Iso-Cost Line and Producers Equilibrium: Solved questions

1- Define iso-cost line or price line.

Ans:The iso-cost line is the locus of all the combinations of two inputs, say capital and labor, which can be hired (purchased) by a producer from his total cost outlay, given the prices of inputs. In other words, the iso-cost line or price line is the line which represents all the combinations of two inputs, say capital and labor, that a producer can hire with his or her given budget or total cost outlay at given prices of inputs (say rate of interest and wages).

Iso-cost line can be expressed in an equation form as;

C=PK.QK+PL.QL

Here, C = total cost outlay,

PK = price of capital good

QK = quantity of capital

PL = price or wages of labor

QL = quantity or number of labor input

The equation can be expressed as; `C=rK+wL......(i)`

Dividing the equation (i) both sides by r we get;

`Or, (C)/(r)=(rK)/(r)+(wL)/(r)`

`Or, (C)/(r)=K+(wL)/(r)`

`Or, (C)/(r)-(wL)/(r)=K`

`\therefore\ K=(C)/(r)-(wL)/(r).......(ii)`

In a similar manner labor function can also be expressed as,

`L=(C)/(w)-(rK)/(w).......(iii)`

By applying capital function `K=(C)/(r)-(wL)/(r)` and labor function `L=(C)/(w)-(rK)/(w),` the units of capital as well as units of labor applied within the given amount of budget or total cost outlay.

Given the total cost `C=$100` price of labor `w=10` and price of capital `r=20`, we get the following combinations of two inputs labor and capital.

Combinations	Capital(r=$20)	Labor(w=$10)	Total cost outlay (C=100)
A	0	10	0×$20+10×10=$100
B	1	8	1×$20+8×10=$100
C	2	6	2×$20+6×10=$100
D	3	4	3×$20+4×10=$100
E	4	2	4×$20+2×10=$100
F	5	0	5×$20+0×10=$100

By plotting the combinations in a graph, we can derive an iso-cost line as shown in the following diagram.

In the above diagram, AF is the iso-cost line. Combination A contains 10 units of labor and and zero units of capital. Similarly combination B contains 8 units of labor and 1 unit of capital. Likewise combinations C, D, E and F contain, 6L+2K, 4L+3K, 2L+4K and 0L+5K. Any combination outside the iso-cost line is not attainable because of insufficient total cost outlay. Similarly any combination inside the iso-cost line is not desirable because some of the budget remains idle.

2- Explain the causes of shift in iso-cost line.

Ans: The iso-cost line may shift outward or inward depending on the change in total cost outlay of the producer or the change in the prices of inputs.

a) Change in total cost outlay

When the total cost outlay (budget) increases, given the prices of inputs, the iso-cost or price line shifts outward and vice versa. On the other hand, given the total cost outlay, with the fall in the prices of inputs, the iso-cost line shifts outward and it shifts inward with the increase in the prices of inputs. The shift in iso-cost line is shown in the following diagram.

In the above diagram, let us suppose that A1B1 is the initial iso-cost line at $100 total cost outlay. When the total cost outlay (producer's budget) increases from $100 to $200, more units of capital and labor can be hired. As a result, iso-cost line shifts outward from A1B1 to A2B2 as shown in the diagram. Conversely, if the total cost outlay or the producer's budget falls, comparatively only a few units of labor and capital can be hired. Consequently, iso-cost line shifts inward.

b) Change in the prices of inputs

When the prices of inputs change, the iso-cost line rotates or swings outward or inward depending on an decrease or increase in the price of an input keeping the price of another input constant. As the price of any one of the inputs increases, keeping the price of another input constant, the iso-cost line rotates in or swings inward. On same condition, if its price decreases, it rotates out or swings outward. The rotating or swinging of iso-cost line is shown in the following figure.

In the above figure, AB in the initial iso-cost line. When the wage rate or price of labor input falls, with constant price of capital input, the producer can hire more labor input. As a result iso-cost line rotates out or swings outward from AB to BD. In the similar manner, if the price of capital input increases with constant wage rate, the producer can comparatively hire only a few capital input. Consequently, the iso-cost line rotates in or swings inward from AB to AC as shown in the above figure.

