Download - Midterm Exam Review AAE 575 Fall 2012
Midterm Exam ReviewAAE 575Fall 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 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
• 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 f12 = 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 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
0
5,000
10,000
15,000
20,000
0 2 4 6 8 10 12 14 16
0
500
1000
1500
2000
2500
3000
0 2 4 6 8 10 12 14 16
x
y
MP
1)Output max is where MP = 0, x = xymax
2)Profit Max is where MP = r/p, x = xopt
r/p
x xopt xymax
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
x1
x2
Isoquant y = y0
-r1/r2
x1*
x2*
= -MP1/MP2
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