Energy saving innovations, non-exhaustible sources of energy and long-run: what would happen if we run out of oil?

We formulate and solve a model of factor saving technological improvement considering three factors of production: labor, capital and energy. The productive activities have three main characteristics: first, in order to use capital goods firms need energy; second, there are two sources of energy: non-exhaustible and exhaustible; third, capital goods can be of different qualities and the quality of these goods can be changed along two dimensions -reducing the need of energy or changing the source of energy used in the production process. The economy goes through three stages of development after industrialization. In the first, firms make use of exhaustible energy and the e¢ ciency in the use of energy is constant. In the second stage, as the price of energy grows the e¢ ciency in its use is increased. In the third stage, the price of exhaustible sources is so high that firms have incentives to use non-exhaustible sources of energy. During this stage the price of energy is constant. In this set up, the end of the oil age has level effects on consumption and output but it does not cause the collapse of the economic system.