Theory of Production CH-3 PDF

CHAPTER THREE THEORY OF PRODUCTION LESSON STRUCTURE 3.1 Introduction 3.2 Chapter Objectives 3.3 The Production Function 3.4 The Short Run Production Function and Stages of Production 3.5 Laws of Production 3.6 Returns to Scale and Homogeneity of the Production Function 3.7...

CHAPTER THREE THEORY OF PRODUCTION LESSON STRUCTURE 3.1 Introduction 3.2 Chapter Objectives 3.3 The Production Function 3.4 The Short Run Production Function and Stages of Production 3.5 Laws of Production 3.6 Returns to Scale and Homogeneity of the Production Function 3.7 Equilibrium of the Firm: Choice of Optimal Combination of Factors of Production 3.8 Lesson Summary 3.9 Review Questions 3.1 INTRODUCTION Production is the process of conversion of inputs (factors of production) into a consumable form (goods and services). In this regard, in the production process or activity, firms turn inputs into output. This transformation of inputs (factors of production) into output is defined at a particular time period and at a given technology. By technology we mean the state of knowledge about the various methods that might be used to transform inputs into outputs, and it is described by a production function. 3.2 CHAPTER OBJECTIVES After studying this lesson thoroughly, you would be able to: ¾ understand the context of the production function ¾ illustrate the short-run production function and stages of production 103  ¾ understand and explain the laws of production ¾ explain the returns to scale and homogeneity of the production function ¾ have a knowledge of equilibrium theory of firms and optimum combination of factors of production. 3.3 THE PRODUCTION FUNCTION The production function is a function that shows the highest output that a firm can produce for every specified combination of inputs. It is a purely technical relation which connects factor inputs to outputs. Assuming labor (L) and capital (K) as the only inputs, the production function can be written as: Q = f(L, K); where Q stands for the total quantity produced of an output/product. The production function allows inputs to be combined in varying proportions so that output can be produced in many ways (say, using either more capital and less labor, or more labor and less capital). For example, a unit of commodity X may be produced by the following processes: Table 3.1: Three Processes for Producing a Unit of X Process 1 (P1) Process 2 (P2) Process 3 (P3) Units of Labor 2 3 1 Units of Capital 3 2 4 These activities or methods of production can be shown by lines from the origin to the point determined by the labor and capital inputs combination. 104  K 3 P1 P2 2 1 P3 0 2 3 4 L Figure 3.1: Alternative Production Processes The production function, which is a purely technical relationship that connects factor inputs and outputs, includes all the technically efficient methods of production. The technically inefficient methods are not included in the production function. A method of production ‘A’ is technically efficient if it uses less of at least one input and no more of the other factors to produce a given level of output as compared with any other method ‘B’. For example, suppose commodity Y can be produced by two methods (Method A and Method B) as shown below: Method A Method B Labor 2 3 Capital 3 3 If these are considered to be the only methods of production, method A is considered as technically the efficient method. This is because the two methods, A and B, use the same amount of capital (3 each), but method A uses less units of labor (2) than B does (3). 105 The basic theory of production concentrates only on the efficient methods, and thus inefficient methods are excluded as a rational producer will not used them. If a process A uses less of one factor (say, L) and more of another (say, K) as compared to B, then A and B cannot be directly compared on the criterion of technical efficiency. For example, the two activities in the table below are not directly comparable. Method A Method B Labor 2 1 Capital 3 4 In such cases, both processes are considered as technically efficient and included in the production function. Which one of them will be chosen at any particular time depends on the price of factors (inputs). The choice of any particular technique among the set of technically efficient processes is an economic one, which is based on the price of factors of production. Note that a technically efficient method is not necessarily economically efficient. Isoquants