Hoorcollege
1
‘Market
equilibrium
and
efficiency’
o Key
assumptions
behind
the
models:
1. Perfect
competition:
consumers
and
firms
take
prices
as
given
2. Symmetric
information:
everybody
had
the
same
information
3. No
externalities:
transactions
only
affect
the
transacting
parties
4. Well-‐defined
property
rights:
negligible
(=
te
verwaarlozen)
transaction,
legal
and
contract
costs.
5. Perfect
rationality:
consumers
maximize
their
utility
and
firms
their
profits.
o The
exchange
economy
∗ The
model:
-‐ 2
households
and
2
goods:
all
results
generalize
to
settings
with
more
of
both.
-‐ Only
exchange:
the
total
stocks
of
the
2
goods
are
fixed.
ℎ ℎ
-‐ Household
h’s
endowment
of
goods
1
and
2:
𝜔
and
𝜔
1 2
ℎ ℎ
-‐ Prices
for
goods
1
and
2:
ρ1
and
ρ2,
so
household
h
could
effectively
spend:
𝜌1𝜔 +
𝜌2𝜔
1 2
ℎ ℎ
-‐
Consumption
on
the
two
good:
𝑥
and
𝑥 ,
so
budget
constraint:
1 2
-‐ Utility
depends
on
the
consumption
of
two
goods:
∗ Equilibrium
behaviour
-‐ Utility
maximization:
-‐ There
are
two
ways
of
solving
maximization
problems:
use
Langrange
or
BC.
-‐ Maximization
yields
following
expression
for
equilibrium
behaviour:
-‐ LHS:
the
marginal
rate
of
substitution
of
good
2
for
good
1
=
how
many
units
of
good
1
am
I
willing
to
give
up
to
get
one
more
unit
of
2?
→
Increasing
in
the
MU
(marginal
utility)
of
good
2
and
decreasing
in
the
MU
of
good
1
-‐ RHS:
the
relative
price
of
good
2
vs
good
1
=
how
many
units
of
good
1
do
I
need
to
give
up
to
get
one
more
unit
of
good
2?
→
Increasing
in
the
price
of
good
2
and
decreasing
in
the
price
of
good
1
-‐ We
can
illustrate
equilibrium
behaviour
in
an
Edgeworth
box.
∗
∗ Equilibrium
prices
-‐ Market-‐clearing
prices:
prices
are
in
equilibrium
when
demand
equals
supply
for
both
goods.
-‐ Disequilibrium:
, -‐
Je
ziet
dus
dat
de
budget
line
(slope:
-‐
ρ1/ρ2)
veranderd.
Market
mechanism
ensures
that
ρ1/ρ2
declines
(=dalen)
as
long
as
there
is
excess
supply
of
good
1
relative
to
good
2.
All
that
matters
for
equilibrium
are
relative
prices
ρ1/ρ2,
so
the
general
price
level
is
irrelevant
(doubling
all
prices
does
not
affect
market
equilibrium).
∗ Equilibrium
and
efficiency
-‐ Ensures
market
equilibrium
a
Pareto
efficient
allocation?
→
Pareto
efficiency:
an
allocation
is
Pareto
efficient
if
no
person
can
be
made
better
off,
without
making
someone
else
worse
off.
It
requires
that
utility
of
household
1
is
maximized
for
any
given
utility
of
household
2.
So
Pareto
efficiency
requires
that
the
allocation
satisfies:
Solving
the
above
maximization
problem
yields
the
following
condition
for
Pareto
efficiency:
→
LHS:
household
1’s
marginal
rate
of
substitution
of
good
2
for
good
1
=
how
many
units
of
good
1
is
hh1
willing
to
give
up
to
get
one
more
unit
of
good
2.
→
RHS:
household
2’s
marginal
rate
of
substitution
of
good
2
for
good
1
-‐ Recall
that
in
market
equilibrium,
every
household
equalizes
its
MRS
with
relative
price:
→
First
fundamental
theorem
of
welfare
economics:
the
allocation
of
commodities
at
a
competitive
equilibrium
is
Pareto
efficient.
o The
production
and
exchange
economy
→
How
do
results
carry
over
to
an
economy
with
production
sector?
∗ The
model
-‐ Economic
agents:
treat
as
if
there
is
only
one
consumer
and
one
firm.
-‐ There
are
two
goods:
1. Good
1:
x1<0,
is
defined
as
a
negative
and
represents
the
inputs
that
the
household
provides
to
the
firm
(the
more
negative
x1,
the
more
inputs
provided).
2. Good
2:
x2>0,
represents
the
firm’s
output
that
is
to
be
consumed
by
the
household.
-‐ Endowments:
ω1
=
ω2
=
0
∗ The
representative
firm
-‐ Production:
the
firm
transforms
inputs
into
outputs
according
to
the
production
function,
x2
=
f(-‐x1)
-‐ Inputs
are
sold
at
price
ρ1
and
outputs
are
sold
at
price
ρ2.
