Midterm Exam Review AAE 575 Fall 2012

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

6. 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)

7. 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

8. 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

9. 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

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

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

12. 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

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

14. 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

15. 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

16. 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

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

18. 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

19. 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

20. 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