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In economic theory, we are concerned with three types of production functions, viz.:- 1. Production Functions with One Variable Input 2. Production Function with Two Variable Inputs 3. Production Function with all Variable Inputs.
Types # 1. Production Functions with One Variable Input:
The Law of Variable Proportions:
If one input is variable and all other inputs are fixed, the firm’s production function exhibits the law of variable proportions. If the number of units of a variable input is increased, keeping other inputs constant, how output changes is the concern of this law. Suppose land, plant, and equipment are the fixed factors, and labor the variable factor. When the number of laborers is increased successively to have larger output, the proportion between fixed and variable factors is altered and the law of variable proportions sets in.
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The law states that as the quantity of a variable input is increased by equal doses, keeping the quantities of other inputs constant, total product will increase, but after a point, at a diminishing rate. This principle can also be defined thus – When more and more units of the variable factor are used, holding the quantities of fixed factors constant, a point is reached beyond which the marginal product, then the average, and finally the total product will diminish.
The law of variable proportions (or the law of non-proportional returns) is also known as the law of diminishing returns. But, as we shall see below, the law of diminishing returns is only one phase of the more comprehensive law of variable proportions.
Let us illustrate the law with the help of Table 3.3, where on the fixed input land of 4 acres, units of the variable factor labor are employed and the resultant output is obtained.
An analysis of Table 3.3 shows that the total, average, and marginal products increase at first, reach a maximum when seven units of labor are used and then it declines. The average product continues to rise till the fourth unit while the marginal product reaches its maximum at the third unit of labor, then they also fall.
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The marginal product starts declining first, then the average product, and finally the total product. It should be noted that the point of falling output is not the same for total, average, and marginal product. This observation points out that the tendency to diminishing returns is ultimately found in the three productivity concepts.
The law of variable proportions is presented diagrammatically in Fig. 3.11. The TP curve first rises at an increasing rate and then reaches the highest point at a decreasing rate and then starts falling slowly. The slope of the TP curve at any point can be known by drawing a tangent at that point. In Fig. 13.11, there are three such points. At point a the slope of TP is the highest, at b it is less than a and at c it becomes zero. The marginal product curve (MP) and the average product curve (AP) also rise with TP.
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The MP curve reaches its maximum point d when the slope of the TP curve is the maximum at point a and then starts falling. The maximum point on the AP curve is e where it coincides with the MP curve. This point also coincides with b on the TP curve from where the total product starts a gradual rise.
When the TP curve reaches its maximum point c, the MP curve becomes zero at point f, and when the former starts declining, the latter becomes negative. It is only when the total product is zero that the average product also becomes zero. The rising, the falling, and the negative phases of the total, marginal, and average products are in fact the different stages of the law of variable proportions.
In Stage I, the average product reaches the maximum and equals the marginal product when four workers are employed, as shown in Table 3.3. In this stage, the total product curve also increases rapidly. Thus, this stage relates to increasing average returns. Here, land is too much in relation to the workers employed. It is, therefore, uneconomical to cultivate land in this stage.
The main reason for increasing returns in the first stage is that in the beginning the fixed factors are larger in quantity than the variable factors. When more units of variable factors are applied to a fixed factor, the fixed factor is used more intensively and production increases rapidly.
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It can also be explained in another way. In the beginning, the fixed factor cannot be put to the maximum use due to the non-applicability of sufficient units of variable factors. But when units of variable factors are applied in sufficient quantities, division of labor and specialization lead to per unit increase in production and the law of increasing returns operates.
When more units of the variable factor are applied on such a fixed factor, production increases more than proportionately. This cause points toward the law of increasing returns.
Production cannot take place in Stage III either. For, in this stage, total product starts declining and the marginal product becomes negative. The employment of the eighth worker actually causes a decrease in total output from 60 to 56 units and makes the marginal product –4.
In the figure, this stage starts from the dotted line of where the MP curve is below the X-axis. Here the workers are too many in relation to the available land, making it absolutely impossible to cultivate it.
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When production takes place to the left of point e, the fixed input is in excess quantity in relation to the variable input. To the right of point f, the variable input is used excessively. Therefore, production will always take place within these stages.
In between Stages I and III is the most important stage of production that of the law of diminishing returns. Stage II starts when the average product is at its maximum to the zero point of the marginal product. At the latter point, the total product is the highest. Table 3.3 shows this stage when the workers are increased from four to seven to cultivate the given land. In the figure, it lies between be and cf.
Here land is scarce and is used intensively. More and more workers are employed in order to have a larger output. Thus, the total product increases at a diminishing rate and the average and marginal products decline. Throughout this stage, the marginal product is below the average product. This is the only stage in which production is feasible and profitable.
This law is based on the following assumptions:
1. It is possible to vary the proportions in which the various productive services (inputs) are combined.
2. Only one input is variable while others are held constant.
3. All units of the variable inputs are homogeneous.
4. There is no change in technology. If the technique of production undergoes a change, the product curves will be shifted accordingly but the law will ultimately operate.