and an Isoquant Map In addition to defining the production function mathematically, it is also common to depict the technically efficient production processes with the help of isoquants. Assuming that labor (L) and capital (K) are the only two inputs used to produce an item, the output achievable for various combinations of inputs can be shown by using isoquants. An isoquant: is the locus of all the technically efficient methods (or all the technically efficient combinations of factors of production) for producing a given level of output. It is a curve showing all the possible combinations of inputs that yield the same level of output. Isoquants may assume different shapes depending on the degree of substitutability between the factors of production. The following are the common ones: 106 1. Linear isoquant: this type assumes perfect substitutability of factors: a given output may be produced by using only labor, or only capital, or by an infinite number of combinations of K and L. See Panel (A) of Figure 3.2 below. 2. Input-output isoquant: this assumes strict complementarily (i.e., zero substitutability) of the factors of production. There is only one method of production for producing any particular level of a commodity. The isoquant takes the shape of a right-angle. This type of isoquant is also called “Leontief Isoquant” after the name Leontief who invented the input output analysis. Panel (B) of the figure below depicts such isoquants. 3. Kinked isoquant: this assumes limited substitutability between factors of production, say K and L, and that there are only few processes for producing a particular amount of a commodity. Substitutability of the factors is possible only at the kinks. See Panel (C) of Figure 3.2 below. 4. Smooth or convex isoquant: this form assumes a continuous (and a less than perfect) substitutability between factors (K and L) only over a certain range, beyond which factors cannot substitute each other. The isoquant is a smooth curve which is convex to the origin. This is depicted in Panel (D) of the figure below. Even though the kinked isoquant is more realistic, most of the time the smooth or convex isoquant is used in the traditional economic theory because it is mathematically simpler to handle by the simple rules of calculus. 107  K K X (Level of Output) X O L O L Panel (A): A Linear Isoquant Panel (B): A Leontief Isoquant K P1 P2 X P3 X P4 O L O L Panel (C): A Kinked Isoquant Panel (D): A Convex Isoquant Figure 3.2: Isoquants of Different Shapes An Isoquant map: is simply a set of several isoquants. An isoquant map is another way of describing a production function, just as an indifference map (discussed in Chapter Two) is a way of describing a utility function. The level of output increases as we move upward to the right where as it remains constant along an isoquant (See points A, B and C in the 108 figure below; 100 units of good X are produced both at A and C while 50 units are produced at B). K A C B X = 100 X = 50 O L Figure 3.3: Movement on an Isoquant versus Movement from an Isoquant to Another Check Your Progress 1. What is meant by production function? What is the use of production function in production analysis? 2. Explain some important/common types of production function. 3. What are isoquants? Explain their main properties. 4. What are the differences between isoquant curves and indifference curves? 5. What is the reason behind an isoquant curve that is convex to the origin? When will an isoquant be straight-line, and when will it be right-angled? 3.4 THE SHORT RUN PRODUCTION FUNCTION AND STAGES OF PRODUCTION The production function in the traditional theory generally assumes the form: X = f(L, K, r, y) 109 Where L is labor, K is capital, r is returns to scale which refers to the long run analysis of the laws of production since it assumes change in the plant, and y is the efficiency parameter related to the organizational and entrepreneurial aspect of the production. We usually abstract from the availability of many factors of production to two factors of production (L and K) only in order to simplify things. In this simplified case, any change in the amount of factors other than L and K is considered to shift a production function. The short run production function and its behavior for a change in amount of the fixed factors (as time goes to the long run) can be shown graphically, as follows: X X’’ = f(L)k3,r3,y3 X’ = f(L)k2,r2,y2 X = f(L)k1,r1,y1 O L Figure 3.4: