<p>I will answer this because it is relatively easy and because I assume future students might be curious about how to show this. However, you really don't deserve for me to just answer your homework question and so I will give an outline rather than a step-by-step answer. </p>

<p>Suppose the following CD production function (I am omitting labor so that you can't just copy the parts of this that I complete): </p>

<p>$$U(K) =A K^{\alpha}$$</p>

<p>To show this production function has diminishing marginal returns we must show: </p>

<p>1) That the first derivative of this function is positive</p>

<p>2) That the second derivative of this function is negative </p>

<p>For an example of 1: </p>

<p>$$U'() = \alpha AK^{\alpha-1}$$</p>

<p>this is clearly a positive number and so 1 is satisfied. </p>

<p>Now, what about 2? When we take the second derivative here, our leading term is now $(\alpha-1)\alpha$ What do we know about alpha? Is our knowledge of alpha enough to know that the second derivative is negative, as desired? </p>

CC BY-SA 3.0
2017-03-22T19:36:12.137Z
Thanks. It's not homework its a sample exam question which I have soon. Its not in my macroeconomics book or anywhere on the internet and i have no teacher so its hard to find answers
yoyo2000
CC BY-SA 3.0
2017-03-22T20:49:34.383Z
Fair enough. You'll have better luck in the future if you make that explicit and also show us your thoughts about how to solve the question. That said, if you edit this post with your attempt to solve this question, I am happy to give you feedback on your solution and to help you achieve a correct solution.
123
CC BY-SA 3.0
2017-03-22T17:16:28.433Z
2017-03-22T17:41:39.650Z
2017-03-22T17:41:39.650Z
123
<p>Use mathematical analysis to show that the Cobb-Douglas production function is consistent with the law of diminishing returns in the short run. Assume that capital is the variable input. </p>

CC BY-SA 3.0
2017-03-22T16:57:30.463Z
Welcome to Economics SE. Please read our [policy](https://economics.meta.stackexchange.com/questions/1465/economics-se-policy-on-homework-questions) on homework questions and our general [FAQ](http://economics.stackexchange.com/help/on-topic).
Oliv
CC BY-SA 3.0
2017-03-23T10:10:59.653Z
I'm voting to close this question as off-topic because you have not indicated what you have tried and what your particular difficulty is.
Adam Bailey
<macroeconomics>
Cobb Douglas production function question
yoyo2000
# Cobb Douglas production function question

(Asked by: username_0 on 2017-03-22)

Use mathematical analysis to show that the Cobb-Douglas production function is consistent with the law of diminishing returns in the short run. Assume that capital is the variable input.

**Tags:** macroeconomics

> **Comments:**
> **username_1**: Welcome to Economics SE. Please read our [policy](https://economics.meta.stackexchange.com/questions/1465/economics-se-policy-on-homework-questions) on homework questions and our general [FAQ](http://economics.stackexchange.com/help/on-topic).  
> **username_2**: I'm voting to close this question as off-topic because you have not indicated what you have tried and what your particular difficulty is.  

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## 1 answers

### Answer by username_3 on 2017-03-22. Score: 1

I will answer this because it is relatively easy and because I assume future students might be curious about how to show this. However, you really don't deserve for me to just answer your homework question and so I will give an outline rather than a step-by-step answer.

Suppose the following CD production function (I am omitting labor so that you can't just copy the parts of this that I complete):

$$U(K) =A K^{\alpha}$$

To show this production function has diminishing marginal returns we must show:

1) That the first derivative of this function is positive

2) That the second derivative of this function is negative

For an example of 1:

$$U'() = \alpha AK^{\alpha-1}$$

this is clearly a positive number and so 1 is satisfied.

Now, what about 2? When we take the second derivative here, our leading term is now $(\alpha-1)\alpha$ What do we know about alpha? Is our knowledge of alpha enough to know that the second derivative is negative, as desired?

> **Comments:**
> **username_0**: Thanks. It's not homework its a sample exam question which I have soon. Its not in my macroeconomics book or anywhere on the internet and i have no teacher so its hard to find answers  
> **username_3**: Fair enough. You'll have better luck in the future if you make that explicit and also show us your thoughts about how to solve the question. That said, if you edit this post with your attempt to solve this question, I am happy to give you feedback on your solution and to help you achieve a correct solution.  

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