Production function


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Micro Economics

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Production function

  2. 2. Production  An entrepreneur must put together resources -- land, labour, capital -- and produce a product people will be willing and able to purchase
  3. 3. Theory of Production and Costs Focus- mainly on the the firm.  We will examine  ◦ Its production capacity given available resources ◦ the related costs involved
  4. 4. What is a firm? A firm is an entity concerned with the purchase and employment of resources in the production of various goods and services.  Assumptions:  ◦ the firm aims to maximize its profit with the use of resources that are substitutable to a certain degree ◦ the firm is" a price taker in terms of the resources it uses.
  5. 5. What Is Production Function  Production function deals with the maximum output that can be produced with a limited and given quantity of inputs.  The production function is dependent on different time frames. Firms can produce for a brief or lengthy period of time.
  6. 6. Production Function Mathematical representation of the relationship:  Q = f (K, L, La)  Output (Q) is dependent upon the amount of capital (K), Land (L) and Labour (La) used 
  7. 7. ASSUMPTIONS THE PRODUCTION FUNCTIONS ARE BASED ON CERTAIN ASSUMPTIONS 1. Perfect divisibility of both inputs and outputs 2. Limited substitution of one factor for another 3. Constant technology 4. Inelastic supply of fixed factors in the short run
  8. 8. THE LAWS OF PRODUCTION LAWS OF VARIABLE PROPORTIONS LAWS OF RETURNS TO SCALE Relates to the study of input output relationship in the short run with one variable input while other inputs are held constant Relates to the study of input output relationship in the long run assuming all inputs to be variable
  9. 9. Firm’s Inputs  Inputs - are resources that contribute in the production of a commodity.  Most resources are lumped into three categories: ◦ Land, ◦ Labor, ◦ Capital.
  10. 10. Fixed vs. Variable Inputs Fixed inputs -resources used at a constant amount in the production of a commodity.  Variable inputs - resources that can change in quantity depending on the level of output being produced.  The longer planning the period, the distinction between fixed and variable inputs disappears, i.e., all inputs are variable in the long run. 
  11. 11. Production Analysis with One Variable Input Total product (Q) refers to the total amount of output produced in physical units (may refer to, kilograms of sugar, sacks of rice produced, etc)  The marginal product (MP) refers to the rate of change in output as an input is changed by one unit, holding all other inputs constant.  MPL TPL L
  12. 12. Total vs. Marginal Product Total Product (TPx) = total amount of output produced at different levels of inputs  Marginal Product (MPx) = rate of change in output as input X is increased by one unit, ceteris paribus.  MPX TPX X
  13. 13. Production Function of a Rice Farmer Units of L Total Product (QL or TPL) Marginal Product (MPL) 0 0 - 1 2 2 2 6 4 3 12 6 4 20 8 5 26 6 6 30 4 7 32 2 8 32 0 9 30 -2 10 26 -4
  14. 14. QL 32 30 Total product 26 QL 20 12 6 2 L 0 1 2 3 4 5 6 7 8 9 10 Labor FIGURE 5.1. Total product curve. The total product curve shows the behavior of total product vis-a-vis an input (e.g., labor) used in production assuming a certain technological level.
  15. 15. Marginal Product The marginal product refers to the rate of change in output as an input is changed by one unit, holding all other inputs constant.  Formula:  MPL TPL L
  16. 16. Marginal Product Observe that the marginal product initially increases, reaches a maximum level, and beyond this point, the marginal product declines, reaches zero, and subsequently becomes negative.  The law of diminishing returns states that "as the use of an input increases (with other inputs fixed), a point will eventually be reached at which the resulting additions to output decrease" 
  17. 17. Total and Marginal Product 35 30 25 TPL 20 15 10 5 MPL 0 0 -5 -10 1 2 3 4 5 6 7 8 9
  18. 18. Law of Diminishing Marginal Returns As more and more of an input is added (given a fixed amount of other inputs), total output may increase; however, the additions to total output will tend to diminish.  Counter-intuitive proof: if the law of diminishing returns does not hold, the world’s supply of food can be produced in a hectare of land. 