3- What do you mean by producer's equilibrium? Explain the method of least cost combination of factor inputs maximizing level of output for a given level of outlay or cost.

Ans: A producer is said to be in equilibrium when he is hiring such a combination of two inputs, say capital and labour, that leaves him with no tendency to rearrange the inputs. In other words, producer’s equilibrium occurs when he yields maximum possible output with optimal combination of factors.

A profit maximizing producer or firm faces two choices of optimal combination of factors or inputs.

1-Maximization of output for given total cost outlay

2-Minimization of costs for given output

An iso-quant map is a set of different iso-quants each representing various levels of output. On the other hand, an iso-cost or price line represents different combinations of two inputs, capital and labor, that the producer can hire from his given total outlay (budget), given the prices of inputs. A producer is said to be in equilibrium when he is able to maximize his output with his given total outlay.

A producer will be able to attain equilibrium when the following conditions are fulfilled. They are;

a) Iso-cost or price line should be tangent to the iso-quant or slope of IQ should be equal to slope of iso-cost line. It is expressed as,

`MRTS_(LK)=(w)/(r)`

b) Iso-quant must be convex to the origin.

It is assumed that a producer is a rational person so he always tries to maximize his output with his given total cost outlay. His tendency to maximize his output, encourages him to reach his iso-cost or price line to a higher possible iso-quant which represents a higher level of output. This process makes the iso-cost line to be tangent to the higher possible iso-quant and the producer attains his point of equilibrium.

The method of maximizing output with total cost outlay is explained with the help of the following figure.

In the figure, it is seen that AB is the iso-cost line representing different combinations of inputs that a producer or firm can hire from the total cost outlay. Similarly, IQ1, IQ2 & IQ3 is an iso-quant map that represents different levels of output such as 100, 200 & 300 units. The producer can choose a factor combination C or E or D that are lying on the iso-cost line AB. But the output maximizing producer chooses the factor combination E, which consists of 0K of capital and 0L of labor, because this one enables the producer to reach the higher possible iso-quant IQ2 at which both the necessary conditions for equilibrium have fulfilled. So, the point E is his equilibrium point where he can produce 200 units of output. All other combinations of inputs such as C and D lie on lower iso-quant IQ1, representing lower level of output than IQ2. Thus, it is clear from the above explanation that the tangency of iso-quant with the given ios-cost line represents the least cost combination of factors maximizing level of output for a given level of total cost outlay.

4- Explain the method of least cost combination of factor inputs minimizing cost for a given level of output.

Ans: An iso-quant map is a set of different iso-quants each representing various levels of output. On the other hand, an iso-cost or price line represents different combinations of two inputs, capital and labor, that the producer can hire with his given total outlay (budget), given the prices of inputs. A producer is said to be in equilibrium when he is able to minimize his total cost outlay to produce given level of output

A producer will be able to attain equilibrium when the following conditions are fulfilled. They are;

a) Iso-cost or price line should be tangent to the iso-quant, i.e. slope of IQ = to slope of iso-cost line. It is expressed as,

`MRTS_(LK)=(w)/(r)`

b) Iso-quant must be convex to the origin.

In order to minimize total cost outlay for the production of the given level of output, the producer chooses that combination of inputs at which the IQ curve, representing given level of output, is tangent to the lower possible iso-cost line. This point of tangency is the point of equilibrium which enables the producer to minimize outlay. It is explained by the help of the following figure.

In the figure, it is seen that there is a single iso-quant which represents 200 units of output but there is a set of different iso-cost lines each of them representing various combinations of inputs that a producer can hire from different amounts of total cost outlay. The combinations of inputs C & D which lie on iso-cost line MN, can produce 200 units of output whereas the combination E that lies on lower iso-cost line can also produce 200 units of output. The combination E requires less total cost outlay than that of combinations C & D. So the rational producer or firm chooses the least cost combination of factor inputs E to minimize cost for a given level of output 200 units. Hence E is the point of equilibrium at which the conditions for equilibrium have fulfilled..