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Midterm Exam Review AAE 575 Fall 2012

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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Midterm Exam Review AAE 575 Fall 2012

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  1. Midterm Exam ReviewAAE 575Fall 2012

  2. Goal Today • Quickly review topics covered so far • Explain what to focus on for midterm • Review content/main points as we review it

  3. 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?

  4. 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

  5. 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)

  6. 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

  7. 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

  8. 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

  9. Factor Interdependence: Technical Substitution/Complementarity • Technically Competitive f12 < 0 • Substitutes • Technically Complementary f12> 0 • Complements • Technically Independent f12 = 0 • Independent

  10. What to Skip • Returns to scale, partial input elasticity, elasticity of scale, homogeneity • Quasi-concavity • Input elasticity • Elasticity of factor substitution • Isoclines and ridgelines

  11. 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

  12. Production Functions • Linear, Quadratic, Cubic • LRP, QRP • Negative Exponential • Hyperbolic • Cobb-Douglas • Square root • Intercept = ?

  13. 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

  14. 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

  15. 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

  16. Graphics x2 Isoquant y = y0 -r1/r2 = -MP1/MP2 x2* x1 x1*

  17. 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

  18. 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

  19. 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

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