Hotelling rule

This article covers Hotelling's contribution to resource economics; for his theorem from the field of competitive analysis, see Hotelling's law .

The Hotelling rule is an important theorem in microeconomics . It was first introduced by Harold Hotelling in his 1931 article The Economics of Exhaustible Resources . Hotelling's rule states that the price of an exhaustible resource must increase over time with the interest rate.

definition

Simplified representation of the scarcity pension and the marginal costs of funding

The price for exhaustible resources cannot be identical to the marginal costs , as it would, for example , result from the full competition model . If this were the case, then it would be optimal to extract the entire inventory of resources as quickly as possible and invest the profits in other, higher-yielding projects. An owner of a resource inventory is therefore only willing not to sell resources if he can expect the value of the resource to increase over time with the market interest rate. A lower increase in value would induce him to sell more in the current period, a higher increase in value would be an incentive to reduce supply. The scarcity pension indicates the opportunity costs of selling an additional resource unit. The development of the scarcity rent with the market interest rate is known as the Hotelling rule. Many models in resource economics are based on this principle.

Mathematical derivation

A non-renewable resource is available in limited quantities and there are no storage costs. In every period a certain benefit arises from the consumption of the resource. Future benefits can be discounted. Thus there is a welfare function over T periods:

${\displaystyle W=\sum _{t=0}^{T}{\frac {1}{(1+r)^{t}}}u_{t}(x_{t})}$

With

• ${\displaystyle W}$ the welfare
• ${\displaystyle r}$ the discount rate of benefits
• ${\displaystyle u_{t}(x_{t})}$the benefit in period , depending on the delivery rate in period ${\displaystyle t}$ ${\displaystyle x_{t}}$ ${\displaystyle t}$
• ${\displaystyle T}$ Time at which the resource was exhausted.

The welfare maximization function can be represented as follows:

${\displaystyle W\rightarrow max!_{x_{0},x_{1},...,x_{T}}}$

under the following conditions:

• that sales in all periods together less than / equal to the total available inventory of the resource must be ${\displaystyle {\bar {x}}}$

${\displaystyle \sum _{t=0}^{T}x_{t}\leq {\bar {x}}}$

• and that there is no negative degradation (non-negativity condition)

${\displaystyle x_{t}\geq 0\quad \forall t}$

To derive the optimality condition, all utility functions must be equal in every period.

${\displaystyle u_{t}(x_{t})=u_{s}(x_{s})\quad \forall t\neq s}$

At the same time, the benefit in each period must be equal to the maximum willingness to pay.

${\displaystyle u_{t}(x_{t})=ZB(x_{t})\quad .}$

For the sake of simplicity, it is assumed in the following that there are only two periods, that something is mined in each period and that in the end the resource is completely mined:

${\displaystyle x_{0}>0,x_{1}>0,x_{0}+x_{1}={\bar {x}}}$

Then the optimality condition follows from the maximization problem (Hotelling's rule):

${\displaystyle u^{\prime }(x_{0}^{*})={\frac {u^{\prime }(x_{1}^{*})}{(1+r)}}\Leftrightarrow r={\frac {u^{\prime }(x_{1}^{*})-u^{\prime }(x_{0}^{*})}{u^{\prime }(x_{0}^{*})}}}$

Current state of research

It has repeatedly been found that the Hotelling rule is incompatible with the actual development of world market prices for natural resources . One of the reasons for this is that the original formulation of the Hotelling rule is based on a partial analysis; a derivation of the rule within the framework of a general equilibrium model predicts constant prices for finite resources. Nevertheless, the rule in its simple form is still used in many models of resource and climate economics.

literature

• S. Devarajan and AC Fisher, (1981): Hotelling's "Economics of Exhaustible Resources": Fifty Years Later . Journal of Economic Literature, Vol. 19 (1): 65-73.
• LC Gray, (1914): Rent under the Assumption of Exhaustibility . Quart. J. Econ., Vol 28: 466-489.
• H. Hotelling, (1931): The Economics of Exhaustible Resources . J. Polit. Econ., Vol. 39: 137-175.

See also

• Solow-Stiglitz efficiency condition

Individual evidence

1. ↑ J. Hassler, P. Krusell (2012): Economics and Climate Change: Integrated Assessment in a Multi-Region World . NBER Working Papers 1/2012, p. 25.
2. ↑ Hans-Werner Sinn (2008): The green paradox: Why you shouldn't forget what you have to offer when it comes to climate policy . Perspectives of Economic Policy, Vol. 9 (Special Issue), pp. 125–126.
3. ^ B. Gaitan, Richard Tol, I. Yetkiner (2006): The Hotelling's Rule Revisited in a Dynamic General Equilibrium Model . In: O. Esen, A. Ogus: Proceedings of the International Conference on Human and Economic Resources . Izmir: Izmir University of Economics.