Chap IV Theories of Production and Cost (2).pptx
- 1. Chapter Four: The Theory of Production and Cost This chapter has two major sections: The first part will introduce you to the basic concepts of production and production function, classification of inputs, essential features of short run production functions and the stages of short run production. The second part mainly deals with the difference between economic cost and accounting cost, the characteristics of short run cost functions, and the relationship between short run production functions and short run cost functions.
- 2. 4.1 Theory of production in the short run 4.1.1 Definition of production Raw materials yield less satisfaction to the consumer by themselves. For a better utility, raw materials has to be transformed into outputs. However, transforming raw materials into outputs requires inputs such as land, labour, capital and entrepreneurial ability. Production is the process of transforming inputs into outputs. The end products of the production process are outputs which could be tangible (goods) or intangible (services). 4.1.2 Production function Production function is a technical relationship between inputs and outputs. It shows the maximum output that can be produced with fixed amount of inputs and the existing technology.
- 3. A production function may take the form of an algebraic equation, table or graph. A general equation for production function can, for instance, be described as: 𝑸 = 𝒇(𝑿𝟏, 𝑿𝟐, 𝑿𝟑, … , 𝑿𝒏) Where, Q is output and 𝑿𝟏, 𝑿𝟐, 𝑿𝟑, … , 𝑿𝒏) are different types of inputs. Inputs are commonly classified as fixed inputs or variable inputs. Fixed inputs are those inputs whose quantity cannot readily be changed when market conditions indicate that an immediate adjustment in output is required. In fact, no input is ever absolutely fixed but may be fixed during an immediate requirement. Examples: Buildings, land and machineries etc. Their quantity cannot be manipulated easily in a short period of time.
- 4. Variable inputs are those inputs whose quantity can be altered almost instantaneously in response to desired changes in output. That is, their quantities can easily be diminished when the market demand for the product decreases and vice versa. The best example of variable input is unskilled labour. In economics, short run refers to a period of time in which the quantity of at least one input is fixed. In other words, short run is a time period which is not sufficient to change the quantities of all inputs so that at least one input remains fixed. Note: short run periods of different firms have different durations. Some firms can change the quantity of all their inputs within a month while it takes more than a year for other types of firms.
- 5. Consider a firm that uses two inputs: capital (fixed input) and labour (variable input). Given the assumptions of short run production, the firm can increase output only by increasing the amount of labour it uses. Hence, its production function can be given by: 𝑸 = 𝒇(𝑳) where, Q is output and L is the quantity of labour. The production function shows different levels of output that the firm can produce by efficiently utilizing different units of labour and the fixed capital. In the above short run production function, the quantity of capital is fixed. Thus, output can change only when the amount of labour changes.
- 6. 4.1.3 Total, average, and marginal product In production, the contribution of a variable input can be described in terms of total, average and marginal product. A) Total product (TP) it is the total amount of output that can be produced by efficiently utilizing specific combinations of the variable input and fixed input. Increasing the variable input (while some other inputs are fixed) can increase the total product only up to a certain point. Initially, as we combine more and more units of the variable input with the fixed input, output continues to increase. But eventually, if we employ more and more unit of the variable input beyond the carrying capacity of the fixed input, output tends to decline.
- 7. In general, the TP function in the short-run follows a certain trend: it initially increases at an increasing rate, then increases at a decreasing rate, reaches a maximum point and eventually falls as the quantity of the variable input rises. This tells us what shape a total product curve assumes. B) Marginal Product (MP) it is the change in output attributed to the addition of one unit of the variable input to the production process, other inputs being constant. For instance, the change in total output resulting from employing additional worker (holding other inputs constant) is the marginal product of labour (MPL). In other words, MPL measures the slope of the total product curve at a given point. 𝑴𝑷𝑳 = ∆𝑸 ∆𝑳 = 𝒅𝑻𝑷 𝒅𝑳
- 8. In the short run, the marginal product of the variable input first increases, reaches its maximum and then decreases to the extent of being negative. That is, as we continue to combine more and more of the variable input with the fixed input, the marginal product of the variable input increases initially and then declines. C) Average Product (AP) is the level of output that each unit of input produces, on the average. It tells us the mean contribution of each variable input to the total product. Mathematically, the average product of labour (APL), for instance, is given by: 𝑨𝑷𝑳 = 𝑻𝑷 𝑳 𝑨𝑷𝑳 first increases, reaches its maximum value and eventually declines.
