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Advanced Microeconomics

Ivan Etzo

University of Cagliari

ietzo@unica.itDottorato in Scienze Economiche e Aziendali, XXXIII ciclo

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Overview

1 Firms behavior and technology

2 Production Plan and Technology set

3 Isoquants

4 Marginal (Physical) Products

5 Returns-to-Scale

6 The technical rate of substitution (TRS)

7 The elasticity of substitution

8 The Constant Elasticity of Substitution (CES) production function

9 Well-Behaved Technologies

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Firms behavior and technological constraints

The firms behavior is expressed by the choices made by the firms

Choices are not free, there are some constraints coming from:

Customers (Demand);Competitors (Market structure);Nature (Technology);

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Technology

At the moment the firm makes the production decisions it faces sometechnological constraints

The technology describes the feasible ways to produce the outputsfrom inputs at a certain point in time ( t ).

Inputs or factors of production: land, labor, capital, raw materials(e.g. energy)

Often, different technologies will produce the same product

Usually capital and labor are considered as the two main broadcategories of inputs

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Defining capital

Capital goods (or Physical capital): inputs which are themselvesan output (i.e. they have been produced too).

Financial capital: the amount of money which is required to start (ormaintain) a business.

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The production set

Technological constraints imply that a given amount of output canbe produced only with certain combination of inputs.

The production set represents all combinations of inputs andoutputs that are technologically feasible.

Which technology is best?

How do we compare technologies?

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Input bundles

xi denotes the amount used of input i ( i.e. the level of input i).

An input bundle is a vector of the input levels; (x1, x2, · · · , xn).

E.g. (x1, x2, x3, , · · · , xn) = (6, 0, 9, · · · , 3).

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Production Functions

y denotes the output level.

The technology’s production function states the maximum amountof output possible from an input bundle.

y = f (x1, · · · xn)

Inputs and outputs are usually measured in flow units (i.e., hoursworked per week, machines hours per week → units of output perweek)

yt = f (x1t , · · · xnt)

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Production FunctionsOne input, one output

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Technology Sets

A production plan is an input bundle and an output level;(x1, · · · , xn, y).

A production plan is feasible if

y ≤ f (x1, · · · xn)

The collection of all feasible production plans is the technology set.

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Technology SetOne input, one output

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Technology SetOne input, one output

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Technologies with Multiple Inputs

What does a technology look like when there is more than one input?

The two input case:Input levels are x1 and x2.Output level is y .

Suppose the production function is

y = f (x1, x2) = 2x1/31 x

1/32

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Technologies with Multiple Inputs

E.g. the maximal output level possible from the input bundle

(x1, x2) = (1, 8) is

y = f (x1, x2) = 2× 11/3 × 81/3 = 2× 1× 2 = 4

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Technologies with Multiple Inputs

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Technologies with Multiple Inputs

The y output unit isoquant is the set of all input bundles that yieldexactly the same output level y .

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Isoquants with Two Variable Inputs

Isoquants can be graphed by adding an output level axis anddisplaying each isoquant at the height of the isoquants output level.

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More isoquants tell us more about the technology.

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More isoquants tell us more about the technology.

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Technologies with Multiple Inputs

The complete collection of isoquants is the isoquant map.

The isoquant map is equivalent to the production function – each isthe other.

E.g.

y = f (x1, x2) = 2x1/31 x

1/32

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Technologies with Multiple Inputs

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Technologies with Multiple Inputs

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Technologies with Multiple Inputs

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Technologies with Multiple Inputs

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Technologies with Multiple Inputs

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Technologies with Multiple Inputs

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Technologies with Multiple Inputs

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Technologies with Multiple Inputs

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Technologies with Multiple Inputs

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Technologies with Multiple Inputs

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Technologies with Multiple Inputs

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Technologies with Multiple Inputs

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Technologies with Multiple Inputs

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Technologies with Multiple Inputs

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Technologies with Multiple Inputs

Stop

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Cobb-Douglas Technologies

A Cobb-Douglas production function is of the form

y = Axa11 xa2

2 ·, · · · · xann

E.g.

y = x1/31 x

1/32

with n = 2, A = 1, a1 = 1/3, a2 = 1/3

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Cobb-Douglas Technologies

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Fixed-Proportions Technologies

A fixed-proportions production function is of the form

y = min{a1x1, a2x2, · · · anxn}

E.g.

y = min{x1, 2x2}with n = 2, A = 1, a1 = 1, a2 = 2

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Fixed-Proportions Technologies

y = min{x1, 2x2}

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Perfect-Substitutes Technologies

A perfect-substitutes production function is of the form

y = a1x1 + a2x2 + · · · + anxn

E.g.y = x1 + 3x2

with n = 2, a1 = 1, a2 = 3

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Perfect-Substitutes Technologies

y = x1 + 3x2

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Marginal (Physical) Products

The marginal product of input i is the rate-of-change of the output

level as the level of input i changes, holding all other input levels fixed.

MPi =∂y

∂xi

The marginal product of input i is diminishing if it becomes smaller

as the level of input i increases. That is, if

∂MPi

∂xi=

∂

∂xi

(∂y

∂xi

)=

∂2

∂x2i

< 0

Remember the law of diminishing marginal product.

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Returns-to-Scale

Marginal products describe the change in output level as a singleinput level changes.

Returns-to-scale describes how the output level changes as all inputlevels change in direct proportion (e.g. all input levels doubled, orhalved).