  19. 19. Average Product (AP) Average product is a concept commonly associated with efficiency.  The average product measures the total output per unit of input used.  ◦ The "productivity" of an input is usually expressed in terms of its average product. ◦ The greater the value of average product, the higher the efficiency in physical terms.  Formula: APL TPL L
  20. 20. TABLE 5.2. Average product of labor. Labor (L) Total product of labor (TPL) Average product of labor (APL) 0 0 0 1 2 2 2 6 3 3 12 4 4 20 5 5 26 5.2 6 30 5 7 32 4.5 8 32 4 9 30 3.3 10 26 2.6
  21. 21. The slope of the line from the origin is a measure of the AVERAGE Y Slope = rise run Y L b a Y Rise = Y 0 Run = L L1 L2 L
  22. 22. Total Product The average product at b is highest. AP at c is less than at a. Q AP at d is less than at c. b c d QL a 0 L
  23. 23. Q Highest Slope of Line from Origin Max APL Inflection point TPL Max MPL 0 L1 L2 L3 L
  24. 24. Relationship between Average and Marginal Curves: Rule of Thumb When the marginal is less than the average, the average decreases.  When the marginal is equal to the average, the average does not change (it is either at maximum or minimum)  When the marginal is greater than the average, the average increases 
  25. 25. AP,MP At Max AP, MP=AP Max MPL Max APL APL 0 L1 L2 L3 L MPL
  26. 26. TP TPL 0 L1 L2 L3 Stage I MP>AP AP increasing AP,MP Stage II MP<AP AP decreasing MP still positive L Stage III MP<0 AP decreasing APL 0 L1 L2 L3 L MPL
  27. 27. Three Stages of Production  In Stage I ◦ APL is increasing so MP>AP. ◦ All the product curves are increasing ◦ Stage I stops where APL reaches its maximum at point A. ◦ MP peaks and then declines at point C and beyond, so the law of diminishing returns begins to manifest at this stage
  28. 28. Three Stages of Production  Stage II ◦ starts where the APL of the input begins to decline. ◦ QL still continues to increase, although at a decreasing rate, and in fact reaches a maximum ◦ Marginal product is continuously declining and reaches zero at point D, as additional labor inputs are employed.
  29. 29. Three Stages of Production Stage III starts where the MPL has turned negative. ◦ all product curves are decreasing. ◦ total output starts falling even as the input is increased
  30. 30. The Law of Variable Proportions • • Elaborately stating the Law : In the short run, as the amount of variable factors increases, other things remaining equal, OP(or the returns to the factors varied will increase more than proportionally to the a amount of the variable inputs in the beginning than it may increase in the same proportion and ultimately it will increase less proportionately. Assuming that the firm only varies the labour (L), it alters the proportion between the fixed input and the variable input. As this altering goes on, the firm experiences the Law of Diminishing Marginal Returns.
  31. 31. Using the concept of MP, During the SR, under the given state of technology and other conditions remaining unchanged, with the given fixed factors, when the units of a variable factor are increased in the production function in order to increase the TP, the TP initially may rise at an increasing rate and after a point, it tends to increase at a decreasing rate because the MP of the variable factor in the beginning may Production Schedule Units of Variable Input (Labour) (n) Total Product (TP) Average Product (AP) (TPn) Marginal Product (TPn- TPn-1) 1 20 20 20 2 50 25 30 3 90 30 40 4 120 30 30 5 135 27 15 6 144 24 9 7 147 21 3 8 148 18.5 1 9 148 16.4 0 10 145 14.5 -3 STAGE I STAGE II STAGE III
  32. 32. TP
  33. 33. Stages Diminishing Total returns -implies reduction in total product with every additional unit of input. Diminishing Average returns -which refers to the portion of the Average Physical Product curve after its intersection with MPP curve. Diminishing Marginal returns refers to the point where the MPP curve starts to slope down and travels all the way down to the x-axis and beyond. Putting it in a chronological order, at first the marginal returns start to diminish, then the average returns, followed finally by the total returns.
