Week 1. INTRODUCTION AND NON-RENEWABLE RESOURCES 1
Explain intuitively: under this assumed inverse demand function
−a R
Pt =K∗e . When would substitution to an alternative resource
t
occur?
o Substitution will take place if the price of the resource rises
to such an extent that it makes alternatives economically
more attractive. When P=K (choke-off price), the demand is
driven to 0 and people switch to an alternative.
Given the resource-depletion model set-up, show the Hotelling rule by constructing the
Hamiltonian. Deriving the Hotelling rule by maximizing social welfare.
t=T
We want to choose Rt that max social welfare: W = ∫ U ( Rt ) e dt s.t. Ṡt =−Rt
ipt
o
t =0
(change in stock is the quantity of resources you deplete).
o Hamiltonian: H=U ( R t ) + μt (−Rt ) , where μt is the shadow price: which shows how
much W increases when the constraint is relaxed by 1 unit.
dH dU ( Rt ) dU ( R t )
=0 → + μ t∗(−1 )=0 → μ t = ≡ Pt , as
dR d Rt d Rt
t =T
U ( Rt )= ∫ P( R)d R
t=0
dH μ̇ Ṗ
= pμ− μ̇ → 0= p μ− μ̇ → p= , fill above result in: p= t , which is the
dS μ Pt
hotelling rule. Pt =price and p=social discount rate.
What does the Hotelling efficiency rule say? Explain in words and write the mathematical
equations.
o If the social value of the resource is to be maximised: the growth rate of price ( Pt ),
should rise at the same rate as the social (or private) discount rate.
Explain intuitively why prices of a non-renewable resource growing faster or slower than
determined by the Hotelling rule will not be
Ṗ t
o If the price grows more slowly ( < p ): then it would be optimal to extract all the
Pt
resources, sell them and invest its revenue in the market with a higher return.
Keep in mind that we assume constant extraction costs and there is no
storage.
Ṗ t
o If the price grows faster ( > p ): then it would be optimal to extract nothing, as it’s
Pt
more profitable to keep it in the ground (so its price increases) than any outside
investment option.
, Draw a graph of how net prices of a non-renewable resource should change over time given a
constant discount rate under the Hotelling rule.
o The hotelling rule can also be written in a different way: Pt =P0 e pt
Which is the same as: Ṗt =P0 p e pt (derivative w.r.t. to t) ->
pt
Ṗ t P 0∗p e
= =p
P t P 0∗e pt
Perfectly competitive and monopolistic firms: will they deliver the socially optimum rate of
extraction? If so, under which conditions?
o Optimum means that it maximises the objective function, as the objective functions
contain a discount rate: perfectly competitive and monopolistic firms select an
extraction ( Rt ) so that its discounted marginal profit is the same at any point in time
t!
o For perfect competitive firms: they DO deliver the socially optimum rate of
extraction.
We then have m firms that max profits over time:
t =T t=T j=m
∫ ∏e −¿
dt s . t . Ṡ j , t=−R j ,t ∧S= ∫ ∑ R j ,t dt , where ∏ ¿ P t∗R j , t
t=0 j ,t t =0 j=1
(Profit==revenue as we don’t have extraction cost) and S=total quantity of the
resource.
As we don’t have extraction costs here, we can’t solve this by saying
profit==0. So, we say ha there are m firms that all face the same net price Pt .
Unlike the monopolistic scenario where there is only 1 firm and where the
price depends on the extraction.
d∏ ;
Marginal profit= M ∏ ;= j ,t
j ,t
d R j ,t
, We need: M ∏ e to be constant (discounted marginal profit must be
¿
j ,t
constant!), so:
e ∗d ∏ ;
¿
e¿∗d ( Pt∗R j ,t ) d R j ,t ¿
∗e =Pt e ≡ constant .
j ,t ¿
= =Pt
d R j ,t d R j ,t d R j ,t
Because we found that the discounted marginal profit is constant (derivative
to R j ,t is 0, so there is no change in its value) we know that perfectly
competitive firms deliver the social optimum rate of extraction.
o For monopolistic firm: They DON’T deliver the socially optimum rate of extraction.
We then have the above situation but then with only one firm and now the price
depends on the extraction!