-‐ The
firm’s
profits
are
given
by:
π
=
ρ2
f(-‐x1)
+
ρ1x1
(or
π
=
ρ2
f
(L)
–
wL)
→
Equilibrium
firm
behaviour:
f’(-‐x1)
=
ρ1/ρ2
, LHS:
the
marginal
rate
of
transformation
of
inputs
(good
1)
into
outputs
(good
2)
=
how
many
units
of
output
can
I
produce
with
one
more
unit
of
input?
(decreasing
in
the
amount
of
inputs
already
employed
–x1)
RHS:
the
relative
price
of
inputs
vs
outputs
=
how
many
units
of
output
do
I
need
to
produce
to
be
able
to
afford
one
more
unit
of
input?
(increasing
in
the
price
of
inputs
ρ1
and
decreasing
in
the
price
of
outputs
ρ2)
-‐ Door
te
kijken
naar
de
production
en
de
profit
kom
je
tot
de
iso-‐profit
line:
∗ Equilibrium
production
∗ The
representative
household
-‐ Utility:
U
=
u
(x1,
x2)
-‐ Households
owns
the
firm
and
earns
its
profits
as
dividends,
so
budget
constraint:
(budget
constraint
=
iso-‐profit
line).
Maximization
of
utility
w.r.t.
x1
and
x2:
u1/u2
=
ρ1/ρ2
∗ Equilibrium
prices
-‐ Market-‐clearing
prices:
prices
are
in
equilibrium
when
demand
equals
supply
for
both
inputs
and
outputs.
-‐ In
the
next
figure
there
is
an
example
of
disequilibrium,
with
excess
demand
for
the
output
and
excess
supply
for
the
input.
→
Market
mechanism
ensures
that
ρ1/ρ2
declines
as
long
as
there
is
excess
demand
for
output
(good
2)
relative
to
input
(good
1).
→
Adjustments
lead
to
a
market-‐clearing
equilibrium.
∗ Equilibrium
and
efficiency
→
Does
market
equilibrium
in
a
model
with
production
yield
a
Pareto
efficient
allocation?
-‐ Efficiency:
that
the
combination
of
household
and
firm
decisions
lead
to
the
highest
possible
utility
for
the
representative
household.
→
Solving
for
the
efficient
allocation
yields:
LHS:
the
household’s
marginal
rate
of
substitution
RHs:
the
firm’s
marginal
rate
of
transformation
If
MRS>MRT,
utility
could
be
increased
by
producing
less.
If
MRS<MRT,
utility
could
be
increased
by
producing
more.
-‐ Market
equilibrium
must
satisfy
efficiency:
→
This
confirms
the
First
fundamental
theorem
of
welfare
economics
in
a
production
model
of
the
economy.
1
‘Market
equilibrium
and
efficiency’
o Key
assumptions
behind
the
models:
1. Perfect
competition:
consumers
and
firms
take
prices
as
given
2. Symmetric
information:
everybody
had
the
same
information
3. No
externalities:
transactions
only
affect
the
transacting
parties
4. Well-‐defined
property
rights:
negligible
(=
te
verwaarlozen)
transaction,
legal
and
contract
costs.
5. Perfect
rationality:
consumers
maximize
their
utility
and
firms
their
profits.
o The
exchange
economy
∗ The
model:
-‐ 2
households
and
2
goods:
all
results
generalize
to
settings
with
more
of
both.
-‐ Only
exchange:
the
total
stocks
of
the
2
goods
are
fixed.
ℎ ℎ
-‐ Household
h’s
endowment
of
goods
1
and
2:
𝜔
and
𝜔
1 2
ℎ ℎ
-‐ Prices
for
goods
1
and
2:
ρ1
and
ρ2,
so
household
h
could
effectively
spend:
𝜌1𝜔 +
𝜌2𝜔
1 2
ℎ ℎ
-‐
Consumption
on
the
two
good:
𝑥
and
𝑥 ,
so
budget
constraint:
1 2
-‐ Utility
depends
on
the
consumption
of
two
goods:
∗ Equilibrium
behaviour
-‐ Utility
maximization:
-‐ There
are
two
ways
of
solving
maximization
problems:
use
Langrange
or
BC.
-‐ Maximization
yields
following
expression
for
equilibrium
behaviour:
-‐ LHS:
the
marginal
rate
of
substitution
of
good
2
for
good
1
=
how
many
units
of
good
1
am
I
willing
to
give
up
to
get
one
more
unit
of
2?
→
Increasing
in
the
MU
(marginal
utility)
of
good
2
and
decreasing
in
the
MU
of
good
1
-‐ RHS:
the
relative
price
of
good
2
vs
good
1
=
how
many
units
of
good
1
do
I
need
to
give
up
to
get
one
more
unit
of
good
2?
→
Increasing
in
the
price
of
good
2
and
decreasing
in
the
price
of
good
1
-‐ We
can
illustrate
equilibrium
behaviour
in
an
Edgeworth
box.
∗
∗ Equilibrium
prices
-‐ Market-‐clearing
prices:
prices
are
in
equilibrium
when
demand
equals
supply
for
both
goods.
-‐ Disequilibrium:
, -‐
Je
ziet
dus
dat
de
budget
line
(slope:
-‐
ρ1/ρ2)
veranderd.
