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AAEC 3315 Agricultural Price Theory

AAEC 3315 Agricultural Price Theory. CHAPTER 5 Theory of Production The Case of One Variable Input in the Short-Run. Objectives. To gain understanding of: Theory of Production Production Curves Total Physical Product Average Physical Product Marginal Physical Product

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AAEC 3315 Agricultural Price Theory

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  1. AAEC 3315Agricultural Price Theory CHAPTER 5 Theory of Production The Case of One Variable Input in the Short-Run

  2. Objectives To gain understanding of: • Theory of Production • Production Curves • Total Physical Product • Average Physical Product • Marginal Physical Product • Law of Diminishing Physical Product • Stages of Production • Production Functions

  3. Production Relationships • Definition: The technical relationship between inputs & output indicating the maximum amount of output that can be produced using alternative amounts of variable inputs in combination with one or more fixed inputs under a given state of technology. Or, simply speaking, it is the technical relationship between inputs & output

  4. Product Curves The Case of One Variable Input • Total Physical Product (TPP) - illustrates the technological or physical relationship that exists between output and one variable input, ceteris paribus • Starts increasing at an increasing rate. • Continues to increase but at a decreasing rate • Reaches the maximum, then decreases • The functional form of a production function is: Y = f (X), where Y is the quantity of output and X is the quantity of input Y TPP X

  5. Y TPP X Product Curves • The point where TPP changes from increasing at an increasing rate to increasing at a decreasing rate is called the Inflection Points. • Points A, B, and C Indicate total amount of output produced at each level of input use Maximum Point Y3 C Y2 B Inflection Point Y1 A X1 X2 X3

  6. Y TPP Y X APP X Product Curves (Cont.) • Average Physical Product (APP) - shows how much production, on average, can be obtained per unit of the variable input with a fixed amount of other inputs • Indicates average productivity of the inputs being used - how productive is each input level on average • APP = Y / X • Drawing a line from the origin which is tangent to the TPP curve gives APP max

  7. Y TPP Y X APP X Product Curves (Cont.) • Marginal Physical Product (MPP) - represents the amount of additional (i.e., marginal) output obtained from using an additional unit of variable input (X). • MPP = ΔY / ΔX = ∂Y/∂X or the slope of the TPP curve. Thus, MPP represents the rate of change in output resulting from adding one more unit of input • Since MPP is the slope of TPP, it reaches a maximum at inflection point • It reaches zero at the maximum point of TPP MPP

  8. Y TPP Y X APP X MPP Product Curves MPP is negative

  9. Y TPP Y X APP X MPP Relationships between Product Curves • MPP reaches a maximum at inflection point • MPP = 0 occurs when TPP is maximum • MPP is negative beyond TPP max • Drawing a line from the origin which is tangent to the TPP curve gives APP max • At point where APP is max, MPP crosses APP (MPP=APP) • When MPP > APP, APP is increasing • When MPP = APP, APP is at a max • When MPP < APP, APP is decreasing The relationship between TPP, APP, & MPP is very specific. If we have COMPLETE information about one curve, the other two curves can be derived. MPP is negative

  10. Law of Diminishing Marginal Physical Product • Law of Diminishing Marginal Physical Product: As additional units of one input are combined with a fixed amount of other inputs, a point is always reached where the additional product received from the last unit of added input (MPP) will decline • This occurs at the inflection point

  11. Y TPP Y X APP X MPP Stages of Production:Rational & Irrational • The stage I of the production function is between 0 and X1 units of X. • In stage I: • TPP is increasing • APP is increasing • MPP increases, reaches a maximum & decreases to APP • Stage I is an irrational stage because APP is still increasing I 0 X1

  12. Y TPP Y X APP X 0 X1 MPP Stages of Production:Rational & Irrational • The stage II of the production function is between X1 and X2 units of X. • In Stage II: • TPP is increasing • APP is decreasing • MPP is decreasing and less than APP, but still positive • RATIONAL STAGE BECAUSE TPP IS STILL INCREASING I II X2