The monopolistic firm max’s profit over time:
t =T t =T
∫ ∏e −¿
dt s . t . Ṡ t=−Rt ∧S= ∫ Rt dt , where ∏ ¿ P t (Rt )∗Rt
t=0 t t=0
d∏ ;
M ∏ ;= t and again we need e ∗M ∏ ; to be constant for all t,
¿
t t
d Rt
so:
e ∗d ∏ ;
¿
e¿∗d ( Pt ( R t )∗R t )
d Rt
t
→
d Rt
→use product rule :
( d Pt
d Rt )
∗R t + Pt∗1 ∗e ¿
This is not constant! As taking the derivative to Rt of the above result doesn’t
equal zero! Therefore, monopolistic firms don’t deliver the socially optimum
rate of extraction!
o Comparing the competitive and monopoly cases:
A monopolistic firm will take loner to fully deplete the non-renewable
resource.
The initial price will be higher in monopolistic markets, and the rate of price
increase will be slower
Extraction of the resource will be slower at fist in monopolistic markets, but
faster towards the end of the depletion horizon.
, Empirically, are the prices of non-renewable resources generally evolving according to the
Hotelling rule? Discuss why or why not.
o NO, hotelling predicts that the price of a non-renewable resource will rise at the rate
of interest. We would then expect that relative commodity prices would increase, but
empirically we see that these prices have only declined for long periods of time.
o Why? Because we made certain assumptions for the hotelling rule, which aren’t
realistic:
Perfectly competitive markets with enforced property rights
Stock of resource fixed and known ex-ante -> so, no frequent new
discoveries
Demand for resource is fixed
Constant extraction costs are known with certainty
, Resources extracted are used with no waste (there is no storage)
o When should we worry about fossil fuel prices not following the Hotelling rule?
If fossil fuel prices increase less than optimal, this could delay transition away
form carbon-based energy!
If the reason for failure is uncertain property rights or strategic
interaction: market will fail to provide optimal solution.
If the reason for failure is extraction cost, exploration or
technological progress-> a market failure is not implied and
optimality may still be achieved.
The hotelling model is nevertheless useful for studying the effect of policy
changes, where we rely on the assumption of profit maximisation.
If you are given the graphical summary of the social planner solution of the non-renewable
resource depletion optimization model, you should be able to describe how 𝑃0 and 𝑇 will
change if there is (assuming in all cases that everything else remains equal): (Section
15.6 Perman, R., Ma, Y., McGilvray, J., & Common, M.. Natural resource and environmental
economics.)
Explain intuitively: under this assumed inverse demand function
−a R
Pt =K∗e . When would substitution to an alternative resource
t
occur?
o Substitution will take place if the price of the resource rises
to such an extent that it makes alternatives economically
more attractive. When P=K (choke-off price), the demand is
driven to 0 and people switch to an alternative.
Given the resource-depletion model set-up, show the Hotelling rule by constructing the
Hamiltonian. Deriving the Hotelling rule by maximizing social welfare.
t=T
We want to choose Rt that max social welfare: W = ∫ U ( Rt ) e dt s.t. Ṡt =−Rt
ipt
o
t =0
(change in stock is the quantity of resources you deplete).
o Hamiltonian: H=U ( R t ) + μt (−Rt ) , where μt is the shadow price: which shows how
much W increases when the constraint is relaxed by 1 unit.
dH dU ( Rt ) dU ( R t )
=0 → + μ t∗(−1 )=0 → μ t = ≡ Pt , as
dR d Rt d Rt
t =T
U ( Rt )= ∫ P( R)d R
t=0
dH μ̇ Ṗ
= pμ− μ̇ → 0= p μ− μ̇ → p= , fill above result in: p= t , which is the
dS μ Pt
hotelling rule. Pt =price and p=social discount rate.
What does the Hotelling efficiency rule say? Explain in words and write the mathematical
equations.
o If the social value of the resource is to be maximised: the growth rate of price ( Pt ),
should rise at the same rate as the social (or private) discount rate.
Explain intuitively why prices of a non-renewable resource growing faster or slower than
determined by the Hotelling rule will not be
Ṗ t
o If the price grows more slowly ( < p ): then it would be optimal to extract all the
Pt
resources, sell them and invest its revenue in the market with a higher return.
Keep in mind that we assume constant extraction costs and there is no
storage.
Ṗ t
o If the price grows faster ( > p ): then it would be optimal to extract nothing, as it’s
Pt
more profitable to keep it in the ground (so its price increases) than any outside
investment option.
