Production theory Harald Wiese
Transcript of Production theory Harald Wiese
MicroeconomicsProduction theory
Harald Wiese
Leipzig University
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Structure
Introduction
Household theoryTheory of the firm (SH 43)
Production theoryCostProfit maximization
Perfect competition and welfare theory
Types of markets
External effects and public goods
Pareto-optimal review
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IntroductionProduction process
production
factorsproduction output
(SH 29)
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Overview
Introduction
Production function
Partial factor variation
Proportional factor variation
Isoquants and marginal rate of technical substitution
Overview: factor variations
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Production function
states the maximum output of a good that can be producedusing given quantities of production factors:
y = f (x1, x2) .
y : produced quantity
x1, x2 : quantities of the (two) production factors
ProblemWhat is the difference between ordinal and cardinal utility theory? Isproduction theory ordinal or cardinal?
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Production function
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Production functionAxioms
ProblemCan the completeness axiom be transferred to production theory?
ProblemIs transitivity satisfied in production theory?
Monotonicity is satisfied if throwing away production factors isfor free.
Convexity can also be transferred to production theory.
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Partial factor variationTerms
Total factor variation: All factors are varied.Partial factor variation: Only one factor is varied.
Marginal productivity (MP):
MP1 =∂y
∂x1Analogy in utility theory?Average productivity (AP):
AP1 =y
x1
Problem1000 workers produce 5000 cars in one month.Average productivity? Unit of measurement?
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Partial factor variation
ProblemHow should one define the production elasticity of a factor?
ProblemExpress the production elasticity as a function of average productivityand marginal productivity!
Problem
Production elasticity of the first factor for y = cxa1xb2 with
a, b, c > 0?
ProblemUnder which conditions does average productivity increase?
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Partial factor variationSato production function
1
1
1
1
average productivity
marginal productivity
output
ProblemWhere do you see:
marginal productmax.
average productmax.
marginal product> average product
marginal product= average productmarginal product 0(SH 45f)
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Partial factor variationDiminishing returns
The marginal product of any production factor increases, remainsconstant and then decreases (can be negative).
ExampleSato production function
y = f (x1, x2) =xa1x
b2
(x1 + x2)a+b−1
,
where a, b > 1
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Proportional factor variationReturns to scale
Definition (constant returns to scale)
f (tx1, tx2) = tf (x1, x2) (t > 1)
Definition (increasing returns to scale)
f (tx1, tx2) > tf (x1, x2) (t > 1)
Definition (decreasing returns to scale)
f (tx1, tx2) < tf (x1, x2) (t > 1)
ProblemReturns to scale for f (x1, x2) = 2x1 + x2 or f (x1, x2) = x1x2?
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Proportional factor variationScale elasticity
Definition (scale elasticity)
εy ,t =
df (tx1,tx2)f (tx1,tx2)
dtt
∣∣∣∣∣∣t=1
=df (tx1, tx2)
dt
t
f (tx1, tx2)
∣∣∣∣t=1
Problem
Scale elasticity for a Cobb-Douglas production function y = xa1xb2 ?
ProblemFor a Cobb-Douglas production function the scale elasticity equalsthe sum of ...?
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Proportional factor variationReturns to scale and scale elasticity
decreasing returns to scale increasing returns to scale1
,
constant
returns to scale
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Proportional factor variationHomogeneity
A production function is called homogeneous of degree ν if
f (tx1, tx2) = tνf (x1, x2) .
Homogeneous production functions with ν = 1 are calledlinearly homogeneous. (= constant returns to scale)
Scale elasticity for homogeneous production functions = ν.
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Isoquants
1
2
3
2
1
ProblemIllustrate increasing returns to scale!Illustrate technological progress! (K 295)
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Marginal rate of technical substitution
is the absolute value of the slope of an isoquant.
states how many units of factor 2 can be waived if oneadditional unit of factor 1 is used and if output is held constant.
ProblemHousehold theory:
MRS =MU1
MU2
Hence, production theory
MRTS =
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Production functions without factor substitution
correspond to perfect complements in household theory
ProblemBartender Harley needs
2 deciliter rum (x1) and
6 deciliter cola (x2)
for a big cola with rum (y)
1 Isoquants for 2 big colas with rum?
2 Production function?
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Overview factor variations
isoquant partial
isoclineproportional
1
2
1
2
1
2
1
2
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Central tutorial I
Problem I.7.1.Constant returns to scale?
a) y = f (K , L) = K12L
23
b) y = f (K , L) = 3K12L
12
c) y = f (K , L) = K12 + L
14
d) y = f (K , L) = 2K + 3L
Problem I.7.2.Production function f (x1, x2) = (2x1 + 4x2)
12
Marginal rate of technical substitution?
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Central tutorial II
Problem I.7.3.Cobb-Douglas production function y = f (x1, x2) = Axa1x
b2 with
A, a, b > 0
a) Marginal product of factor 1?
b) Production elasticity for factor 1?
c) Scale elasticity?
d) MRTS?
e) Parameter values for
constant,decreasing, orincreasing returns to scale?
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