LONG RUN LAW OF PRODUCTION
In the long run expansion of output may be achieved by varying all factors. In the long run all factors are variable. The law of returns to scale refers to the effects of scale relationships. In the long run output may be increased by changing all factors by the same proportion or by different proportions. The term ‘returns to scale ‘refers to the changes in output as all factors change by the same proportion
In the long run expansion of output may be achieved by varying all factors. In the long run all factors are variable. The law of returns to scale refers to the effects of scale relationships. In the long run output may be increased by changing all factors by the same proportion or by different proportions. The term ‘returns to scale ‘refers to the changes in output as all factors change by the same proportion
Suppose a firm starts from an initial level of inputs and output and increases all the inputs proportionately, there are three technical possibilities:
(i) If output increases more than proportionally, we have increasing returns to scale;
(ii) If output increase by the same proportion as the inputs increase, we have constant returns to scale; (iii) If output increases less than proportionally with the increase in the factors, we have decreasing returns to scale.
(i) If output increases more than proportionally, we have increasing returns to scale;
(ii) If output increase by the same proportion as the inputs increase, we have constant returns to scale; (iii) If output increases less than proportionally with the increase in the factors, we have decreasing returns to scale.
Increasing Returns to Scale:
If a firm increases both the inputs in a certain proportion and the output results in more than proportionately, it exhibits increasing returns to scale. For example, if both the inputs K and L are consecutively doubled, output is more than doubled, the returns to scale is said to be increasing. Fig illustrates the increasing results to scale.
When the combination of inputs at point a (1K + 1L), is doubled to a combination of inputs at point b (2K + 2L), the output is more than doubled as it increases from 100 units to 225 units. Further successive doubling the inputs causes more than double increase in output as is illustrated by the movement from point b to c increasing the inputs from (2 K + 2 L) to (4K + 4L) output increases to 500 units. This sort of relationship between inputs and output demonstrates increasing returns to scale.
If a firm increases both the inputs in a certain proportion and the output results in more than proportionately, it exhibits increasing returns to scale. For example, if both the inputs K and L are consecutively doubled, output is more than doubled, the returns to scale is said to be increasing. Fig illustrates the increasing results to scale.
Constant Returns to Scale
When a certain proportionate change in input causes a corresponding proportionate change in output, it exhibits constant returns to scale. For example, doubling the factor inputs achieves double the level of the initial output; trebling inputs achieves treble output. Following Fig illustrates constant returns to scale. When inputs are doubled from 1K+1L to 2K+2L the output also increases from 100 to 200 and when inputs are again doubled to 4K+4L, the output is doubled to 400.
When a certain proportionate change in input causes a corresponding proportionate change in output, it exhibits constant returns to scale. For example, doubling the factor inputs achieves double the level of the initial output; trebling inputs achieves treble output. Following Fig illustrates constant returns to scale. When inputs are doubled from 1K+1L to 2K+2L the output also increases from 100 to 200 and when inputs are again doubled to 4K+4L, the output is doubled to 400.
Decreasing Returns to Scale
When a certain proportionate change in inputs causes a less than proportionate change in output, it is the decreasing returns to scale. For example, when both inputs are doubled and output increases less than double, then the decreasing returns to scale operate. Following fig.illustrates the decreasing returns to scale. The figure shows that the inputs are increased from 1K+1L to 2K+2L, the output increases from 100 to 175 units.
When a certain proportionate change in inputs causes a less than proportionate change in output, it is the decreasing returns to scale. For example, when both inputs are doubled and output increases less than double, then the decreasing returns to scale operate. Following fig.illustrates the decreasing returns to scale. The figure shows that the inputs are increased from 1K+1L to 2K+2L, the output increases from 100 to 175 units.
Reasons for increasing returns to scale
(a) Technical and/or managerial indivisibilities
In general, indivisibility means that certain inputs, particularly equipments and managerial skills are available only in minimum sizes or in definite ranges of sizes. They cannot be divided into parts to suit production of small scale. As a firm’s scale of operations increases, it can use the minimum sizes and then the larger sizes of more efficient equipment. For example, half a typewriter or half a turbine cannot be used. Similarly, half a manager cannot be employed, if possibilities of part-time employment are ruled out. Due to the indivisibility of equipments, machinery and managers, they are employed in a certain minimum size/quantity even if the scale of production is much less than the capacity output. Therefore, when the scale of production is increased, the productivity of indivisible inputs causes exponential increase in output due to technological advantage. This results in increasing returns to scale. But the indivisibility quickly exhausts itself as a cause of increasing returns to scale.
