In his seminal 1931 paper, Harold Hotelling demonstrates that in a competitive market for a nonrenewable resource, the price of the resource changes at a rate equal to the interest rate, or to the return on capital. This analysis augments and further justifies Hotelling’s Rule by demonstrating that it holds within a multisector optimization model with human and physical capital, and with both renewable and non-renewable resources. When consumers and producers engage in optimizing behavior, on the margin the net return to physical capital equals the return to harvesting a renewable resource or extracting a nonrenewable resource. Moreover, this analysis reveals that the alleged inconsistencies of Hotelling’s Rule with empirical findings are likely the result of market characteristics specific to each empirical study, not the foundational logic of Hotelling’s rule.
This analysis demonstrates competitive criteria for market equilibrium when considering renewable or nonrenewable natural resources as factors of production in conjunction with an endogenously derived human capital component. Production depends upon choosing labor, physical capital, and the resource harvest rate in order to maximize intertemporal profits. As intertemporal utility maximizers, consumers choose both the rates of consumption and human capital accumulation. The consumer owns both the stock of physical capital and the resource stock. Thus, rent from physical capital and revenues from resource extraction are paid to the consumer. In turn, the consumer chooses consumption and time devoted to the accumulation of human capital so as to maximize intertemporal utility. Specifically, human capital formation follows the rule postulated by Lucas [
This analysis merges endogenous growth literature with that of both renewable and non-renewable resources. The conditions facing the producer and the consumer utilize the perfect foresight competitive approach of Becker [
The literature describing the optimal extraction of nonrenewable resources is extensive (Krautkraemer [
Finally, much has been made of the lack of empirical support for Hotelling’s Rule. Krautkraemer [
Gaudet [
In a study of the application of Hotelling’s rule to old growth timber, Livernois et al. [
The reality is that Cairn and Davis, Gaudet, and Kronenberg address a fundamentally different issue. Rather than providing instances where Hotelling’s rule fails, they are building upon the foundation that is Hotelling’s rule in a manner that is very similar to the way in which researchers use pure competition as a reference point. Whereas these and other authors focus on a subset of exogenous factors which may exist in some markets at certain times, Hotelling’s rule provides the foundation for the determination of non-renewable resource prices, ceteris paribus. Thus, the apparent inconsistency with empirical findings is the result of the original rule’s simplicity and scope, not its logic. Similar to assumptions of perfect competition, Hotelling’s rule serves as a foundational starting point for all subsequent discussions involving variations and characteristics specific to targeted non-renewable resource markets. In contrast, we ask a different question. Does Hotelling’s rule hold in a multisector optimization model with both non-renewable and renewable resources? To this end, we create a model that both augments and further justifies Hotelling’s Rule by demonstrating that it holds within a multisector optimization model with renewable and non-renewable resources, and human and physical capital.
Define N ( t ) = N o e n t to represent the fully employed population at time t. In the spirit of Lucas [
Define effective labor as
L t = ∫ 1 N ( t ) γ t ( i ) μ t ( i ) h t ( i ) d i
where again, μ t i represents time devoted to working in a productive capacity, h t i represents the skill level of the ith individual at time t, and γ i is a coefficient indicating the ability of the ith individual to convert skill-hours into labor. We adopt Lucas’ assumption that individuals possess identical skill levels, they choose their productive hours similarly, and δ = δ i and γ = γ i are constant. Thus, effective labor reduces to L t = N o e n t γ μ t h t = N t γ μ t h t . In the spirit of Lucas, we assume that each consumer devotes the same proportion of time to work each period. Thus, μ i = μ and L t = N t γ μ h t with μ determined endogenously.
These assumptions allow us to focus on a representative consumer. Let C t denote consumption of the capital-consumption good at time t. In each period the representative consumer receives benefits according to the utility function, u ( C t , μ ) : [ ℝ + \ ( 0 , 1 ] ] → ℝ + where u ∈ C 2 depends on personal consumption of the capital-consumption good and, the amount of time one chooses to devote to employment (or alternatively, the accumulation of human capital). We assume that u C > 0 and u C C < 0 . According to the model the relationship between effective labor and work-time is L t > 0 ⇔ μ > 0 . In addition, an increase in μ will increase consumer income. Thus, we assume that u μ > 0 and u μ μ < 0 . Assume the transversality condition, lim C → 0 + u C = + ∞ , also holds.
The consumer receives the following stream of revenues from the firm,
r t K t + q t R t + w t L t
where L t is effective labor, R t is the resource extraction (harvest) rate and K t is physical capital. The prices associated with each of these inputs are, respectively, w t , q t and r t . In return, the consumer must compensate the firm for consumption and new units of physical capital. Physical capital depreciates at the rate, ζ . Assuming that the consumption good and the physical capital good are the same, and that the price of that good is the numeraire ( p t ≡ 1 ) , this compensation amounts to
C t + ( K ˙ t + ζ K t ) = F ( L t , R t , K t )
where production is defined by Y t = F ( L t , R t , K t ) and K ˙ t is investment in physical capital.
