Hotelling Model

Overview


The Hotelling model is a simple and often quoted model which gives a basic understanding of the drivers of a commodities price.
(see ahn)

Assumptions


  • The amount of commodity is fixed and known
  • Extracting the commodity requires a fixed marginal cost
  • The firm extracting the commodity is a price taker and the demand for the commodity is a linear downward sloping function.
  • There is a single constant interest rate on capital

Intuition


The owner of the commodity mine must decide how much commodity to extract and sell in each period. The price received for selling the commodity is given by
{% a + b \times q %}
where q is the quantity extracted and sold.

If there were no time value of money, the producer would seek to extract an equal amount of commodity on each day. If the producer produces more commodity today than tomorrow, she will receive a lower price for the commodity today. However, because the producer can take the money and invest it, lower prices today can be optimal if the the interest earned on money today makes up for the lost revenue. Hotelling argues that the producer should produce just enough commodity today that makes her indifferent between producing more today or tomorrow.

In other words, if {% \pi_t %} is the profit in period t, and r is the interest rate, we should have
{% \pi_1 = \pi_0 (1 + r) %}

Mathematical Formulation


{% max \int_0^{\infty} [p(t) q(t) - c q(t)] e^{-rt} dt %}
  • p is the price
  • q is the quantity extracted
  • c is the marginal cost of extraction
  • r is the constant rate of interest

Extensions


Contents