Recent research has been able to measure two forms of technical change---one (fossil) energy-saving and one saving on capital/labor. The results first show strong evidence for "directed technical change" in the sense that the total resources devoted to saving on the inputs responds endogenously to incentives and that that the two aggregate technology series display a negative medium-run correlation. Second, the elasticity of substitution between these inputs is close to zero (in contrast to the standard assumption of 1). Against this background we set up a simple and almost-tractable model of directed technical change where a final good is produced with capital and finite fossil energy. Specifically, the model features log utility, Leontief production, full depreciation and zero-cost fossil-fuel extraction. We calibrate the model and show that it can simultaneously capture important features of the U.S. post-war period. First, if the capital-augmenting technology starts below its steady state value, the model features peak oil and a relatively fast growth in the capital-augmenting technology and a low growth rate of the energy-saving technology as in the U.S. up to the first oil-price shock. Second, monopoly power (modeled as a reduction in the natural resource) causes an increase in the rate of energy-saving technical change and slow growth in the technology that saves on capital (i.e., a productivity slowdown) as in the period after the first oil shock.
