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(x 1, x 2 ), 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(x 1, x 2 ) as x 2 = g(x 1, y)

Factor-Factor Relationships: Isoquants Isoquant slope dx 2 /dx 1 = Marginal rate of technological substitution (MRTS) How calculate MRTS? Ratio of Marginal production MRTS = dx 2 /dx 1 = –f 1 /f 2 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 x 1 decreases marginal product of x 2 Technically Complementary: increasing x 1 increases marginal product of x 2 Technically Independent: increasing x 1 does not affect marginal product of x 2

Factor Interdependence: Technical Substitution/Complementarity Technically Competitivef 12 < 0 – Substitutes Technically Complementaryf 12 > 0 – Complements Technically Independentf 12 = 0 – Independent

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 f 12 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):  (x) = pf(x) – rx – K First Order Condition (FOC) –  ’(x) = 0 and solve for x – Get pMP = r or MP = r/p Second Order Condition (SOC) –  ’’(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

x y MP 1)Output max is where MP = 0, x = x ymax 2)Profit Max is where MP = r/p, x = x opt r/p x x opt x ymax

Economics of Optimal Input Use Multiple Inputs  x 1,x 2  = pf(x 1,x 2 ) – r 1 x 1 – r 2 x 2 – K FOC’s: d  /dx 1 = 0 and d  /dx 2 = 0 and solve for pair (x 1,x 2 ) – d  /dx = pf 1 (x 1,x 2 ) – r 1 = 0 – d  /dy = pf 2 (x 1,x 2 ) – r 2 = 0 SOC’s: more complex f 11 0 Be able to implement this model for simple production function Read discussion in notes: what it all means

Graphics x1x1 x2x2 Isoquant y = y 0 -r 1 /r 2 x1*x1* x2*x2* = -MP 1 /MP 2

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 –  (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  no = PY(1 – no ) – G  trt = PY(1 – trt ) – C trt – G  no = PY no (1 –  N) – G  trt = PY trt (1 –  N(1 – k)) – C trt – G Set  no =  trt and solve for N EIL = C trt /(PY  k) Treat if N > N EIL, 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 = –f 1 /f 2 vs f 12 Economic optimal input use – Single and multiple inputs (continuous) – Discrete, mixed inputs, and thresholds