EIRP Proceedings, Vol 15, No 1 (2020)
The Determination of a Company Production under the Conditions of Minimizing the Production Costs, but Also Profit Maximization
Ctlin Angelo Ioan^{1}, Gina Ioan^{2}
Abstract. The paper deals with the problem of determining the production of a company under the conditions in which it wants both the minimization of the production costs and the maximization of the profit.
Keywords: production function; Cobb-Douglas; profit
1. Introduction
Let us consider a firm F whose activity is formalized using a production function Q which depends on a number of production factors x_{1},...,x_{n}, n2. In order to ensure its competitiveness on the market, its main purpose is to reduce its total cost which will implicitly lead to the output of its products at the lowest possible cost. On the other hand, the company wants to maximize its profit. For example, we will consider the production function as Cobb-Douglas type, which is equivalent to a constancy of the elasticities of production in relation to the factors of production, which is not restrictive, at least for a limited time.
The Cobb-Douglas function has the following expression:
Q:D -{0}R_{+}, (x_{1},...,x_{n})Q(x_{1},...,x_{n})= R_{+} (x_{1},...,x_{n})D, A , _{1},...,_{n} .
,i=
The main indicators are:
= = = , i= ;
= = = , i= ;
RMS(i,j)= , i,j= ;
= = , i= ;
_{ij}=-1, i,j= .
2. The costs of the Cobb-Douglas production function
Considering now the problem of minimizing costs for a given production Q_{0}, where the prices of inputs are p_{i}, i= , we have:
From the obvious relations: we obtain: and from the second equation: . Noting r= 0, we finally obtain:
= , k=
The total cost is:
TC(Q_{0})= = .
The marginal cost is:
TC_{m}(Q_{0})= = .
Noting, for simplicity: = 0 and s= , follows: TC_{m}(Q_{0})= , TC(Q)=r = because sr=1-r r(s+1)=1 r= .
Let’s note that, because r0, we have s(-1,).
Consider the profit of the company: (Q)=p(Q)Q-TC(Q)
If p(Q)=a-bQ, a,b0, we have: (Q)=aQ-bQ^{2}-TC(Q)=aQ-bQ^{2}- Q^{s+1}.
hence, the extremely necessary condition of profit becomes:
‘(Q)=a-2bQ-TC_{m}(Q)=0
otherwise: a-2bQ- =0.
Also, the necessary and sufficient condition for maximization is:
“(Q)=-2b-TC_{m}(Q)’0
that is:
-2b- 0 if s0 which is obvious because Q0 and if s(-1,0).
If r=1, namely s=0, then: a--2bQ=0 Q= .
In this case, the maximization condition returns to: -2b0 which is true.
If r1, namely s0, results:
a-2bQ-Q^{s}=0, s(-1,0)(0,).
Let the functions ‘:(0,)R, ‘(Q)=a-2bQ-Q^{s} and “(Q)=-2b-sQ^{s-1}, ‘“(Q)=-s(s-1)Q^{s-2}.
Case 1: s0
As in this case, “(Q)0 we have that ‘ it is strictly decreasing. Because =a0, =- it turns out that the equation ‘(Q)=0 it has only one strictly positive root Q_{1}(0,) which, by virtue of the above, is a local maximum point.
Case 2: s0
In this case, ‘“(Q)0 therefore “is strictly decreasing. But we have: =, =-2b0 so the equation “(Q)=0 has a single positive root Q^{*} that satisfies the relationship: -2b-s =0 or otherwise: Q^{*}= . Thus, “(Q)0 Q(0,Q^{*}) and “(Q)0 Q(Q^{*},), namely ‘ is strictly increasing on (0,Q^{*}) and strictly decreasing on (Q^{*},).
Because: =-, ‘(Q^{*})=a-2bQ^{*}- =a-2b - =
-(2b+) , =- we have:
Case 2.1
If -(2b+) 0 then the equation ‘(Q)=0 has no positive roots. In this case, has constant monotony. How s(-1,0) =0, =- so the profit being negative, the company is at a loss and therefore the only option is to stop production.
Case 2.2
If -(2b+) =0 then the equation ‘(Q)=0 has the root Q^{*}= .
But “(Q^{*})=0 and ‘“(Q)=-s(s-1) 0 so has no extreme point. On the other hand, in this case 0 so is decreasing. In this case, as production increases, profit will decrease. The maximum profit will therefore be recorded for Q = 0, meaning the company will not produce.
Case 2.3
If -(2b+) 0 then the equation ‘(Q)=0 has two positive roots: Q_{1}(0,Q^{*}). Q_{2}(Q^{*},). How “(Q_{1})0 follows that Q_{1} is a local minimum point, and how “(Q_{2})0 it turns out that Q_{2} is a local maximum point.
So let the equation: 0=‘(Q)=a-2bQ-Q^{s} with the solution . Thus: or otherwise: .
We have = = =
= 0.
Therefore, for production which satisfies the equation: the company will record a maximum profit.
3. Partial Conclusions
r=1 s=0 implies Q= ;
r1 s0 implies that Q is the root of the equation ;
If -(2b+) 0, s= then the company ceases its activity;
If -(2b+) 0, s= , s(-1,0) implies that Q is the root of the equation which additionally satisfies the condition Q .
4. The Solution of the Nonlinear Equation
Let the equation: a-2bQ- =0, s(-1,), Q0 and f:(0,)R, f(Q)=a-2bQ- , f’(Q)=-2b-sQ^{s-1}, f”(Q)=-s(s-1)Q^{s-2}.
For convergence, the function must have the same monotony and concavity over the interval (a, b) in which the root is found. The starting point is the one for which f(Q_{0})f”(Q_{0})0.
Case 1: s1
How f” (Q)=-s(s-1)Q^{s-2}0 it turns out that Q_{0} is chosen so that f(Q_{0})0. On the other hand, it turns out that f’ is strictly decreasing. How f’ (0)=-2b0 it follows that f is strictly decreasing. But =- implies that we will choose Q_{0} large enough. How = = 0 we have that Q_{0}= .
Case 2: s=1
The equation becomes a-(2b+)Q=0 from where Q= .
Case 3: s(0,1)
How f”(Q)=-s(s-1)Q^{s-2}0 it turns out that Q_{0} is chosen so that f(Q_{0})0. In this case, f’ is strictly increasing and how =-, =-2b0 it turns out that f’ is strictly negative, so f is strictly decreasing. How f(0)=a0 follows that Q_{0}=0.
Case 4: s(-1,0)
How f”(Q)=-s(s-1)Q^{s-2}0 it turns out that Q_{0} is chosen so that f(Q_{0})0. is chosen so that, f’ is strictly decreasing and how =-, =-2b0 it turns out that f’ is strictly negative, so f is strictly decreasing. How =- implies that we will choose Q_{0} large enough. On the other hand, Q_{0}Q^{*} that is Q_{0} .
Applying Newton’s recurrence formula, it results:
Therefore: , n0.
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1 Associate Professor, PhD, Danubius University of Galati, Department of Economics, Romania, Address: 3 Galati Blvd., Galati 800654, Romania, Corresponding author: catalin_angelo_ioan@univ-danubius.ro.
2 Senior Lecturer, PhD, Danubius University of Galati, Department of Economics, Romania, Address: 3 Galati Blvd., Galati 800654, Romania, E-mail: ginaioan@univ-danubius.ro.
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