The Short Run Production Function and How It Shifts as the Amount of Fixed Factors of Production Change with the Passage of Time In Figure 3.4, the quantity of output X produced is drawn as a function of the amount of labor, for fixed amounts of the other factors. For a given curve (X), as labor increases, ceteris paribus (the others factors fixed at k1, r1 and y1), output increases and we move along the curve depicting the production function. If any one or all of the fixed factors (K, r, y) increases, the production function shifts upwards. If, for instance, the levels of the three fixed inputs rise from (k1, r1 and y1) to (k2, r2 and y2), the curve X shifts upward to X’’, and so on. 110 The time period for which we assume that some factors are fixed in amount is called the short run. Thus, curve X in the figure above is drawn for the short run. If we could increase (or change in general) the amount of all factors, then we are in a long run. The slope of the production function (say, X = f(L)) is the marginal product of the factor of production L (MPL). Similarly, the slope of the production function X = f(K) is the marginal product of capital (MPK). The marginal product of a factor is defined as the change in output resulting from the change in the factor by a unit, keeping all other factors constant. That is: ∂X ∂X MPL = and MPK =. ∂L ∂K Graphically, the MPL is shown by the slope of the production function X = f(L) and the MPK is shown by the slope of the production function X = f(K). As you remember from chapter two, the slope of a curve at any one point is the slope of a tangent line at that point. The average product of an input is the total product divided by the units of the input used to produce it. Graphically, the average product of a factor at a given point is given by the slope of a straight line from the origin to the point. Let’s derive the average product and marginal product of labor from the total product of labor graphically. By doing so, we will also distinguish among three stages of production. As shown in Panel (A) of Figure 3.5, as the units of labor used in the production process goes on increasing, the output initially increases at an increasing rate (up to point A), then rises at a decreasing rate (from point A to point C), reaches a maximum (at point C), and then starts falling. As a result, since marginal product is the slope of the total product curve, the marginal product of labor initially increases, reaches maximum, and then starts declining. The marginal product of labor (MPL) is even negative when the total product declines (beyond 111 point C). The average product of labor (APL), which is the slope of the line drawn from the origin to the corresponding point on the total product (TPL) curve, initially increases, reaches maximum (at point Z) and then starts declining. The APL and MPL curves are shown in Panel (B) of the same figure. The following points are clearly reflected in Figure 3.5 below: # Before point Z is reached, in Panel A of the figure, the slope of a tangent line at a point on the TPL curve is greater than the slope of a line from the origin to the point. In other words, the MPL is above the APL. # At point Z, where the APL reaches its maximum, the slope of a tangent line at a point on the TPL curve is greater than the slope of a line from the origin to the point. That is, APL and MPL are equal at the maximum of the APL (Panel B of Figure 3.5). # When TPL curve reaches its maximum (point C in Panel A), the MPL equals zero. 112  X C D B Z TP = f(L) A Panel (A): The TPL Curve O LA LZ LB LC LD L MPL APL A’ Z’ Panel (A): The MPL and APL Curves C’ APL O LZ’ LC’ L MPL Figure 3.5: The Relationship among TPL, MPL and APL Now, let us study the three stages of production. Figure 3.6 below is partly reproduced from Panel B of the above figure to assist us to this end. Accordingly, we divide this production function into three stages as: Stage I (from zero TPL up to the maximum of APL), Stage II (from the maximum of APL to zero MPL), and Stage III (from zero MPL onwards). 