<p>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.</p>

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

<p>The assumptions made are:</p>

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

<p>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:</p>

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

<p>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:</p>

<p>$y_{t}=A_{t}k_{t}^{\alpha}x_{t}^{\beta}e_{t}^{\eta}$</p>

<p>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.</p>

CC BY-SA 4.0
2018-05-23T09:15:42.477Z
$$y_{t}=A_{t}k_{t}^{\alpha}x_{t}^{\beta}e_{t}^{\eta}$$ in order to rewrite this in terms of growth rate, I will take log $$lny_{t}=lnA_{t}+\alpha ln k_{t}+{\beta}lnx_{t}+{\eta}lne_{t}$$ and then I will take derivative with respect to t; $$\frac{\dot{y_t}}{y_t}=\frac{\dot{A_t}}{A_t}+\alpha \frac{\dot{k_t}}{k_t}+\beta \frac{\dot{x_t}}{x_t} +\eta \frac{\dot{e_t}}{e_t}$$ so , $$\frac{\dot{y_t}}{y_t}=g+\alpha * g + \beta  *h + \eta * \epsilon$$  so do I have shown the Groth rate expression correctly for this case? The production function is really difficult.
mnm123
CC BY-SA 4.0
2018-05-23T08:01:00.847Z
2018-05-23T08:01:00.847Z
Maarten Punt
<blockquote>
  <p>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.</p>
</blockquote>

<p>What I think is that </p>

<p>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:</p>

<p>$$I=R/yL$$
$$R=IyL$$</p>

<p>Now we can rewrite this equation in terms of growth rates </p>

<p>$$\hat{R} = \hat {I}+ \hat {y} + \hat {L}$$ .</p>

<p>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:
$$ \hat {y} =-\hat {I}— \hat {L}$$ .</p>

<p>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.</p>

<p>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.</p>

<p>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. </p>

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

<p>Thank you </p>

CC BY-SA 4.0
2018-05-22T19:33:23.520Z
What exactly is your question? You post a question and then copy paste what I assume is an excerpt from a book without further specifying where exactly  you are stuck
Maarten Punt
CC BY-SA 4.0
2018-05-22T19:57:05.893Z
I couldn’t exactly do this question in the yellow box. And I searched this question from some books. And I found such an explanation. But I exactly not sure and i’m Waiting what you think about this question or this explanation @MaartenPunt
mnm123
CC BY-SA 4.0
2018-05-22T21:05:59.743Z
Well given that these partial answers provided come from economics textbook I think most economists would agree that this is true (within the framework of assumptions made, with which they may disagree). The original question though asks you to prove something. To be able to prove something like that we'd need additional info or assumptions. Like what is the production function? Cobb-Douglas, CES, Leontief or a general one? Is the labour force growing? Is technology constant? Etc etc
Maarten Punt
CC BY-SA 4.0
2018-05-22T21:27:47.310Z
Okay thank you. This question in the yellow box is past preliminary exam question. And all question is this. That is, there is no additional information that you said. How can I prove this question? Can you help me to do this? I will become really happy. @MaartenPunt
mnm123
CC BY-SA 4.0
2018-05-23T08:00:42.050Z
I have no doubt that you will become happy. However I already said that I cannot "prove" this without additional details. I mean the question doesn't even specify the growth of what (presumably income per capita). Given that this is a homework question I will however provide a hint using the "standard" framework.
Maarten Punt
CC BY-SA 4.0
2018-05-23T08:56:15.317Z
Okay I see what you mean. Yes this is general question. And this is not exactly homework. Past phd exam question as I said. But Thank for helps. Your answer really helpful. I will try to improve it. @MaartenPunt
mnm123
<macroeconomics><microeconomics><economic-growth><solow>
Sustainable growth in the long run
mnm123
# Sustainable growth in the long run

(Asked by: username_0 on 2018-05-21)

> 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

**Tags:** macroeconomics, microeconomics, economic-growth, solow

> **Comments:**
> **username_1**: What exactly is your question? You post a question and then copy paste what I assume is an excerpt from a book without further specifying where exactly  you are stuck  
> **username_0**: I couldn’t exactly do this question in the yellow box. And I searched this question from some books. And I found such an explanation. But I exactly not sure and i’m Waiting what you think about this question or this explanation @MaartenPunt  
> **username_1**: Well given that these partial answers provided come from economics textbook I think most economists would agree that this is true (within the framework of assumptions made, with which they may disagree). The original question though asks you to prove something. To be able to prove something like that we'd need additional info or assumptions. Like what is the production function? Cobb-Douglas, CES, Leontief or a general one? Is the labour force growing? Is technology constant? Etc etc  
> **username_0**: Okay thank you. This question in the yellow box is past preliminary exam question. And all question is this. That is, there is no additional information that you said. How can I prove this question? Can you help me to do this? I will become really happy. @MaartenPunt  
> **username_1**: I have no doubt that you will become happy. However I already said that I cannot "prove" this without additional details. I mean the question doesn't even specify the growth of what (presumably income per capita). Given that this is a homework question I will however provide a hint using the "standard" framework.  
> **username_0**: Okay I see what you mean. Yes this is general question. And this is not exactly homework. Past phd exam question as I said. But Thank for helps. Your answer really helpful. I will try to improve it. @MaartenPunt  

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## 1 answers

### Answer by username_1 on 2018-05-23. Score: 1

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.

> **Comments:**
> **username_0**: $$y_{t}=A_{t}k_{t}^{\alpha}x_{t}^{\beta}e_{t}^{\eta}$$ in order to rewrite this in terms of growth rate, I will take log $$lny_{t}=lnA_{t}+\alpha ln k_{t}+{\beta}lnx_{t}+{\eta}lne_{t}$$ and then I will take derivative with respect to t; $$\frac{\dot{y_t}}{y_t}=\frac{\dot{A_t}}{A_t}+\alpha \frac{\dot{k_t}}{k_t}+\beta \frac{\dot{x_t}}{x_t} +\eta \frac{\dot{e_t}}{e_t}$$ so , $$\frac{\dot{y_t}}{y_t}=g+\alpha * g + \beta  *h + \eta * \epsilon$$  so do I have shown the Groth rate expression correctly for this case? The production function is really difficult.  

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