5. It assumes a short-runs situation, for in the long-run all productive services are variable.
6. The product is measured in physical units, i.e., in quintals, tonnes, etc. The use of money in measuring the product may show increasing rather than decreasing returns if the price of the product rises, even though the output may have declined.
Marshall applied the operation of the law to agriculture, mining, forests, fisheries, and the building industry.
The last segment of the theory of production is the problem of determining the least- cost combination of factors for a given output. The aim of every producer is to get maximum profits, and to achieve this he combines the various resources in such a proportion that a given output is manufactured at the least cost.
This problem is similar to the problem faced by the consumer who allocates his money income among several commodities for obtaining maximum satisfaction. The consumer is in equilibrium when the marginal utilities and the price ratios of the goods bought become equal.
To achieve this equilibrium position, the consumer acts on the principle of substitution. Similarly, the producer will be in equilibrium when the marginal productivities of the various factor units employed by him are equal to their prices. To achieve the least-cost combination of a given output, he substitutes a cheap input for a costly input.
If he finds that the marginal product of a rupee’s worth of factor A is greater than that of factor B, he will spend less on B and more on A. He will continue to spend like this with the consequence that the marginal product of a rupee’s worth of factor B will steadily rise, while that of factor A will fall, until the least-cost combination is achieved.
Suppose that a producer uses three inputs A, B, and C in the production of commodity X. The price of A is Rs.3 per unit, of B Rs.2 per unit, and of C Rs.1 per unit. The cost outlay on the three factors is Rs.61 per day. The daily marginal productivities of the different units of these factors resources are shown in Table 3.4.
The price (Pa) of A being Rs.3 per unit, of B (Pb) Rs.2 per unit, and that of C (Pc) Rs.1 per unit, in equilibrium the marginal product of A (M Pa) should be 1.5 of B (M Pb) and twice that of C (M Pc). When the consumer continues to use more units of factors A, B, and C to produce a fixed quantity of X (columns 1, 3, 5 of the table, their marginal productivities continue to decline (columns 2, 4, 6).
Ultimately, the equilibrium position is reached when the marginal productivity of factor A (M Pb½ = 6) and the marginal productivity of B is twice that of C (M Pc = 3). This position is attained at 9 units of A, 11 units of B, and 12 units of C, where the marginal product per rupee’s worth of each input is equalized.
The proportionality rule for the equality of marginal productivity price ratios of different factor units can be expressed as –
MPa/Pa = MPb/Pb = MPc/Pc = – – – – = MPn/Pn
Where MPa, MPb, and MPc are the marginal products of inputs A, B, and C and Pa, Pb, and Pc are the prices of A, B, and C, respectively.
In the above numerical example, this proportionality rule is satisfied when the marginal product per rupee’s worth of each factor unit is three units of the good X, i.e.,
9/3 = 6/2 = 3/1 = 3
Another condition for the least-cost combination is that the entire cost outlay for the given period should be fully spent at the level of equilibrium. This condition is also satisfied at the above combination of 9 units of A, 11 units of B, and 12 units of C. This is shown below.
Total outlay to be spent per day is Rs.61
Outlay on 9 units of A @ Rs.3 per unit = 9 x 3 = Rs.27
Outlay on 11 units of B @ Rs.2 per unit = 11 x 2 = Rs.22
Outlay on 12 units of C @ Re.1 per unit = 12 x 1 = Rs.12
Total outlay on A + B + C = Rs.61
If a combination, other than the above is taken, the two conditions will not be satisfied. Suppose Re.1 is withdrawn from input B and spent on input C, it will mean loss of 3 units and an addition of only one unit when the thirteenth unit of C is acquired. This reallocation leads to a fall in the firm’s total product by two units. The producer is also not in a position to utilize the entire cost outlay. He spends Rs.1 less than before –
[(9 x 3) + (10 x 2) + (13 x 1) = 60]
The principle of least-cost combination also implies that each factor unit will be so employed as to equate its marginal product per rupee’s worth in every use or occupation. If the marginal product of labor is greater in cotton textile industry than in the jute industry, labor will move from the latter to the former till marginal productivity of labor becomes equal in both the industries. Equality between different units of capital, labor, etc., is also established in a similar manner.
Assumptions:
The analysis given above is based on the following assumptions:
1. There is perfect competition in the factor market.
2. There is perfect mobility of factor units.
3. The prices of factor services are given and constant.
4. The marginal productivity of each factor is independent of the other.
To conclude, the principle of least-cost combination is an important tool in production theory. It points out that the efficient combination of variable factors which the producer should use depends upon the marginal productivities and prices of the respective factors.
Types # 2. Production Function with Two Variable Inputs:
Isoquants:
To understand a production function with two variable inputs, it is necessary to explain what an isoquant is. An isoquant is also known as iso-product curve or equal-product curve or a production-indifference curve. These curves show the various combinations of two variable inputs resulting in the same level of output. Table 3.5 shows how different pairs of labor and capital result in the same output.