  34. 34. Observations The L of DMR becomes evident in the marginal product column. Initially MP of Labor rises . The TP rises at an increasing rate (= MP). Average Product also rises. Stage of increasing Returns certain point (4th unit of Labour), the MP begins to diminish. Rate of increase in the TP slows down. Stage of diminishing returns. When AP is max, AP=MP=30 at 4th unit of labour. After AS MP diminishes, it becomes zero and negative thereafter (Stage III) When MP is zero, TP is maximum. (148 is the highest amount of TP, when MP is equal to 0 when 9 units of labour are employed. When MP becomes negative, TP also starts to diminish in the same proportion but AP declines after being positive up to a certain
  35. 35. Question Following data relates to the quantity of tuna that could be caught with different crew sizes. No. of fisherme n Daily Tuna Catch 3 4 5 6 7 8 9 300 450 590 665 700 725 710 Indicate the points that delineate the three stages of production .
  36. 36. Explanation of the stages The operation of the law of diminishing returns in three stages is attributed to two fundamental characteristics of factors of production: i) Indivisibility of certain fixed factors. ii) Imperfect substitutability between factors. 
  37. 37. Marginal Revenue Productivity   The marginal revenue productivity , also referred to as the marginal revenue product of labor and the value of the marginal product or VMPL, is the change in total revenue earned by a firm that results from employing one more unit of labor. It determines, under some conditions, the optimal number of workers to employ at an exogenously determined market wage rate. In a competitive firm, the marginal revenue product will equal the product of the price and the marginal product of labor. It is not efficient for a firm to pay its workers more than it will earn in profits from their labor
  38. 38. Production function with one variable input Total Product: Q = 30L+20L2-L3  Average Product : Q /L  Marginal Product : MP = dQ/dL = 30+40L3L2 
  39. 39. What is long run production function ?  Long run refers to that time in the future when all inputs are variable inputs.  In the long run both capital and labour are included  Output can be varied by changing the levels of both L & K and the long run production function is expressed as: Q = f (L, K)
  41. 41. LONG RUN TOTAL PRODUCTIONReturns to scale  During the short period, some factors of production are relatively scarce, therefore , the proportion of the factors may be changed but not their scale. But in the long run, all factors are variable, therefore, the scale of production can be changed in the long run  Returns to scale is a factor that is studied in the long run.  Returns to scale show the responsiveness of total product when all the inputs are increased proportionately.
  42. 42. Returns to Scale When all inputs are changed in the same proportion (or scale of production is changed),the total product may respond in three possible ways: 1) Increasing returns to scale 2) Constant returns to scale, and 3) Diminishing returns to scale 
  43. 43. INCREASING RETURNS TO SCALE    The law of increasing returns to scale operates when the percentage increase in the total product is more than the percentage increase in all the factor inputs employed in the same proportion. Many economies set in and increase in return is more than increase in factors. For e.g 10 percent increase in labour and capital causes 20 percent increase in total output. Similarly, 20 percent increase in labour and capital causes 45 percent increase in total output.
  44. 44. CONSTANT RETURNS TO SCALE   Law of constant returns to scale operates when a given percentage increase in the factor inputs in the same proportion causes equal percentage increase in total output. Economies of scale are counter balanced by diseconomies of scale.
