# Factor and product relationship problems

The objective of factor-product relationship is to determine the optimum quantity of the variable input that will be used in combination with fixed inputs in order to. How should the business produce (Factor-Factor) Profit maximizing quantity of variable input (Marginal value product and marginal input cost). The factor product relationship is one of the three basic relationships in The formulation of production efficiency problems of technical efficiency such as.

The of farm management and production economics is normally carried out by using various production functions.

These are constant returns, increasing marginal returns etc. These functions are depicted in the classical production function in our figure. In real life input-output relationships usually take on one or a combination of the following three forms; - Constant returns.

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Constant returns constant marginal returns Assuming that the factor costs are constant, under perfect competition in all input markets, a firm experiencing constant returns, will have constant long run average costs, and a firm experiencing increasing returns will have decreasing long run average costs Fisch,Ferguson,Gelles, Gregory M. The production function that has constant returns may be obtained when the quantity of TP increases by the same amount for each additional unit of variable input.

This relationship between the input and output that has constant returns is also termed as linear relationship. The result curve when plotted is a straight line hence the name linear.

A hypothetical example Yield of maize at varying levels of nitrogen fertilizer application per hectare. Nitrogen is varied in 50kg units bags and the maize yield increases by 8 bags for each additional unit of bag. In the production function we have equal increments in total increments for every extra unit of input, have linear functions that experience constant increasing returns.

These production functions are typical of the first stage of the classical production function where the firm experiences increase in TP production at an increasing scale.

The linear production functions exist in agriculture where technical units of plants are readily divisible. Farmers with limited capital are often faced with a linear production function or constant returns to scale with a relevant range of credit or funds which they can obtain. If a farmer has 10 hectares and a homogeneous land, he may apply 50kg of fertilizer input.

## Theory of production

This is possible because the 10 hectare farm is divisible. Although constant productivity of the nature mentioned above is possible, a linear production function does not generally exist when inputs per hectare or per animal is intensified.

Increasing returns to a single factor exist when each successive unit of input for a variable resource adds more to the total product than the previous unit as shown below; illustration not visible in this excerpt Land is fixed at one hectare while labor is varied in person hours. A production function showing increasing returns of maize yield per person hour of weeding labor applied per hectare. Examples of increasing returns in agriculture include; - Increasing quality of fertilizer from the previously low levels.

Diminishing returns to a variable factor exist when each additional unit of input adds less output than the previous unit. This is illustrated in the following table. Yield of maize at varying rates of labor application per hectare hypothetical data. The law of diminishing returns is encountered in practically all forms of production use e. Elasticity of production can be applied to the production function or to the input-output relationship. It is defined as a percentage change in output resulting from a percentage change in input.

The amount of variable input at the point of diminishing returns is the minimum that would be rationally used because the efficiency of the input is maximum at that point. Thus using the definition of the diminishing returns it can be said that even without input and output prices, input use will always be extended to the point of diminishing returns.

The factor product relationship. The classical production function gives the relationship between one variable input and other fixed inputs. In reality however more than one variable inputs are usually varied in production. The purpose of the classical production however is that it helps to illustrate the basic input output concepts. The farmer does not only vary the inputs but sometimes produces more than one product using the same resources e.

A two variable input and one product. The basic assumption that was made in the input output relationship where one input was varied was that the farmer has no control over the market prices. It was also assumed that at least one productive input was fixed in quantity so that all production processes are short run and subject to the law of diminishing returns. A production function relating output of maize in bags to various combinations of land and labor.

These figures represent the production function relating the output of maize to the inputs of land and labor. The production function in this table can be regarded as a series of sub production functions. A production function in each row gives the amount of maize that can be produced with the fixed amount of land at that level.

The data used in the top row is the same as was used in the classical production function, so the MPPL is the difference between the two consecutive figures in the same row. On the other hand each column gives the amount of maize that can be produced by a fixed number of laborers when varying the hectares of land. The marginal product per unit of land is the difference between two consecutive figures in the same column.

The study of the factor-factor relationship one of the five economic problems which confront agriculture producers, and which is how much to produce. In other words which method of production should be used or specifically what is the best method of combining the inputs in producing products at a target level of output.

In using the economic approach to the solution of how to produce, how to produce, we select a method of production combination of inputs which would be the cheapest less costly in producing the chosen products.

In this case maize are the target quantities. The following selection looks at the various combinations of two variable inputs which will produce a given target level of output and how to select the least costly combination.