(b) Mass Production
One of the basic characteristics of advanced industrial technology is the existence of ‘mass-production’ methods over large sections of manufacturing industry. ‘Mass – production’ methods are processes available only when the level of output is large. They are more efficient than the best available processes for producing small levels of output.
(c) Dimensional Relations
Dimensional relationships are considered to be the important causes of increasing returns to scale. If we double the diameter of a pipe, the flow through it gets more than doubled. For example 3 x 3 = 9, when the two digits are doubled 6x6 = 36, which is four times of 9. In accordance with the dimensional relationship, when the inputs are doubled, the output increases more than double.
(d) Higher Degree of Specialization
Higher degree of specialization is also pointed out as a cause of increasing returns to scale. With more labor, the firm can divide and subdivide tasks and thus gain efficiency of labour. With more machinery, the firm can buy special types and also assign special jobs even to standardized kinds of machinery.
Reasons for Constant Returns to Scale
The phase of increasing returns to scale cannot go on indefinitely. There is a limit to the economies of scale. When inputs are perfectly divisible the capital- labour ratio is fixed, and the production function is homogeneous of degree 1, constant returns to scale takes place. The phase of constant returns to scale can be brief, before decreasing returns to scale set in. The Cobb-Douglas production function is linearly homogeneous exhibiting constant returns to scale.
Reasons for Decreasing Returns to Scale
(a)Diminishing returns to management
The most common causes of diminishing returns to scale are ‘diminishing returns to management’. The management is responsible for the co-ordination of the activities of the various sections of the firm. Even when authority is delegated to individual managers (production manager, sales manager, etc.) the final decisions have to be taken by the top management. As the output grows, top management becomes eventually overburdened and hence less efficient in its role as coordinator and ultimate decision-maker. In this view, decreasing returns to scale are actually a special case of variable proportions.
(b)Exhaustible natural resources
As the scale of production is increased, it does not result in increase of output in the same proportion due to the exhaustible nature of natural resources. For example, doubling the fishing fleet may not lead to doubling of the catch of fish; or doubling the plant in mining or an oil-extraction field may not lead to a doubling of output.
(c) Organizational Problems
Certain organizational problems creep in to the large- scale production. A plant with a very large number of workers is more difficult to manage than a smaller one. The larger the number of workers, the larger becomes the number of foremen and supervisors creating several layers of middle management causing problems of communication. As a result it becomes difficult for manager to make sharp decisions. Owing to the organizational dis-economies that accompany large plant, firms frequently build multiple plants Automobile assembly plants are kept relatively small by having most components made elsewhere.
PRODUCTION FUNCTION THROUGH ISO-QUANT CURVE
A firm may decide to maximize output for a given cost or minimize cost subject to a given output. An iso-quant exhibits that a given output can be produced with various different input-combinations, given the input prices. However, there is only one input combination that provides us with least –cost criteria. This section illustrates the identification of the least-cost input combination.
Figure shows that 100 units of output can be produced with various different combinations of K and L located on IQ1.Points a, b and c represents three different combinations of capital and labour. Point a represent K3 +L1, point b represent K2 + L2 and point c represent K1+L3.
All the three combinations can produce 100 units of output. Technically, any of these input combinations can be chosen for producing 100 units of output, but all are not economically efficient. With given input prices the total cost of production differs from point to point. There is only one point on the isoquant IQ1 that produce 100 units of output at minimum cost. Any firm’s best interest is to identify the input combination that produces a given level of output at minimum cost. An upper or lower isoquant represent higher or lower levels of output that can be produced with greater and lesser and different combinations of inputs with varying total costs.
The combination of inputs with which a firm produces the product also depends on the prices of inputs and the amount of money that a firm has to spend. Information concerning relative input prices enables a producer to determine operating input combination and minimize the cost of producing a given output or maximize output for a given level of costs.
Input prices are determined by the forces of demand and supply in the market. If a firm uses only labor and capital, the total cost of the firm can be denoted by
C = wL + rK
Where,
C = total cost
W = wage rate of labour
L = quantity of labour used
R = rental price of capital
K = quantity of capital used.
