Report

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 • 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 • 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 • 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: 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 • 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/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/Complementarity • Technically Competitive f12 < 0 – Substitutes • Technically Complementary f12 > 0 – Complements • Technically Independent – Independent f12 = 0 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 • 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 • • • • • • • Linear, Quadratic, Cubic LRP, QRP Negative Exponential Hyperbolic Cobb-Douglas Square root Intercept = ? 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 1) Output max is where MP = 0, x = xymax 20,000 r/p 15,000 y 10,000 5,000 0 0 2 4 6 0 2 4 6 x 8 10 12 10 12 14 16 14 16 3000 2500 MP 2000 1500 1000 500 0 x 8 xopt xymax 2) Profit Max is where MP = r/p, x = xopt Economics of Optimal Input Use Multiple 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 x2 Isoquant y = y0 -r1/r2 = -MP1/MP2 x2* x1* x1 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 • 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 • 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/f2 vs f12 • Economic optimal input use – Single and multiple inputs (continuous) – Discrete, mixed inputs, and thresholds