Directed technical change and the British Industrial Revolution

We build a directed technical change model of the British Industrial Revolution where one intermediate goods sector uses a fixed renewable energy (“wood”) quantity, and another uses coal at a fixed price. With a high enough elasticity of substitution between the two goods in producing final output, an industrial revolution, where over time the coal-using sector grows relative to the wood-using sector and its growth accelerates, is not inevitable. However, greater initial scarcity of wood and/or higher population growth puts the economy on a path to an industrial revolution. The converse slows industrialization, or even prevents it forever.