<p>The types of production processes that are considered in the Cambridge Capital Controversies are ones which are <strong>dynamic</strong>.</p>
<p>This means that production does not take place in a single period but over multiple periods.</p>
<p>With this structure the costs for a particular production process wont be paid all at the same time but rather at different points in time. (i.e. when you pay your workers is not at the same time you pay for your machinery).</p>
<p>One cost function which generates a concave factor price frontier (assuming <span class="math-container">$L=1$</span> and <span class="math-container">$K=1$</span>) is:
<span class="math-container">$$w+(1+r)^2=\bar{C}$$</span></p>
<p>If we are thinking about this in terms of timing we have:</p>
<ul>
<li>In the first period workers are paid their wage <span class="math-container">$w$</span>.</li>
<li>In the second period the owner of capital is paid <span class="math-container">$(1+r)^2$</span>.</li>
</ul>
<p>A concave factor price frontier can arise when there is interest and timing associated with payments.</p>
<p>In traditional economic models in which production functions do all the production in one period we will only see linear factor price frontiers.  If our production process is dynamic however(i.e. where capital is not paid for in the first period but later), then the factor price frontier will be concave.</p>
<p><strong>TLDR:</strong> The shape of your factor price frontier depends on if your costs are paid all in one period or over multiple periods. If production occurs in one period and is paid in one period then your factor price frontier will be linear otherwise it will be non linear/concave.</p>

CC BY-SA 4.0
2024-11-18T17:04:46.713Z
2024-11-18T18:47:37.200Z
2024-11-18T18:47:37.200Z
EconJohn
<p>As I continue my decent into madness of looking at the Cambridge Capital Controversy, I've been taking some time to look at examples of reswitching from the factor price frontier. Knowing that this is a bit of an obscure topic Ive taken some time to provide some background before asking my question.</p>
<h4>Background</h4>
<p>Typically when you are introduced to the factor price frontier in the context of the cambridge capital controversy you are shown three pictures, illustrating three cases:</p>
<p><strong>(a)</strong> The case of production with one technique.<br />
<strong>(b)</strong> The case of production with two techniques with switching.<br />
<strong>(c)</strong> The case of production with two techniques with reswitching.</p>
<p>The level curves in this space corresponds to a constant cost of production and what mix of factor prices will preserve a constant cost. In the case of switching and reswitching the frontier is defined by the outer envelope (as highlighted by the green dotted line) with switch points occurring at <span class="math-container">$\alpha$</span> and <span class="math-container">$\beta$</span> in their respective graphs.</p>
<p><a href="https://i.sstatic.net/V0dkRiht.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/V0dkRiht.png" alt="The Factor Price Frontier with one technique, Two Techniques with Switching and  Two Techniques Reswithing" /></a></p>
<p>Understanding these charts with factor prices was initially confusing for me specifically why we would consider the outer envelope  (shouldn't we consider the inner envelope?). To make this easier we replace <span class="math-container">$w$</span> and <span class="math-container">$r$</span> with <span class="math-container">$MPL$</span> and <span class="math-container">$MPK$</span>. giving us an updated figure below:</p>
<p><a href="https://i.sstatic.net/nH7u8UPN.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/nH7u8UPN.png" alt="Replacing w and r with MPL and MPK" /></a></p>
<p>This makes things much clearer as now as it makes sense to choose the technique which will provide you a higher marginal products of labor and capital assuming the costs are the same. Further it makes explicitly clear that we can interpret each point along the level curves as the slope of our Isoquant for an underlying production function<sup>1</sup>. this works well for Cobb Douglas type of level curves.</p>
<h4>Main question</h4>
<p>I understand how we can have a linear factor price frontier and the case of switching. What technology must we have for reswitching to occur as it implies that the slope of our isoquants changes in a concave manner?</p>
<p><sub>1. The math on this is: <span class="math-container">$\frac{1}{MRTS}=\frac{1}{-\frac{\frac{\partial F}{\partial L}}{\frac{\partial F}{\partial K}}}=-\frac{\frac{\partial F}{\partial K}}{\frac{\partial F}{\partial L}}=-\frac{r}{w}$</span></sub></p>

<cambridge-capital-controversy><reswitching>
Cambridge Capital Controversy: what technology do we need to have for "reswitching" to occur according to the factor price frontier?
EconJohn
# Cambridge Capital Controversy: what technology do we need to have for "reswitching" to occur according to the factor price frontier?

