isoquant of a leontief production function

Question

Consider a firm that can produce q units of good G using two technologies and two production factors, $z_1$ and $z_2$. There are two ways how a firm can produce the good G: It can use 2 units of $z_1$ and 1 unit of $z_2$ to produce one unit of good G, or it can use 1 unit of $z_1$ and 2 units of $z_2$ to also produce one unit of the same good. Using these proportions of inputs, the firm is able to replicate any amount of good in question. The production set of the firm displays (among other properties) free disposal property and is convex.

My question is whether the production function is expressed like this $$f(z)=\min\{ z_1 + 2z_2 ; 2z_1 + z_2\}$$ and how would we express the isoquant function? Would it be like this?

Answer 1

I think it is simpler to analyze the production function first, and then use the definition of an isoquant.

Production function

Let's rewrite the production function in a more convenient way \begin{align*} f\left( z_1, z_2 \right) &= \min\left\lbrace z_1 + 2z_2, 2z_1 + z_2 \right\rbrace \\ &= \begin{cases} z_1 + 2z_2 && \text{if } z_1 + 2z_2 \leq 2z_1 + z_2 \\ 2z_1 + z_2 && \text{if } z_1 + 2z_2 > 2z_1 + z_2 \\ \end{cases} \\ &= \begin{cases} z_1 + 2z_2 && \text{if } z_2 \leq z_1 \\ 2z_1 + z_2 && \text{if } z_2 > z_1 \\ \end{cases} \end{align*} So the typical isoquant has a kink at $z_1 = z_2$, that is on along the 45° line.

Isoquant

Let $q$ be a fixed quantity. The level-q isoquant is given by the $z_1$ and $z_2$ that satisfy $f(z_1, z_2) = q$. Therefore, \begin{cases} z_1 + 2z_2 = q && \text{if } z_2 \leq z_1 \\ 2z_1 + z_2 = q && \text{if } z_2 > z_1 \\ \end{cases} Then, \begin{cases} z_2 = \frac{q - z_1}{2} && \text{if } z_2 \leq z_1 \\ z_2 = q - 2z_1 && \text{if } z_2 > z_1 \\ \end{cases} We end up with an equation $$ z_2 = \begin{cases} \frac{q - z_1}{2} && \text{if } z_2 \leq z_1 \\ q - 2z_1 && \text{if } z_2 > z_1 \\ \end{cases} $$

In words, the level-q isoquant is made of:

• the line $z_2 = \frac{q - z_1}{2}$ in the space $z_2 \leq z_1$ (below the 45° line)
• the line $z_2 = q - 2z_1$ in the space $z_2 > z_1$ (above the 45° line)

That is something we can draw. For example, for $q = 9$ we get