Technical Aspects of Production
One Output, One (Variable) Input Production Function
Reading: Beattie and Taylor, Chapter 2.1 (pp. 9-16)
Notation
Output y What the firm sells or uses as input in another production process
(e.g. crop, livestock)
Input xi What the firm buys and/or uses to produce y
Variable inputs: fertilizer, pesticide, labor
Fixed inputs: land, human capital
Functional Representation
For various analyses we often simplify problem by ignoring fixed inputs, assume all variable inputs are used at optimal levels, and focus on one variable input x1, to write:
Examples without plateaus
Quadratic:
Polynomial:
Square Root
Cobb-Douglas
Examples with plateaus
(Negative) Exponential Mitscherlich-Baule
Exponential-Power generalized Mitscherlich-Baule
Spillman
von Liebig (LRP or QRP)
Hyperbolic
Questions to think about/discuss
Where get functional forms? Biological or Ecological theory? Let data speak?
Does an Intercept make sense or not?
Should there be a maximum output/yield?
Typical Properties of Production functions
1. Non-Negative Output: y ≥ 0 for all x [Level]
Strict case: Positive output y > 0 for all x
y(x0) ≥ 0 for all x0 ≥ 0 (strict case: y(x0) > 0 for all x0 > 0)
2. Non-Negative Slope , at least in region where operate [Slope]
Strict case: positive slope
Non-differentiable: y(x1) ≥ y(x0) for all x1 ≥ x0 (strict case: y(x1) > y(x0) for all x1 > x0)
Exception: phyotoxic effects, nutrient toxicity, etc.
3. Concavity: as use more input, eventually extra output generated by extra input begins to decrease: Law of Diminishing Marginal Product [Curvature]
, strict case
Non-differentiable:
for all x0 and x1 and 0 ≤ a ≤ 1 (Concave)
for all x0 and x1 and 0 ≤ a ≤ 1 (Strict Case)
Concave, differentiable Concave, non-differentiable
Strictly concave linear section: only concave, not strictly concave
Quasi-concavity: less strict than concavity, allows convex regions, so sigmoidal functions acceptable. Allows biologically relevant cases that will still be economically meaningful
For all f(x0) ≥ = t and f(x1) ≥ = t, for 0 ≤ a ≤ 1 (quasi-concave)
For all f(x0) ≥ = t and f(x1) ≥ = t, for 0 < a < 1 (strict case)
Quasi-concave not quasi-concave
Convex section okay convex section make sit non-quasi-concave
Technical Terms
Total Product: Another term for output (yield or meat)
Marginal Product: “Output from last unit of input”
Derivative formulation: MP =
Discrete formulation: MP =
Average Product: AP = “Average output per unit of input”
MP at x0 = Slope of the TP function at x = x0
AP at x0 = Slope of the line from origin to TP function at x = x0 (Prove graphically)
AP: Input use efficiency: nutrient use efficiency, water use efficiency, etc.
Numerical Example:
Polynomial: = 2.5x + 3.5x2 – 0.1x3
MP = = 2.5 + 7x – 0.3x2
AP = = 2.5 + 3.5x – 0.1x2
Graphical Presentation of “Classical Production Function”
Evaluate MP and AP at x = 10
y = 2.5x + 3.5x2 – 0.1x3 = 25 + 350 – 100 = 275
AP = 2.5 + 3.5x – 0.1x2 = 2.5 + 35 – 10 = 27.5
At x = 10, on average 27.5 units of output are generated per unit of input.
Input or Factor Elasticity
Economists like to use elasticities: unitless measures of responsiveness
· If A increases by z %, what % does B increase?
· Numerical measure of how responsive B is to changes in A
· Unitless: so does not matter what units used to measure A and B, get same result
If input use increases z%, what % does output increase?
Implications:
· MP > AP > 1 Output increases more than proportionally with x
· MP = AP = 1 Output increases proportionally with x
· MP < AP < 1 Output increases less than proportionally with x
MP = Apply Product Rule: MP =
Implications:
· When AP rising MP > AP
· When AP flat MP = AP
· When AP falling MP < AP
Relationship between AP and MP will matter for economics and input use efficiency
(Nutrient use efficiency, water use efficiency, etc.)
We will find that it is only economical to use inputs at levels only where AP is flat or decreasing (where MP < AP). AP maximum will occur at competitive optimum of zero economic profit, so if farmers earning above normal rates of return (positive economic profit), then MP < AP, AP falling, output price increases will decrease input use efficiency, etc.
