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 Law of Diminishing Physical Product Stages of Production Production Functions

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

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 X Y TPP XTPP=Y

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 X Y TPP Inflection Point Maximum Point A B C X1 X2X3 Y1 Y2 Y3

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 X Y TPP X Y APP XTPP=YAPP

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 X Y TPP X Y APP MPP XTPP=YMPP

Product Curves X Y TPP X Y APP MPP MPP is negative XTPP=YAPPMPP

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. X Y TPP X Y APP MPP MPP is negative

Law of Diminishing Marginal Physical Product L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

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 X Y TPP X Y APP MPP 0 X1 I

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 X Y TPP X Y APP MPP 0 X1 X2 II I

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 X Y TPP X Y APP MPP 0 X1 X2 III I II

A Hypothetical Production Function Schedule Input (X) TPP (Y) APP (Y/X) MPP (ΔTPP/ ΔX) Stage I Stage II Stage III

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. X Y TPP TPP1

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.

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

A Hypothetical Production Function A Mathematical Example X Y TPP X Y APP MPP 0 10 III I II o Given o TP = X 2 – (1/30)X 3, o AP = TP/X = X – (1/30)X 2 o MP = ∂TP/∂X = 2X – (1/10)X 2 o At what levels of X does the MP reach its maximum? o MP reaches its maximum where ∂MP/∂X = 0 o That is, where o 2 – (2/10)X = 0 o Or, 0.2 X = 2 o Or, X = 10

A Hypothetical Production Function A Mathematical Example X Y TPP X Y APP MPP 0 10 III I II o Given o TP = X 2 – (1/30)X 3, o AP = TP/X = X – (1/30)X 2 o MP = ∂TP/∂X = 2X – (1/10)X 2 o At what levels of X does the AP reach its maximum? o AP reaches its maximum where ∂AP/∂X = 0 o That is, where o 1 – (2/30)X = 0 o Or, (1/15) X = 1 o Or, X = 15 15

A Hypothetical Production Function A Mathematical Example X Y TPP X Y APP MPP 0 10 III I II o Given, o TP = X 2 – (1/30)X 3, o AP = TP/X = X – (1/30)X 2 o MP = ∂TP/∂X = 2X – (1/10)X 2 o At what levels of X does the TP reach its maximum? o TP reaches its maximum where ∂TP/∂X = MP = 0 o That is, where o MP = 2x – (1/10)X 2 = 0 o Using the quadratic formula of o X =

A Hypothetical Production Function A Mathematical Example X Y TPP X Y APP MPP 0 10 III I II o Given o TP = X 2 – (1/30)X 3, o AP = TP/X = X – (1/30)X 2 o MP = ∂TP/∂X = 2X – (1/10)X 2 o What is the range of X values for Stage II? o Stage II is the stage that begins where AP is at its maximum and ends where TP is at its maximum. o Thus, the range of X values or Stage II is 15 and

A Hypothetical Production Function A Mathematical Example X Y TPP X Y APP MPP 0 10 III I II o Given o TP = X 2 – (1/30)X 3, o AP = TP/X = 2X – (1/30)X 2 o MP = ∂TP/∂X = 2X – (1/10)X 2 o At what level of X does the Law of Diminishing Returns set in? o It sets in where MP reaches its maximum. o Thus at X = 10 the law of Diminishing returns sets in. 1520

A Hypothetical Production Function A Mathematical Example o Given TP = X 2 – (1/30)X 3, o At Y = and X = 15, what is the elasticity of production? o Applying o E p = (2X – (1/10) X 2 ) * (X/Q) o E p = ((2*15) – (225/10)) * (15/112.5) o E p = (30 – 22.5) * (0.133) o E p = 7.5 * = 0.997