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JOURNAL OF RESEARCH IN NATIONAL DEVELOPMENT VOLUME 7 NO 2, DECEMBER, 2009

CALCULUS OF MULTIVARIATE FUNCTIONS: IT’S APPLICATION IN BUSINESS

Emmanuel I. Awen
Department of Business Management
Michael T. Iber
Department of Accounting, and
              Fatima T.  Iordyeh
Department of Business Management, University of Mkar, Mkar, Benue State
Nigeria
E-mail: mike_iber@yahoo.com

 

Abstract
Calculus of multivariate functions is a mathematical concept that has to do with two sets of variables, the dependent variables and the independent variables.  Multivariate functions can be applied to situations in business organizations like a manufacturer’s profit (which is a dependent variable) depending on sales, overhead costs, the cost of raw materials used, and so on (which are independent variables); output of a factory may depend on the amount of capital invested in the plant, the size of the labour force and the cost of raw materials, and so on.  The paper dwells on differential calculus of multivariate functions and has considered types of multivariate differentiation, higher order to partial differentiation, maxima and minima for multivariate functions without constraints, application of calculus of multivariate functions in business and finally, conclusion.


Introduction
Calculus has two parts, differentiation and integration.  Differentiation is concerned with determining the rate of change.  As Lucey (1996) puts it, the process of differentiation establishes the slope of a function at a particular point.  Alternatively, this can be described as establishing the rate of change of the dependent variable (say, cost) with respect to an infinitesimally small increment in the value of the independent variable (say, activity). Differential calculus can be applied to functions that have one dependent variable and one independent variable, for example:


            y=x3 + 4x2……….. (1)
where y is dependent upon the value of the independent variable, x.


However, there are occasions where the functions contain two or more independent variables, for example, when the cost function of a firm depends on both labour hours and machine hours.  A function of differential calculus that has more than one independent variable is known as a multivariate function.  According to Budnick (1993), functions which involve more than one independent variable are referred to as multivariate functions, or functions of several variables.  A hypothetical multivariate function that has a dependent variable and independent variables is the one that Varberg and Purcell (1992) defined as; if z=f(x,y) we call x and y the independent variables and z the dependent variable.  An example of a multivariate function (or a cost function) is as follows:

y=10x2 + 5z2 – 4xz + 14………… (2)

where   y = total cost (dependent variable)
x=Labour hours (an independent variable).
z=Machine hours (an independent variable).

Types of Multivariate Differentiation
In the differentiation of multivariate functions, two processes are used, which are known as partial differentiation and total differentiation.

  1. Partial differentiation

Partial derivative (differentiation) of a multivariate function gives the instantaneous rate of change of the dependent variable with respect to an independent variable while other independent variables are kept constant

(Agbadudu, 1987).  Partial derivatives of multivariate functions can be determined from formulae for derivatives.  This is done by differentiating in the normal

 

way for one of the independent variables, say x, whilst at the same time treating the other variable, say y, as a constant.  For example, let y be a function of two variables x, and x2; that is


            y=f(x1,x2)………….. (3)

∂y
∂x1

  Then the partial derivative of y with respect to x1, denoted by                    
         is

∂y
∂x1

  



               = lim f(x1 +  ∆x1, x2)  -  f(x1,x2),
adadf                        ∆x1         0    ∆x1

y
∂x2

  If the limit exists.
Similarly,                     = lim f(x1, x2 +  ∆x1, x2)  -  f(x1,x2),
adad                                        ∆x2         0    ∆x2
            If the limit exists.
Note: lim is the short form of limit.

 

  1. Total differentiation

As Agbadudu (1987) puts it, the differentials of the independent variables, say K and L, shall be denoted by dk and dL and defined as dk=∆k and dL=∆L respectively.  In total differentiation all the independent variables are differentiated simultaneously.

∂Q

 

∂Q
∂L

  



From the above definition of total deferentiation, the following is the differential formula using quantity Q:

dQ=           dk    +                      dL  …………. (4)

∂z
∂y

  Similarly, if Z=f(x,y),
Then dz=                     dx  +                   dy  ………… (5)


Higher order  partial differentiation

∂   2z
∂  y2

 

∂2z
∂x2

  Partial derivatives of higher order are similar to the single variable.  For example if z=f(x,y), the second partial derivative of z with respect to x is the rate of change of the slope of the original function in the plane parallel to the z and x axes (Budnick,1993).  It is computed by viewing the first partial derivative as a function in itself and then merely partially differentiating z with respect to y.  The symbols for the second partial derivative of z with respect to x and y are respectively
adadadafdfa                                    and

