<p>I'm not sure this answers your question, but if I understand your question correctly, you are just asking what Frischian demand functions are. Very short: Frischian demands are the demands expressed as a function of prices and marginal utility of income.</p>
<p>For a thorough intro to Frischian demand functions, see the great paper of <a href="https://doi.org/10.2307/1911653" rel="nofollow noreferrer">Browning, Deaton and Irish (1985)</a> or the nice working paper of <a href="https://ideas.repec.org/p/ivi/wpasad/1993-10.html" rel="nofollow noreferrer">Browning (1993)</a></p>
<p>In demand theory, one usually expresses demand as a function of prices <span class="math-container">$p$</span> and total income <span class="math-container">$m$</span>. This gives the Marshallian demand functions <span class="math-container">$q_i(p,m)$</span>. It is the solution to the following utility maximization problem:
<span class="math-container">$$
v(p,m) = \max_{q} u(q) \text{ s.t. } pq \le m.
$$</span>
The function <span class="math-container">$v(p,m)$</span> is the indirect utility function.</p>
<p>There are alternatives, however. For example, one could express everything in terms of prices and utility. This corresponds to the problem of minimizing expenditures given some threshold utility level <span class="math-container">$u$</span>:
<span class="math-container">$$
e(p,u) = \min_{q} pq \text{ s.t. } u(q) \ge u.
$$</span>
The function <span class="math-container">$e(p,u)$</span> is the expenditure function and the solution to the expenditure minimization problem gives the Hicksian demand functions <span class="math-container">$h_i(p,u)$</span>, which, using the envelope theorem, can be obtained by taking the partial derivatives of <span class="math-container">$e(p,u)$</span> with respect to the prices.
<span class="math-container">$$
\frac{\partial e(p,u)}{\partial p_i} = h_i(p,u).
$$</span></p>
<p>There are some well known dualities between the Marshallian and Hicksian framework. For example, for all <span class="math-container">$(p,u)$</span>:
<span class="math-container">$$
v(p,e(p,u)) = u
$$</span>
and for all <span class="math-container">$(p,m)$</span>
<span class="math-container">$$
e(p,v(p,m)) = m.
$$</span></p>
<p>There is a third (somewhat less known) alternative framework that expresses demand as a function of prices and the marginal utility of income. Actually, it is more convenient to use the reciprocal of the marginal utility of income, which I will denote by <span class="math-container">$r$</span>. The underlying optimization problem is the so called &quot;profit&quot; maximization problem:
<span class="math-container">$$
\pi(p,r) = \max_{u,q} r u - p q \text{ s.t. } u(q) = u.
$$</span>
Intuitively, the consumer purchases a bundle <span class="math-container">$q$</span> and sells the resulting utility at a price <span class="math-container">$r$</span>.</p>
<p>Notice the similarity with producer theory. The utility maximization problem corresponds to the output maximization problem (given a certain expenditure level for the inputs). The expenditure minimization problem corresponds to the cost minimization problem where we minimize costs given some desired level of output. The profit maximization problem corresponds to, well, the profit maximization problem of producer theory.</p>
<p>The first order conditions of this problem gives:
<span class="math-container">$$
r \frac{\partial u(q)}{\partial q_i} = p_i.
$$</span>
Note the similarity between these and the first order conditions for the utility maximisation problem, when substituting <span class="math-container">$r$</span> with the reciprocal of the Lagrange multiplier (which indeed give the marginal utility of income).</p>
<p>The resulting demand functions, say <span class="math-container">$f_i(p,r)$</span> are called the Frischian demand functions. By the envelope theorem, we have:
<span class="math-container">$$
\frac{\partial \pi(p,r)}{\partial p_i} = -f_i(p,r).
$$</span>
As demands are the negatives of the derivatives of the profit function, we also obtain (by Young's theorem) that the cross price effects are symmetric:
<span class="math-container">$$
\frac{\partial f_i(p,r)}{\partial p_j} = \frac{\partial f_j(p,r)}{\partial p_i}.
$$</span>
This is similar to Hicksian symmetry.</p>
<p>The function <span class="math-container">$\pi(p,r)$</span> can be shown to be convex (if utility is concave), so the Hessian (which is the matrix of the negatives of the cross partial derivatives) is positive semi-definite. This means that Frischian demands are downward sloping: <span class="math-container">$\frac{\partial f_i(p,r)}{\partial p_i} &lt; 0$</span>. Also note that <span class="math-container">$f_i(p,r)$</span> is homogeneous of degree zero in <span class="math-container">$p$</span> and <span class="math-container">$r$</span>.</p>
<p>Note that also:
<span class="math-container">$$
\begin{align*}
\pi(p,r) &amp;= \max_{u,q} ru - pq \text{ s.t. } u(q) = u,\\
&amp;= \max_u \left\{ru - \min_{q} \left\{pq | u(q) = u\right\}\right\},\\
&amp;= \max_u ru - c(p,u). 
\end{align*}
$$</span></p>
<p>This gives the relationship between the profit function and the cost function (the profit function is the concave conjugate of the cost function). By conjugacy:
<span class="math-container">$$
c(p,u) = \max_{r} ru - \pi(p,r).
$$</span></p>
<p>Next,
<span class="math-container">$$
\begin{align*}
\pi(p,r) &amp;= \max_{q} r u(q) - pq,\\
&amp;= \max_{m}\left\{-m + r \max_{q} \{u(q)| pq = m\}\right\},\\
&amp;= \max_m rv(p,m) - m
\end{align*}
$$</span>
This gives the relationship between the profit and indirect utility function.
Frischian demands are quite convenient to work with if utility functions are additively separable.</p>
<p>Consider the case where <span class="math-container">$u(q) = \sum_t v^t(q^t)$</span> where <span class="math-container">$t$</span> denotes the subgroups. Then:
<span class="math-container">$$
\begin{align*}
\pi(p,r) &amp;= \max_{u, q_t} r u - \sum_t p^t q^t \text{ s.t. } \sum_t v^t(q^t) = u,\\
&amp;= \sum_t \max_{v^t, q^t} \left\{r v^t(q^t) - p^t q^t\right\},\\
&amp;= \sum_t \pi(p^t, r).
\end{align*}
$$</span>
So the profit function is also additively decomposable. The Frischian demand of good <span class="math-container">$i$</span> in subgroup <span class="math-container">$t$</span>, will only be a function of prices <span class="math-container">$p^t$</span> in group <span class="math-container">$t$</span> and <span class="math-container">$r$</span>,  <span class="math-container">$f^t_i(p^tt, r)$</span>.</p>
<p>If we take, for example <span class="math-container">$t$</span> to denote time (so utility is additively separable over time), then we know that, if there is no uncertainty, the marginal utility of income (<span class="math-container">$1/r$</span>) remains constant. As such, the Frischian demands show that demand at period <span class="math-container">$t$</span>, will only depend on the prices in this period <span class="math-container">$p^t$</span> once controlling for the value of <span class="math-container">$r$</span>. So all intertemporal effects (from outside period <span class="math-container">$t$</span>) are captured by the single sufficient (but unobservable) statistic <span class="math-container">$r$</span>.</p>

