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Midterm Exam Review AAE 575 Fall 2012. Goal Today. Quickly review topics covered so far Explain what to focus on for midterm Review content/main points as we review it. Technical Aspects of Production.

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Presentation Transcript
goal today
Goal Today
  • Quickly review topics covered so far
  • Explain what to focus on for midterm
  • Review content/main points as we review it
technical aspects of production
Technical Aspects of Production
  • What is a production function? What do we mean when we write y = f(x), y = f(x1, x2), etc.?
  • What properties do we want for a production function
    • Level, Slope, Curvature
    • (Don‘t worry about quasi-concave)
    • (Don’t worry about input elasticity)
  • Marginal product and average product
    • Definition/How to calculate
    • What’s the difference?
technical aspects of production multiple inputs
Technical Aspects of ProductionMultiple Inputs
  • Three relationships discussed
    • Factor-Output (1 input production function)
    • Factor-Factor (isoquants)
    • Scale relationship (proportional increase inputs)
    • (Don’t worry about scale relationship)
  • How do marginal products and average products work with multiple inputs?
    • MPs and APs depend on all inputs
factor factor relationships isoquants
Factor-Factor Relationships: Isoquants
  • What is an isoquant?
    • Input combinations that give same output (level surface production function)
    • Graphics for special cases: imperfect substitution, perfect substitution, no substitution
  • How to find isoquant for a production function?
    • Solve y = f(x1, x2) as x2 = g(x1, y)
factor factor relationships isoquants1
Factor-Factor Relationships: Isoquants
  • Isoquant slope dx2/dx1 = Marginal rate of technological substitution (MRTS)
  • How calculate MRTS? Ratio of Marginal production MRTS = dx2/dx1 = –f1/f2
  • Don’t worry about elasticity of factor substitution
  • Don’t worry about isoclines and ridgelines
factor interdependence technical substitution complementarity
Factor Interdependence: Technical Substitution/Complementarity
  • What’s the difference between input substitutability and technical substitution/complementarity?
  • Input Substitutability
    • Concerns substitution of inputs when output is held fixed along an isoquant
    • Measured by MRTS
    • Inputs must be substitutable along a “well-behaved” isoquant
  • Technical Substitution/Complementarity
    • Concerns interdependence of input use
    • Does not hold output constant
    • Measured by changes in marginal products
factor interdependence technical substitution complementarity1
Factor Interdependence: Technical Substitution/Complementarity
  • Indicates how increasing one input affects marginal product (productivity) of another input
  • Technically Competitive: increasing x1 decreases marginal product of x2
  • Technically Complementary: increasing x1 increases marginal product of x2
  • Technically Independent: increasing x1 does not affect marginal product of x2
factor interdependence technical substitution complementarity2
Factor Interdependence: Technical Substitution/Complementarity
  • Technically Competitive f12 < 0
    • Substitutes
  • Technically Complementary f12> 0
    • Complements
  • Technically Independent f12 = 0
    • Independent
what to skip
What to Skip
  • Returns to scale, partial input elasticity, elasticity of scale, homogeneity
  • Quasi-concavity
  • Input elasticity
  • Elasticity of factor substitution
  • Isoclines and ridgelines
problem set 1
Problem Set #1
  • What parameter restriction on a standard production function ensure desired properties for level, slope and curvature?
  • How to derive formula for MP and AP for single & multiple input production functions?
  • Deriving isoquant equation and/or slope of isoquant
  • Calculate cross partial derivative f12 and interpret meaning: Factor Interdependence
production functions
Production Functions
  • Linear, Quadratic, Cubic
  • LRP, QRP
  • Negative Exponential
  • Hyperbolic
  • Cobb-Douglas
  • Square root
  • Intercept = ?
economics of o ptimal i nput use
Economics of Optimal Input Use
  • Basic model (1 input): p(x) = pf(x) – rx – K
  • First Order Condition (FOC)
    • p’(x) = 0 and solve for x
    • Get pMP = r or MP = r/p
  • Second Order Condition (SOC)
    • p’’(x) < 0 (concavity)
    • Get pf’’(x) < 0 (concave production function)
  • Be able to implement this model for standard production functions
  • Read discussion in notes: what it all means
slide15

Output max is where MP = 0, x = xymax

  • Profit Max is where MP = r/p, x = xopt

r/p

y

x

MP

xopt

xymax

x

economics of optimal input use multiple i nputs
Economics of Optimal Input UseMultiple Inputs
  • p(x1,x2) = pf(x1,x2) – r1x1 – r2x2 – K
  • FOC’s: dp/dx1 = 0 and dp/dx2 = 0 and solve for pair (x1,x2)
    • dp/dx = pf1(x1,x2) – r1 = 0
    • dp/dy = pf2(x1,x2) – r2 = 0
  • SOC’s: more complex
  • f11 < 0, f22 < 0, plus f11f22 – (f12)2 > 0
  • Be able to implement this model for simple production function
  • Read discussion in notes: what it all means
graphics
Graphics

x2

Isoquant y = y0

-r1/r2

= -MP1/MP2

x2*

x1

x1*

special cases discrete inputs
Special Cases: Discrete Inputs
  • Tillage system, hybrid maturity, seed treatment or not
  • Hierarchical Models: production function parameters depend on other inputs: can be a mix of discrete and continuous inputs
    • Problem set #2: ymax and b1 of negative exponential depending on tillage and hybrid maturity
    • p(x,T,M) = pf(x,T,M) – rx – C(T) – C(M) – K
  • Be able to determine optimal input use for x, T and M
  • Calculate optimal continuous input (X) for each discrete input level (T and M) and associated profit, then choose discrete option with highest profit
special cases thresholds
Special Cases: Thresholds
  • When to use herbicide, insecticide, fungicide, etc.
    • Input used at some fixed “recommended rate”, not a continuous variable
  • pno = PY(1 – lno) – G
  • ptrt = PY(1 – ltrt) – Ctrt – G
  • pno = PYno(1 – aN) – G
  • ptrt = PYtrt(1 – aN(1 – k)) – Ctrt – G
  • Set pno = ptrt and solve for NEIL = Ctrt/(PYak)
  • Treat if N > NEIL, otherwise, don’t treat
final comments
Final Comments
  • Expect a problem oriented exam
  • Given production function
    • Find MP; AP; parameter restrictions to ensure level, slope, and curvature; isoquant equation
  • Input Substitution vs Factor Interdependence
    • MRTS = –f1/f2vs f12
  • Economic optimal input use
    • Single and multiple inputs (continuous)
    • Discrete, mixed inputs, and thresholds