More on this later when we cover economics of production.
One Output, Two (Variable) Inputs
(Beattie and Taylor Section 2.2: pp. 16 ff)
Notation: , again assuming all other inputs fixed or at optimal levels
Three types of relationships:
a) Factor-Output Relationship (One Output, One Input Production Function))
b) Factor-Factor Relationship (Isoquants)
c) Scale Relationship (Proportional increase of all inputs versus output)
See text Figure 2.3
Figures from 317 Text:
Graphical Intuition for:
One Output, One Input Production Function and “classical” shape
Isoquants as “Level Sets,” i.e. contour lines
Scale Relationship
Factor-Output Relationships
Average Product: One for each input
Marginal Product: One for each input
Note: In general the AP and MP functions are depend on both inputs
Example: Generalized Cobb-Douglas
Text works through some other examples: linear, quadratic, transcendental, CES
Good problem set or exam questions
Factor-Factor Relationships
Isoquant: curve or function showing all combinations of inputs that can produce the same level of output (level set).
Show 317 figure deriving isoquants graphically from 3-diminensional production surface
Intuition: An isoquant is x2 as a function of x1 and output y: Solve for x2
For some functions, can solve for equation for isoquant directly, the
Generalized Cobb-Douglas:
Numerical example: Isoquant for y = 10 and b1 = b2 = ½ and a = 1
i.e. isoquant is rectangular hyperbola
Negative exponential gives linear isoquant
(do math on own)
Can I find one for which you can only get implicit equation for x2 as function of x1?
Marginal Rate of Technical Substitution: MRTS:
Rate at which one input must be substituted for the other input to keep output unchanged.
MRTS12 = Rate at which x1 can be substituted for x2 and leave output unchanged:
MRTS12 = = Slope of the isoquant < 0
MRTS12 is negative because isoquant has negative slope
Intuitively: if x2 decreases (a negative change), then x1 must increase to keep output unchanged, thus MRTS12 is negative.
Often texts/articles multiply MRTS by minus 1 so it is a positive number
Use a graph to show intuition of MRTS as
More rigorous mathematical formulation:
Totally differentiate production function:
Along an isoquant, by definition output does not change, so dy = 0
Rearrange to solve for , the slope of isoquant:
Summary: MRTS12 =
Notice derivative wrt x1 in numerator, and derivative wrt x2 in denominator
Generalized Cobb-Douglas:
MRTS12 =
MRTS12 = Implies constant MRTS along proportional expansion path
Numerical example: b1 = b2 = ½ and a = 1
MRTS12 =
Mathematics of Curvature
Production Functions should have positive Marginal Products for each input: fi > 0
Restrictions are placed on the curvature of production functions to obtain economically sensible results for “rational” decision-makers (second order conditions for profit max).
Often assume strict concavity, which in terms of derivatives for two input production function requires , , and .
Most of the production functions presented the first day of class satisfy these properties with simple restrictions on the parameters: e.g. Cobb-Douglas , a > 0, b £ 1.
I will use this and expect everyone to know how to check the second order conditions.
Graphically, concavity implies a production surface that is a “bubble” or “dome.”
In some situations, less restrictive quasi-concavity is assumed, which in terms of derivatives for two input production function requires .
I will not explain where this comes from, nor will I use this.
Graphical intuition is sufficient in this class: quasi-concavity implies “bell-shaped” hill that flares out at the bottom.
Note:
· Concave functions are also quasi-concave
· Both strict concavity and strict quasi-concavity imply isoquants convex to the origin.
Discussion of Curvature/Convexity of Isoquants:
Text shows derivation of the curvature (second derivative) of an isoquant using the total differential. You cannot simply take derivative of –MP1/MP2 because must assume y is held constant. See discussion and footnote, pp. 24-26.
Assuming the standard strictly quasi-concave production function (or more restrictive: strictly concave production function) implies that isoquants are convex to the origin (the way they are usually drawn).
Show with graph why the other cases (positive sloping and/or concave isoquants) do not make economic sense as long as have free disposal of inputs.
Three Cases that matter economically
1) Imperfect Substitutability (isoquants convex to origin)
2) Perfect Substitutability (linear isoquants)
3) No Substitutability (right angle isoquants)
Case 1: Factors can be substituted, but the MRTS is diminishing.
Implies that the “return” to using an input as a substitute decreases, or
as you want to substitute away from one of the inputs, you have to use more and more of the substituting input.