  45. 45. DIMINISHING RETURNS TO SCALE  The law of diminishing returns to scale occurs when a given percentage increase in all factor inputs in equal proportion causes less than percentage increase in output.  Output increases in a smaller proportion.  Diseconomies
  46. 46. Graphically, the returns to scale concept can be illustrated using the following graphs Q IRTS Q X,Y Q CRTS X,Y DRTS X,Y
  47. 47. Production Isoquants/ isoquant curve/iso-product curve • In the long run, all inputs are variable & isoquants are used to study production decisions – An isoquant or iso-product curve is a curve showing all possible input combinations capable of producing a given level of output – Isoquants are downward sloping; if greater amounts of labor are used, less capital is required to produce a given output 47
  48. 48. Isoquant a curve showing all possible efficient combinations of input that are capable of producing a certain quantity of output (Note: iso means same, so isoquant means same quantity)
  49. 49. Isoquant for 100 units of output 100 units of output can be produced in many different ways including L1 units of labor & K1 units of capital, L2 units of labor & K2 units of capital, L3 units of labor & K3 units of capital, & L4 units of labor & K4 units of capital. Quantity of capital used per unit of time K1 K2 K3 100 K4 L1 L2 L3 L4 Quantity of labor used per unit of time
  50. 50. Isoquants for different output levels Quantity of capital used per unit of time As you move in a northeasterly direction, the amount of output produced increases, along with the amount of inputs used. 125 100 50 Quantity of labor used per unit of time
  51. 51. It is possible for an isoquant to have positively sloped sections. Quantity of capital used per unit of time In these sections, you’re increasing the amounts of both inputs, but output is not increasing, because the marginal product of one the inputs is negative. Quantity of labor used per unit of time
  52. 52. The lines connecting the points where the isoquants begin to slope upward are called ridge lines. Quantity of capital used per unit of time ridge lines Quantity of labor used per unit of time
  53. 53. No profit-maximizing firm will operate at a point outside the ridge lines, since it can produce the same output with less of both outputs. Quantity of capital used per unit of time K2 B A K1 L1 L2 Notice, for example, that since points A & B are on the same isoquant, they produce the same amount of output. However, point B is a more expensive way to produce since it uses more capital & more labor. Quantity of labor used per unit of time
  54. 54. Marginal rate of technical substitution (MRTS) The slope of the isoquant The rate at which you can trade off inputs and still produce the same amount of output. For example, if you can decrease the amount of capital by 1 unit while increasing the amount of labor by 3 units, & still produce the same amount of output, the marginal rate of technical substitution is 1/3.
  55. 55. Marginal Rate of Technical Substitution (MRTS) or slope of an isoquant ΔK/ΔL = the - MPL/MPK negative of the ratio of the marginal products of the inputs, with the input on the horizontal axis in the numerator.
  56. 56. Other types of Isoquants Linear Isoquants  L- shaped Isoquants  Kinked Isoquants 
  57. 57. How does output respond to changes in scale in the long run? Three possibilities:  1. Constant returns to scale  2. Increasing returns to scale  3. Decreasing returns to scale 
  58. 58. Constant returns to scale  Doubling inputs results in double the output.
  59. 59. Constant returns to scale Attributed to the limits of the economies of scale.  When economies of scale reach their limits and diseconomies of scale are yet to begin, returns to scale become constant. 
  60. 60. Increasing returns to scale Doubling inputs results in more than double the output. One reason this may occur is that a firm may be able to use production techniques that it could not use in a smaller operation.
  61. 61. Decreasing returns to scale Doubling inputs results in less than double the output. One reason this may occur is the difficulty in coordinating large organizations (more paper work, red tape, etc.)
  62. 62. Graphs of Constant, Increasing, & Decreasing Returns to Scale Capital Capital Capital 150 150 100 50 Labor Constant Returns to Scale: isoquants for output levels 50, 100, 150, etc. are evenly spaced. 150 100 50 Labor Increasing Returns to Scale: isoquants for output levels 50, 100, 150, etc. get closer & closer together. 100 50 Labor Decreasing Returns to Scale: isoquants for output levels 50, 100, 150, etc. become more widely spaced.