Market
mechanism
ensures
that
ρ1/ρ2
declines
(=dalen)
as
long
as
there
is
excess
supply
of
good
1
relative
to
good
2.
All
that
matters
for
equilibrium
are
relative
prices
ρ1/ρ2,
so
the
general
price
level
is
irrelevant
(doubling
all
prices
does
not
affect
market
equilibrium).
∗ Equilibrium
and
efficiency
-‐ Ensures
market
equilibrium
a
Pareto
efficient
allocation?
→
Pareto
efficiency:
an
allocation
is
Pareto
efficient
if
no
person
can
be
made
better
off,
without
making
someone
else
worse
off.
It
requires
that
utility
of
household
1
is
maximized
for
any
given
utility
of
household
2.
So
Pareto
efficiency
requires
that
the
allocation
satisfies:
Solving
the
above
maximization
problem
yields
the
following
condition
for
Pareto
efficiency:
→
LHS:
household
1’s
marginal
rate
of
substitution
of
good
2
for
good
1
=
how
many
units
of
good
1
is
hh1
willing
to
give
up
to
get
one
more
unit
of
good
2.
→
RHS:
household
2’s
marginal
rate
of
substitution
of
good
2
for
good
1
-‐ Recall
that
in
market
equilibrium,
every
household
equalizes
its
MRS
with
relative
price:
→
First
fundamental
theorem
of
welfare
economics:
the
allocation
of
commodities
at
a
competitive
equilibrium
is
Pareto
efficient.
o The
production
and
exchange
economy
→
How
do
results
carry
over
to
an
economy
with
production
sector?
∗ The
model
-‐ Economic
agents:
treat
as
if
there
is
only
one
consumer
and
one
firm.
-‐ There
are
two
goods:
1. Good
1:
x1<0,
is
defined
as
a
negative
and
represents
the
inputs
that
the
household
provides
to
the
firm
(the
more
negative
x1,
the
more
inputs
provided).
2. Good
2:
x2>0,
represents
the
firm’s
output
that
is
to
be
consumed
by
the
household.
-‐ Endowments:
ω1
=
ω2
=
0
∗ The
representative
firm
-‐ Production:
the
firm
transforms
inputs
into
outputs
according
to
the
production
function,
x2
=
f(-‐x1)
-‐ Inputs
are
sold
at
price
ρ1
and
outputs
are
sold
at
price
ρ2.
-‐ The
firm’s
profits
are
given
by:
π
=
ρ2
f(-‐x1)
+
ρ1x1
(or
π
=
ρ2
f
(L)
–
wL)
→
Equilibrium
firm
behaviour:
f’(-‐x1)
=
ρ1/ρ2
, LHS:
the
marginal
rate
of
transformation
of
inputs
(good
1)
into
outputs
(good
2)
=
how
many
units
of
output
can
I
produce
with
one
more
unit
of
input?
(decreasing
in
the
amount
of
inputs
already
employed
–x1)
RHS:
the
relative
price
of
inputs
vs
outputs
=
how
many
units
of
output
do
I
need
to
produce
to
be
able
to
afford
one
more
unit
of
input?
(increasing
in
the
price
of
inputs
ρ1
and
decreasing
in
the
price
of
outputs
ρ2)
-‐ Door
te
kijken
naar
de
production
en
de
profit
kom
je
tot
de
iso-‐profit
line:
∗ Equilibrium
production
∗ The
representative
household
-‐ Utility:
U
=
u
(x1,
x2)
-‐ Households
owns
the
firm
and
earns
its
profits
as
dividends,
so
budget
constraint:
(budget
constraint
=
iso-‐profit
line).
Maximization
of
utility
w.r.t.
x1
and
x2:
u1/u2
=
ρ1/ρ2
∗ Equilibrium
prices
-‐ Market-‐clearing
prices:
prices
are
in
equilibrium
when
demand
equals
supply
for
both
inputs
and
outputs.
-‐ In
the
next
figure
there
is
an
example
of
disequilibrium,
with
excess
demand
for
the
output
and
excess
supply
for
the
input.
→
Market
mechanism
ensures
that
ρ1/ρ2
declines
as
long
as
there
is
excess
demand
for
output
(good
2)
relative
to
input
(good
1).
→
Adjustments
lead
to
a
market-‐clearing
equilibrium.
∗ Equilibrium
and
efficiency
→
Does
market
equilibrium
in
a
model
with
production
yield
a
Pareto
efficient
allocation?
-‐ Efficiency:
that
the
combination
of
household
and
firm
decisions
lead
to
the
highest
possible
utility
for
the
representative
household.
→
Solving
for
the
efficient
allocation
yields:
LHS:
the
household’s
marginal
rate
of
substitution
RHs:
the
firm’s
marginal
rate
of
transformation
If
MRS>MRT,
utility
could
be
increased
by
producing
less.
If
MRS<MRT,
utility
could
be
increased
by
producing
more.
-‐ Market
equilibrium
must
satisfy
efficiency:
→
This
confirms
the
First
fundamental
theorem
of
welfare
economics
in
a
production
model
of
the
economy.