113  MPL APL Stage I Stage II Stage III Z’ C’ APL O LZ’ LC’ L MPL Figure 3.6: The Three Stages of Production At stage I, MPL > APL, and both of them are rising initially while MPL falls latter on. Since each additional unit of labor is coming up with a contribution larger than the average (MPL > APL), it is rational to hire more labor and produce more output. Thus, it is not reasonable to produce at this stage. A rational producer (firm) observes that using more labor is preferred to the existing situation and thus moves out of this stage. At the third stage, where both APL and MPL are declining and MPL < APL, it is not rational to produce at all because each additional unit of labor makes the total product to decline (i.e. its contribution is negative). Thus, it is in the second stage that a rational firm operates. Here each additional labor contributes positively to the production but less than the average. Where exactly in this stage does a rational firm produce? The answer is, it depends on factor prices. At this stage as the use of a variable input (labor) increases with other inputs (like capital) being fixed, the resulting additions to output (MPL) will eventually decrease. This manner is captured by a principle known as the law of variable proportions or the law of diminishing marginal returns. 114 In summary, the production theories concentrate only on the efficient part of the production function, that is, on the ranges of output over which the marginal productivities are positive but declining. The second stage of production in the above analysis corresponds to this efficient stage in the short run. No rational firm would employ labor less than OLZ’ or beyond OLC’ (in Figure 3.6). This means over the range ∂ ( MPL ) where MPL > 0 but 1, then m2 > m. This implies that our new production has increased by more than m. so we have increasing returns to scale. 3. Q = K0.3L0.2. Again we put in our multipliers and create our new production function. 122  Q* = (mK)0.3(mL)0.2 = K0.3L0.2m0.5 = Qm0.5. Since m > 1, then m0.5 < m. So we have decreasing returns to scale. 3.6 RETURNS TO SCALE AND HOMOGENEITY OF THE PRODUCTION FUNCTION Suppose we increase both factors of production in the function X = f(L,K) by the same proportion m, and we observe the resulting new level of output X* as X* = f(mK,mL). If m can be factored out (that is, can be taken out of the bracket as a common factor), then the new level of output can be expressed as a function of m (to the power n) and the initial level of output X as follows: X* = mnf(L,K) or X* = mnX. If so, the function is called homogeneous. If m cannot be factored out, the production function is called non- homogeneous. The above three examples are homogeneous functions since m can be factored out in each case. Thus, a homogeneous function is a function such that if each of the inputs is multiplied by m, then m can be completely factored out of the function. The power n of m is called the degree of homogeneity and is a measure of the returns to scale. ¾ If n = 1, we have a constant returns to scale. ¾ If n < 1, we have a decreasing returns to scale. ¾ If n > 1, we have an increasing returns to scale. Given a Cobb-Douglas production function X = ALbKc, returns to scale is measured by the sum of the powers of the factors. That is, ¾ If b + c = 1, then there is a constant returns to scale. ¾ If b + c > 1, then there is an increasing returns to scale. ¾ If b + c < 1, then there is a decreasing returns to scale. Proof Let L and K increases by m. The new level of output is 123  X* = A(mL)b(mK)c = AmbLbmcKc = Amb+cLbKc = mb+c(ALbKc) X* = mb+c(X) This implies the function is homogeneous of degree b+c and the type of the returns to scale depends on the sum. Product Line: It shows a physical movement from one isoquant to another as we change either both factors and a single factor. It describes the technically possible alternative paths of expanding output. What path will actually chosen by the firm will depend on the prices of factors. The product curve passes through the origin if both factors are variable. But if only one factor is variable (the other being kept constant), the product line is a straight line parallel to the axis of the variable factor. K K K Product Lines Product Lines Product Line K O L O L O L Panel (A): Product Lines Panel (B): Product Lines for Panel (C): A Product Line for a Homogenous Function a Non-Homogenous Function where K is Fixed Figure 3.9: Different Kinds of Product Lines 124 A special type of product line which is the locus of points of different isoquants at which the MRTS of factors is constant is called an isocline. For homogeneous production functions, the isoclines are straight lines through the origin. In such a case, the K/L ratio is constant along any isocline (refer to the Panel A of Figure 3.9). Check Your Progress 1. What is the law of variable proportions? How does it differ from the laws of returns to scale? 