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Lecture # 09 Inputs and Production Functions Lecturer: Martin Paredes

Lecture # 09 Inputs and Production Functions Lecturer: Martin Paredes. Outline. The Production Function Marginal and Average Products Isoquants Marginal Rate of Technical Substitution Returns to Scale Some Special Functional Forms Technological Progress. Definitions.

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Lecture # 09 Inputs and Production Functions Lecturer: Martin Paredes

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  1. Lecture # 09 Inputs and Production Functions Lecturer: Martin Paredes

  2. Outline • The Production Function • Marginal and Average Products • Isoquants • Marginal Rate of Technical Substitution • Returns to Scale • Some Special Functional Forms • Technological Progress

  3. Definitions • Inputs or factors of production are productive resources that firms use to manufacture goods and services. • Example: labor, land, capital equipment… • The firm’s output is the amount of goods and services produced by the firm.

  4. Definitions Production transforms a set of inputs into a set of outputs Technology determines the quantity of output that is feasible to attain for a given set of inputs.

  5. Definitions The production function tells us the maximum possible output that can be attained by the firm for any given quantity of inputs. Q = F(L,K,T,M,…)

  6. Definitions • A technically efficient firm is attaining the maximum possible output from its inputs (using whatever technology is appropriate) • The firm’s production set is the set of all feasible points, including: • The production function (efficient point) • The inefficient points below the production function

  7. Q Example: The Production Function and Technical Efficiency C • L

  8. Q Example: The Production Function and Technical Efficiency D • C • L

  9. Q Example: The Production Function and Technical Efficiency Production Function Q = f(L) D • C • L

  10. Q Example: The Production Function and Technical Efficiency Production Function Q = f(L) D • C • • B • A L

  11. Q Example: The Production Function and Technical Efficiency Production Function Q = f(L) D • C • • B • A Production Set L

  12. Notes: • The variables in the production function are flows (amount of input per unit of time), not stocks (the absolute quantity of the input). • Capital refers to physical capital (goods that are themselves produced goods) and not financial capital (money required to start or maintain production).

  13. Comparison between production function and utility function

  14. Comparison between production function and utility function

  15. Marginal Product Definition: The marginal product of an input is the change in output that results from a small change in an input E.g.: MPL = Q MPK = Q L K • It assumes the levels of all other inputs are held constant.

  16. Marginal Product Example: Suppose Q = K0.5L0.5 Then: MPL = Q = 0.5 K0.5 L L0.5 MPK = Q = 0.5 L0.5 K K0.5

  17. Average Product Definition: The average product of an input is equal to the total output to be produced divided by the quantity of the input that is used in its production E.g.: APL = Q APK = Q L K

  18. Average Product Example: Suppose Q = K0.5L0.5 Then: APL = Q = K0.5L0.5=K0.5 L L L0.5 APK = Q = K0.5L0.5=L0.5 K K K0.5

  19. Law of Diminishing Marginal Returns Definition: The law of diminishing marginal returns states that the marginal product (eventually) declines as the quantity used of a single input increases.

  20. Example: Total and Marginal Product Q Q= F(L,K0) L

  21. Example: Total and Marginal Product Q Q= F(L,K0) Increasing marginal returns Diminishing marginal returns MPL maximized L

  22. Example: Total and Marginal Product Q Q= F(L,K0) MPL = 0 when TP maximized Increasing total returns Diminishing total returns L

  23. Q Example: Total and Marginal Product L MPL maximized TPL maximized where MPL is zero. TPL falls where MPL is negative; TPL rises where MPL is positive. MPL L

  24. Marginal and Average Products • There is a systematic relationship between average product and marginal product. • This relationship holds for any comparison between any marginal magnitude with the average magnitude.

  25. Marginal and Average Products • When marginal product is greater than average product, average product is increasing. • E.g., if MPL > APL , APL increases in L. • When marginal product is less than average product, average product is decreasing. • E.g., if MPL < APL, APL decreases in L.

  26. APL MPL Example: Average and Marginal Products MPL maximized APL maximized L

  27. Q Example: Total, Average and Marginal Products L MPL maximized APL MPL APL maximized L

  28. Isoquants Definition: An isoquant is a representation of all the combinations of inputs (labor and capital) that allow that firm to produce a given quantity of output.

  29. K Example: Isoquants Q = 10 Slope=dK/dL L L 0

  30. K Example: Isoquants All combinations of (L,K) along the isoquant produce 20 units of output. Q = 20 Q = 10 Slope=dK/dL L 0

  31. Isoquants Example: Suppose Q = K0.5L0.5 • For Q = 20 => 20 = K0.5L0.5 => 400 = KL => K =400/L • For Q = Q0 => K = (Q0)2 /L

  32. Marginal Rate Of Technical Substitution Definition: The marginal rate of technical substitution measures the rate at which the firm can substitute a little more of an input for a little less of another input, in order to produce the same output as before.

  33. Marginal Rate Of Technical Substitution Alternative Definition : It is the negative of the slope of the isoquant: MRTSL,K = — dK (for a constant level of dL output)

  34. Marginal Product and the Marginal Rate of Technical Substitution • We can express the MRTS as a ratio of the marginal products of the inputs in that basket • Using differentials, along a particular isoquant: MPL . dL + MPK . dK = dQ = 0 • Solving: MPL = _ dK = MRTSL,K MPK dL

  35. Marginal Product and the Marginal Rate of Technical Substitution Notes: • If we have diminishing marginal returns, we also have a diminishing marginal rate of technical substitution. • In other words, the marginal rate of technical substitution of labour for capital diminishes as the quantity of labour increases along an isoquant.

  36. Marginal Product and the Marginal Rate of Technical Substitution Notes: • If both marginal products are positive, the slope of the isoquant is negative • For many production functions, marginal products eventually become negative. Then: • MRTS < 0 • We reach an uneconomic region of production

  37. K Example: The Economic and the Uneconomic Regions of Production Isoquants Q = 20 Q = 10 L 0

  38. K Example: The Economic and the Uneconomic Regions of Production Q = 20 B • A • Q = 10 L 0

  39. K Example: The Economic and the Uneconomic Regions of Production Q = 20 B • A • Q = 10 MPL < 0 L 0

  40. K Example: The Economic and the Uneconomic Regions of Production MPK < 0 Q = 20 B • A • Q = 10 MPL < 0 L 0

  41. K Example: The Economic and the Uneconomic Regions of Production MPK < 0 Uneconomic Region Q = 20 B • A • Q = 10 MPL < 0 L 0

  42. K Example: The Economic and the Uneconomic Regions of Production MPK < 0 Uneconomic Region Q = 20 B • A • Q = 10 Economic Region MPL < 0 L 0

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