Case 2: Factors can be perfectly substituted, the MRTS remains constant.
Once put into economic decision making (profit max) situation, leads to corner solutions where only one of the inputs is used.
Case 3: (Linear) Leontief production function or fixed proportion production function:
: No input substitutability.
Inputs are used in constant proportions and output follows an expansion path that is a ray from origin with slope a/b.
Work through example with a = b = 1.
Note: Calculus cannot be used because the function is non-differentiable.
Minor Topics Concerning Isoquants:
1) Elasticity of Factor Substitution (s) SKIP/very fast
2) Isoclines and Ridgelines
Elasticity of Factor Substitution (s) SKIP
Measures the substitutability between inputs along the isoquant, which is inversely related to the curvature of isoquant (remember MRTS measures its slope).
Definition: percent change in factor ratio (x1/x2) over the percent change in RTS:
How does the factor ratio change as the isoquant’s slope changes?
Show graphically difference between large and small s:
Small change in slope associated with a large change in factor ratio gives a very flat isoquant (little curvature) with lots of substitutability between inputs: large s
Large change in slope associated with a small change in factor ratio gives a very curved isoquant with little substitutability between inputs: small s
Linear isoquants: little curvature, lots input substitutability, large s
Right-angle isoquants: lots of curvature, little input substitutability, small s
Text derives the general formula (page 30): quite complex function of first and second partial derivatives and input levels. We will not do so here.
Note: in general s depends on level of inputs (and thus on output) so that should expect variable elasticity of factor substitution as you change the input mix and/or output.
However, there is a special class of production functions that exhibit constant elasticity of substitution (CES). Most common example is the Cobb-Douglas, for which s = 1.
What good is this elasticity s?
How does changing the MRTS affect the input ratio?
We will show that the profit maximizer will equate the MRTS to the input price ratio
–(r1/r2), so s then is the elasticity of the input ratio x1/x2 to the input price ratio r1/r2, or the responsiveness of relative input use to relative factor prices.
Flat/Linear Isoquants: Very responsive to input price changes: large s
Sharp/Right Angle Isoquants: Unresponsive to input price changes: small s
Note: in general s depends on level of inputs (and thus on output) so that should expect variable elasticity of factor substitution as you change the input mix and/or output.
However, there is a special class of production functions that exhibit Constant Elasticity of Substitution (CES). Most common example is the Cobb-Douglas, for which s = 1.
Isoclines and Ridge Lines
Isocline: locus of points (here a line) for which isoquants have the same MRTS (i.e. slope).
Traces out input combinations for which isoquant slopes are all the same.
An “expansion path” for points having the same MRTS.
Mathematically: hold MRTS fixed and solve for x2 as function of MRTS and x1
MRTS12 = Solve for MRTS and x1
Show graphically what this looks like
Generalized Cobb-Douglas:
Last class it was shown MRTS12 =
Solve for x2: the isocline is A ray from the origin
Ridge Line: isocline for MRTS of zero or -¥ (i.e. undefined 1/0).
Defines the economically meaningful portion of the input space (x1,x2)
Basic Intuition:
· Ridge Lines “cut off” the portion of isoquants that bend backward or bend upward
· Needed for production functions that eventually have areas with negative marginal products (e.g. polynomial).
· “Well behaved” production functions have the axes as their ridge lines.
See the text Figures 2.9 and 2.12.
Factor Interdependence: Technical Substitutes and Complements
Difference between input substitutability and technical substitution/complementarity between inputs.
Input Substitutability:
· Concerns the substitution of inputs when output is held fixed along an isoquant
· Measured by MRTS (and/or Elasticity of Factor Substitution s)
· Inputs must be substitutable along a “well-behaved” isoquant
Technical Substitution/Complementarity:
· Concerns the interdependence of input use
· Does not hold output constant
· Measured by changes in marginal products
Technically Competitive/Complementary indicates how increasing the level of one input affects the marginal product of the other input
1) Technically Competitive: as x1 increases the marginal product of x2 decreases.
2) Technically Complementary: as x1 increases the marginal product of x2 increases.
3) Technically Independent: as x1 increases the marginal product of x2 does not change.
Competitive = Substitutes
Mathematical Condition: Determined by the sign of the cross partial derivative f12
Technically Competitive f12 < 0
Technically Complementary f12 > 0
Technically Independent f12 = 0
Intuition:
What does using more of x1 do to the productivity of x2?
Assume price x1 falls, so use more of it, how will use of x2 respond?