<p>You can set up the cost minimization problem explicitly, but it might be easier to consider the similarity with the utility maximising consumer problem and first look at the dual utility maxmisation problem.</p>
<p>If so, the utility (output) maximisation problem takes the form:
<span class="math-container">$$
\max_{s_j, \ell} \left(\left(\sum_{j= 1}^N s_j^\mu\right)^{1/\mu}\right)^\beta \ell^{1 - \beta} \text{ s.t. } \sum_j p_j s_j + w \ell = m.
$$</span>
This utility function is weakly separable, so we can use two stage budgetting where in the second stage one optimally chooses <span class="math-container">$s_1, \ldots, s_N$</span> given the expenditures on these goods. Let <span class="math-container">$z = \sum_{j} p_j s_j$</span> be the expenditure on the goods <span class="math-container">$s_j$</span>. Then the second stage can be written as:
<span class="math-container">$$
\max_{s_j} \left(\sum_{j = 1}^N s_j^\mu\right)^{1/\mu} \text{ s.t. } \sum_j p_j s_j = z.
$$</span>
This production function is a standard CES. So the indirect utility function takes the form (double check):
<span class="math-container">$$
\left(\sum_{j = 1}^N s_j^\mu\right)^{1/\mu}= \underbrace{\left(\sum_j p_{j}^{1 - \mu}\right)^{1/(\mu - 1)}}_{=P} z.
$$</span>
Substituting into the utility function gives the first stage problem:
<span class="math-container">$$
\max_{z, \ell}  (Pz)^\beta \ell^{1-\beta} \text{ s.t. } z + y = m
$$</span>
Now, reparametrize <span class="math-container">$x = P z$</span> and <span class="math-container">$p = 1/P$</span> to obtain the following problem:
<span class="math-container">$$
\max_{z, \ell}  (x)^\beta \ell^{1-\beta} \text{ s.t. } px + w \ell = m
$$</span>
This is a standard Cobb-Douglass utility maximisation problem, so the indirect utility is (double check):
<span class="math-container">$$
v(p,w,m) = \beta^\beta (1-\beta)^{1-\beta} \left(1/p\right)^{\beta} \left(1/w\right)^{1-\beta} m
$$</span>
Given this, we can subsitute out <span class="math-container">$p= 1/P$</span> to get:
<span class="math-container">$$
v(p, w, m) = \beta^\beta (1-\beta)^{1-\beta} \left(\sum_j p_j^{1 - \mu}\right)^{\beta/(\mu - 1)} \left(1/w\right)^{1-\beta} m
$$</span>
This gives the indirect utility function. The expenditure (cost) function can be obtained by inverting this with respect to <span class="math-container">$m$</span>:
<span class="math-container">$$
c(p,w, y) = \left(1/\beta\right)^\beta \left(1/(1-\beta)\right)^{1-\beta} \left(\sum_j p_j^{1-\mu}\right)^{-\beta/(\mu - 1)} \left(w\right)^{1-\beta} y.
$$</span>
Note that this is linear in <span class="math-container">$y$</span> as the production function is homogeneous of degree 1.</p>

CC BY-SA 4.0
2023-09-21T05:12:54.893Z
2023-09-21T05:17:59.443Z
2023-09-21T05:17:59.443Z
tdm
<p>Assume that there is a production function as follow:</p>
<p><span class="math-container">$$h=\left\{ \left[ \sum_{j=1}^N{s_{j}^{\mu}} \right] ^{\frac{1}{\mu}} \right\} ^{\beta}l^{1-\beta}$$</span></p>
<p>The price of <span class="math-container">$s_j$</span> is <span class="math-container">$p_j$</span> and the wage of <span class="math-container">$l$</span> is <span class="math-container">$w_l$</span>. <span class="math-container">$h$</span> is the final good and <span class="math-container">$\mu ,\beta ,N$</span> are parameters. I tried use Lagrange method to calculate cost function but I failed. How to solve this kind of production function's cost function?</p>

<microeconomics>
How can I write the unit cost function of this nesting production function
Wang Simon
# How can I write the unit cost function of this nesting production function

(Asked by: username_0 on 2023-09-20)

Assume that there is a production function as follow:

$$h=\left\{ \left[ \sum\_{j=1}^N{s\_{j}^{\mu}} \right] ^{\frac{1}{\mu}} \right\} ^{\beta}l^{1-\beta}$$

The price of $s\_j$ is $p\_j$ and the wage of $l$ is $w\_l$. $h$ is the final good and $\mu ,\beta ,N$ are parameters. I tried use Lagrange method to calculate cost function but I failed. How to solve this kind of production function's cost function?

**Tags:** microeconomics


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## 1 answers

### Answer by username_1 on 2023-09-21. Score: 2

You can set up the cost minimization problem explicitly, but it might be easier to consider the similarity with the utility maximising consumer problem and first look at the dual utility maxmisation problem.

If so, the utility (output) maximisation problem takes the form:
$$
\max\_{s\_j, \ell} \left(\left(\sum\_{j= 1}^N s\_j^\mu\right)^{1/\mu}\right)^\beta \ell^{1 - \beta} \text{ s.t. } \sum\_j p\_j s\_j + w \ell = m.
$$
This utility function is weakly separable, so we can use two stage budgetting where in the second stage one optimally chooses $s\_1, \ldots, s\_N$ given the expenditures on these goods. Let $z = \sum\_{j} p\_j s\_j$ be the expenditure on the goods $s\_j$. Then the second stage can be written as:
$$
\max\_{s\_j} \left(\sum\_{j = 1}^N s\_j^\mu\right)^{1/\mu} \text{ s.t. } \sum\_j p\_j s\_j = z.
$$
This production function is a standard CES. So the indirect utility function takes the form (double check):
$$
\left(\sum\_{j = 1}^N s\_j^\mu\right)^{1/\mu}= \underbrace{\left(\sum\_j p\_{j}^{1 - \mu}\right)^{1/(\mu - 1)}}\_{=P} z.
$$
Substituting into the utility function gives the first stage problem:
$$
\max\_{z, \ell} (Pz)^\beta \ell^{1-\beta} \text{ s.t. } z + y = m
$$
Now, reparametrize $x = P z$ and $p = 1/P$ to obtain the following problem:
$$
\max\_{z, \ell} (x)^\beta \ell^{1-\beta} \text{ s.t. } px + w \ell = m
$$
This is a standard Cobb-Douglass utility maximisation problem, so the indirect utility is (double check):
$$
v(p,w,m) = \beta^\beta (1-\beta)^{1-\beta} \left(1/p\right)^{\beta} \left(1/w\right)^{1-\beta} m
$$
Given this, we can subsitute out $p= 1/P$ to get:
$$
v(p, w, m) = \beta^\beta (1-\beta)^{1-\beta} \left(\sum\_j p\_j^{1 - \mu}\right)^{\beta/(\mu - 1)} \left(1/w\right)^{1-\beta} m
$$
This gives the indirect utility function. The expenditure (cost) function can be obtained by inverting this with respect to $m$:
$$
c(p,w, y) = \left(1/\beta\right)^\beta \left(1/(1-\beta)\right)^{1-\beta} \left(\sum\_j p\_j^{1-\mu}\right)^{-\beta/(\mu - 1)} \left(w\right)^{1-\beta} y.
$$
Note that this is linear in $y$ as the production function is homogeneous of degree 1.


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