For example, suppose capital costs Rs.1000 per unit (r = Rs.1000) and labour receives a wage of Rs.2500 per labor month (w = Rs.2, 500). If a total of Rs.15000 is to be spent for inputs, the following combinations are possible. Rs.15000 = Rs.1000K + 2.500L, or K = 15 – 2.5 L
More generally, the producer can choose among the combinations given by the equation
K=C/r–w/r
Fig.illustrates it more clearly. If Rs.15000 is spent for inputs and no labor is used, 15 units of capital may be bought. If C is to be spent and r is the unit cost, C/r units of capital may be purchased. This is the vertical-axis intercept of the line. If 1 unit of labour is purchased at Rs.2500, 2.5 units of capital must be sacrificed; if 2 units of labour are bought, 5 units of capital must be sacrificed; and so on. Thus as the purchase of labour is increased, the purchase of capital must be reduced. For each additional unit of labour, w/r units of capital must be forgone. The isocost line shows the various combinations of inputs that may be purchased for a stipulated amount of expenditure.
The slope of the isocost line represents the ratio of the price of a unit of input to the price of a unit of another input. In case the price of any one of them changes, there would be a corresponding shift in the slope of the isocost curve and the equilibrium would shift too.
Finding the Least-Cost Technology with Isoquants and Isocosts
A firm spends fixed amount of money (C) on two inputs only capital (K) and labor (L). It decides its budget and knows the prices of each of the inputs that are assumed to remain constant. With given input prices, a producer wishes to operate efficiently by producing the maximum attainable output. Thus, among all input combinations that can be purchased for the fixed amount (C), the producer seeks one that results in the largest level of output. The given level of cost (C) is represented by the isocost line KL in the accompanying
Fig. The slope of the KL is equal to the (negative) ratio of the price per unit of labor to the price per unit of capital. IQ1, IQ2, IQ3 are isoquants representing various levels of output. IQ3 level of output is not obtainable because the available input combinations are limited to those lying on or beneath the isocost cost KL.
The producer could operate at points such as R and S At these two points, the input combinations required to produce the IQ1 level of output are available for a given cost represented by the isocost KL. However, the output can be increased without incurring additional cost by the selection of a more appropriate input combination. Indeed, output can be expanded until the IQ2 level is reach- the level at which an isoquant is just tangent to the specified isocost curve. A greater output is not obtainable for the given level of expenditure; a lesser output is inefficient because production can be expanded at no additional cost. Hence the input combination represented by the slope of ray OQ is optimal because it is the combination that maximizes output for the given level of cost. At point R, the marginal rate of technical substitution of labor for capital is relatively high as 3 units of capital are foregone for one addition unit of labor. The relative input price, given by the slope of KL, is much less, say, 1:1 In this case, 1 unit of labor costs the same as 1 unit of capital but it can replace 3 units of capital in production. The producer would be better off by substituting labor for capital. The opposite argument holds for point S, where the MRTS is less than the input-price ratio.
The producer, therefore, attains equilibrium only when the MRTS of labor for capital is equal to the ratio of the price of labor to the price of capital. This equilibrium point is where the producer maximizes output for a given level of cost. The market input- price ratio tells the producer the rate at which one input can be substituted for another in purchasing. So long as the two are not equal, a producer can achieve either a greater output or a lower cost by moving in the direction of equality.
An entrepreneur may alternatively seek to minimize the cost of producing a stipulated level of output. The concept is explained in the Fig .The KL1, KL2, KL3 are isocost curves and IQ represents the stipulated level of output. The level of cost shown by KL1 is not feasible because the IQ level of output is not physically producible by any input combination available for this outlay. The IQ level of output could be produced by the input combinations represented by the point R and S, both at the cost level KL3. By moving either from R to Q or from S to Q, the entrepreneur can obtain the same output at lower cost. A position of equilibrium is attained only at point Q where the isoquant is just tangent to an isocost curve. Thus in equilibrium the marginal rate of technical substitute of inputs must be equal to the ratio of the prices of inputs.
For the achievement of least-cost combination of inputs two conditions must be fulfilled. The first-order condition requires that the marginal rate of exchange of one input for the other (i.e. of capital for labour or vice-versa) must equal the ratio of their marginal physical product.
MPl / MPk measures the slope of the isoquant.
Therefore at the point of least cost input combination slope of the isoquant must equal the slope of the isocost line. The least cost input combination occurs at a point where isoquant is tangent to the isocost. In fig. the point of tangency is at point e where the input combination of K and L is equal to OK’ of capital and OL’ of labour. This input combination is optimal as at this point K/L= MPl / MPk.
The second order condition requires that the marginal rate of exchange between labour and capital (K/L) must equal the ratio of the marginal physical product (MPl / MPk) at the highest possible isoquant. In the fig. the first order condition is fulfilled on points R, S and e where the isoquant IQ1 intersects isocost line KL. But R and S are not on the highest possible isoquant. The second order condition is fulfilled at point e only. Point R and S satisfy only first order condition, but point e satisfies both first-order and second order conditions. Therefore, least-cost factor combination occurs at point e.
