# Law of Substitution or Principle of Least Cost Combination

Law of Substitution or Principle of Least Cost Combination

The objective of profit maximization can be achieved by two ways, one by increasing output and other by minimizing the cost. The minimization of cost can be possible by deciding the use of more than one resource in substitution of other resources.

The objective of factor-factor relationship is two fold:
1) Minimization of cost at a given level of Output.
2) Optimization of output to the fixed factors through alternative resource use combinations.

y =f (x1, x2, x3, x4…………….. xn)

Y is the function of x1 and x2 while other inputs are kept at constant. The relationship can be better explained by the principle of least cost combination.

Principle of Least Cost combination:
A given level of output can be produced using many different combinations of two variable inputs. In choosing between the two completing resources, the saving in the resource replaced must be greater than the cost of resource added.
The principle of least cost combination states that if two factor inputs are considered for a given output the least cost combination will be such where their inverse price ratio is equal to their marginal rate of substitution.

1. Marginal Rate of substitution:
MRS is defined as the units of one input factor that can be substituted for a single unit of the other input factor. So MRS of x2 for one unit of x1 is
Number of unit of replaced resource (x2)
=——————————————————–
Number of unit of added resource (x1)

2. Price Ratio (PR) =
Cost per unit of added resource
= ————————————————–
Cost per unit of replaced resource
Price of x1
= ———————–
Price of x2

Therefore the least cost combination of two inputs can be obtained by equating MRS with inverse price ratio.

i.e. x2  *  Px2 = x1  *  Px1

This combination can be obtained by following algebraic method or Graphic method.

A) Isoquant (Iso product) curve:
Iso means equal and quant means quantity.
An Isoquant represents the different combinations of two variable inputs used in the production of a given amount of output.

Properties of Isoquant:

1) They slope down ward to the right:
If more of one is used less of another input will be employed at the given level of output.
2) They are convex to the origin: It is because of diminishing MRS of one input for another. The additional units of an input will replace less and less units of another input.
3) Isoquant does not intersect: It is not possible to have different outputs from a single combination of inputs.
4) Slope of Isoquant represents the MRS.

B) Iso-Cost line: An Iso-cost line indicates all possible combinations of two inputs which can be purchased with a given amount of investment fund (outlay)
Each combination of inputs has same total cost which includes the cost of two inputs. (X1 and X2) combined.
Total cost = Px1. x1 + Px2. x2

Properties of Iso-cost line:

1) As total outlay increases, the Iso- cost line moves higher and higher away from the origin and vis- a-visa.
2) The Iso- cost lines are straight
3) Slope of Iso-cost line represents price ratio i.e. Px1/ Px2 when x1 is taken on x axis and x2 on y axis.

Least cost combination point: One Iso-cost and Iso-quant curves are depicted, it is now easy to locate the point of least cost combination. The slope of Isoquant and Iso-cost line represent the MRS and Price ratio respectively. The criteria for obtaining least and combination is MRS = PR and hence graphically it can be obtained where slope of Isoquant = slope of Iso-cost lines. This is found whore Iso-quant and Iso-cost lines tangent to each other. From this tangency point takes perpendiculars to both axis and obtains units of and x2. That combination is having least cost combination.
As discussed, Isoquants represent combination of inputs which can be produced the given level of output. Therefore different Isoquants represent different quantities of output. Isoquants nearer to origin represent less quantity of output and vis-a-visa. If we have numerous Isoquants then we can depict the isocline, ridge line and expansion path. These are depicted in the above diagram. The meaning of these concepts is given below.

1) Iso-cline: It is a line passes through the points of equal slope or MRS on an Isoquant surface. With the input price ratio being constant for each Isoquant the MRS between the inputs is the same for each level of output.

2) Ridge line:
These are also called as border line. Ridge lines join the end points of Isoquants. The area within the ridge lines is rational region of production arid beyond that the two regions are irrational. Therefore these lines represent the limits of economic relevance.

3) Expansion Path:
It is Isoclines for one set of prices for a given period. It connects the points of least cost combinations of inputs for all output level. As such, the MRS must be equal to the input price ratio.