Production theory Harald Wiese

MicroeconomicsProduction theory

Harald Wiese

Leipzig University

Harald Wiese (Leipzig University) Production theory 1 / 21

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


IntroductionProduction process

production

factorsproduction output

(SH 29)


Overview

Introduction

Production function

Partial factor variation

Proportional factor variation

Isoquants and marginal rate of technical substitution

Overview: factor variations


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?


Production function


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.


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?


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?


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)


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


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?


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 ...?


Proportional factor variationReturns to scale and scale elasticity

decreasing returns to scale increasing returns to scale1

,

constant

returns to scale


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.


Isoquants

1

2

3

2

1

ProblemIllustrate increasing returns to scale!Illustrate technological progress! (K 295)


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 =


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?


Overview factor variations

isoquant partial

isoclineproportional

1

2

1

2

1

2

1

2


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?


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?


Documents

Production theory Harald Wiese