Changes in Input Prices
If the price of one input, say labour, changes, the firm will adjust the input mix by substituting one input for the other. If the price of labour increases, the firm substitutes capital for labour. On the contrary if the price of labour decreases, making labour relatively cheaper, labour will be substituted for capital. In general, if the relative prices of inputs change, managers will respond by substituting the input that has become relatively less expensive for the input that has become relative more expensive.
The figure illustrates the effect of change in price of input on the input-mix.
Suppose the firm currently is operating at point a where 100 units of output are produced using the resource combination (K=10, L=2). This is an efficient resource mix because the 100 unit isoquant is tangent to the isocost line cc at point a. If the price of labour falls, the price of capital remaining constant the labour has become relatively cheaper. The isocost line swings to the right from cc to the isocost cc’ and the firm moves from point a to point b, which is a new efficient resource combination. The new isocost line is tangent to the 120-unit isoquant at point b. Now nine units of capital and six units of labour are employed. Note that at point a, the efficient ratio of capital to labour was 5:1. Now the efficient ratio of the two inputs is 3:2. The reduction in the price of labour has caused the firm to substitute that relatively less expensive input for capital. This change in input combination is price-effect, which combines substitution and budget effects. The price and budget effects can be separated in the same manners as the substitution and income effects are separated.
Thus, we find that as a result of changes in the price of an input, input combination of the firm changes leading the firm to employ more of cheaper input and less of the costlier one. Besides, the level of output also changes. If price of an input decreases the level of output increases, and vice versa.
Expansion Path:
In the long run all inputs are variable without any limitation (technical or financial) to the expansion of output The firm’s objective is the choice of the optimal way of expanding its output, so as to maximize its profits.
Let us take the case of a firm producing 100 units of output using ten units of capital and ten units of labour, and the firm being in equilibrium at point ‘a’. The input prices are w=20 and r =20 as shown in the accompanying Fig
Thus the cost of this input combination is Rs.400. At point a , the 1000 unit isoquant is tangent to the Rs.40 isocost line. If the firm decides to increase its output, it will move to point b if, 1500 units are to be produced and then to point c if, 1750 units of output are to be produced. In general, the firm expands by moving from one tangency or efficient production point to another. These efficient points represent expansion path. An expansion path may be defined as the set of combinations of capital and labour that meet the efficiency condition MPL /w = MPK/r.
We can determine an equation for the expansion path by substituting the marginal production functions and input prices into the efficiency condition, and then by solving for capital as a function of labour if the production function is
Q = 100K½ L½
The corresponding marginal product function are
MPL = dQ/DL = 50K½ /L½
and
MPK = dQ/DK = 50L½ /K½
Substituting the marginal product equations in the efficient condition (MPL/ MPK = w/r) gives,
Solving for K gives:
This expression is the equation for the expansion path for the production function Q = 100K½ L½. If w and r are known, equation (5.10) defines the efficient combination of capital and labour for producing any rate of output. If w = 1 and r= 1, the expansion path would be,
K = L
If w = 2 and r= 1, the equation for expansion would be
K = 2L
If the expansion path is known, the knowing the isoquant- isocost system is not necessary to determine efficient production points. The firm will only produce at those points on the expansion path.
The expansion path indicates optimal input combinations, but it does not indicate the specific rate of output associated with that rate of input use. The output rate is determined by substituting the equation for the expansion path into the original production function.
For example, if we substitute the equation for the expansion path, K = (w/r) L, into the production function, Q = 100K½ L½ yields
Q = 100(w/r. L) ½ L½
or
Q = 100L (w/r)½
The two equations have three unknown: K, L, and Q. We assume the prices of labour (w) and capital (r) to be known. If the value of K, L or Q is given, the efficient rate of the other two variables can be calculated.
Let us consider the problem of determining the efficient input combination for producing 1,000 units of output if w = 4 and r = 2. Therefore Q = 1000, w = Rs.4, and r = Rs.2. Let us substitute these values into equation (5.12) and solve for L.
1000 = 100L (4/2) ½
L = 7.07
The, substitute L = 7.07 into equation (5.11), the equation for the expansion path to find K.
K = (4/2) 7.07
K = 14.14
Thus the input combination (K=14.14, L = 7.07) is the most efficient way to produce 1,000 units of output.