## Principles of Agricultural Economics

Let our target level of combination be 40 bags of maize. This level of output can be produced by a combination of 5 laborers and 1 hectare of land.

These are various combinations which are not specifically precisely in the table but which would also produce 40 bags of maize. These intermediate combinations are given in the following table.

Isoquant and least cost combination for 40 bags of maize. The curve is sometimes called the isoquality curve or an isoquant for each level of output as demonstrated in the figure below.

This figure is similar to a contour map used in carrying out terraces in plotting topography in an area of land. Whereas contours on land represent different heights isoquants are product contours. In our figure the slope of an isoquant at a given point shows the quality of land replaced by one extra laborer. The calculated rate of substitution of labor for land is given in the third column of the table.

It is obtained by dividingthe increase in land i. Isoquants generally have a negative slope. So the slope of the curve is negative of the marginal rate of input substitution.

This is so because when we use more of one input we would expect to use less of other variable input in producing a given level of output. Isoquants for higher levels of output normally lie above and to the right of isoquant for lower levels of output.

This means that it requires more of either or both inputs to produce more output. Isoquants are convex to the origin. This means that the marginal rate of input substitution diminishes as more of one factor is used to replace the other.

In our example each added unit of labor substitutes or replaces less land than the previous unit. The reason for diminishing rate of substitution is that one input is rarely a perfect substitute for the other.

In our example labor cannot land and land cannot substitute labor to produce maize. Some land and some labor must be used to produce maize. The spacing of the isoquant tells us the effects on TP of increasing inputs of both factors together.

If isoquants are equally spaced it means that there is a constant returns as the two inputs are increased together. Where isoquants get closer there are increasing returns due to increased inputs for both factors. Conversely where isoquants become further apart implies diminishing returns to increases in both inputs. Least cost combination of inputs. Economic optimum of two variable inputs. The least cost combination of inputs is the combination of variable input which will produce a given level of output at a minimum cost.

The least cost combination can be estimated by using the MRT between two inputs as well as the respective prices of each input in order to determine the least cost combination which gives the least cost.

This method is satisfactory when a few combinations produce a given output. In our example approximately 7 combinations of land and labor are needed to produce 40 bags of maize. The exact location of the least cost combination can be determined geometrically using the concept of MRTS and the least cost. Every output level can be represented by an isoquant and any possible outlay cost can be represented by an isocost line.

Given that the least cost criterion is where the isocost line is tanget to the isoquant we can draw the isoquant and the isocost line and determine the amount of land and labor which wll give the least cost combination.

Thus the exact cost combination of the inputs occurs as the point where the isocost line is tangent ot the isoquant given that the isoquant is convex to the origin. In any other combination of various inputs on this isoquant TC is higher. For a given set of relative prices you can stress out the least cost combinations for increasing levels of output. Thus the exact combination of the input is where the isocost is tangent to the isoquant. For every level of output, X2 will decrease as X1 is increased.

For a small increase in the amount of X1 labor the change in output generated by a change in labor is referred to as MPP1. Like wise a small decrease in the amount of X2 land the change in the output will be —MPP2. But if total output remains constant those two changes must be equal.

Thus a change in X1 into MPP2. If we divide a change in change in X into MPP2. It follows from this ratio that the least cost combination occurs when the MPP per unit expenditure is the same for each unit e. If we multiply both sides be the same price PY, may not change the equality. This implies that; MPPX1. Profit will be maximized at this point. Profit will be maximized with the improved and increased use of technology. In some agricultural enterprises like water-short irrigated and rain fed areas a lot needs to be done to reduce water requirements and increase the productivity of the factors of production.

Researchers are developing water saving technologies such as alternate wetting and drying, continuous soil saturation Borell, et al,direct seed drying, ground cover systems Lin et al, and a system of rice intensification Stoop et al, ; but it is important to not that these systems have a loophole of prolonged periods of flooding and therefore water losses still remain high. In this system, rice is sown directly into dry soil and irrigation is given to keep the soil sufficiently moist for good plant growth, but the soil is never flooded Bouman, In many societies however, especially the tropics, agriculture tends to depend on weather.

