<p>The min operation is known as a Leontief production function <a href="http://en.wikipedia.org/wiki/Leontief_production_function" rel="nofollow">http://en.wikipedia.org/wiki/Leontief_production_function</a></p>

<p>You're right that with Leontief production functions, the equilibrium is always aE=uK, which is to say that production is linear, and factors are consumed in fixed proportion. There are some real world examples for this: to drive, for example, you need a car and wheels in an exact 1:4 proportion, and having 6 wheels per car doesn't let you produce anything more than 4 wheels per car. But for the most part the aim is just to motivate a linear approximation to a multi-sector economy, which we can easily plug into a computer and run economy-wide computations with. We didn't have high powered computers in those days (c. 1950), so computing multi-sector non-linear production technologies was simply out of the question.</p>

CC BY-SA 3.0
2015-04-23T22:20:38.830Z
2015-04-23T22:20:38.830Z
Matthew
<p>I've been reading a book on a Kaleckian economics model, but there is a recursive function that has been bothering me. Can anyone provide some insight into why the equation why designed like this?</p>

<p>Quote from the book:</p>

<blockquote>
  <p>$Y = \min\{aE, uK\} \quad (1.1)$  </p>
  
  <p>Equation (1.1) denotes the fixed-coefficients production function, where $u$ denotes output; $E$, employment; $K$, capital stock; $a = Y/E$, labor productivity; and $u = Y/K$, the output-capital ratio. In the following analysis, we assume that the capital-potential output ratio is unity. From this, we can regard the output-capital ratio $u = Y/K$ as the capacity utilization rate.$^{4)}$
  ...</p>
  
  <p>$4)$ The capacity utilization rate $u$ is defined as $u = Y/Y^*$, where $Y$ denotes the actual output and $Y^*$ denotes the potential output. The capacity utilization rate is decomposed into $u = (Y/K)(K/Y^*)$, where $K/Y^*$ denotes the capital/potential output ratio and captures the production technology. If we assume that $K/Y^*$ is constant, then $u$ and $Y/K$ change in the same direction. From this, we can regard the output/capital ratio as the capacity utilization rate. In this chapter, for simplicity, we assume that $K/Y^* = 1$. Therefore, we obtain $u = Y/K.$</p>
</blockquote>

<p>I assume $a$ and $u$ are the fixed coefficients. But even if we suppose $uK &lt; aE $ then since $u = Y/K$, we should have $uK = aE$, or is that wrong? If correct, why do we use use the minimum operation?</p>

<p><a href="https://books.google.com/books?id=mZtH3gI61KcC&amp;pg=PA20&amp;lpg=PA20&amp;dq=From%20this,%20we%20can%20regard%20the%20output/capital%20ratio%20as%20the%20capacity%20utilization%20rate.%20In%20this%20chapter,%20for%20simplicity,%20we%20assume%20that%20K/Y%E2%88%97%3D1.%20Therefore,%20we%20obtain&amp;source=bl&amp;ots=PlBhuc9Mnd&amp;sig=siMN_IbN4hjYzuMXH3IBP5fkPCo&amp;hl=en&amp;sa=X&amp;ei=sXw4VfnbAdPhoATxp4D4Cw&amp;ved=0CCEQ6AEwAA#v=onepage&amp;q=From%20this%2C%20we%20can%20regard%20the%20output%2Fcapital%20ratio%20as%20the%20capacity%20utilization%20rate.%20In%20this%20chapter%2C%20for%20simplicity%2C%20we%20assume%20that%20K%2FY%E2%88%97%3D1.%20Therefore%2C%20we%20obtain&amp;f=false" rel="nofollow">Here</a> is the digital book. Quote from page 20.</p>

CC BY-SA 3.0
2015-04-23T13:40:13.720Z
There seems to be a typo: $Y$ should denote output, not $u$.
FooBar
<mathematical-economics><economic-growth>
A recursive production function
Jack
# A recursive production function

(Asked by: username_0 on 2015-04-21)

I've been reading a book on a Kaleckian economics model, but there is a recursive function that has been bothering me. Can anyone provide some insight into why the equation why designed like this?

Quote from the book:

> $Y = \min\{aE, uK\} \quad (1.1)$
>
> Equation (1.1) denotes the fixed-coefficients production function, where $u$ denotes output; $E$, employment; $K$, capital stock; $a = Y/E$, labor productivity; and $u = Y/K$, the output-capital ratio. In the following analysis, we assume that the capital-potential output ratio is unity. From this, we can regard the output-capital ratio $u = Y/K$ as the capacity utilization rate.$^{4)}$
> ...
>
> $4)$ The capacity utilization rate $u$ is defined as $u = Y/Y^\*$, where $Y$ denotes the actual output and $Y^\*$ denotes the potential output. The capacity utilization rate is decomposed into $u = (Y/K)(K/Y^\*)$, where $K/Y^\*$ denotes the capital/potential output ratio and captures the production technology. If we assume that $K/Y^\*$ is constant, then $u$ and $Y/K$ change in the same direction. From this, we can regard the output/capital ratio as the capacity utilization rate. In this chapter, for simplicity, we assume that $K/Y^\* = 1$. Therefore, we obtain $u = Y/K.$

I assume $a$ and $u$ are the fixed coefficients. But even if we suppose $uK < aE $ then since $u = Y/K$, we should have $uK = aE$, or is that wrong? If correct, why do we use use the minimum operation?

[Here](https://books.google.com/books?id=mZtH3gI61KcC&pg=PA20&lpg=PA20&dq=From%20this,%20we%20can%20regard%20the%20output/capital%20ratio%20as%20the%20capacity%20utilization%20rate.%20In%20this%20chapter,%20for%20simplicity,%20we%20assume%20that%20K/Y%E2%88%97%3D1.%20Therefore,%20we%20obtain&source=bl&ots=PlBhuc9Mnd&sig=siMN_IbN4hjYzuMXH3IBP5fkPCo&hl=en&sa=X&ei=sXw4VfnbAdPhoATxp4D4Cw&ved=0CCEQ6AEwAA#v=onepage&q=From%20this%2C%20we%20can%20regard%20the%20output%2Fcapital%20ratio%20as%20the%20capacity%20utilization%20rate.%20In%20this%20chapter%2C%20for%20simplicity%2C%20we%20assume%20that%20K%2FY%E2%88%97%3D1.%20Therefore%2C%20we%20obtain&f=false) is the digital book. Quote from page 20.

**Tags:** mathematical-economics, economic-growth
> **Comments:**
> **username_1**: There seems to be a typo: $Y$ should denote output, not $u$.  

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## 1 answers

### Answer by username_2 on 2015-04-23. Score: 2 | Accepted answer

The min operation is known as a Leontief production function <http://en.wikipedia.org/wiki/Leontief_production_function>

You're right that with Leontief production functions, the equilibrium is always aE=uK, which is to say that production is linear, and factors are consumed in fixed proportion. There are some real world examples for this: to drive, for example, you need a car and wheels in an exact 1:4 proportion, and having 6 wheels per car doesn't let you produce anything more than 4 wheels per car. But for the most part the aim is just to motivate a linear approximation to a multi-sector economy, which we can easily plug into a computer and run economy-wide computations with. We didn't have high powered computers in those days (c. 1950), so computing multi-sector non-linear production technologies was simply out of the question.


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