, or three firms starting ateither firm 1, 2, or 3; the simulation with two or four firms used a similar

procedure. The simulation examined N at 2 then 3 then 4 firms, M at 100

customers, fixed costs F as zero (which is generally recognized as important

only for a firm’s decision to enter or exit a market), and V as 3 then 30

dollars (or euros, yen, pounds, francs, etc.). Initial prices were set at 19 then

18 then 17 etc. (P=20-f where f is the number of the firm). For N firms,

1/Nth of the customers examined one firm’s price and likewise the last 1/Nth

of the customers examined prices of all the firms (with exception when the

simulation was set to select only one firm for each customer).

Results

Firms 2 3 4

Variable Cost

3 3.03 3.10 3.07

30 30.17 30.11 30.21

3 Firms P=20-f P=80-20f One Search

5 5.03 5.09 62.25

10 9.99 10.01 63.4315 15.06 15.04 59.72

20 20.03 19.96 61.79

25 25.01 25.01 62.57

30 30.01 30.04 63.63



Computer program

simulating competitive pricing

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