If the price of capital increase to r = 4 and the firm still wanted to produce 1,000 units of output. Then
1000 = 100 L √4/4
L = 10
Now substitute L = 10 into equation5.11
K = (4/4) L
LK = 10
The new efficient combination is (K= 10, L=10). The firm responded to the higher price of capital by substituting labor for capital. That is, the capital input was reduced from 14.14 to 10 and the labour input was increased from 7.07 to 10.
Expenditure Elasticity
The expenditure elasticity of a factor of production is analogous to income elasticity. Income elasticity is related to the income-consumption curve; and commodities are classified as superior, normal or inferior as income elasticity exceeds unit, lies in the unit interval, or is negative. Similarly, factors of production are classified as superior, normal, or inferior according as the corresponding expenditure elasticity exceeds unit, lies in the unit interval, or is negative.
The expenditure elasticity of a factor of production may be defined as the relative responsiveness of the usage of this factor to changes in total expenditure.
In other words, the expenditure elasticity of X is the proportional change in the usage of X divided by the proportional change in total expenditure. The changes in total expenditure are restricted to movements along the expansion path.
Symbolically, the formula for the expenditure elasticity is
where,
x is the usage of factor X
c is the total expenditure on factors of production.
In general, indivisibility means that certain inputs, particularly equipments and managerial skills are available only in minimum sizes or in definite ranges of sizes. They cannot be divided into parts to suit production of small scale. As a firm’s scale of operations increases, it can use the minimum sizes and then the larger sizes of more efficient equipment. For example, half a typewriter or half a turbine cannot be used. Similarly, half a manager cannot be employed, if possibilities of part-time employment are ruled out. Due to the indivisibility of equipments, machinery and managers, they are employed in a certain minimum size/quantity even if the scale of production is much less than the capacity output. Therefore, when the scale of production is increased, the productivity of indivisible inputs causes exponential increase in output due to technological advantage. This results in increasing returns to scale. But the indivisibility quickly exhausts itself as a cause of increasing returns to scale.
(b) Mass Production
One of the basic characteristics of advanced industrial technology is the existence of ‘mass-production’ methods over large sections of manufacturing industry. ‘Mass – production’ methods are processes available only when the level of output is large. They are more efficient than the best available processes for producing small levels of output.
(c) Dimensional Relations
Dimensional relationships are considered to be the important causes of increasing returns to scale. If we double the diameter of a pipe, the flow through it gets more than doubled. For example 3 x 3 = 9, when the two digits are doubled 6x6 = 36, which is four times of 9. In accordance with the dimensional relationship, when the inputs are doubled, the output increases more than double.
(d) Higher Degree of Specialization
Higher degree of specialization is also pointed out as a cause of increasing returns to scale. With more labor, the firm can divide and subdivide tasks and thus gain efficiency of labour. With more machinery, the firm can buy special types and also assign special jobs even to standardized kinds of machinery.
Reasons for Constant Returns to Scale
The phase of increasing returns to scale cannot go on indefinitely. There is a limit to the economies of scale. When inputs are perfectly divisible the capital- labour ratio is fixed, and the production function is homogeneous of degree 1, constant returns to scale takes place. The phase of constant returns to scale can be brief, before decreasing returns to scale set in. The Cobb-Douglas production function is linearly homogeneous exhibiting constant returns to scale.
Reasons for Decreasing Returns to Scale
(a)Diminishing returns to management
The most common causes of diminishing returns to scale are ‘diminishing returns to management’. The management is responsible for the co-ordination of the activities of the various sections of the firm. Even when authority is delegated to individual managers (production manager, sales manager, etc.) the final decisions have to be taken by the top management. As the output grows, top management becomes eventually overburdened and hence less efficient in its role as coordinator and ultimate decision-maker. In this view, decreasing returns to scale are actually a special case of variable proportions.
(b)Exhaustible natural resources
As the scale of production is increased, it does not result in increase of output in the same proportion due to the exhaustible nature of natural resources. For example, doubling the fishing fleet may not lead to doubling of the catch of fish; or doubling the plant in mining or an oil-extraction field may not lead to a doubling of output.
(c) Organizational Problems
Certain organizational problems creep in to the large- scale production. A plant with a very large number of workers is more difficult to manage than a smaller one. The larger the number of workers, the larger becomes the number of foremen and supervisors creating several layers of middle management causing problems of communication. As a result it becomes difficult for manager to make sharp decisions. Owing to the organizational dis-economies that accompany large plant, firms frequently build multiple plants Automobile assembly plants are kept relatively small by having most components made elsewhere.