- 10. The AP curve can be measured by the slope of rays originating from the origin to a point on the TP curve (see figure 4.1). For example, the 𝑨𝑷𝑳 at 𝑳𝟐 is the ratio of 𝑻𝑷𝟐 to 𝑳𝟐 = Slope of ray a. The RELATIONSHIP between 𝑴𝑷𝑳 and 𝑨𝑷𝑳 can be stated as follows. When APL is increasing, 𝑴𝑷𝑳 >𝑨𝑷𝑳. When APL is at its maximum, 𝑴𝑷𝑳 = 𝑨𝑷𝑳. When APL is decreasing, 𝑴𝑷𝑳 <𝑨𝑷𝑳. Example: Suppose that the short-run production function of certain cut-flower firm is given by: 𝑸 = 𝟒𝑲𝑳 − 𝟎. 𝟔𝑲𝟐 − 𝟎. 𝟏𝑳𝟐, where, Q is quantity of cut- flower produced, L is labour input and K is fixed capital input (K=5). a) Determine the 𝑨𝑷𝑳 function. b) At what level of labour does the total output of cut-flower reach the maximum? c) What will be the maximum achievable amount of cut-flower production?
- 11. Solution: a)𝑨𝑷𝑳 = 𝑸 𝑳 = 𝟒𝑲𝑳 −𝟎.𝟔𝑲𝟐 −𝟎.𝟏𝑳𝟐 𝑳 = 𝟒𝑲 − 𝟎.𝟔𝑲𝟐 𝑳 − 𝟎. 𝟏𝑳 = 𝟐𝟎 − 𝟏𝟓 𝑳 − 𝟎. 𝟏𝐋 = 𝟐𝟎𝑳−𝟏𝟓−𝟎.𝟏𝑳𝟐 𝑳 b)The total product level(Max Q) occurs when 𝑴𝑷𝑳 = 0. 𝑴𝑷𝑳 = 𝑑𝑇𝑃 𝑑𝐿 = 𝑑(20𝐿 − 15 − 0.1𝑳𝟐) 𝑑𝐿 = 20 − 0.2𝐿 = 0 0.2𝐿 = 20; 𝑳∗ = 𝟏𝟎𝟎 Hence, total output will be the maximum when 100 workers are employed. c) Substituting the optimal values of labor (L=100) and capital (K=5) into the original production function (Q): 𝑸𝒎𝒂𝒙 = 𝟒𝑲𝐿∗ − 𝟎. 𝟔𝑲𝟐 − 𝟎. 𝟏𝐿∗𝟐 = 𝟒 𝟓 100 ∗ − 𝟎. 𝟔 𝟓 𝟐 − 𝟎. 𝟏 100 ∗𝟐 = 𝟐𝟎𝟎𝟎 − 𝟏𝟓 − 𝟏𝟎𝟎𝟎 = 𝟗𝟖𝟓 𝑸𝒎𝒂𝒙 = 𝟗𝟖𝟓
- 12. 4.1.4 The law of variable proportions/LDMR The law of variable proportions states that as successive units of a variable input(say, labour) are added to a fixed input (say, capital or land), beyond some point the extra, or marginal, product that can be attributed to each additional unit of the variable resource will decline. For example, if additional workers are hired to work with a constant amount of capital equipment, output will eventually rise by smaller and smaller amounts as more workers are hired. Assumptions of LDMR technology is fixed and thus the techniques of production do not change. all units of labour are assumed to be of equal quality. Each successive worker is presumed to have the same innate ability, education, training, and work experience.
- 13. Marginal product ultimately diminishes not because successive workers are less skilled or less energetic rather it is because more workers are being used relative to the amount of plant and equipment available. The law starts to operate after the marginal product curve reaches its maximum (this happens when the number of workers exceeds 𝑳𝟏 in figure 4.1). This law is also called the law of diminishing marginal returns(LDMR). 4.1.5 Stages of production We are not in a position to determine the specific number of the variable input (labour) that the firm should employ because this depends on several other factors than the productivity of labour. However, it is possible to determine the ranges over which the variable input (labour) be employed. To this end, economists have defined three stages of short run production.
- 14. Stage I: This stage of production covers the range of variable input levels over which the average product (APL) continues to increase. It goes from the origin to the point where the APL is maximum, which is the equality of MPL and APL (up to 𝑳𝟐 level of labour employment in figure 4.1). This stage is not an efficient region of production though the MP of variable input is positive. The reason is that the variable input (the number of workers) is too small to efficiently run the fixed input so that the fixed input is under- utilized (not efficiently utilized). Stage II: It ranges from the point where 𝑨𝑷𝑳 is at its maximum (MPL=APL) to the point where MPL is zero (from 𝑳𝟐 to 𝑳𝟑 in figure 4.1). Here, as the labour input increases by one unit, output still increases but at a decreasing rate.
