# A recursive production function

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 is the digital book. Quote from page 20.

• There seems to be a typo: $Y$ should denote output, not $u$. Apr 23, 2015 at 13:40

## 1 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.