About Short-Term Costs and Long-Term Costs
Cătălin Angelo IOAN
1Alin Cristian IOAN
2Abstract.The paper treats the theory of short-term costs and long-term costs from an axiomatic perspective.
Keywords:cost, short-term, long-term
1 Introduction
Fixed costs (CF) are those costs that are independent of the value of production (rent, lighting costs, heating costs, interest etc.) and are paid whether or not production.
Quasi-fixed costs (CCF) are those costs which are also independent of the value of production, but are paid only if it exists production (e.g. advertising expenses).
Variable costs (CV) represents the total production costs that vary with output in the same direction (raw materials costs, wages, energy costs in the production process etc.).
The total cost (CT) of production is the sum of fixed and variable costs: CT=CF+CV.
Considering a level of output, we shall call average fixed cost (CFM) the value:
CFM= Q CF and represents the fixed cost per unit.
At a level Q of the output, the average variable cost (CVM) is the value:
CVM= Q CV and represents the variable cost per unit.
Similarly, the average total cost (CTM) is:
CTM= Q CT and represents the total cost of a unit of product.
From the expressions above, it follows:
CTM=CT =Q CF +Q
CV =CFM+CVMQ
1 Danubius University of Galati, Department of Finance and Business Administration, catalin_angelo_ioan@univ- danubius.ro
Considering a level of output Q, we call marginal cost (Cm) the value:
Cm= Q
CT
and represents the trend of variation of the total cost to a given production.
In terms of discrete, for two times t1and t2with values Q1and Q2of production, CT1 and CT2- total costs at that times,we define:
Cm=
1 2
1 2
Q Q
CT CT
=
Q CT
meaning the variation of total cost required to change a unit of production.
2 Short-term costs and long-term costs
A production process is said to take place in a short term if at least one factor of production remains constant.
Typically, on short-term, we assume that the capital (all resources that contribute to the deployment of production without consumption record, but suffered a depreciation in time) is constant, as the land (geographical locations, mineral deposits etc.), the only significant variable being the labor. From the analysis of short-term production follows, in this case, that after a number of workers it manifests the law of decreasing marginal returns.
We shall note, to emphasize the special nature of these costs: CVS – the variable cost in the short term, CTS – the total cost of short-term, CVMS – the average variable cost in the short term, CTMS – the average total cost in the short term, CmS – the marginal cost on short-term, fixed costs fixed or quasi-fixed cost keeping the above notations: CF, CFM, respectively CCF because they can appear only in this situation.
We have therefore:
CTS=CF+CVS;
CFM=
CF ;Q
CVMS=
CVS ;Q
CTMS=
CTS =CFM+CVMS;Q
CmS=
Q CTS
=
Q CVS
.
A production process is said to take place in the long term if all inputs are variable. Unlike the previous situation, in the long-term production, due to technological change or to improve the management, productivity may increase. On long-term, fixed costs are not recorded, so we consider CF=0, CFM=0 and CCF=0.
We note, in this case: CVL – the variable cost in the long term, CTL – the total cost in the long term, CVML – the average variable cost in the long term, CTML – the long-term average total cost and CmL – the long-term marginal cost.
We have now:
CTL=CVL;
CVML=
CVL ;Q
CTML=
CTL =CVML;Q
CmL=
Q CTL
=
Q CVL
.
From the monotonicity property of the cost function with respect to the production, follows that CTL is increased relative to the output Q.
Relative to the two types of production we might make a few comments.
Let's consider for this question, two sets of inputs: fixed – 1,...,k and variable – k+1,...,n, k1 with prices p1,...,pn. It is obvious that this division into categories of factors makes sense only in the short term, because at long term they are all variable. We note also minimal consumption of fixed factors with x1,...,xk which means that for any amount consumed by factor xixi, i=1,k, the fixed cost will be pi x (for example the electricity bill relative to lighting for a production is the same regardless ofi
the level of production achieved).
Fixed costs would be:
CF=
k 1 i pixi
The total cost on short-term is the sum of fixed cost and the variable cost:
CTS=CF+CVS=
k 1
i pixi +
n
1 k j pjxj
On long-term, fixed cost becomes variable, obtaining:
CTL=CVL=
k 1
i pixi +
n
1 k j pjxj
The difference between the two periods is therefore:
CTL-CTS=
k 1
i pixi +
n
1 k
j pjxj-
k 1
i pixi -
n
1 k
j pjxj=
k
1
i pi xi xi
Therefore, if it is consumed the total amount of inputs 1,...,k then: xixi, i= k1, so in this case:
CTLCTS. If from each factor of production 1,...,k remain unused amounts, then: xixi, i=1,k therefore: CTLCTS.
On the other hand, for a fixed period, because the needs of the production process, it is natural to consider that fixed inputs not consumed in larger quantities than they are currently available. The cost on long-term should not be mixed with some cost on short-term relative to a production which was done at a time when there were no changes in technology or other factors that contribute to reducing costs.
For this reason, we always have at a fixed time: xi xi, i=1,k therefore CTLCTS.
Absolutely natural, dividing costs at the production follows:
CTML=
Q CTL
CTS =CTMSQ
3 An axiomatic approach to cost
We shall impose to all these costs a number of axioms, namely:
C.1. The marginal cost CmL (CmS) is positive, convex and has a unique local minimum.
