Download Non-renewable resource exploitation: basic models NRE - Lecture 2 Aaron Hatcher

Non-renewable resource exploitation:
basic models
NRE - Lecture 2
Aaron Hatcher
Department of Economics
University of Portsmouth
Introduction
I
General rule for e¢ cient exploitation of a non-renewable
resource
vt0 (qt ) =
1
λ t +1 ,
1+δ
λ t +1 =
1
λ t +2
1+δ
Introduction
I
General rule for e¢ cient exploitation of a non-renewable
resource
vt0 (qt ) =
I
1
λ t +1 ,
1+δ
Thus
vt0 (qt ) =
λ t +1 =
1
λ t +2
1+δ
1
v 0 (qt +1 )
1 + δ t +1
Introduction
I
General rule for e¢ cient exploitation of a non-renewable
resource
vt0 (qt ) =
I
1
λ t +1 ,
1+δ
Thus
1
λ t +2
1+δ
1
v 0 (qt +1 )
1 + δ t +1
The shadow price is the marginal value of stock left in situ
vt0 (qt ) =
I
λ t +1 =
Introduction
I
General rule for e¢ cient exploitation of a non-renewable
resource
vt0 (qt ) =
I
1
λ t +1 ,
1+δ
λ t +1 =
Thus
1
λ t +2
1+δ
I
1
v 0 (qt +1 )
1 + δ t +1
The shadow price is the marginal value of stock left in situ
I
In continuous time (without costs)
vt0 (qt ) =
p (t ) = λ (t )
and
ṗ
λ̇
=
=r
p (t )
λ (t )
Hotelling’s Rule in competitive markets
I
With zero extraction costs, we require
ṗ = rp (t ) > 0
Hotelling’s Rule in competitive markets
I
With zero extraction costs, we require
ṗ = rp (t ) > 0
I
How? Downward-sloping demand curve and decreasing supply
Hotelling’s Rule in competitive markets
I
With zero extraction costs, we require
ṗ = rp (t ) > 0
I
How? Downward-sloping demand curve and decreasing supply
I
Arbitrage maintains the equilibrium rate of price increase
Hotelling’s Rule in competitive markets
I
With zero extraction costs, we require
ṗ = rp (t ) > 0
I
How? Downward-sloping demand curve and decreasing supply
I
Arbitrage maintains the equilibrium rate of price increase
I
Recall
p (t ) = p (T ) e
r [T
t]
Hotelling’s Rule in competitive markets
I
With zero extraction costs, we require
ṗ = rp (t ) > 0
I
How? Downward-sloping demand curve and decreasing supply
I
Arbitrage maintains the equilibrium rate of price increase
I
Recall
p (t ) = p (T ) e
I
r [T
t]
The …nal price p (T ) is the “backstop” or “choke” price
where q (T ) = 0
Hotelling’s Rule in competitive markets
I
With zero extraction costs, we require
ṗ = rp (t ) > 0
I
How? Downward-sloping demand curve and decreasing supply
I
Arbitrage maintains the equilibrium rate of price increase
I
Recall
p (t ) = p (T ) e
r [T
t]
I
The …nal price p (T ) is the “backstop” or “choke” price
where q (T ) = 0
I
Without costs, we would expect x (T ) = 0
p(T)
p(t)
q(t)
0
Tc
Socially optimal extraction
I
A social planner seeks to maximise total social welfare
Socially optimal extraction
I
A social planner seeks to maximise total social welfare
I
Discounted sum of CS and PS
Socially optimal extraction
I
A social planner seeks to maximise total social welfare
I
Discounted sum of CS and PS
I
With zero costs,
W (q (t ))
Z q (t )
0
p (q (t )) dq
Socially optimal extraction
I
A social planner seeks to maximise total social welfare
I
Discounted sum of CS and PS
I
With zero costs,
W (q (t ))
Z q (t )
p (q (t )) dq
0
I
A competitive …rm maximises π (t )
p (t ) q (t )
Socially optimal extraction
I
A social planner seeks to maximise total social welfare
I
Discounted sum of CS and PS
I
With zero costs,
W (q (t ))
Z q (t )
p (q (t )) dq
0
I
A competitive …rm maximises π (t )
I
Since
p (t ) q (t )
dW (q (t ))
∂π (t )
=
= p( )
dq
∂q
pro…t maximisation by competitive …rms maximises social
welfare
Socially optimal extraction
I
A social planner seeks to maximise total social welfare
I
Discounted sum of CS and PS
I
With zero costs,
W (q (t ))
Z q (t )
p (q (t )) dq
0
I
A competitive …rm maximises π (t )
I
Since
p (t ) q (t )
dW (q (t ))
∂π (t )
=
= p( )
dq
∂q
pro…t maximisation by competitive …rms maximises social
welfare
I
Assuming interest rates equal the social discount rate
Extraction by a monopoly
I
A monopoly producer maximises
Z T
0
p (q (t )) q (t ) e
rt
dt
Extraction by a monopoly
I
A monopoly producer maximises
Z T
p (q (t )) q (t ) e
rt
dt
0
I
But now
d
dp ( )
q (t )
[p (q (t )) q (t )] = p ( ) +
dq
dq
Rq < p ( )
Extraction by a monopoly
I
A monopoly producer maximises
Z T
p (q (t )) q (t ) e
rt
dt
0
I
But now
d
dp ( )
q (t )
[p (q (t )) q (t )] = p ( ) +
dq
dq
I
Rq < p ( )
The monopolist has a downward-sloping marginal revenue
curve Rq
Extraction by a monopoly
I
A monopoly producer maximises
Z T
p (q (t )) q (t ) e
rt
dt
0
I
But now
d
dp ( )
q (t )
[p (q (t )) q (t )] = p ( ) +
dq
dq
Rq < p ( )
I
The monopolist has a downward-sloping marginal revenue
curve Rq
I
Marginal revenue is less than the market price, except at
p (T ) where q (T ) = 0
Extraction by a monopoly contd.