CC BY-SA 4.0
2024-03-04T08:58:59.447Z
2024-03-04T08:58:59.447Z
tdm
<p>I understand that Frisch elasticity is the &quot;elasticity of hours worked to the wage rate&quot;.<br />
I wonder why then we use this specific naming only for this?<br />
Is this to make it clear what this means?<br />
or is there any other special meaning or characteristics on &quot;Frisch elasticity&quot;?</p>

CC BY-SA 4.0
2024-02-29T08:46:15.377Z
Does the [Wikipedia article on the subject](https://en.wikipedia.org/wiki/Frisch_elasticity_of_labor_supply) answer your question? If not, could you elaborate on what it does not answer?
Giskard
<macroeconomics><microeconomics>
What is the frisch elasticity and its meaning?
student1
# What is the frisch elasticity and its meaning?

(Asked by: username_0 on 2024-02-29)

I understand that Frisch elasticity is the "elasticity of hours worked to the wage rate".  
I wonder why then we use this specific naming only for this?  
Is this to make it clear what this means?  
or is there any other special meaning or characteristics on "Frisch elasticity"?

**Tags:** macroeconomics, microeconomics

> **Comments:**
> **username_1**: Does the [Wikipedia article on the subject](https://en.wikipedia.org/wiki/Frisch_elasticity_of_labor_supply) answer your question? If not, could you elaborate on what it does not answer?  

---

## 1 answers

### Answer by username_2 on 2024-03-04. Score: 3

I'm not sure this answers your question, but if I understand your question correctly, you are just asking what Frischian demand functions are. Very short: Frischian demands are the demands expressed as a function of prices and marginal utility of income.

For a thorough intro to Frischian demand functions, see the great paper of [Browning, Deaton and Irish (1985)](https://doi.org/10.2307/1911653) or the nice working paper of [Browning (1993)](https://ideas.repec.org/p/ivi/wpasad/1993-10.html)

In demand theory, one usually expresses demand as a function of prices $p$ and total income $m$. This gives the Marshallian demand functions $q\_i(p,m)$. It is the solution to the following utility maximization problem:
$$
v(p,m) = \max\_{q} u(q) \text{ s.t. } pq \le m.
$$
The function $v(p,m)$ is the indirect utility function.