PRODUCTION FUNCTION THROUGH ISO-QUANT CURVE
A firm may decide to maximize output for a given cost or minimize cost subject to a given output. An iso-quant exhibits that a given output can be produced with various different input-combinations, given the input prices. However, there is only one input combination that provides us with least –cost criteria. This section illustrates the identification of the least-cost input combination.
Figure shows that 100 units of output can be produced with various different combinations of K and L located on IQ1.Points a, b and c represents three different combinations of capital and labour. Point a represent K3 +L1, point b represent K2 + L2 and point c represent K1+L3.
All the three combinations can produce 100 units of output. Technically, any of these input combinations can be chosen for producing 100 units of output, but all are not economically efficient. With given input prices the total cost of production differs from point to point. There is only one point on the isoquant IQ1 that produce 100 units of output at minimum cost. Any firm’s best interest is to identify the input combination that produces a given level of output at minimum cost. An upper or lower isoquant represent higher or lower levels of output that can be produced with greater and lesser and different combinations of inputs with varying total costs.
The combination of inputs with which a firm produces the product also depends on the prices of inputs and the amount of money that a firm has to spend. Information concerning relative input prices enables a producer to determine operating input combination and minimize the cost of producing a given output or maximize output for a given level of costs.
Input prices are determined by the forces of demand and supply in the market. If a firm uses only labor and capital, the total cost of the firm can be denoted by
C = wL + rK
C = total cost
W = wage rate of labour
L = quantity of labour used
R = rental price of capital
K = quantity of capital used.
For example, suppose capital costs Rs.1000 per unit (r = Rs.1000) and labour receives a wage of Rs.2500 per labor month (w = Rs.2, 500). If a total of Rs.15000 is to be spent for inputs, the following combinations are possible. Rs.15000 = Rs.1000K + 2.500L, or K = 15 – 2.5 L
More generally, the producer can choose among the combinations given by the equation
K=C/r–w/r
Fig.illustrates it more clearly. If Rs.15000 is spent for inputs and no labor is used, 15 units of capital may be bought. If C is to be spent and r is the unit cost, C/r units of capital may be purchased. This is the vertical-axis intercept of the line. If 1 unit of labour is purchased at Rs.2500, 2.5 units of capital must be sacrificed; if 2 units of labour are bought, 5 units of capital must be sacrificed; and so on. Thus as the purchase of labour is increased, the purchase of capital must be reduced. For each additional unit of labour, w/r units of capital must be forgone. The isocost line shows the various combinations of inputs that may be purchased for a stipulated amount of expenditure.
Finding the Least-Cost Technology with Isoquants and Isocosts
A firm spends fixed amount of money (C) on two inputs only capital (K) and labor (L). It decides its budget and knows the prices of each of the inputs that are assumed to remain constant. With given input prices, a producer wishes to operate efficiently by producing the maximum attainable output. Thus, among all input combinations that can be purchased for the fixed amount (C), the producer seeks one that results in the largest level of output. The given level of cost (C) is represented by the isocost line KL in the accompanying
The producer could operate at points such as R and S At these two points, the input combinations required to produce the IQ1 level of output are available for a given cost represented by the isocost KL. However, the output can be increased without incurring additional cost by the selection of a more appropriate input combination. Indeed, output can be expanded until the IQ2 level is reach- the level at which an isoquant is just tangent to the specified isocost curve. A greater output is not obtainable for the given level of expenditure; a lesser output is inefficient because production can be expanded at no additional cost. Hence the input combination represented by the slope of ray OQ is optimal because it is the combination that maximizes output for the given level of cost. At point R, the marginal rate of technical substitution of labor for capital is relatively high as 3 units of capital are foregone for one addition unit of labor. The relative input price, given by the slope of KL, is much less, say, 1:1 In this case, 1 unit of labor costs the same as 1 unit of capital but it can replace 3 units of capital in production. The producer would be better off by substituting labor for capital. The opposite argument holds for point S, where the MRTS is less than the input-price ratio.
The producer, therefore, attains equilibrium only when the MRTS of labor for capital is equal to the ratio of the price of labor to the price of capital. This equilibrium point is where the producer maximizes output for a given level of cost. The market input- price ratio tells the producer the rate at which one input can be substituted for another in purchasing. So long as the two are not equal, a producer can achieve either a greater output or a lower cost by moving in the direction of equality.