Even when there is lot rain water during the dry weather, agriculture is greatly affected during drought. Irrigation is not used in these societies. Other notable technologies, such as in areas where farmers have been able to grow more than one crop as a result of direct seeding, the benefits have been even more pronounced Pandey and Velasco, Direct seeding offers such advantages as faster and easier sowing, reduced labor and less drudgery, earlier crop maturity, by days, more efficient water use, less methane emission and often higher profit in areas with assured water supply than that of transplanting Balasubramanian and Hill, It is reported in Brazil that high yields can be sustained when aerobic rice is grown once in four crops, but not under continuous mono cropping Guimaraes and Stone, Aerobic soil conditions and dry tillage practices, beside alternate wetting and drying conditions are conducive for germination and growth of highly competitive weeds, which causes grain yield losses of percent Singh et al, In the tropics, upland price grown under favorable conditions typically reaches maximum yields of just over 4 tones per hectare George et al, Technology therefore plays a big role to maximize productivity of the production factors.

Perfect subsititutes and complements. There are inputs which might be perfect substitutes e. In such cases it is rational to use one of the inputs which is cheaper than the other.

If compost manure is cheaper than fertilizers then compost manure should be used instead of fertilizers. Similary whichever cheaper labor of men or female should be used. This can be illustrated by the following figure. Similarly 3 units of female labor may substitute 2 units of male labor. Since male labor and female labor are considered as perfect substitutes the isoquants are straight lines.

If we assume that men are paid twice the hourly wages of women; women should be employed for weeding other than men. The relative cost of weeding by women is 1. And thus total cost is higher if men are used. In this case the isocost lines are never equal to price ratios.

The least cost method of production is therefore found at the end of an isoquant where it touches the lowest isocostline. All other points on AB lie on higher isocostlines and point B represent the most costly method of production. Isoquant diagram for two factors of production, x1 and x2 see text. Three isocost lines are shown, corresponding to variable costs amounting to v1, v2, and v3.

If units are to be produced, expenditure of v1 on variable factors will not suffice since the v1-isocost line never reaches the isoquant for units. An expenditure of v3 is more than sufficient; and v2 is the lowest variable cost for which units can be produced. It may be noted that the cheapest combination for the production of any quantity will be found at the point at which the relevant isoquant is tangent to an isocost line.

Thus, since the slope of an isoquant is given by the marginal rate of substitution, any firm trying to produce as cheaply as possible will always purchase or hire factors in quantities such that the marginal rate of substitution will equal the ratio of their prices. The isoquant—isocost diagram or the corresponding solution by the alternative means of the calculus solves the short-run cost minimization problem by determining the least-cost combination of variable factors that can produce a given output in a given plant.

The variable cost incurred when the least-cost combination of inputs is used in conjunction with a given outfit of fixed equipment is called the variable cost of that quantity of output and denoted VC y.

The total cost incurred, variable plus fixed, is the short-run cost of that output, denoted SRC y. Marginal cost Two other concepts now become important. The average variable cost, written AVC yis the variable cost per unit of output.

The marginal variable cost, or simply marginal cost [MC y ] is, roughly, the increase in variable cost incurred when output is increased by one unit; i. Though for theoretical purposes a more precise definition can be obtained by regarding VC y as a continuous function of output, this is not necessary in the present case.

The usual behaviour of average and marginal variable costs in response to changes in the level of output from a given fixed plant is shown in Figure 3. In this figure costs in dollars per unit are measured vertically and output in units per year is shown horizontally. The figure is drawn for some particular fixed plant, and it can be seen that average costs are fairly high for very low levels of output relative to the size of the plant, largely because there is not enough work to keep a well-balanced work force fully occupied.

People are either idle much of the time or shifting, expensively, from job to job. As output increases from a low level, average costs decline to a low plateau. But as the capacity of the plant is approached, the inefficiencies incident on plant congestion force average costs up quite rapidly. Overtime may be incurred, outmoded equipment and inexperienced hands may be called into use, there may not be time to take machinery off the line for routine maintenance; or minor breakdowns and delays may disrupt schedules seriously because of inadequate slack and reserves.

Thus the AVC curve has the flat-bottomed U-shape shown. Maximization of short-run profits The average and marginal cost curves just deduced are the keys to the solution of the second-level problem, the determination of the most profitable level of output to produce in a given plant. The only additional datum needed is the price of the product, say p0. The most profitable amount of output may be found by using these data.

If the marginal cost of any given output y is less than the price, sales revenues will increase more than costs if output is increased by one unit or even a few more ; and profits will rise. Contrariwise, if the marginal cost is greater than the price, profits will be increased by cutting back output by at least one unit.

