<p>Here <span class="math-container">$a_{iL}(w,r)$</span> and <span class="math-container">$a_{iK}(w,r)$</span> is just notation for the solutions to the cost minimization problem. There is no need for any derivation. It could only be possible to find a specific solution (i.e. solve the cost minimization problem to find the functions <span class="math-container">$a_{iL}$</span> and <span class="math-container">$a_{iK}$</span>) if you were given a specific production function <span class="math-container">$f_i$</span>.</p>
<p>[I suppose here <span class="math-container">$f_i$</span> must be constant returns to scale to make the idea of a unit cost function useful.]</p>

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2023-11-02T09:33:19.853Z
2023-11-02T09:33:19.853Z
smcc
<p>Unit cost function is the minimum cost to produce <strong>one unit</strong> of output.</p>
<p><a href="https://i.sstatic.net/0aIku.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/0aIku.png" alt="enter image description here" /></a></p>
<p>The solution to the above problem is:</p>
<p><a href="https://i.sstatic.net/1qZC7.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/1qZC7.png" alt="enter image description here" /></a></p>
<p>where ai(w,r) is the amount of labour needed to produce 1 unit of good i at factor prices (w,r).</p>
<p>How do we arrive at this specific solution by finding the optimal values of K and L?</p>

<macroeconomics><microeconomics><international-trade>
Deriving Unit cost function
user205152
# Deriving Unit cost function

(Asked by: username_0 on 2023-10-28)

Unit cost function is the minimum cost to produce **one unit** of output.

[![enter image description here](https://i.sstatic.net/0aIku.png)](https://i.sstatic.net/0aIku.png)

The solution to the above problem is:

[![enter image description here](https://i.sstatic.net/1qZC7.png)](https://i.sstatic.net/1qZC7.png)

where ai(w,r) is the amount of labour needed to produce 1 unit of good i at factor prices (w,r).

How do we arrive at this specific solution by finding the optimal values of K and L?

**Tags:** macroeconomics, microeconomics, international-trade

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## 1 answers

### Answer by username_1 on 2023-11-02. Score: 1

Here $a\_{iL}(w,r)$ and $a\_{iK}(w,r)$ is just notation for the solutions to the cost minimization problem. There is no need for any derivation. It could only be possible to find a specific solution (i.e. solve the cost minimization problem to find the functions $a\_{iL}$ and $a\_{iK}$) if you were given a specific production function $f\_i$.

[I suppose here $f\_i$ must be constant returns to scale to make the idea of a unit cost function useful.]


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