An entrepreneur may alternatively seek to minimize the cost of producing a stipulated level of output. The concept is explained in the Fig .The KL1, KL2, KL3 are isocost curves and IQ represents the stipulated level of output. The level of cost shown by KL1 is not feasible because the IQ level of output is not physically producible by any input combination available for this outlay. The IQ level of output could be produced by the input combinations represented by the point R and S, both at the cost level KL3. By moving either from R to Q or from S to Q, the entrepreneur can obtain the same output at lower cost. A position of equilibrium is attained only at point Q where the isoquant is just tangent to an isocost curve. Thus in equilibrium the marginal rate of technical substitute of inputs must be equal to the ratio of the prices of inputs.
For the achievement of least-cost combination of inputs two conditions must be fulfilled. The first-order condition requires that the marginal rate of exchange of one input for the other (i.e. of capital for labour or vice-versa) must equal the ratio of their marginal physical product.
MPl / MPk measures the slope of the isoquant.
The second order condition requires that the marginal rate of exchange between labour and capital (K/L) must equal the ratio of the marginal physical product (MPl / MPk) at the highest possible isoquant. In the fig. the first order condition is fulfilled on points R, S and e where the isoquant IQ1 intersects isocost line KL. But R and S are not on the highest possible isoquant. The second order condition is fulfilled at point e only. Point R and S satisfy only first order condition, but point e satisfies both first-order and second order conditions. Therefore, least-cost factor combination occurs at point e.
Changes in Input Prices
If the price of one input, say labour, changes, the firm will adjust the input mix by substituting one input for the other. If the price of labour increases, the firm substitutes capital for labour. On the contrary if the price of labour decreases, making labour relatively cheaper, labour will be substituted for capital. In general, if the relative prices of inputs change, managers will respond by substituting the input that has become relatively less expensive for the input that has become relative more expensive.
Thus, we find that as a result of changes in the price of an input, input combination of the firm changes leading the firm to employ more of cheaper input and less of the costlier one. Besides, the level of output also changes. If price of an input decreases the level of output increases, and vice versa.
Expansion Path:
In the long run all inputs are variable without any limitation (technical or financial) to the expansion of output The firm’s objective is the choice of the optimal way of expanding its output, so as to maximize its profits.
Let us take the case of a firm producing 100 units of output using ten units of capital and ten units of labour, and the firm being in equilibrium at point ‘a’. The input prices are w=20 and r =20 as shown in the accompanying Fig
We can determine an equation for the expansion path by substituting the marginal production functions and input prices into the efficiency condition, and then by solving for capital as a function of labour if the production function is
Q = 100K½ L½
The corresponding marginal product function are
MPL = dQ/DL = 50K½ /L½
and
MPK = dQ/DK = 50L½ /K½
Substituting the marginal product equations in the efficient condition (MPL/ MPK = w/r) gives,
Solving for K gives:
This expression is the equation for the expansion path for the production function Q = 100K½ L½. If w and r are known, equation (5.10) defines the efficient combination of capital and labour for producing any rate of output. If w = 1 and r= 1, the expansion path would be,
K = L
If w = 2 and r= 1, the equation for expansion would be
K = 2L
If the expansion path is known, the knowing the isoquant- isocost system is not necessary to determine efficient production points. The firm will only produce at those points on the expansion path.
The expansion path indicates optimal input combinations, but it does not indicate the specific rate of output associated with that rate of input use. The output rate is determined by substituting the equation for the expansion path into the original production function.
For example, if we substitute the equation for the expansion path, K = (w/r) L, into the production function, Q = 100K½ L½ yields
Q = 100(w/r. L) ½ L½
or
Q = 100L (w/r)½
The two equations have three unknown: K, L, and Q. We assume the prices of labour (w) and capital (r) to be known. If the value of K, L or Q is given, the efficient rate of the other two variables can be calculated.
Let us consider the problem of determining the efficient input combination for producing 1,000 units of output if w = 4 and r = 2. Therefore Q = 1000, w = Rs.4, and r = Rs.2. Let us substitute these values into equation (5.12) and solve for L.
1000 = 100L (4/2) ½
L = 7.07
The, substitute L = 7.07 into equation (5.11), the equation for the expansion path to find K.
K = (4/2) 7.07
K = 14.14
Thus the input combination (K=14.14, L = 7.07) is the most efficient way to produce 1,000 units of output.
If the price of capital increase to r = 4 and the firm still wanted to produce 1,000 units of output. Then
1000 = 100 L √4/4
L = 10
Now substitute L = 10 into equation5.11
K = (4/4) L
LK = 10
The new efficient combination is (K= 10, L=10). The firm responded to the higher price of capital by substituting labor for capital. That is, the capital input was reduced from 14.14 to 10 and the labour input was increased from 7.07 to 10.