∂ 2z
∂ xy

 

∂ 2z
∂ y    ∂ x

  The major difference in higher order derivatives in multivariate functions is the presence of cross partial derivatives.  This is because if z = f(x,y), the cross partial derivative is either
adfadaadadf         or                                    and


For a continuous functions, they are equal


Maxima and minima for multivariate function without constraints

adfadfad

∂z
∂x

  The methods for determining points at which there are extreme points for multivariate functions are similar to a single variable.  This is because a function of two independent variables, say, z-f(x,y), reaches a maximum or minimum when the slope of the function in both x and y direction is zero (Varberg and Purcell, Ibid).  consequently, the necessary condition will be as follows: a maximum or minimum of the function z=f(x,y) can occur only where the partial

derivative of the function with respect to x and the partial derivative with respect to y are simultaneously zero.  That is       or  


fx(x*=a,y*=b)=0 …………… (6)

∂z
∂x

  



adfadfaand                  or   fy(x*=a,y*=b)=0 …………… (7)

The two equations are solved simultaneously for x*=a and y*=b.  The values of x and y for which the derivatives are zero are the critical points.  The sufficient condition, like the calculus of a single variable, uses the second derivatives.

Application of calculus of multivariate functions in business
The calculus of multivariate functions can be applied to practical business activities and in daily human activities.  In the submissions of Edwards and Penny (1994), many real-world functions depend on two or more variables.  For example:

  1. The altitude above sea level at a particular location on the earth’s surface depends on the latitude and longitude of the location.
  2. A manufacturer’s profit depends on sales, overhead costs, the cost of each raw material used, and, in some cases, other additional variables.
  3. The amount of usable energy a solar panel can gather depends on its efficiency, its angle of inclination to the sun’s rays, the angle of elevation of the sun above the horizon, and other factors.

In business, if a manufacturer determines the particular units of a commodity, say x, that can be sold domestically for say, N 120 per unit, and another unit of a commodity, say y, that can be sold to foreign markets for say, N 200 per unit, then the total revenue obtained from all sales is given by

R=120x + 200y ……….. (8)

It shows from the above example that revenue R, depends on the prices and quantities of commodities x and y, which are independent of R. Hoffmann etal (2005) has also accepted with the practicality of multivariate functions by saying that in psychology, a person’s intelligence quotient (1Q) is measured by the ratio
1Q = 100m    ………. (9)
              a
Where a and m are the person’s actual age and mental age, respectively.  This shows that the level of a person’s intelligence depends on the person’s mental age and actual age.  A carpenter constructing a storage box x feet long, y feet wide, and z feet deep knows that the box will have volume V and surface area S, where
V=xyz and S=2xy +2xz + 2yz,
(Hoffmann et al, 2005).

These are typical; of practical situations in which a quantity of interest depends on the values of two or more variables.  other examples include the output of a factory, which may depend on the amount of capital invested in the plant, the size of the labour force, and the cost of raw materials; also the volume of water in a community’s reservoir at any point in time may depend on the amount of rainfall as well as the population of the community.

A business organization that is planning for an expansion may depend on increasing their workforce, acquiring more facilities, injecting in more money, and any other factors.

(a)        Application of partial differentiation
            Partial derivatives, like the single variables, have almost the same applications, especially in marginal analysis.  For instance, if a utility function u. is given by

∂u
∂x2

             U=U(x1x2) ……….. (10)
adfadfaadfafThen    and             are marginal utility functions.

Illustration 1:

 

∂y
∂x1

  If y=x12 x2 + x22 + 8    

adfdafThen          = 2x1 x2 + 0 + 0
                                 = 2x1 x2

∂y      ∂ x2

             It can be seen that the second term x22 in the expression as well as the constant 8 are regarded as constants as far x1 is concerned.  Similarly,
adfadf                  = x21 + 2x2

∂y
∂x1

  Let y= 2x21 x32 + x51x2x3 + 6x3 +4

∂y
∂x2

 dfadfThen           =4x1 x32 + 5x41x2x3
afad                      =6x21 x22 + x51x3

∂y
∂x3

  

fadfaand           =x5x2 + 6

∂z
∂y

 

∂z
∂x

  Illustration 2:
adfafadfFind the partial derivatives of          and          if z=f(x,y)=tx2y.