If consumption is discounted by e − ω ( t ) with ω > 0 , the consumer’s problem becomes,
max C , μ ∫ 0 ∞ e − ω ( t ) u ( C t , μ ) d t (CP)
subject to the budget constraint,
r t K t + q t R t + w t L t = C t + ( K ˙ t + ζ K t )
and the human capital constraint,
h ˙ t = h t δ ( 1 − μ )
In this optimal control problem the control variables are C t and μ . The corresponding state variables are K t and h t .
Proposition 1. Consumer optimization requires the net return to capital to equal the percentage change in effective labor,
r t − ζ = n + δ ( 1 − μ ) (1)
Proof. The Hamiltonian associated with this problem is
H = e − ω ( t ) [ u ( C t , μ ) ] + η 1 [ r t K t + q t R t + w t L t − C t − K ˙ t − ζ K t ] + η 2 h t δ ( 1 − μ )
where η 1 and η 2 are the present value multipliers of this Hamiltonian system.
The first order conditions for a maximum are given by,
∂ H ∂ C = e − ω ( t ) [ u ′ ( C t , μ ) ] − η 1 = 0 and ∂ H ∂ μ = η 1 γ h t N t − η 2 δ h t = 0
Thus,
η 1 = e − ω ( t ) [ u ′ ( C t , μ ) ] (2)
and
η 2 = γ N t δ e − ω ( t ) [ u ′ ( C t , μ ) ] (3)
Next, differentiate Equations (2) and (3) with respect to time to obtain,
η ˙ 1 = e − ω ( t ) [ ( − ω ˙ ) u ′ ( C t , μ ) + u ″ ( C t , μ ) C ˙ t ]
and
η ˙ 2 = γ N t δ e − ω ( t ) [ ( − ω ˙ + n ) u ′ ( C t , μ ) + u ″ ( C t , μ ) C ˙ t ]
Hence,
η ^ 1 = − ω ˙ + u ″ ( C t , μ ) u ′ ( C t , μ ) C ˙ t (4)
and
η ^ 2 = − ω ˙ + n + u ″ ( C t , μ ) u ′ ( C t , μ ) C ˙ t (5)
Next, find the costate equations. These are,
− ∂ H ∂ K = η ˙ 1 = − η 1 ( r t − ζ )
and
− ∂ H ∂ h = η ˙ 2 = − η 2 δ ( 1 − μ )
Note the costate equations,
η ^ 1 = − ( r t − ζ ) (6)
and
η ^ 2 = − δ ( 1 − μ ) (7)
Equating Equation (4) with (6), and (5) with (7), one sees,
− ω ˙ + u ″ ( C t , μ ) u ′ ( C t , μ ) C ˙ t = − ( r t − ζ ) (8)
and
− ω ˙ + u ″ ( C t , μ ) u ′ ( C t , μ ) C ˙ t = − ( n + δ ( 1 − μ ) ) (9)
Equating (8) and (9), one obtains the necessary condition for consumer optimization (1). □
We assume the producer is an intertemporal profit maximizer. This approach is quite common in both the continuous time models of capital accumulation and the renewable resource models, although this model differs from other renewable resource models in that the resource is a factor of production1.
The producer chooses the level of output and the resource extraction rate consistent with intertemporal profit maximization. Production, Y t = F ( L t , R t , K t ) , depends on effective labor, the resource extraction (harvest) rate and physical capital, again with prices w t , q t and r t . If discounting future profits depends on perfect knowledge of the stream of future interest rates such as,
I ( t ) = ∫ 0 t i ( σ ) d σ
then the firm’s problem is defined to be,
max R t , Y t ∫ 0 ∞ e − I ( t ) [ F ( L t , R t , K t ) − w t L t − q t R t − r t K t ] d t (PP)
subject to two dynamic constraints,
K ˙ t = Y t − C t − ζ K t (10)
X ˙ t = G ( X t ) − R t , with X t ≥ 0 (11)
If Π ( t ) = F ( L t , R t , K t ) − w t L t − q t R t − r t K t , the producer problem can be expressed as:
max R t , Y t ∫ 0 ∞ e − I ( t ) Π ( t ) d t
subject to Equations (10) and (11).
Since physical capital is also assumed to be the consumption good, consumers purchase the new stock at the numeraire price. Effective labor and physical capital are fully employed. The firm hires labor at the going wage, rents all physical capital, and purchases the natural resource from consumers.
Proposition 2. Necessary conditions for intertemporal profit maximization are,
F K = r t (12)
and
i ( t ) = G ′ ( X t ) + Π ^ R (13)
where i(t) is the competitive interest rate in period, t.