  63. 63. ISOQUANT MAP- A family or a group of isoquants is called an ISOQUANT MAP K4 A Units of K Iq4 B K3 Iq3 = 400 = 300 C K2 Iq2 = 200 Iq1 = 100 D K1 0 L1 L2 L3 Units of L L4
  64. 64. Capital, K (machines rented) The Isocost Line A a 10 b 8 c 6 Cost = Rs50 Per unit price of labor input = Rs10/hour Per unit price of capital input = Rs5/machine d 4 e 2 f 0 1 2 3 4 B 5 6 7 8 Labor, L (worker-hours employed) 9 10 66
  65. 65. Slope of isocost line M=PL.QL+PK.QK Where, M=total outlay PL= price per unit of labor PK= price per unit of capital QL= units of labor QK= units of capital   Slope of isocost line= OA/OF price per unit of labour input price per unit of capital input Slope of isocost line can be changed in two ways: 1) Change in the factor price, and 2) Change in total outlay or total cost 
  66. 66. Changes in One factor Price Capital, K (machines rented) Decrease in the factor price causes rightward shift and increase in factor price causes leftward shift in iso-cost line. Cost = 500; labor,R = 16.5 or 10or 1/ hour The money wage, W = Rs5/machine a 10 8 6 A Change in unit price of labor 4 …Rs10 2 Rs16.5 0 10 h f …Rs1 1 2 3 4 5 6 7 Labor, L (worker-hours employed) 8 9 68
  67. 67. Change in total outlay or total cost Direction of increase in total cost capital (r) K of Slope = -w/r TC= Rs. 100 Units TC= Rs. 75 TC=Rs. 50 L Units of labour(w) 69
  68. 68. Isoquants and Cost Minimization K IQ 3 IQ 2 M • 4 6 • N P • P” TC=Rs1 00 TC=Rs=75 2 • Q=300 Q=200 P’ TC=Rs5 0 Q=100 0 10 Units of Cap[ital IQ 1 8 • 0 2 4 6 8 10 12 Units of 14 16 18 L 20 70
  69. 69. Optimization & Cost • Expansion path gives the efficient (least-cost) input combinations of labor and capital needed for every level of output. Derived for a specific set of input prices Along expansion path, input-price ratio is constant & equal to the marginal rate of technical substitution • It is defined as the locus of tangency points between iso-cost lines and isoquants. 71
  70. 70. EXPANSION PATH •It Capital input implies to Long run because:  No input is fixed. Path starts from origin indicating that if output is zero costs are zero. •Expansion path gives us the level of output & one least combination that can produce this level of output. •Movement along the line gives the costs at which output can be expanded •So called Expansion Path. Labor input
  71. 71. Estimation of production function – Cobb Douglas Production Function The function used to model production is of the form: Q(L,K) = ALaKb where: Q = total production L = labor input K = capital input A = total factor productivity a and b are the output elasticities of labor and capital, respectively. These values are constants determined by available technology.
  72. 72. Output elasticity measures the responsiveness of output to a change in levels of either labor or capital used in production, ceteris paribus. E.g. if a= 0.15, a 1% increase in labor would lead to approximately a 0.15% increase in output. Total Factor productivity :TFP tries to assess the efficiency with which both capital and labour are used. Once a country's labour force stops growing and an increasing capital stock causes the return on new investment to decline, TFP becomes the main source of future economic growth. It is calculated as the percentage increase in output that is not accounted for by changes in the volume of inputs of capital and labour. So if the capital stock and the workforce both rise by 2% and output rises by 3%, TFP goes up by 1%.
  73. 73. Returns to scale based on Cobb Douglas function If a+b = 1,the production function has constant returns to scale (CRTS). That is, if L and K are each increased by 20%, then Q increases by 20%.  If output increases by less than that proportional change, there are decreasing returns to scale (DRS). i.e. a+b<1   If output increases by more than that proportion, there are increasing returns to scale (IRS) ). i.e. a+b>1
  74. 74. Leontif Production function Capital and labor are perfect complements.  Capital and labor are used in fixedproportions.  Q = min {bK, cL}  Since capital and labor are consumed in fixed proportions there is no input substitution along isoquants (hence, no MRTSKL). 