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"?


1 Answer 1


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) or the nice working paper of Browning (1993)

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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