Sustainable growth in the long run Suppose that, in addition to physical capital and labour, an economy requires a fixed factor and a non-renewable resource to produce the final good. Prove that an economy with these input requirements may still be able to sustain growth in the long run.


What I think is that 

Using the concept of resource intensity, we can examine how economic growth, population growth, and the growth of non-renewable resource use are related. Define y as GDP per capita, L as the size of the population, I as resource intensity, and R as resource consumption. The definition of resource intensity is:

$$I=R/yL$$
$$R=IyL$$

Now we can rewrite this equation in terms of growth rates 

$$\hat{R} = \hat {I}+ \hat {y} + \hat {L}$$ .

This equation says, for example, that if output per capita is growing at 1% per year, population is growing at 1% per year, and resource intensity is constant, then total resource use will grow at 2% per year.
The equation can also be turned around and used to illustrate how resource limitations affect economic growth. For example, assume that the quantity of a nonrenewable resource available for use is constant, as in the case of a renewable resource that is al- ready being exploited at the maximum sustainable yield. This assumption implies that Rn = 0. The previous equation can then be rewritten as:
$$ \hat {y} =-\hat {I}— \hat {L}$$ .

In this form, the equation says that for output per capita to have a positive growth rate, resource intensity must be falling faster than population is growing.

For other nonrenewable resources, the ratio of the ultimately recoverable quantity of the resource to current use is higher, but the same result holds: Eventually, if use continues as its current rate, the re- source will run out. Indeed, a moment’s thought makes it obvious that no rate of depletion of a nonrenewable resource is sustainable indefinitely. In the case of renewable resources, it is possible to sustain use indefinitely, but only at a limited level. If GDP is growing and resource use is already at the maximum sustainable yield, then resource intensity (the quantity of resource used per unit of output) must fall over time.

These considerations would seem to suggest that our current level of resource consumption may indeed not be sustainable. The question is: Does that mean that our level of income is not sustainable?
The answer to this question is a definitive no, for two reasons. First, even though the resources that we use now exist in fixed supply, there are frequently possibilities for substituting different resources. Second, although resource deple- tion is a drag on growth, other factors, most notably technological progress, may be sufficient to overcome this drag and allow income to keep rising. We address these issues in turn. 

I think so. What do you think about this question? And what is fixed factor? I cannot include it.  

Thank you
As I said in the comments, without additional details proving anything is either impossible or trivial. Impossible if I do not make any assumptions, trivial if I am allowed to make any assumption I like.

Let me give you a hint on how to solve this in the framework how it is usually taught.

The assumptions made are:


There are decreasing returns to scale to any single production factor
There are constant returns to scale to all production factors together
Technology and the labour force grow at an exogenous rate (which may be 0)
The production function is cobb-douglas form


Then, denoting national income at time $t$ as $Y_{t}$, capital as $K_{t}$, labour as $L_{t}$, technology as $A_{t}$, a fixed factor such as land as $X$ (which is independent of time) and a non-renewable resource extraction as $E_{t}$ we can formulate the production function as:

$Y_{t}=A_{t}K_{t}^{\alpha}X^{\beta}E_{t}^{\eta}L_{t}^{1-(\alpha+\beta+\eta)}$ 

We can determine the income per capita (or more precisely the income per labourer) by dividing both sides by $L_{t}$. Using small letters to denote a variable in per labourer terms we get:

$y_{t}=A_{t}k_{t}^{\alpha}x_{t}^{\beta}e_{t}^{\eta}$

Note that whereas $X$ is independent of time, $x$ is not because the labour force grows over time so the amount of land per labourer changes over time. You can now rewrite this equation in terms of growth rates to determine how it is (im)possible to continue to grow income per capita even in the presence of a non-renewable and fixed production factor.