This is the second basic finding: Such a conclusion is shown in Figure 3. Marginal cost and price The conclusion that marginal cost tends to equal price is important in that it shows how the quantity of output produced by a firm is influenced by the market price. At any higher market price, the firm will produce the quantity for which marginal cost equals that price. Thus the quantity that the firm will produce in response to any price can be found in Figure 3 by reading the marginal cost curve, and for this reason the marginal cost curve is said to be the short-run supply curve for the firm.

The short-run supply curve for a product—that is, the total amount that all the firms producing it will produce in response to any market price—follows immediately, and is seen to be the sum of the short-run supply curves or marginal cost curves, except when the price is below the bottoms of the average variable cost curves for some firms of all the firms in the industry.

This curve is of fundamental importance for economic analysis, for together with the demand curve for the product it determines the market price of the commodity and the amount that will be produced and purchased.

One pitfall must, however, be noted. In the demonstration of the supply curves for the firms, and hence of the industry, it was assumed that factor prices were fixed. Though this is fair enough for a single firm, the fact is that if all firms together attempt to increase their outputs in response to an increase in the price of the product, they are likely to bid up the prices of some or all of the factors of production that they use.

In that event the product supply curve as calculated will overstate the increase in output that will be elicited by an increase in price. A more sophisticated type of supply curve, incorporating induced changes in factor prices, is therefore necessary. Such curves are discussed in the standard literature of this subject. Marginal product It is now possible to derive the relationship between product prices and factor prices, which is the basis of the theory of income distribution.

To this end, the marginal product of a factor is defined as the amount that output would be increased if one more unit of the factor were employed, all other circumstances remaining the same. Algebraically, it may be expressed as the difference between the product of a given amount of the factor and the product when that factor is increased by an additional unit. The marginal products are closely related to the marginal rates of substitution previously defined.

It has already been shown that the marginal rate of substitution also equals the ratio of the prices of the factors, and it therefore follows that the prices or wages of the factors are proportional to their marginal products.

This is one of the most significant theoretical findings in economics. To restate it briefly: This is not a question of social equity but merely a consequence of the efforts of businessmen to produce as cheaply as possible. Further, the marginal products of the factors are closely related to marginal costs and, therefore, to product prices. This, also, is a fundamental theorem of income distribution and one of the most significant theorems in economics.

Its logic can be perceived directly. If the equality is violated for any factor, the businessman can increase his profits either by hiring units of the factor or by laying them off until the equality is satisfied, and presumably the businessman will do so. The theory of production decisions in the short run, as just outlined, leads to two conclusions of fundamental importance throughout the field of economics about the responses of business firms to the market prices of the commodities they produce and the factors of production they buy or hire: The first explains the supply curves of the commodities produced in an economy.

Though the conclusions were deduced within the context of a firm that uses two factors of production, they are clearly applicable in general. Maximization of long-run profits Relationship between the short run and the long run The theory of long-run profit-maximizing behaviour rests on the short-run theory that has just been presented but is considerably more complex because of two features: It is of the essence of long-run adjustments that they take place by the addition or dismantling of fixed productive capacity by both established firms and new or recently created firms.

At any one time an established firm with an existing plant will make its short-run decisions by comparing the ruling price of its commodity with cost curves corresponding to that plant.

If the price is so high that the firm is operating on the rising leg of its short-run cost curve, its marginal costs will be high—higher than its average costs—and it will be enjoying operating profits, as shown in Figure 3. The firm will then consider whether it could increase its profits by enlarging its plant. The effect of plant enlargement is to reduce the variable cost of producing high levels of output by reducing the strain on limited production facilities, at the expense of increasing the level of fixed costs.

In response to any level of output that it expects to continue for some time, the firm will desire and eventually acquire the fixed plant for which the short-run costs of that level of output are as low as possible. This leads to the concept of the long-run cost curve: These result from balancing the fixed costs entailed by any plant against the short-run costs of producing in that plant.

The long-run costs of producing y are denoted by LRC y. The marginal long-run cost is the increase in long-run cost resulting from an increase of one unit in the level of output.

It represents a combination of short-run and long-run adjustments to a slight increase in the rate of output. It can be shown that the long-run marginal cost equals the marginal cost as previously defined when the cost-minimizing fixed plant is used.

Long-run cost curves Cost curves appropriate for long-run analysis are more varied in shape than short-run cost curves and fall into three broad classes.