Using Calculus of Variations to optimize the cost of production: Ericsson case

Today I will try to combine calculus of variations problem solving methods with strategic decision making in corporate finance. But before I begin the analysis, I will briefly describe all the elements and concepts I am going to use.

Firstly, for those who are now familiar with it, calculus of variations is a branch of mathematics which is devoted to problem solving in infinite-dimensional spaces of functions (as opposed to R1 or Rn). To explain it simply, regular calculus (single or multivariable) seeks to find values of xn for which some function f(x1, x2, x3���xn) is maximized. However, in calculus of variation (CV) we are trying to find the function itself rather than its elements. The most famous and widely used concept from the CV is the Euler���s equation, which we will also use in the Ericsson case analysis.

Secondly, I will also use such concepts as breakeven point, inventory and production cost which I learned at the Corporate Finance course this year. The main goal of this short interdisciplinary-synthesis exercise is to show how mathematics can be used to make well-though management decisions. More precisely, I will try to find and prove mathematically which production strategy should the company (Ericsson in this case) choose in order to minimize its production cost.

Consider Ericsson Company in 2003. According to Vernimmen Corporate Finance textbook, their total breakeven was 23,156 that year. Let���s try to find out what production model should the company has adopted so that their costs were minimized:

Let x(t)=inventory, with x(0)=0 and x(T)=B where B=breakeven point and T=365 days. Then, x'(t)=���production rate per day���. We assume that production cost of one unit is proportional to production rate, since higher PR requires more expensive machines, more working hours etc. Therefore, c1x'(t)=���production cost��� and c1x'2(t)=���production cost per day���. We also assume that inventory cost per day is proportional to amount of inventories in storage: c2x(t)=���cost of inventories���. Our goal is to minimize total cost, which can be written as:

We assume that the first and the second partial derivatives are continuous; therefore we can use the Euler���s condition:

Here we have, F(t, x, x')= c1x'^2+c2x, therefore solving the system gives us:

This second degree derivative can be easily solved by integrating it twice, and therefore, giving us a result:
We use our initial conditions to find constants:

Therefore, we can find our function x(t):
Also, we know that T=365 and B=23,156. We get:

Although, this formula could be practically applied only if we knew constants c1 and c2 (which would vary from one industry to another due to different nature of the production processes and companies' sales models), we can still draw some important conclusions. One of them is that the cost function is quadratic, meaning that the firm should reduce its rate of production in the early phases of production, but increase it at the latter ones. This finding is also supported by the statement of Prof. Salvi (i.e. my Finance professor) that ���companies should not produce rapidly in the early-life stages of the product, but rather postpone it until it gains the position in the market.��� Could you name another implication of this finding?