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
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•
•
•
•
•
•
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