Proof. Use the methods of optimal control to solve the producer problem. The present value Hamiltonian is,
H = e − ∫ 0 t i ( σ ) d σ [ Π ( t ) ] + λ 1 [ Y t − C t − ζ K t ] + λ 2 [ G ( X t ) − R t ]
The control variables for the producer problem are R t and Y t . The corresponding state variables are X t and K t . Thus, the first order conditions for the producer’s problem are,
∂ H ∂ Y = λ 1 = 0
and
∂ H ∂ R = e − ∫ 0 t i ( σ ) d σ [ F R − q t ] + λ 1 F R − λ 2 = 0
Thus,
λ 1 = 0
and
λ 2 = e − ∫ 0 t i ( σ ) d σ [ F R − q t ]
or,
λ 2 = e − ∫ 0 t i ( σ ) d σ Π R
Differentiating these one sees that,
λ ^ 1 = 0 (14)
Also,
λ ˙ 2 = d [ e − ∫ 0 t i ( σ ) d σ ] d t [ F R − q t ] + e − ∫ 0 t i ( σ ) d σ d [ F R − q t ] d t
so,
λ ^ 2 = − i ( t ) + [ F R − q t ] ∧
or,
λ ^ 2 = − i ( t ) + Π ^ R (15)
The first costate equation is,
λ ˙ 1 = − ∂ H ∂ K = − e − ∫ 0 t i ( σ ) d σ [ F K − r t ] (16)
or,
λ ˙ 1 = − ∂ H ∂ K = − e − ∫ 0 t i ( σ ) d σ Π K
The second costate equation is,
λ ˙ 2 = − ∂ H ∂ X = − λ 2 G ′ (Xt)
or,
λ ^ 2 = − G ′ ( X t ) (17)
Combining Equations (14) and (16), and Equations (15) with (17), one obtains the necessary conditions for producer optimization,
F K = r t
and
i ( t ) = G ′ ( X t ) + Π ^ R 2 □
Equation (13) represents the total return on natural resource use. In particular, G ′ ( X t ) represents the own return to the resource. The term Π ^ R = [ p t F R − q t ] ∧ is the percentage change in the marginal profit associated with natural resource use. Equation (13) is a generalization of the optimization condition derived by Clark [
A competitive equilibrium is one in which the conditions satisfying the consumer problem (CP) and the producer problem (PP) coincide.
Theorem 1. A complete competitive solution requires that the net return to physical capital equal the rate at which effective labor is augmented, as well as the return to the natural resource:
F K − ζ = n + δ ( 1 − μ ) = G ′ ( X t ) + Π ^ R (18)
Proof. Combine Equations (8), (9), and (12) to obtain F K − ζ = n + δ ( 1 − μ ) . If one assumes perfect foresight, i ( t ) = ω ˙ − u ″ ( C t , μ ) u ′ ( C t , μ ) C ˙ t 3. Equation (13) provides the right hand side of (18). □
This analysis demonstrates a necessary condition for a competitive solution when a natural resource is an input and human capital accumulation is endogenous. The term n + δ ( 1 − μ ) represents the development of effective labor (the sum of the rates of population growth and human capital accumulation). Therefore, Equation (18) states that, on the margin, the net return to physical capital must equal the rate at which effective labor is augmented (the sum of the rates of population growth and human capital accumulation), as well as the return to harvesting-or extracting-the natural resource. If the resource is nonrenewable [ G ( X t ) ≡ 0 ] , then in the absence of discounting, population growth, and human capital accumulation, Equation (18) reduces to F K = q ^ t . That is, the price of the resource varies according to the return to capital. Thus, this condition is an extension of Hotelling’s Rule when resources are used as inputs.
This is where some mistakenly see a refutation of Hotelling’s rule in industry specific cases. Hotelling’s Rule is a generalization, but not inconsistent with industry specific examples. For example, Krautkraemer [
Π ^ R = [ F R − q t ] ∧ = F ˙ R − q ˙ t F R − q t > 0 . This condition is not violated by falling resource
prices. In fact, according to this specification, one would expect falling resource prices whenever the marginal product of the resource is decreasing ( F ˙ R < 0 ) .
Harold Hotelling [
In contrast, this study answers a significantly different question. Does Hotelling’s rule hold in a multisector optimization model with both non-renewable and renewable resources? In doing so, we augment and further justify Hotelling’s Rule within a multisector optimization model with human and physical capital, and with both renewable and non-renewable resources. Within this framework, a competitive solution requires the net return to physical capital equal (1) the rate at which effective labor increases and (2) the return to the renewable or non-renewable resource. Moreover, a simpler version of the model yields results that are consistent with Hotelling’s original rule.
Raymond, F.E. (2017) A Modern Validation of Hotelling’s Rule. Theoretical Economics Letters, 7, 2070-2080. https://doi.org/10.4236/tel.2017.77140