I
Hotelling’s Rule for a monopoly becomes
Ṙq
=r
Rq
Extraction by a monopoly contd.
I
Hotelling’s Rule for a monopoly becomes
Ṙq
=r
Rq
I
The monopolist controls the market price
Extraction by a monopoly contd.
I
Hotelling’s Rule for a monopoly becomes
Ṙq
=r
Rq
I
The monopolist controls the market price
I
If discounted marginal revenue is constant, the discounted
market price is decreasing
Extraction by a monopoly contd.
I
Hotelling’s Rule for a monopoly becomes
Ṙq
=r
Rq
I
The monopolist controls the market price
I
If discounted marginal revenue is constant, the discounted
market price is decreasing
I
The current price is increasing at less than the interest rate
Extraction by a monopoly contd.
I
Hotelling’s Rule for a monopoly becomes
Ṙq
=r
Rq
I
The monopolist controls the market price
I
If discounted marginal revenue is constant, the discounted
market price is decreasing
I
The current price is increasing at less than the interest rate
I
The initial monopoly price is higher than the initial
competitive price
Extraction by a monopoly contd.
I
Hotelling’s Rule for a monopoly becomes
Ṙq
=r
Rq
I
The monopolist controls the market price
I
If discounted marginal revenue is constant, the discounted
market price is decreasing
I
The current price is increasing at less than the interest rate
I
The initial monopoly price is higher than the initial
competitive price
I
The initial quantity extracted is smaller but declines more
gradually
Extraction by a monopoly contd.
I
Hotelling’s Rule for a monopoly becomes
Ṙq
=r
Rq
I
The monopolist controls the market price
I
If discounted marginal revenue is constant, the discounted
market price is decreasing
I
The current price is increasing at less than the interest rate
I
The initial monopoly price is higher than the initial
competitive price
I
The initial quantity extracted is smaller but declines more
gradually
I
Monopoly extraction is more gradual and extended but not
“better” for social welfare
p(T)
p(t)
q(t)
0
Tc
Tm
Costly extraction
I
Let …rms face a variable cost function
c (t )
c (q (t ) , x (t ))
Costly extraction
I
Let …rms face a variable cost function
c (t )
I
Here cq > 0 and cx
0
c (q (t ) , x (t ))
Costly extraction
I
Let …rms face a variable cost function
c (t )
I
Here cq > 0 and cx
I
Equivalently, π x
0
0
c (q (t ) , x (t ))
Costly extraction
I
Let …rms face a variable cost function
c (t )
I
Here cq > 0 and cx
I
Equivalently, π x
I
Now e¢ ciency implies
c (q (t ) , x (t ))
0
0
πq = λ
and
λ̇ = λr
πx
Costly extraction
I
Let …rms face a variable cost function
c (t )
I
Here cq > 0 and cx
I
Equivalently, π x
I
Now e¢ ciency implies
c (q (t ) , x (t ))
0
0
πq = λ
and
λ̇ = λr
I
πx
Hotelling’s Rule for a competitive …rm becomes
π̇ q
=r
πq
πx
πq
Costly extraction contd.
I
If π x > 0 then
π̇ q
<r
πq
Costly extraction contd.
I
If π x > 0 then
π̇ q
<r
πq
I
Extraction costs moderate the rate of price rise
Costly extraction contd.
I
If π x > 0 then
π̇ q
<r
πq
I
Extraction costs moderate the rate of price rise
I
Otherwise, the e¢ cient extraction path depends on the cost
function
Costly extraction contd.
I
If π x > 0 then
π̇ q
<r
πq
I
Extraction costs moderate the rate of price rise
I
Otherwise, the e¢ cient extraction path depends on the cost
function
I
Extraction may terminate before x (T ) = 0 and p (T ) may
not reach the backstop price