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