It will be seen that output is the same either by employing 4L + 1C or by 5L + OC (and so on). This relationship, when shown graphically, results in an isoquant.
Thus, by graphing a production function with two variable inputs, one can derive the isoquant tracing all the combinations of the two factors of production that yield the same output. An isoquant is defined as the curve passing through the plotted points representing all the combinations of the two factors of production which will produce a given output.
Figure 3.12 gives a typical isoquant diagram where as one moves upward to the right, higher levels of outputs are obtained, using larger quantities of output.
For each level of output there will be a different isoquant. When the whole array of isoquants are represented on a graph, it is called as an isoquant map.
An important assumption in the isoquant diagram is that the inputs can be substituted for each other. Let us take a particular combination of X and Y resulting in an output Q 600, one finds other quantities of the inputs resulting in the same output. Let us suppose that X represents labor and Y, machinery. If the quantity of the labor (X) is reduced, the quantity of machinery (Y) must be increased in order to produce the same output.
MRTS:
The slope of the isoquant has a technical name- marginal rate of technical substitution (MRTSw), or sometimes, the marginal rate of substitution in production. Thus, in terms of inputs of capital services K and labor L.
MRTS = dK/dL
(MARTS is similar to MRS, i.e., Marginal Rate of Substitution, which is the slope of an indifference curve.)
Isoquants assume different shapes depending upon the degree of substitutability of inputs under consideration.
i. Linear Isoquants:
In this type, there is perfect substitutability of inputs. For example, a given output say 100 units can be produced by using only capital or only labor or by a number of combinations of labor and capital, say 1 unit of labor and 5 units of capital, or 2 units of labor and 3 units of capital, and so on.
Likewise, given a power plant equipped to burn either oil or gas, various amounts of electric power can be produced by burning gas only, oil only, or varying amounts of each. Gas and oil are perfect substitutes only, oil only, or varying amounts of each. Gas and oil are perfect substitutes here. Hence, the isoquants are straight lines (See Fig. 3.13).
ii. Right-Angle Isoquant:
In this type, there is complete non-substitutability between the inputs (or strict complementarity). For example, exactly two wheels and one frame are required to produce a bicycle and in no way can wheels be substituted for frames or vice-versa. Likewise, two wheels and one chassis are required for a scooter. This is also known as Leontief isoquant or input-output isoquant (See Fig. 3.14).
iii. Convex Isoquant:
This form assumes substitutability of inputs but the substitutability is not perfect. For example, a shirt can be made with relatively small amount of labor (L1) and a large amount of cloth (C1). The same shirt can be as well made with less cloth (C2), if more labor (L2) is used because the tailor will have to cut the cloth more carefully and reduce wastage.
Finally, the shirt can be made with still less cloth (C3) but the tailor must take extreme pains so that labor input requirement increases to L3. So, while a relatively small addition of labor from L1 to L2 allows the input of cloth to be reduced from C1 to C2, a very large increase in labor from L2 to L3 is needed to obtain a small reduction in cloth from C2 to C3. Thus, the substitutability of labor for cloth diminishes from L1 to L2 to L3.
The main properties of isoquants are the following:
1. An isoquant is downward sloping to the right, i.e., negatively inclined. This implies that for the same level of output, the quantity of one variable will have to be reduced in order to increase the quantity of other variable.
2. A higher isoquant represents larger output. That is, with the same quantity of one input and larger quantity of the other input. Larger output will be produced.
3. No two isoquants intersect or touch each other. If two isoquants intersect or touch each other, this would mean that there will be a common point on the two curves; and this would imply that the same amount of two inputs can produce two different levels of output (i.e., 400 and 500 units) which is absurd.
4. Isoquant is convex to the origin. This means that its slope declines from left to right along the curve. In other words, when we go on increasing the quantity of one input, say labor by reducing the quantity of other input, say capital. We see that less units of capital are sacrificed for the additional units of labor.
Types # 3. Production Functions with All Variable Inputs:
A closely related question in production economics is how a proportionate increase in all the input factors will affect total production.
This is the question of returns to scale and one can think of three possible situations:
1. If the proportional increase in all inputs is equal to the proportional increase in output, returns to scale are constant. For instance, if a simultaneous doubling of all inputs results in a doubling of production, then returns to scale are constant (Fig. 3.16).
2. If the proportional increase in output is larger than that of the inputs, then we have increasing returns to scale (Fig. 3.17).
3. If output increases less than proportionally with input increase, we have decreasing returns to scale (Fig. 3.18).
The most typical situation is for a production function to have first increasing then decreasing returns to scale as shown in Fig. 3.18.
The increasing returns to scale are attributable to specialization. As output increases, specialized labor can be used and efficient large-scale machinery can be employed in the production process. However, beyond some scale of operations not only are further gains from specialization limited, but also co-ordination problems may begin to increase costs substantially. When coordination costs more than offset additional benefits of specialization, decreasing returns to scale begin.
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