(Asked by: username_0 on 2024-11-14)

As I continue my decent into madness of looking at the Cambridge Capital Controversy, I've been taking some time to look at examples of reswitching from the factor price frontier. Knowing that this is a bit of an obscure topic Ive taken some time to provide some background before asking my question.

#### Background

Typically when you are introduced to the factor price frontier in the context of the cambridge capital controversy you are shown three pictures, illustrating three cases:

**(a)** The case of production with one technique.  
**(b)** The case of production with two techniques with switching.  
**(c)** The case of production with two techniques with reswitching.

The level curves in this space corresponds to a constant cost of production and what mix of factor prices will preserve a constant cost. In the case of switching and reswitching the frontier is defined by the outer envelope (as highlighted by the green dotted line) with switch points occurring at $\alpha$ and $\beta$ in their respective graphs.

[![The Factor Price Frontier with one technique, Two Techniques with Switching and  Two Techniques Reswithing](https://i.sstatic.net/V0dkRiht.png)](https://i.sstatic.net/V0dkRiht.png)

Understanding these charts with factor prices was initially confusing for me specifically why we would consider the outer envelope (shouldn't we consider the inner envelope?). To make this easier we replace $w$ and $r$ with $MPL$ and $MPK$. giving us an updated figure below:

[![Replacing w and r with MPL and MPK](https://i.sstatic.net/nH7u8UPN.png)](https://i.sstatic.net/nH7u8UPN.png)

This makes things much clearer as now as it makes sense to choose the technique which will provide you a higher marginal products of labor and capital assuming the costs are the same. Further it makes explicitly clear that we can interpret each point along the level curves as the slope of our Isoquant for an underlying production function1. this works well for Cobb Douglas type of level curves.

#### Main question

I understand how we can have a linear factor price frontier and the case of switching. What technology must we have for reswitching to occur as it implies that the slope of our isoquants changes in a concave manner?

1. The math on this is: $\frac{1}{MRTS}=\frac{1}{-\frac{\frac{\partial F}{\partial L}}{\frac{\partial F}{\partial K}}}=-\frac{\frac{\partial F}{\partial K}}{\frac{\partial F}{\partial L}}=-\frac{r}{w}$

**Tags:** cambridge-capital-controversy, reswitching

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## 1 answers

### Answer by username_0 on 2024-11-18. Score: 2 | Accepted answer

The types of production processes that are considered in the Cambridge Capital Controversies are ones which are **dynamic**.

This means that production does not take place in a single period but over multiple periods.

With this structure the costs for a particular production process wont be paid all at the same time but rather at different points in time. (i.e. when you pay your workers is not at the same time you pay for your machinery).

One cost function which generates a concave factor price frontier (assuming $L=1$ and $K=1$) is:
$$w+(1+r)^2=\bar{C}$$

If we are thinking about this in terms of timing we have:

* In the first period workers are paid their wage $w$.
* In the second period the owner of capital is paid $(1+r)^2$.

A concave factor price frontier can arise when there is interest and timing associated with payments.

In traditional economic models in which production functions do all the production in one period we will only see linear factor price frontiers. If our production process is dynamic however(i.e. where capital is not paid for in the first period but later), then the factor price frontier will be concave.

**TLDR:** The shape of your factor price frontier depends on if your costs are paid all in one period or over multiple periods. If production occurs in one period and is paid in one period then your factor price frontier will be linear otherwise it will be non linear/concave.


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