econ 4910 seminar 21 econ 4910 spring 2007 second seminar lecturer finn r. førsund

ECON 4910 seminar 2 1

ECON 4910 Spring 2007Second seminar

Lecturer Finn R. Førsund


The Meade externality models Consider the following externality models

found in Meade (1952): a) Unpaid factor b) Reciprocal c) Atmosphere


a) Unpaid factor Output from one activity creates an

externality in another production activity, using the same type of input, example is an apple orchard and a honey producer located together

Discuss the sign of

1 1 1

22 2 2 1

1

( )

( , ), 0

y f x

fy f x y

y

22

2 1

f

x y


Problem formulation for Pareto optimality Problem formulation

1 1 1

2 2 1 2

1 2

( )

( , ) o

o

Max y f x

subject to

f x y y

x x x


Solving the problem

The Lagrangian (eliminating y1, y2)

1 1

2 2 1 1 2

1 2

( )

( ( , ( )) )

( )

o

o

L f x

f x f x y

x x x


Solving the problem, cont.

First-order conditions

Assuming interior solutions for x1,x2

Marginal benefit (creating objective function value) equal to marginal cost (opportunity cost of using the input)

1

2

1 2, 11

2,2

0

0

y

x

Lf f f

x

Lf

x

1 21 2, 1 2,,y xf f f f


Solving the problem, cont. 1

Interpretation of the shadow prices The Envelope Theorem

λ: Increase in y1 when y2 decreases with one unit

γ: Increase in y1 when x increases with one unit

*1

2 2

*1

o o

o o

y L

y y

y L

x x


Solving the problem, cont. 2

Solution for 4 endogenous variables x1,x2, λ,γ: 2 first-order conditions plus 2 constraints

Forming the ratio of productivities

Value of ratio of marginal productivities different from the trade-off λ between y1 and y2

1 2

2 1

1 2, 1 2,

1

2, 2,1

y x

x y

f f f f

f

f f


Competitive market solution

Maximising profit taking output- and input prices p,q as given

1 1 1

1 1 1

2 2 2

2 2 2 1

{ }

.

( )

{ }

.

( ; )

Max p y qx

s t

y f x

Max p y qx

s t

y f x y


Competitive market solution, cont. First-order conditions

Ratio of marginal productivities equal to trade-off between outputs neglects the externality and implies too little of the resource used in activity 1 and too much in activity 2 when the externality is positive

1 1 2 2

1 2

12

,p f q p f q

f p

pf


Demonstration: comparing Pareto optimum with market solution Pareto optimum

λ is the trade-off between outputs

Assuming prices fixed λ = p2/p1

Market solution

p2/p1 is the market trade-off between outputs

2 1

1

2, 2,1x y

f

f f

1 2

12

f p

pf

2 1 1

1 2 1 2

2, 2 1 2, 1 2 2,

/

1 /x y y

f p p p

f p p f p p f


Implementing the Pareto-optimal solution Introducing a Pigou tax (subsidy) on the

activity generating an externality Profit maximising problem with subsidy

1 1 1

1 1 1

1 1

{( ) }

.

( )

( )

Max p t y qx

s t

y f x

p t f q


Implementing the Pareto-optimal solution, cont. The market solution with a Pigou subsidy

Finding the optimal subsidy rate

1 2

12

f p

p tf

1

1

1 2 2

12 1 2 2,

2 2,

y

y

f p p

p tf p p f

t p f


Policy discussion

Alternative: merging the firms The optimisation problem of the merged firm

Solution

1 1 2 2 1 2

1 1 1

2 2 2 1

{ ( )}

.

( )

( ; )

Max p y p y q x x

s t

y f x

y f x y

21 1 2 2, 1 1 2 2

1 2

2 1 2 2, 1

, ,y x

y

p f p f f q p f q

f p

f p p f


The case of reciprocal externality Reciprocal externalities imply that the relative

strength of the externalities determines the resource allocation

Setting up the optimisation problem for finding the Pareto solution we cannot eliminate the outputs y1, y2

An apparently innocent change makes the mathematics much more cumbersome

Two subsidies may be used in principle


The atmosphere externality case The Lagrangian for the Pareto problem

First-order conditions

1 1

1 1 2 2 2

1 2

( )

( ( ( )) ( ) )

( )

o

o

L f x

A f x f x y

x x x

1 1 2 21

22

( ) 0

0

Lf A f f x

x

LAf

x


The atmosphere externality case, cont. Forming the rate of productivities

The competitive market solution

1

2 221 ( )

f A

A f xf

1 2

12

f p A

pf


Implementing the Pareto-optimal solution Introducing a Pigou subsidy on the

externality-generating output from activity 1 The market solution with a subsidy

Finding the optimal subsidy rate setting market solution equal to optimal solution inserting for λ

1 2

12

f p A

p tf

1 2 22 2 2

1 1 2 2 22

( )( )

f p A p At p A f x

p t p p A f xf

econ 4910 seminar 21 econ 4910 spring 2007 second seminar lecturer finn r. førsund

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