- 15. Due to this, the second stage of production is termed as the stage of diminishing marginal returns. The reason for decreasing average and marginal products is due to the scarcity of the fixed factor. That is, once the optimum capital-labour combination is achieved, employment of additional unit of the variable input will cause the output to increase at a slower rate. As a result, the marginal product diminishes. This stage is the efficient region of production. Additional inputs are contributing positively to the total product and MP of successive units of variable input is declining (indicating that the fixed input is being optimally used). Hence, the efficient region of production is where the marginal product of the variable input is declining but positive.
- 16. Stage III: In this stage, an increase in the variable input is accompanied by decline in the total product. Thus, the total product curve slopes downwards, and the marginal product of labour becomes negative. This stage is also known as the stage of negative marginal returns to the variable input. The cause of negative marginal returns is the fact that the volume of the variable inputs is quite excessive relative to the fixed input; the fixed input is over-utilized. Obviously, a rational firm should not operate in stage III because additional units of variable input are contributing negatively to the total product (MP of the variable input is negative). In figure 4.1, this stage is indicated by the employment of labour beyond 𝑳𝟑.
- 17. 4.2 Theory of costs in the short run 4.2.1 Definition and types of costs To produce goods and services, firms need factors of production or simply inputs. To acquire these inputs, they have to buy them from resource suppliers. Cost is, therefore, the monetary value of inputs used in the production of an item. Economists use the term ―profit differently from the way accountants use it. To the accountant, profit is the firm‘s total revenue less its explicit costs (accounting costs). To the economist, profit is total revenue less economic costs (explicit and implicit costs).
- 18. Accounting cost is the monetary value of all purchased inputs used in production; it ignores the cost of non-purchased (self-owned) inputs. It considers only direct expenses such as wages/salaries, cost of raw materials, depreciation allowances, interest on borrowed funds and utility expenses (electricity, water, telephone, etc.). These costs are said to be explicit costs. Explicit costs are out of pocket expenses for the purchased inputs. If a producer calculates her cost by considering only the costs incurred for purchased inputs, then her profit will be an accounting profit. 𝑨𝒄𝒄𝒐𝒖𝒏𝒕𝒊𝒏𝒈 𝒑𝒓𝒐𝒇𝒊𝒕 = 𝑻𝒐𝒕𝒂𝒍 𝒓𝒆𝒗𝒆𝒏𝒖𝒆 – 𝑨𝒄𝒄𝒐𝒖𝒏𝒕𝒊𝒏𝒈 𝒄𝒐𝒔𝒕 = 𝑻𝒐𝒕𝒂𝒍 𝒓𝒆𝒗𝒆𝒏𝒖𝒆 – 𝑬𝒙𝒑𝒍𝒊𝒄𝒊𝒕 𝒄𝒐𝒔𝒕 In the real world economy, entrepreneurs may use some resources which may not have direct monetary expense since the entrepreneur can own these inputs himself or herself.
- 19. Economic cost of producing a commodity considers the monetary value of all inputs (purchased and non-purchased). Calculating economic costs will be difficult since there are no direct monetary expenses for non-purchased inputs. The monetary value of these inputs is obtained by estimating their opportunity costs in monetary terms. The estimated monetary cost for non-purchased inputs is known as implicit cost. Example: if Mr. X quits a job which pays him Birr 10, 000.00 per month in order to run a firm he has established, then the opportunity cost of his labour is taken to be Birr 10,000.00 per month (the salary he has forgone in order to run his own business). Therefore, economic cost is given by the sum of implicit cost and explicit cost.
- 20. 𝑬𝒄𝒐𝒏𝒐𝒎𝒊𝒄 𝒑𝒓𝒐𝒇𝒊𝒕 = 𝑻𝒐𝒕𝒂𝒍 𝒓𝒆𝒗𝒆𝒏𝒖𝒆 – 𝑬𝒄𝒐𝒏𝒐𝒎𝒊𝒄 𝒄𝒐𝒔𝒕 (𝑬𝒙𝒑𝒍𝒊𝒄𝒊𝒕 𝒄𝒐𝒔𝒕 + 𝑰𝒎𝒑𝒍𝒊𝒄𝒊𝒕 𝒄𝒐𝒔𝒕) Economic profit will give the real profit of the firm since all costs are taken into account. Accounting profit of a firm will be greater than economic profit by the amount of implicit cost. If all inputs are purchased from the market, accounting and economic profit will be the same. However, if implicit costs exist, then accounting profit will be larger than economic profit. 4.2.2 Total, average and marginal costs in the short run A cost function shows the total cost of producing a given level of output; can be described using equations, tables or curves. A cost function can be represented using an equation as follows: 𝑪 = 𝒇 (𝑸), where C is the total cost of production and Q is the level of output.