Solution:

f(x+∆x,y) – f(x,y)
        ∆x

 afdaUsing the limit approach to find the partial derivatives,
    ∆f                =

5(x+∆x)2y – 5(x2y)
           ∆x

                        
                        =

5(x2+2x∆x + ∆x2)y –5x2y
                ∆x

  



            =

5x2 + 10x∆x + 5y∆x2 –5x2y
                ∆x

  

            =

∆x (10xy + 5y∆x)
          ∆x

  

            =
            =          10xy + 5y∆x

∂z
∂x

  

adfaThus,          = lim ∆f(x,y)
dfad                   ∆x    0   ∆x
                        = lim(10xy + ty∆x)
fafa                           ∆x      0

∂z
∂y

                         = 10xy
dfadfTo find          , we have

∆f (x,y)
∆y

 

f(x,y+∆x,y) – f(x,y)
           ∆y

  

         0                         =

5x2(y+∆y) – 5x2y
           ∆y

  

                                    =

5x2y+5x2∆y – 5x2y
           ∆y

  

                                    =

∂z
∂y

                                     =          5x2
afdAnd       = lim ∆f(x,y)
dfadf                   ∆y    0   ∆y
                 =lim 5x2
dfadfadf                   ∆y    0  
                 = 5x2

dq   p
dp  q

  The concept of price elasticity of demand can also be treated in a single variable.  We recall that if a demand function is given by q=f(p) then marginal demand is          and elasticity of demand E is             .  Suppose our product q1 does not
only depend on its price, but also on the prices of other product and consumer income, that is,


 

dQ
dP

            q1=f(P1,P2, ……, Pn, y) ……. (ii)
Then price elasticity for this product is given by

q1   p1
∂p1  q1

  E1=                    …………. (12)


This is the same as in single variable derivatives except that we resort to partial differentiation here. However, this is not the only elasticity we can derive here.  This is because the quantity demanded of product one is also linked to the price of products two, three and so on, and to the consumer income.  It will be proper to inquire into the effect that changes in those variables would have upon the quantity demanded for product one.

 ∂ ∂Pj  q1

 

q1  
∂pj 

  Now             (and j ≠  1) is the instantaneous rate of change of quantity demanded of product one with respect to changes in the price of the related product j.  We have cross elasticity of demand, Elj and is given by


Elj=                    …….. (13)


This is the measure that will indicate the responsiveness of quantity demanded of q1 to changes in prices of other products.  The sign Elj is of significance; if it is positive, it means that product j is a substitute to product one but if it is negative, then product j is complementary to product one.

q1   y
∂y   q1

    Another term that comes in is the income elasticity of demand, Ely.  This is a measure that will indicate the responsiveness of quantity demanded of product one (concerned) to changes in consumer income.  It is given by Elgy=     


 

               ……… (14)

 

Illustration 3:

  1
100y

  Find the price elasticity, cross elasticity and income elasticity of the following:

 1
20 P2

 

 1
50y

 

 20
 P3

 

 100
  P1

 adada(i)         q1=200-20p13/2 + 3p2-p3 +             at (4,10,20,1000)
adadfadfaadfdfad(ii)        q1=            +              +              +             at (10,40,20,350)

(iii)       q1=p1-2.5 p2-0.2 p30.5y

Solution:

  1
100

  (i)         At the given points or values, we see that
fadfa            Q1=200-20(4)3/4 + 3(10)-20 +             (1000)

 3
 2

 

Q1 
∂P1 

                 =200-160+30-20+10=60
adfad                     = -         (20)P½  =-30(4)½  =-60

Q1 
∂P2 

 

q1 
∂P3 

 

q1 
∂y 

 

  1
100

  



dfadf                     = 3    and               = -1j            = 
adf

q1   p1
∂p1 q1

  

ad\ E1=         =60                  = -4

q1   p2
∂p2 q1

 fad 

adfa    E12=        =3                    =  ½

q1   p3
∂p3 q1

 fad 

dfadf    E13=        =-1                   =  -1/3 
  

  1
  6

 

 1000
  60

 

  1
100

 

q1   y
∂y   q1

  From the results above, it shows that product two is a substitute to product one and product three is complementary to product one.
fadadfadfadAnd Ely=                =                       =

(ii)        At the given values of p1,p2,p3 and y, q1=20

q1 
∂p3 

 

q1 
∂p2 

 

 100
   P1

            

q1 
∂p1 

                
                     = -             = -1,          = 1/20,           = -1/20

q1 
∂y 

  



and            = 1/50
afda

q1   p1
∂p1 q1

  

fadf\ E1=         =-1                   = -1/2

q1   p2
∂p2 q1

 dfa 

fadf    E12=       =1/20                     =  1/10

q1   p3
∂p3 q1

 dfad 

adfad    E13=        =-1/20              =  -1/20 

  7
  20

 

q1   y
∂y   q1

 fadfadfadf 

adfadfadfadfAnd Ely=                =                       =

q1   y
∂y   q1

 

 y 
 q1 

  