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Returns-to-Scale

Constant returns-to-scale

The technology described by the production function f (x) exhibits

constant returns-to-scale if, for any input bundle (x1, · · · xn),

f (kx1, kx2, · · · kxn) = kf (x1, x2, · · · xn)

i.e. the production function f (x) is homogeneous of degree 1.E.g. (k = 2) doubling all input levels doubles the output level.

Increasing returns-to-scale

f (kx1, kx2, · · · kxn) > kf (x1, x2, · · · xn)

Decreasing returns-to-scale

f (kx1, kx2, · · · kxn) < kf (x1, x2, · · · xn)

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Returns-to-Scale

A single technology can ’locally’ exhibit different returns-to-scale.

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Examples of Returns-to-Scale

Perfect-Substitutes production function

y = a1x1 + a2x2 + · · · + anxn

y = a1 (kx1) + a2 (kx2) + · · · + an (kxn)

= k (a1x1 + a2x2 + · · · + anxn)

= ky

⇒ Constant RtS

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Examples of Returns-to-Scale

Perfect-Complements production function

y = min{a1x1, a2x2, · · · anxn}

min{a1 (kx1) , a2 (kx2) , · · · an (kxn)}

k (min{a1x1, a2x2, · · · anxn})

⇒ Constant RtS

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Examples of Returns-to-Scale

Cobb-Douglas production

y = xa11 xa2

2 · · · xann

(kx1)a1 (kx2)a2 · · · (kxn)an

= ka1+··· +any

Thus:if∑n

i=1 ai = 1⇒ Constant RtSif∑n

i=1 ai > 1⇒ Increasing RtSif∑n

i=1 ai < 1⇒ Decreasing RtS

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Return to Scale and Marginal Product

A technology can exhibit increasing returns-to-scale even if all of itsmarginal products are diminishing.Why?

E.g.

y = x2/31 x

2/32

MPx1 =∂y

∂x1=

2

3x−1/31 x

2/32

∂MP1

∂x1= −1

3

2

3x−4/31 x

2/32 < 0

MPx2 = MPx1 < 0

RtS: a1 + a2 = 43 > 0 ⇒ Increasing RtS

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Return to Scale and Marginal Product

A marginal product is the rate-of-change of output as one input levelincreases, holding all other input levels fixed.

Marginal product diminishes because the other input levels are fixed,so the increasing inputs units have each less and less of other inputswith which to work.

Therefore, diminishing marginal product is a short run concept

RtS imply that all inputs vary, thus it refers to the long run

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The technical rate of substitution (TRS)

Assume y = f (x1, x2) and that the firm is producing y∗ = f (x∗1 , x

∗2 )

At what rate can a firm substitute one input for another withoutchanging its output level?

Using total differential of function f (x1, x2)

dy =∂f

∂x1dx1 +

∂f

∂x2dx2

imposedy = 0

∂f

∂x1dx1 +

∂f

∂x2dx2 = 0

dx2

dx1= −∂f /∂x1

∂f /∂x2

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The technical rate of substitution (TRS)

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The technical rate of substitution (TRS)Example: The Cobb-Douglas technology

Consider f (x1, x2) = xα1 x1−α2

TRS = −∂f /∂x1

∂f /∂x2= − α

1− αx2

x1

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The elasticity of substitution

The elasticity of substitution measures the curvature of an isoquant.

σ =

∆(x2/x1)x2/x1

∆TRSTRS

It measures the percentage change in the factor ratio divided by thepercentage change in TRS.

Using logarithmic derivative

ε =dlny

dlnx

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The elasticity of substitutionExample: The Cobb-Douglas technology

Consider f (x1, x2) = xα1 x1−α2

we know thatTRS = − α

1− αx2

x1

→ x2

x1= −1− α

αTRS

taking logs and deriving with respect to | TRS |

lnx2

x1= ln

1− αα

+ ln | TRS |

σdln (x2/x1)

dln | TRS |= 1

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The CES production function

The Constant Elasticity of Substitution has the following form

y = [a1xρ1 + a2xρ2 ]1ρ

the CES exhibits constant returns to scale

It contains other production functions as a special cases1 If ρ = 1⇒ Perfect-substitutes production function2 If ρ = 0⇒ Cobb-Douglas production function3 If ∞ = − ⇒ Fixed-proportion production function

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The CES production function

1 ρ = 1⇒ y = a1x1 + a2x2

2 ρ = 0 Consider the TRS of the CES produciton function

TRS = −a1

a2

(x1

x2

)ρ−1

when ρ→ 0 then the TRS tends to the following limit

TRS = −a1

a2

(x1

x2

)that is, the TRS of the Cobb-Douglas production function (i.e. theisoquant is the same).

3 ρ =∞ then the TRS of the CESTRS=0 if x2 > x1 ; TRS =∞ if x1 > x2

i.e. Just as the TRS of the Fixed-proportion production function.

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The CES production function

Proof: the CES has a constant elasticity of substitutionWe need to compute the elasticity of substitution σFrom the TRS of the CES (where, for simplicity, a1 = a2 = 1)

TRS = −(

x1

x2

)ρ−1

x2

x1=| TRS |

11−ρ

taking logs

lnx2

x1=

1

1− ρln | TRS |

and applying the definition of σ using the log derivative

σ =dln(x2/x1)

dln | TRS |=

1

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Well-Behaved Technologies

A well-behaved technology is

monotonicconvex.

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Well-Behaved TechnologiesMonotonicity

Monotonicity: More of any input generates more output.

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Well-Behaved TechnologiesConvexity

Convexity: If the input bundles x and x both provide y units ofoutput then the mixture tx + (1− t)x provides at least y units ofoutput, for any 0 < t < 1.

Convexity implies that the | TRS | decreases as x1 increases.

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