<p>The dotted line is just to help you read the scale and compare the isoquantas at each production level. It touches all the isoquantas at the point where you have equal inputs of labour and capital.</p>

<p>The rest of the explanation is on the figure itself - it's just an example of increasing / constant / decreasing returns to scale.</p>

CC BY-SA 4.0
2018-05-08T12:49:26.500Z
2018-05-08T12:49:26.500Z
JoaoBotelho
<p>I've computed <a href="https://www.wolframalpha.com/input/?i=f(K,L)%3DK%5E(0.4)+*+L%5E(0.6)" rel="nofollow noreferrer">this</a> in WolframAlpha and I think it'll help you understand just what exactly a production function is. Remember, the production function $Y=f(K,L)$ is defined by $f:\mathbb{R}^2 \to \mathbb{R}$. When you see a graph with axis in $K$ and $L$, you`re seeing a part of a 3D object, the surface that a function with domain in $\mathbb{R}^2$ generates. The isoquants are just contour lines of it. </p>

<p>The change in the isoquant's steepness is a consequence of the shape a Cobb-Douglas Produtcion Function's surface. The non-linear nature of Cobb-Douglas model makes its contour lines change steepness as $f(K,L)$ gets bigger, as production goes up. There's an interesting economic intuition behind it. The more inputs a firm employs in its production process, the smaller the marginal rate of technical substitution of the optimal plant is because of diminishing marginal returns. As a firm employs more of an input, the smaller is that input`s marginal productvity. As a firms employs more and more of <em>two</em> inputs, capital and labor, both the marginal productivity of capital and labor get smaller, meaning a smaller marginal rate of technical substitution.</p>

CC BY-SA 4.0
2018-05-07T00:27:20.140Z
2018-05-07T01:10:58.027Z
2018-05-07T01:10:58.027Z
Pedro Cavalcante
<p><a href="https://i.sstatic.net/7fhCG.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/7fhCG.png" alt="enter image description here"></a></p>

<p>What does the dotted line in the diagram show and what are they called? </p>

<p>I am also wondering why the steepness of the isoquants changes at Q=3 for the third diagram. </p>

<p>Thanks</p>

<microeconomics><production-function>
Question on returns of scale in isoquants
D.I.N
# Question on returns of scale in isoquants

(Asked by: username_0 on 2018-05-06)

[![enter image description here](https://i.sstatic.net/7fhCG.png)](https://i.sstatic.net/7fhCG.png)

What does the dotted line in the diagram show and what are they called?

I am also wondering why the steepness of the isoquants changes at Q=3 for the third diagram.

Thanks

**Tags:** microeconomics, production-function

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## 2 answers

### Answer by username_1 on 2018-05-08. Score: 1

The dotted line is just to help you read the scale and compare the isoquantas at each production level. It touches all the isoquantas at the point where you have equal inputs of labour and capital.

The rest of the explanation is on the figure itself - it's just an example of increasing / constant / decreasing returns to scale.


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### Answer by username_2 on 2018-05-07. Score: 1

I've computed [this](https://www.wolframalpha.com/input/?i=f(K,L)%3DK%5E(0.4)+*+L%5E(0.6)) in WolframAlpha and I think it'll help you understand just what exactly a production function is. Remember, the production function $Y=f(K,L)$ is defined by $f:\mathbb{R}^2 \to \mathbb{R}$. When you see a graph with axis in $K$ and $L$, you`re seeing a part of a 3D object, the surface that a function with domain in $\mathbb{R}^2$ generates. The isoquants are just contour lines of it.

The change in the isoquant's steepness is a consequence of the shape a Cobb-Douglas Produtcion Function's surface. The non-linear nature of Cobb-Douglas model makes its contour lines change steepness as $f(K,L)$ gets bigger, as production goes up. There's an interesting economic intuition behind it. The more inputs a firm employs in its production process, the smaller the marginal rate of technical substitution of the optimal plant is because of diminishing marginal returns. As a firm employs more of an input, the smaller is that input`s marginal productvity. As a firms employs more and more of *two* inputs, capital and labor, both the marginal productivity of capital and labor get smaller, meaning a smaller marginal rate of technical substitution.


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