(iii)       Ely=               =  p1-2.5 p2-0.2 p30.5y                     =1
(b)       

Application of total differentiation
Illustration 4: Find the total differential of
 
Q=L3/4K1/4

Q 
∂L 

 

Q 
∂k 

  Solution:
                               = 1/4L3/4K-3/4,             = 3/4L-1/4K1/4
∴          dQ=1/4L3/4K-3/4dK + 3/4L-1/4K1/4dL.

  •  

Application of higher order  partial differentiation

Illustrations 5: Find the first and second partial derivatives of the following functions

  • fx=4x+3y2;fy=6xy-10y+3y2

fxx=4; fyy=6x-10+6y and fxy=fyx=6y

  • Zxx=4xy3+y2; Zy=6x2y2+2xy

Zxx=4y3; Zyy=12x2y+2x and
Zxy=Zyx=12xy2+2y
Illustration 6: If utility function U for two commodities x and y is given by
U=5x+10y-3x2-4y2+2xy.
Show that marginal utilities are diminishing. 
Solution:
The marginal utility of each of the goods are
            Ux=5-6x+2y
and Uy=10-8y+2x.


Marginal utilities are decreasing if their slopes are negative.  The slopes of these terms are second partial derivatives of U.
            Now, Uxx=-6, which is negative,

And Uyy=-8, which is also negative.
(d)        Application of maxima and minima for multivariate functions

Illustration 7: The cost of construction of a project depends upon the number of skilled

 

workers x and unskilled workers y.   If the cost is given by c(x,y)=40,000+8x3-72xy+9y2

  • Determine the number of workers in each category that will minimize cost.
  • What is this minimum cost?

Solution:
The first order condition implies that Cx and Cy simultaneously will be zero,
i.e. Cx=0 or 24x2-72y=0
and Cy=0 or -72x2+18y=0
Solving the two functions above, we have the following points: (0,0) and 12,48).
Now Cxx=48x and Cyy=18; Cxy=-72
At (0,0), Cxx=0 and Cyy=48; Cxy=-72
So that at (0,0), Cxx Cyy-C2xy=0-(-72)2 > 0
i.e. we are at a minimum point there.
(i)         The number of skilled and unskilled workers that result in minimum cost are 12 and 48 respectively.
(ii)        The minimum cost is 40,000+8(12)3-72(12)(48)+(48)2=N33,090


Conclusion
Differential functions are made up of those with one dependent variable and one independent variables as well as those with one dependent variable and more than one independent variables.  The functions with more than one independent variables are referred to as multivariate functions.

Single variable functions are those in which one factor depends on just one another factor for a particular result to occur, like total profit being dependent on total cost.  However, multivariate functions are those in which one factor depends on two or more other factors for a particular result to occur, like cost of production being dependent on cost of raw materials, cost of labour hours, cost of machines hours, and so on. Multivariate functions can be applied to profit in terms of what profit is dependent upon, as well as cost and total output variables.  Multivariate functions are also applicable in business in areas like number of employees selected during interviews and in determination of wages and so on. Multivariate functions can be used in transport businesses by considering so many variables before fixing fares for passengers.  It can also be used in hotel business by charging costs that are dependent on many variables.

In most organizations, the aforementioned analysis are undertaken using the rule of thumb approach.  Such analysis of profit, transport cost, wages determination, and others mentioned, can be determined more scientifically and objectively using calculus of multivariate functions.

References

Agbadudu, A.B. (1987): Mathematical methods in Business and Economics 1st Ed., Lagos: Lagos University Press.

Budnick, F.S. (1993): Applied mathematics for Business, Economic and the Social Sciences 4th Ed., New York: McGraw-Hill, p. 930.

Edwards, C.H., and Penny, D. E. (1994): Calculus with analytic geometry 4th Ed., New Jersey: Prentice Hall.

Hoffmann, L.D., Bradley, G.L., and Rosen, K.H. (2005): Applied calculus: for Business, Economics, and the Social and Life Sciences 8th Ed., New York: McGraw-Hill.

Lucey, T. (1996): Quantitative techniques 5th Ed., London: The Tower Building Publishers.
 
Varberg, D., and Purcell, E.J. (1992): Calculus with analytic geometry 6th Ed., New Jersey: Prentice Hall.