2. How is the degree of homogeneity of a production function related to the returns to scale of the production function? 3.7 EQUILIBRIUM OF THE FIRM: CHOICE OF OPTIMAL COMBINATION OF FACTORS OF PRODUCTION A firm is said to be in equilibrium when it employs those levels of inputs that will maximize its profit. This means the goal of the firm is profit maximization (maximizing the difference between revenue and cost). Thus the problem facing the firm is that of constrained profit maximization, which may take one of the following forms: I. Maximizing profit subject to a cost constraint. In this case, total cost and prices are given and the problem may be stated as follows: Max П = R – C = PxX – C Clearly maximization of П (profit) is achieved in this case if X (quantity of output) is maximized, since C (cost) and Px (price of the product) are constants. II. Maximize profit for a given level of output. Max П = R – C = PxX – C Clearly in this case maximization of profit is achieved by minimizing cost, since X and Px are given. To derive the equilibrium of the firm graphically, we will use the isoquant map and the isocost lines. As discussed earlier, an isoquant is a curve that shows the various 125 combinations of K and L that will give the same level of output. It is convex to the origin whose slope is defined as: ∂X − dK = ∂L = MPL dL ∂X MPK ∂K The isocost line is defined by the cost equation: C = rK + wL; where w = wage rate, and r = price of capital services. The isocost line is the locus of all combinations of factors that the firm can purchase with a given monetary cost or outlay. The slope of the isocost line is equal to the ratio of the w prices of the factors of production in absolute terms, −. r From the isocost equation given by: C = wL + rK => rK = C - wL C w => K = − L. r r w From this the slope is − r K C r O C L w Figure 3.10: An Isocost Line Now, let us see how the equilibrium of the firm is determined in the two cases mentioned above. 126 Case 1: Maximization of Output Subject to a Cost Constraint Given the level of cost and the price of the factors and output, the firm will be in equilibrium when it maximizes the quantity of output it produces. This is at the point of tangency of the isocost line to the highest possible isoquant curve. In the following graph (Figure 3.11), the equilibrium of the firm is obtained at point e, where the firm produces X2 with K1 and L1 units of the two inputs. Higher levels of output (to the right of e) are desirable but not attainable due to the cost constraint. Other points below the isocost line lie on a lower isoquant than X2. Hence X2 is the maximum output that can be achieved given the above assumptions (C, w, r and Px being constant). K K1 e X3 X2 X1 O L1 L Figure 3.11: Maximizing Output subject to Cost At the point of tangency: a. Slope of isoquant = slope of isocost w MPL − = = MRTS LK. This is a necessary condition for profit maximization. r MPK b. The isoquant is convex to the origin. This is the sufficient condition for profit maximization. 127 The mathematical derivation of the above equilibrium condition is as follows. A rational producer seeks the maximization of its output, given total cost outlay and the prices of factors. That is, Maximize X = f (K, L) subject to C = wL + rK This is a constrained optimization which can be solved by using the Lagrangean method. The steps are: a. Rewrite the constraint in the form: wL + rK – C = 0 b. Multiply the constraint by a constant λwhich is the Lagrangean multiplier: λ(wL + rK – C) = 0 c. Form the composite function: Z = X – λ(wL + rK – C) d. Partially differentiate the function with respect to each factor as well as the multiplier, and then equate to zero. ∂Z ∂X * = − λw = 0 ∂L ∂L Ö MPL = λw MPL Ö λ= …………………………………………………………………… (1) w ∂Z ∂X * = − λr = 0 ∂K ∂K Ö MPK = λr MPK Ö λ= …………………………………………………………………… (2) r ∂Z * = wL + rK − C = 0 ∂λ Ö wL + rK = C ……………………………………………………………… (3) MPL MPK w MPL From equations (1) and (2) we understand that: = or = = MRTS LK w r r MPK 128 This shows that the firm is in equilibrium when it equates the ratio of the marginal productivity of each factor to its price. It can be shown that the second order conditions for the equilibrium of the firm require that the marginal product curves of the two factors have a negative slope. ∂ ( MPL ) ∂ 2 X Slope of the MPL = = < 0, ∂L ∂L2 ∂ ( MPK ) ∂ 2 X Slope of the MPK = = < 0, and ∂K ∂K 2 ∂2 X ∂2 X ∂2 X 2.