Expenditure Elasticity
The expenditure elasticity of a factor of production is analogous to income elasticity. Income elasticity is related to the income-consumption curve; and commodities are classified as superior, normal or inferior as income elasticity exceeds unit, lies in the unit interval, or is negative. Similarly, factors of production are classified as superior, normal, or inferior according as the corresponding expenditure elasticity exceeds unit, lies in the unit interval, or is negative.
The expenditure elasticity of a factor of production may be defined as the relative responsiveness of the usage of this factor to changes in total expenditure.
In other words, the expenditure elasticity of X is the proportional change in the usage of X divided by the proportional change in total expenditure. The changes in total expenditure are restricted to movements along the expansion path.
Symbolically, the formula for the expenditure elasticity is
where,
x is the usage of factor X
c is the total expenditure on factors of production.
A factor of production is said to be
superior, normal, or inferior according as its expenditure elasticity exceeds
unity, lies in the unit interval, or is negative.
Accompanying figure
illustrates the expenditure elasticity. Consider the expansion path and
concentrate on factor X. Along ray OR both inputs expand proportionally. At
points such as A the usage of factor X expands proportionally more than total
expenditure along the expansion path. At all such points the factor is
superior. At points such as B factor usage expands proportionally less than
total expenditure. Expenditure elasticity lies in the unit interval, and the
factor is said to be normal. At D the change in usage of both inputs is
proportional and the expenditure elasticity is unity.
In certain cases, the
usage of a factor may decline when output and resource expenditure are
increased. At point C the expenditure elasticity of X is instantaneously zero.
Beyond point C, the expansion path ‘bends back’ on itself. The usage of X
diminishes as expenditure is increased beyond point C. Over this range of
expenditure and output, X is an inferior factor. For example, when the level of
farm production goes beyond a certain level, employment of labour might actually
decrease as the farmer switches from manual farming to mechanized farming.
Equilibrium of a Multi Product Firm:
So far the firm
has been described as producing one output, or product. Nearly all firms,
however, produce more than one product. We will now consider a firm with two
products and see how does it choose the proportions in which to produce its two
products.
Suppose a firm
produces two different products X and Y. Also suppose that the firm has some
given quantity of resources – plant, equipment, and labour. . With this given
quantity of resources, the firm can produce two products in different
proportions, subject to the condition that if more of X is produced, less of Y
can be produced, and vice versa.
The monthly production possibilities
for the firm are shown in Fig.5.19. Curve AD is the firm’s monthly production
possibility curve. Point B and C on the curve correspond with possibilities B
and C. if the firm had a larger quantity of resources, the curve would like
farther northeast. Thus for each quantity of resources, the firm has a
different production possibility curve.
The curve is
concave to the origin and its slope being greater at C than at B which means
that, as the output of X is increase, the sacrifice of Y output becomes larger
and larger. Similarly, increases in the production of Y are accompanied by
ever-larger sacrifices of X. The firm’s resources are not equally adaptable in
producing both X and Y, and when they are concentrated mostly on one of the
products, as at point C, the resources are less production. The concavity of
the curve signifies increasing opportunity cost of one output in terms of the
other. For each addition unit of Y output sacrificed, the gain in X output
becomes smaller and smaller and vice versa.
The firm wishes to maximize the revenue it gets from selling its two
products. Besides its production possibilities, the firm has to take into
account the prices it receives for X and Y.
Let us assume that the demand for both products is perfectly elastic to
the firm, then the prices the firm sells at are unaffected by the quantities it
sells.
Isorevenue lines show the
reveneues available to the firm. In Fig.5.20 FG is an isorevenue line. The
quantity OG of product X multiplied by the price of X yields the same revenue
as OF of Y multiplied by the price of Y. The slope of the isorevenue line is
Px/Py, because slope = FO/OG = Y/X, and since Ypy = XPx, Y/X= Px/Py. The
position of an isorevenue line signifies how large the total revenue is. The farthest northeast it lies, the greater
is the total revenue shown by an isorevenue line. Similar to isocost line,
there is a family of isorevenue line.
The optimum for the firm is at point E in Fig.5.20. The firm
produces X and Y in the amounts indicated by point E. In so doing, the firm
maximizes its total revenue because E is on the highest attainable isorevenue
line. If the firm were to produce elsewhere on its curve, at points H or K, it would be on a lower isorevenue
line, the dashed line in the figure.
At point E the isorevenue line is tangent to the production possibility
curve. Therefore, at point E the slopes are equal – that is, the ratio of the
prices Px/Py is equal to the rate of substitution (or transformation) of the
two products. This is another rule of
economic efficiency 
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