Be so Q1 – the local minimum point of marginal cost and CmLm=CmL(Q1). We therefore have:
CmL(Q)CmLm, CmL’(Q)<0 Q<Q1 and CmL’(Q)>0 Q>Q1. Now CTL”(Q)<0 Q<Q1 so the function CTL is concave for QQ1and CTL”(Q)>0Q>Q1therefore the function CTL is convex for QQ1. Due to the fact that in the case of the short-term marginal cost, the derivative of fixed cost is null (being constant) these statements remain valid for variable cost in the short term.
On the other hand, from the axiom C.1 the functions CmL and CmS are convex in the neighbourhood of Q1. Also, the point of minimum of the marginal cost coincides with the inflection point of the curve CTL and that of CTS.
Marginal costs and total cost on long-term Figure 1
Marginal costs and total cost on short-term Figure 2
C.2. The average variable cost CVML (CVMS) is positive, convex and has a unique local minimum point.
Let Q2– the minimum point of CVML and CVMLm=CVML(Q2)=
2 2
Q ) Q (
CVL .
The nature of the point Q2implies that: CVML’(Q)<0Q<Q2and CVML’(Q)>0Q>Q2. On the other hand:
CVML’=
'
Q CVL
= 2
Q CVL Q ' CVL
= Q
CVML '
CVL
= Q
CVML CmL
from where:
CmL<CVMLQ<Q2and CmL>CVMLQ>Q2. Also, CmL(Q2)=CVML(Q2).
Following these considerations, the local minimum point of CVML coincide with the point of intersection of this curve with the long-term marginal cost.
The proof for short-term costs is the same. In this case, because CFM=
CF follows that CFM isQ decreasing (CF being constant), therefore CFM’<0Q0. From CTMS=CFM+CVMS we have that CTMS’=CFM’+CVMS’<0Q<Q2.
Marginal costs and average variable cost on long-term Figure 3
Marginal costs and average variable cost on short-term Figure 4
C.3. The average total cost on short-time CTMS is positive, convex and increasing for a value large enough of production.
From axiom statementQ3such that: CTMS’>0Q>Q3. How CTMS’<0Q<Q2follows that Q2<Q3. We have therefore Q4such that: CTMS’(Q4)=0 therefore Q4is a minimum point. We have therefore Q4(Q2,Q3). Also:
CTMS’=
'
Q CTS
= 2
Q CTS Q ' CTS
= Q
CTMS '
CTS
= Q
CTMS CmS
so in the point Q4we have: CmS(Q4)=CTMS(Q4), CmS<CTMSQ<Q4and CmS>CTMSQ>Q4. From these considerations, the local minimum point of CTMS coincide with the point of intersection of this curve with the short-term marginal cost.
Marginal cost and average total cost on short-term Figure 5
Axioms C.1 – C.3 determined the existence of four points Q1, Q2, Q3and Q4. The question is now the determination of the order of these points in order plotting the graphs of the above curves.
We have therefore:
CTMS’=
CmS CTMS
Q
1 from where:
CTMS”=
Q2
CTMS CmS
Q ' CTMS '
CmS =
Q2
CTMS CmS
Q CTMS Q CmS
' 1
CmS
=
Q2
CTMS CmS
2 Q '
CmS .
We have now: CTMS”(Q1)=
12
1
1 Q
) Q ( CTMS )
Q ( CmS
2
>0 from where: CmS(Q1)CTMS(Q1) therefore Q1<Q4.
Also, CVMS’=
Q CVMS
CmS from where:
CVMS”=
Q2
CVMS CmS
Q ' CTMS '
VCmS
=
Q2
CVMS CmS
Q CVMS Q CmS
' 1
CmS
=
Q2
CVMS CmS
2 Q '
CmS
.
In Q1we have: CVMS”(Q1)=
12
1 1
Q
) Q ( CVMS )
Q ( CmS
2
, and from the convexity of CVMS follows
that CmS(Q1)CVMS(Q1) therefore Q1<Q2. From the fact that Q4(Q2,Q3) we have, finally:
Q1<Q2<Q4<Q3
We now ask the question of determining the order of the curves corresponding to CTMS, CVMS, CFM, CmS.
Since CTMS=CVMS+CFM we have: CTMS>CVMS și CTMS> CFM.
Q<Q1: CmS<CVMS<CTMS, CmS, CTMS, CVMS, CFM
Q=Q1: minimum for CmS
Q1<Q<Q2: CmS<CVMS<CTMS, CmS, CTMS, CVMS, CFM
Q=Q2: minimum for CVMS
Q2<Q<Q4: CVMS<CmS<CTMS, CmS, CTMS, CVMS, CFM
Q=Q4: minimum for CTMS
Q4<Q: CmS>CTMS>CVMS, CmS, CTMS, CVMS, CFM
In addition, CVMS and CTMS are convex curves and their intersections with CmS is performed in the local minimum points of these curves. Also, the graph of CFM=
CF is an equilateral hyperbolaQ (CFMQ=CF=constant).
Costs on short-term Figure 6
Costs on long-term Figure 7
4 Conclusions
We approached the theory of short-term costs and the long-term costs from an axiomatic perspective because in the vast majority of the literature the graphs of costs are presented only after purely economic explanations. The authors have questioned the existence of a minimum cost axioms that generate their behavior.
5 References
1. Ioan C.A., Ioan G. (2011),n-Microeconomics, Zigotto Publishing, Galati
2. Simon C.P., Blume L.E. (2010),Mathematics for Economists, W.W.Norton&Company 3. Stancu S. (2006),Microeconomics, Ed. Economica, Bucharest