  13. Y TPP Y X APP X 0 X1 MPP Stages of Production:Rational & Irrational • Stage III of the production function is beyond X2 level X • In Stage III: • TPP is decreasing • APP is decreasing • MPP is decreasing and negative • IRRATIONAL STAGE BECAUSE TPP IS DECREASING I II III X2

  14. A Hypothetical Production Function Schedule Stage I Stage II Stage III

  15. TPP Effects of Technological Change • We know that the PF gives the max amount of output that can be produced by a firm using a given technology. • The PF can shift over time as a result of a technological change • Technological change refers to the introduction of new technology that increases output with the same amount of resources. Y TPP1 X

  16. Elasticity of Production • The elasticity of production measures the degree of responsiveness between output and input. • Using Calculus: • Like any other elasticity, elasticity of production is independent of units. • It measures the percentage change in production in response to a percentage change in variable input.

  17. Y TPP Y X APP X 0 MPP A Hypothetical Production FunctionA Mathematical Example • Consider a Production Function TP = X2 – 1/30X3, where TP (Y) is quantity of output and X is the quantity of input. • AP = TP/X = X – (1/30)X2 • MP = ∂TP/∂X • = 2X – (3/30)X2 • = 2X – (1/10) X2 I II III

  18. Y TPP Y X APP X MPP A Hypothetical Production FunctionA Mathematical Example • Given • TP = X2 – (1/30)X3, • AP = TP/X = X – (1/30)X2 • MP = ∂TP/∂X = 2X – (1/10)X2 • At what levels of X does the MP • reach its maximum? • MP reaches its maximum • where ∂MP/∂X = 0 • That is, where • 2 – (2/10)X = 0 • Or, 0.2 X = 2 • Or, X = 10 I II III 10 0

  19. Y TPP Y X APP X 10 0 MPP A Hypothetical Production FunctionA Mathematical Example • Given • TP = X2 – (1/30)X3, • AP = TP/X = X – (1/30)X2 • MP = ∂TP/∂X = 2X – (1/10)X2 • At what levels of X does the AP • reach its maximum? • AP reaches its maximum • where ∂AP/∂X = 0 • That is, where • 1 – (2/30)X = 0 • Or, (1/15) X = 1 • Or, X = 15 I II III 15

  20. Y TPP Y X APP X 10 0 MPP A Hypothetical Production FunctionA Mathematical Example • Given, • TP = X2 – (1/30)X3, • AP = TP/X = X – (1/30)X2 • MP = ∂TP/∂X = 2X – (1/10)X2 • At what levels of X does the TP • reach its maximum? • TP reaches its maximum • where ∂TP/∂X = MP = 0 • That is, where • MP = 2x – (1/10)X2 = 0 • Using the quadratic • formula of • X = 20 I II III 15 20

  21. Y TPP Y X APP X 10 0 MPP A Hypothetical Production FunctionA Mathematical Example • Given • TP = X2 – (1/30)X3, • AP = TP/X = X – (1/30)X2 • MP = ∂TP/∂X = 2X – (1/10)X2 • What is the range of X values for • Stage II? • Stage II is the stage that • begins where AP is at its • maximum and ends where TP • is at its maximum. • Thus, the range of X values or • Stage II is 15 and 20. I II III 15 20

  22. Y TPP Y X APP X 10 0 MPP A Hypothetical Production FunctionA Mathematical Example • Given • TP = X2 – (1/30)X3, • AP = TP/X = 2X – (1/30)X2 • MP = ∂TP/∂X = 2X – (1/10)X2 • At what level of X does the Law • of Diminishing Returns set in? • It sets in where MP reaches • its maximum. • Thus at X = 10 the law of • Diminishing returns sets in. I II III 15 20

  23. A Hypothetical Production FunctionA Mathematical Example • Given TP = X2 – (1/30)X3, • At Y = 112.5 and X = 15, what is the elasticity of production? • Applying • Ep = (2X – (1/10) X2) * (X/Q) • Ep = ((2*15) – (225/10)) * (15/112.5) • Ep = (30 – 22.5) * (0.133) • Ep = 7.5 * 0.133 = 0.997

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