, Draw a graph of how net prices of a non-renewable resource should change over time given a
constant discount rate under the Hotelling rule.
o The hotelling rule can also be written in a different way: Pt =P0 e pt
Which is the same as: Ṗt =P0 p e pt (derivative w.r.t. to t) ->
pt
Ṗ t P 0∗p e
= =p
P t P 0∗e pt
Perfectly competitive and monopolistic firms: will they deliver the socially optimum rate of
extraction? If so, under which conditions?
o Optimum means that it maximises the objective function, as the objective functions
contain a discount rate: perfectly competitive and monopolistic firms select an
extraction ( Rt ) so that its discounted marginal profit is the same at any point in time
t!
o For perfect competitive firms: they DO deliver the socially optimum rate of
extraction.
We then have m firms that max profits over time:
t =T t=T j=m
∫ ∏e −¿
dt s . t . Ṡ j , t=−R j ,t ∧S= ∫ ∑ R j ,t dt , where ∏ ¿ P t∗R j , t
t=0 j ,t t =0 j=1
(Profit==revenue as we don’t have extraction cost) and S=total quantity of the
resource.
As we don’t have extraction costs here, we can’t solve this by saying
profit==0. So, we say ha there are m firms that all face the same net price Pt .
Unlike the monopolistic scenario where there is only 1 firm and where the
price depends on the extraction.
d∏ ;
Marginal profit= M ∏ ;= j ,t
j ,t
d R j ,t
, We need: M ∏ e to be constant (discounted marginal profit must be
¿
j ,t
constant!), so:
e ∗d ∏ ;
¿
e¿∗d ( Pt∗R j ,t ) d R j ,t ¿
∗e =Pt e ≡ constant .
j ,t ¿
= =Pt
d R j ,t d R j ,t d R j ,t
Because we found that the discounted marginal profit is constant (derivative
to R j ,t is 0, so there is no change in its value) we know that perfectly
competitive firms deliver the social optimum rate of extraction.
o For monopolistic firm: They DON’T deliver the socially optimum rate of extraction.
We then have the above situation but then with only one firm and now the price
depends on the extraction!
The monopolistic firm max’s profit over time:
t =T t =T
∫ ∏e −¿
dt s . t . Ṡ t=−Rt ∧S= ∫ Rt dt , where ∏ ¿ P t (Rt )∗Rt
t=0 t t=0
d∏ ;
M ∏ ;= t and again we need e ∗M ∏ ; to be constant for all t,
¿
t t
d Rt
so:
e ∗d ∏ ;
¿
e¿∗d ( Pt ( R t )∗R t )
d Rt
t
→
d Rt
→use product rule :
( d Pt
d Rt )
∗R t + Pt∗1 ∗e ¿
This is not constant! As taking the derivative to Rt of the above result doesn’t
equal zero! Therefore, monopolistic firms don’t deliver the socially optimum
rate of extraction!
o Comparing the competitive and monopoly cases:
A monopolistic firm will take loner to fully deplete the non-renewable
resource.
The initial price will be higher in monopolistic markets, and the rate of price
increase will be slower
Extraction of the resource will be slower at fist in monopolistic markets, but
faster towards the end of the depletion horizon.
, Empirically, are the prices of non-renewable resources generally evolving according to the
Hotelling rule? Discuss why or why not.
o NO, hotelling predicts that the price of a non-renewable resource will rise at the rate
of interest. We would then expect that relative commodity prices would increase, but
empirically we see that these prices have only declined for long periods of time.
o Why? Because we made certain assumptions for the hotelling rule, which aren’t
realistic:
Perfectly competitive markets with enforced property rights
Stock of resource fixed and known ex-ante -> so, no frequent new
discoveries
Demand for resource is fixed
Constant extraction costs are known with certainty
, Resources extracted are used with no waste (there is no storage)
o When should we worry about fossil fuel prices not following the Hotelling rule?
If fossil fuel prices increase less than optimal, this could delay transition away
form carbon-based energy!
If the reason for failure is uncertain property rights or strategic
interaction: market will fail to provide optimal solution.
If the reason for failure is extraction cost, exploration or
technological progress-> a market failure is not implied and
optimality may still be achieved.
The hotelling model is nevertheless useful for studying the effect of policy
changes, where we rely on the assumption of profit maximisation.
If you are given the graphical summary of the social planner solution of the non-renewable
resource depletion optimization model, you should be able to describe how 𝑃0 and 𝑇 will
change if there is (assuming in all cases that everything else remains equal): (Section
15.6 Perman, R., Ma, Y., McGilvray, J., & Common, M.. Natural resource and environmental
economics.)