There are alternatives, however. For example, one could express everything in terms of prices and utility. This corresponds to the problem of minimizing expenditures given some threshold utility level $u$:
$$
e(p,u) = \min\_{q} pq \text{ s.t. } u(q) \ge u.
$$
The function $e(p,u)$ is the expenditure function and the solution to the expenditure minimization problem gives the Hicksian demand functions $h\_i(p,u)$, which, using the envelope theorem, can be obtained by taking the partial derivatives of $e(p,u)$ with respect to the prices.
$$
\frac{\partial e(p,u)}{\partial p\_i} = h\_i(p,u).
$$

There are some well known dualities between the Marshallian and Hicksian framework. For example, for all $(p,u)$:
$$
v(p,e(p,u)) = u
$$
and for all $(p,m)$
$$
e(p,v(p,m)) = m.
$$

There is a third (somewhat less known) alternative framework that expresses demand as a function of prices and the marginal utility of income. Actually, it is more convenient to use the reciprocal of the marginal utility of income, which I will denote by $r$. The underlying optimization problem is the so called "profit" maximization problem:
$$
\pi(p,r) = \max\_{u,q} r u - p q \text{ s.t. } u(q) = u.
$$
Intuitively, the consumer purchases a bundle $q$ and sells the resulting utility at a price $r$.

Notice the similarity with producer theory. The utility maximization problem corresponds to the output maximization problem (given a certain expenditure level for the inputs). The expenditure minimization problem corresponds to the cost minimization problem where we minimize costs given some desired level of output. The profit maximization problem corresponds to, well, the profit maximization problem of producer theory.

The first order conditions of this problem gives:
$$
r \frac{\partial u(q)}{\partial q\_i} = p\_i.
$$
Note the similarity between these and the first order conditions for the utility maximisation problem, when substituting $r$ with the reciprocal of the Lagrange multiplier (which indeed give the marginal utility of income).

The resulting demand functions, say $f\_i(p,r)$ are called the Frischian demand functions. By the envelope theorem, we have:
$$
\frac{\partial \pi(p,r)}{\partial p\_i} = -f\_i(p,r).
$$
As demands are the negatives of the derivatives of the profit function, we also obtain (by Young's theorem) that the cross price effects are symmetric:
$$
\frac{\partial f\_i(p,r)}{\partial p\_j} = \frac{\partial f\_j(p,r)}{\partial p\_i}.
$$
This is similar to Hicksian symmetry.

The function $\pi(p,r)$ can be shown to be convex (if utility is concave), so the Hessian (which is the matrix of the negatives of the cross partial derivatives) is positive semi-definite. This means that Frischian demands are downward sloping: $\frac{\partial f\_i(p,r)}{\partial p\_i} < 0$. Also note that $f\_i(p,r)$ is homogeneous of degree zero in $p$ and $r$.

Note that also:
$$
\begin{align\*}
\pi(p,r) &= \max\_{u,q} ru - pq \text{ s.t. } u(q) = u,\\
&= \max\_u \left\{ru - \min\_{q} \left\{pq | u(q) = u\right\}\right\},\\
&= \max\_u ru - c(p,u).
\end{align\*}
$$

This gives the relationship between the profit function and the cost function (the profit function is the concave conjugate of the cost function). By conjugacy:
$$
c(p,u) = \max\_{r} ru - \pi(p,r).
$$

Next,
$$
\begin{align\*}
\pi(p,r) &= \max\_{q} r u(q) - pq,\\
&= \max\_{m}\left\{-m + r \max\_{q} \{u(q)| pq = m\}\right\},\\
&= \max\_m rv(p,m) - m
\end{align\*}
$$
This gives the relationship between the profit and indirect utility function.
Frischian demands are quite convenient to work with if utility functions are additively separable.

Consider the case where $u(q) = \sum\_t v^t(q^t)$ where $t$ denotes the subgroups. Then:
$$
\begin{align\*}
\pi(p,r) &= \max\_{u, q\_t} r u - \sum\_t p^t q^t \text{ s.t. } \sum\_t v^t(q^t) = u,\\
&= \sum\_t \max\_{v^t, q^t} \left\{r v^t(q^t) - p^t q^t\right\},\\
&= \sum\_t \pi(p^t, r).
\end{align\*}
$$
So the profit function is also additively decomposable. The Frischian demand of good $i$ in subgroup $t$, will only be a function of prices $p^t$ in group $t$ and $r$, $f^t\_i(p^tt, r)$.

If we take, for example $t$ to denote time (so utility is additively separable over time), then we know that, if there is no uncertainty, the marginal utility of income ($1/r$) remains constant. As such, the Frischian demands show that demand at period $t$, will only depend on the prices in this period $p^t$ once controlling for the value of $r$. So all intertemporal effects (from outside period $t$) are captured by the single sufficient (but unobservable) statistic $r$.


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