- 21. In the short run, total cost (TC) can be broken down in to two – total fixed cost (TFC) and total variable cost (TVC). By fixed costs we mean costs which do not vary with the level of output. They are regarded as fixed because these costs are unavoidable regardless of the level of output. The firm can avoid fixed costs only if he/she stops operation (shuts down the business). The fixed costs may include salaries of administrative staff, expenses for building depreciation and repairs, expenses for land maintenance and the rent of building used for production. Variable costs, on the other hand, include all costs which directly vary with the level of output. if the firm produces zero output, the variable cost is zero. Example: cost of raw materials, the cost of direct labour and the running expenses of fuel, water, electricity, etc.
- 22. In general, the short run total cost is given by the sum of total fixed cost and total variable cost. That is, TC = TFC + TVC Based on the definition of the short run cost functions, let’s see what their shapes look like. A) Total fixed cost (TFC): Total fixed cost is denoted by a straight line parallel to the output axis. This is because such costs do not vary with the level of output. B) Total variable cost (TVC): The total variable cost of a firm has an inverse S-shape. The shape indicates the law of variable proportions in production. At the initial stage of production with a given plant, as more of the variable factor is employed, its productivity increases. Hence, the TVC increases at a decreasing rate.
- 23. This continues until the optimal combination of the fixed and variable factor is reached. Beyond this point, as increased quantities of the variable factor are combined with the fixed factor, the productivity of the variable factor declines, and the TVC increases at an increasing rate. C) Total Cost (TC): The total cost curve is obtained by vertically adding TFC and TVC at each level of output. The shape of the TC curve follows the shape of the TVC curve, i.e. the TC has also an inverse S-shape. It should be noted that when the level of output is zero, TVC is also zero which implies 𝑻𝑪 = 𝑻𝑭𝑪.
- 25. Per unit costs From total costs functions we can derive per-unit costs. These are even more important in the short run analysis of the firm. a) Average Fixed Cost (AFC) is total fixed cost per unit of output. It is calculated by dividing TFC by the corresponding level of output. The curve declines continuously and approaches both axes asymptotically. 𝐀𝐅𝐂 = 𝑻𝑭𝑪 𝑸 b) Average Variable Cost (AVC) Average variable cost is total variable cost per unit of output. It is obtained by dividing total variable cost by the level of output. 𝐀𝐕𝐂 = 𝑻𝑽𝑪 𝑸
- 26. The short run AVC falls initially, reaches its minimum, and then starts to increase. Hence, the AVC curve has U-shape and the reason behind is the law of variable proportions. c) Average total cost (ATC) or (AC) is the total cost per unit of output. It is calculated by dividing the total cost by the level of output. 𝐀𝐂 = 𝑻𝑪 𝑸 Equivalently, 𝐀𝐕𝐂 = 𝑻𝑭𝑪+𝑻𝑽𝑪 𝑸 = 𝑨𝑭𝑪 + 𝑨𝑽𝑪 Thus, AC can also be given by the vertical sum of AVC and AFC.
- 27. D) Marginal Cost (MC) Marginal cost is defined as the additional cost that a firm incurs to produce one extra unit of output. In other words, it is the change in total cost which results from a unit change in output. Graphically, MC is the slope of TC function. 𝐌𝐂 = 𝒅𝑻𝑪 𝒅𝑸 In fact, MC is also a change in TVC with respect to a unit change in the level of output. M𝐂 = 𝒅𝑻𝑽𝑪+ 𝒅𝑻𝑭𝑪 𝑸 = 𝒅𝑻𝑽𝑪 𝒅𝑸 , since 𝒅𝑻𝑭𝑪 𝒅𝑸 = 𝟎 Given inverse S-shaped TC and TVC curves, MC initially decreases, reaches its minimum and then starts to rise. From this, we can infer that the reason for the MC to exhibit U-shape is also the law of variable proportions.
- 28. In summary, AVC, AC and MC curves are all U-shaped due to the law of variable proportions.
