3 producer update.2015.12.03

Special production functions - CES
CES - constant elasticity of substitution Elasticity of substitution dln(xj /xi ) dln(xj /xi ) = = σij (x ) = dlnMRTSij dln(MPi /MPj )
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Roadmap
Production function
Cost function
Duality
Profit maximization - competitive case
CES, continue
• General form of CES
y σ ρ<1
= = and
α1 x1 + α2 x2 1 1−ρ ρ 0
Roadmap
Production function
Cost function
Duality
Profit maximization - competitive case
Special production functions - Separable
Separable production functions • Classify inputs into a small number of groups g1 = (x1 , x2 , ..., xn1 ), g2 = (xn1 +1 , ..., xn2 ), ..., gm = (xnm−1 +1 , ..., xn ) Then allow within-group substitutability to be different from between-group substitutability. • Weak: within group subst. indpt. of inputs in other groups ∂ MRTSij ∂xk = ∂ MPi /MPj ∂xk = 0 for ∀i, j ∈ gs and k gs
• Continuous • Strictly increasing • f (0) = 0 • Strictly quasi-concave: complementarity between inputs • Strongest complementarity - Leontief production function/fixed proportion technology f (x1 , x2 ) = min(αx1 , βx2 ) - when αx1 = βx2 , one can not produce more by increasing x1 or x2 ALONE. • Weakest complementarity - linear production function/perfectly substitutable inputs f (x1 , x2 ) = αx1 + βx2 - no matter what the combination of inputs, one can always β substitute α units of x1 for 1 unit of x2 .
Roadmap
Production function
Cost function
Duality
Profit maximization - competitive case
Production function
• Technological feasibility: production possibility set • Production plan • a vector of inputs and outputs y = (y1 , .., ym ) ∈ Y • inputs (-) and outputs (+) • Single output production function: f :
Roadmap
Production function
Cost function
Duality
Profit maximization - competitive case
Firm’s problem
A firm − purchases inputs from input markets: cost − produces and sells its product on the output market: revenue − with the goal of maximizing its profit: revenue-cost
another (holding other inputs and output constant). • Relate to the curvature of the isoquants - how fast does MRTS diminish along an isoquant Strong(weak) substitutability: when increasing x1 and reducing x2 along one isoquant, MRTSij - the ability of x1 to substitute for x2 - drops a little(a lot) • σ ∈ [0, +∞)
• Strong: subst. between any two inputs from gs and gt indpt. of
inputs in groups other than gs and gt ∂ MRTSij ∂xk = ∂ MPi /MPj ∂xk = 0 for ∀ i ∈ gs , j ∈ gt and k gs ∪gt
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- when gs = gt , it is the same as the case of weak separation
Roadmap
Production function
Cost function
Duality
Profit maximization - competitive case
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Roadmap
Production function
Cost function
Duality
Profit maximization - competitive case
Graphical presentation of production function
• Isoquant: similar to indifference curves • iso - equal • quant - quantity • take total derivative of the production function at a given output level to find the slope • Quasi-concave ⇒ convex upper contour set
⇒ isoquants convex to the origin ⇒ diminishing slope when increasing along the x-axis
• Marginal rate of technical substitution (MRTS)
MRTSij =
MPi MPj
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n +
→
+
y = f (x) = f (x1 , ..., xn ) where x ≥ 0 and y ≥ 0
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Roadmap
Production function
Cost function
Duality
Profit maximization - competitive case
Properties of promap
Production function
Cost function
Duality
Profit maximization - competitive case
Returns to scale
• Returns to scale: long run concept • Global returns to scale: for all t > 1 and all x • Increasing: f (tx) > tf (x) ⇐ homogeneous of degree α and α > 1 • Constant: f (tx) = tf (x) ⇔ homogeneous of degree 1 • Decreasing: f (tx) < tf (x) ⇐ homogeneous of degree α and α < 1 • Homogeneous production function of degree α ≤ 1 is concave • When α = 1, it is called linear homogeneous function • Linear homogeneous production function is concave • h(x) = g(f (x)) is concave if g and f are both concave functions
MC = MR
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Roadmap
Production function
Cost function
Duality
Profit maximization - competitive case
Profit maximizing rule
MC = MR ∂ ∂ TC(y) = TR(y) ∂y ∂y ∂ ∗ wi (y)xi (y) = ∂y (p(y)y)