Macroeconomic formulas for global limited raw material economy exist?

Macroeconomics basic formulas are referring to a model in which the raw material are unlimited. Also the efficiency of production is a consequence of the pressure done by consumers (that buy the most cheap product, between the ones with the same level of quality).

exist formulas that try to model an economic system that "expand" toward the maximization of efficiency in using the limited quantity of raw resources?

(provided that we can describe the actual basic model like one that "expand" toward the maximization of quantity of production and quantity of selling, by total units or by total revenue).

I am not an expert on the domain. However, a popular extension in various domains of macroeconomics is to replace the standard Cobb-Douglas function

$$Y=AK^{\alpha}L^{1-\alpha}$$

by a production function that take a third resources (typically called land and noted $T$) which is supposed to be in a constant quantity. The production function hence becomes

$$Y=AK^{\alpha}L^{\beta}T^{1-\alpha-\beta}$$

With

• $$\frac{\dot{A}}{A}=g$$
• $$\frac{\dot{L}}{L}=n$$
• $$\dot{K}=sY-\delta K$$
• $$\frac{\dot{T}}{T}=0$$

It is -to my knowledge, typically used in growth theory, to account for malthusian theories, more here for instance, and some empirics Here, but i guess every textbook in growth has a note or more on it.

I also recall seeing it in international trade, but less frequently.