- 29. In the above figure, the AVC curve reaches its minimum point at 𝑸𝟏 level of output and AC reaches its minimum point at 𝑸𝟐 level of output. The vertical distance between AC and AVC, that is, AFC decreases continuously as output increases. It can also be noted that the MC curve passes through the minimum points of both AVC and AC curves. Example: Suppose the short run cost function of a firm is given by: TC = 2𝑸𝟑 –2𝑸𝟐 + Q + 10 a) Find the expression of TFC & TVC b) Derive the expressions of AFC, AVC, AC and MC c) Find the levels of output that minimize MC and AVC and then find the minimum values of MC and AVC
- 30. Solution: Given 2𝑸𝟑 –2𝑸𝟐 + Q + 10 A) TFC = 10; and 𝐓𝐕𝐂 = 2𝑸𝟑 –2𝑸𝟐 + Q B) 𝐀𝐅𝐂 = 𝑻𝑭𝑪 𝑸 = 𝟏𝟎 𝑸 𝐀𝐕𝐂 = 𝑻𝑽𝑪 𝑸 = 2𝑸𝟑 –2𝑸𝟐 + Q 𝑸 = 2𝑸𝟐 –2𝑸 + 1 𝐀𝐂 = 𝑻𝑪 𝑸 = 2𝑸𝟑 –2𝑸𝟐 + Q + 10 𝑸 = 2𝑸𝟐 –2𝑸 + 1 + 𝟏𝟎 𝑸 𝐌𝐂 = 𝒅𝑻𝑪 𝒅𝑸 = 𝒅(2𝑸𝟑 –2𝑸𝟐 + Q + 10) 𝒅𝑸 = 6𝑸𝟐 –𝟒𝑸 + 1 C) To find the minimum value of MC, 𝒅𝑴𝑪 𝒅𝑸 = 𝟎 = 12Q - 4 = 0 𝑸 = 𝟏/𝟑 Thus, MC is minimized when Q = 0.33
- 31. The minimum value of MC will be: MC = 6𝑸𝟐 –𝟒𝑸 + 1 = 6( 𝟏 𝟑 )𝟐 –𝟒( 𝟏 𝟑 ) + 1 = 0.33 To find the minimum value of AVC, 𝒅𝑨𝑽𝑪 𝒅𝑸 = 𝟎 = 4𝑄 − 2= 0 𝑸 = 𝟎. 𝟓 AVC is minimized at 𝑸 = 𝟎. 𝟓. The minimum value of AVC will be: AVC = 2𝑸𝟐 –2𝑸 + 1 AVC = 2 𝟎. 𝟓 𝟐 –2(𝟎. 𝟓) + 1 = 0.5 – 1 + 1 = 0.5
- 32. 4.2.3 The relationship between short run production and cost curves Suppose a firm in the short run uses labour as a variable input and capital as a fixed input. Let the price of labour be given by w, which is constant. Given these conditions, we can derive the relation between MC and MPL as well as the relation between AVC and APL. i) MC and MP of Labour 𝐌𝐂 = 𝒅𝑻𝑪 𝒅𝑸 = ∆𝑻𝑽𝑪 ∆𝑸 , Where 𝑻𝑽𝑪 = 𝒘 ∗ 𝑳 Thus, 𝐌𝐂 = ∆(𝒘∗𝑳) ∆𝑸 = 𝒘 ∆𝑳 ∆𝑸 ; but ∆𝑳 ∆𝑸 = 𝟏 𝑴𝑷𝑳 Therefore, 𝑴𝑪 = 𝒘 𝑴𝑷𝑳
- 33. The expression 𝑴𝑪 = 𝒘 𝑴𝑷𝑳 shows that MC and 𝑴𝑷𝑳 are inversely related. When initially MPL increases, MC decreases; when MPL is at its maximum, MC must be at a minimum and when finally MPL declines, MC increases. II) AVC and AP of Labour AV𝐂 = 𝑻𝑽𝑪 𝑸 , Where 𝑻𝑽𝑪 = 𝒘 ∗ 𝑳 Thus, 𝐀𝐕𝐂 = (𝒘∗𝑳) 𝑸 = 𝒘 𝑳 𝑸 ; but 𝑳 𝑸 = 𝟏 𝑨𝑷𝑳 Therefore, 𝐀𝐕𝐂 = 𝒘 𝑨𝑷𝑳 This expression also shows inverse relation between AVC and APL. When APL increases, AVC decreases; when APL is at a maximum, AVC is at a minimum and when finally APL declines, AVC increases.
- 34. Graphically, From the above figure, we can conclude that the MC curve is the mirror image of MPL curve and AVC curve is the mirror image of APL curve.