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%From: daniel@cabtep2.cnea.gov.ar
%Date: Sat, 6 Jul 2002 16:04:45 -0500

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{\Large \bf Comments on job market models}\\
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{\bf Daniel Badagnani \footnote{email: daniel@cabtep2.cnea.gov.ar}}\\
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{\bf abstract}\\
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We point out that the condition of maximization of the benefit is not
$Y'(n)=\omega$ in general, due to the eventual dependence of the salary
$\omega$ on the employement $n$. We show that in keynessian models, where
the relation between $\omega$ and $n$ is given, we need to correct the
relation between the salary and rent, but in classical models, where we
use the benefit to find a demand function, the maximization of the 
benefit do not permit it. This is a case in wich, regardless the interplay 
between offer and demand is present, the price and level of interchange of a 
good (human work) cannot be determined by the intersection of offer and 
demand functions.

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\section{review of the simplest model}

Let us start by reviewing the job market in the simplest classical model.
Let $n$ be the employement, $Y(n)$ the rent, and $\omega$ the salary. Define
the benefit as $B(n)=Y(n)-n \omega$. We want to find out what salary would
be payed to n contracted workers, $\omega(n)$. That should be the salary
such that the benefit is maximized. That happens when $B'(n)=0$, that is,
$\omega(n)=Y'(n)$. This is the demand for jobs. The jobs market completes
with the offer curve, wich is supposed to be a growing function of the
salary representing the number of workers willing to work for that salary.
At equilibrium, the offer and demand should be equal, determining the
rent, the employement and the salary.

This setting follows the paradigm of free market ideology, in wich the
interplay between offer and demand is supposed to
self regulate markets. Under this scheme there is no room for unemployement
at equilibrium, because if too many people seek for jobs, employers lower
the salary and so the offer get lower. Recall that in the keynessian
approach the relation between employement and salary is supposed to be
determined outside market, for instance by means of minimum salary
regulations. In the next two sections we will not enter in any controversy
about werther a model describes or not accurately any actual present or past
economy or even if some model feature is or is not desirable, and will
concentrate in the mathematical structure of the model and its internal
consistency. In section \ref{employement_models} we will discuss some
semirealistic modelling in order to establish connections with keynessian
models. In the last section we give the conclussions.

\section{The mistake}

Lets see again the expression for the benefit

\begin{equation}
B(n)=Y(n)-n\omega
\label{benefit}
\end{equation}

We found its maximum by deriving it respect to $n$ and setting the result to
zero. Now, the hypotesis made is that $\omega$ is an unknown function of
$n$, the 'demand curve', and we forgot completely about $\omega'(n)$. The
actual equation for $\omega$ is thus a first order differential equation

\begin{equation}
0=Y'(n)-\omega(n)-n\omega'(n)
\label{ODE}
\end{equation}

If we forget about $\omega'$, we are considering $\omega$ as a
given parameter, so we would be in the simplest keynesian hypotesis. If
we were considering a keynessian model, where the relation between $\omega$
and $n$ is given in advance, then we are done, having in mind that now the
condition for maximum benefit is \ref{ODE} and not $\omega=Y'(n)$. Let's try
to correct the classical model and find a demand curve solving
\ref{ODE}. As it is a first order equation, it needs a boundary condition.
We choose $B(0)=0$ (no work, no benefit). Let us replace in \ref{ODE}
$\omega$ by $g/n$; then it becomes

\begin{equation}
g'(n)=Y'(n)
\label{ODE2}
\end{equation}

The most general solution for \ref{ODE2} is g(n)=Y(n)+C. Replacing it back
into \ref{benefit} we get $B(n)=-C$, so by the boundary condition we get
$C=0$, and the solution is

\begin{equation}
\omega(n)=\frac{Y(n)}{n}
\label{soln}
\end{equation}

The salary would be then the rent divided among the workers, and the benefit
is zero. We got not the maximum but the minimum of the benefit, wich is the
other condition for the anihilation of the derivative. Why \ref{ODE} didn't
show the maximum as another solution? Well, the maximum can be found by
inspection, and correspond to $\omega=0$ and $n=\infty$, so there is not a
function $\omega(n)$ that maximizes $B(n)$.

\section{What happened?}

The last section was a cumbersome way to show something that could be told
by simple inspection: if the only criterion to choose the salary is to
maximize the benefit as given by \ref{benefit}, then it would be set to
zero. So, there is not such a thing like a 'demand curve' $n(w)$. Of course,
it doesn't mean that there is no job demand: if we want a benefit we have to
employ workers. We can talk about offer curves: for instance the number 
$n$ of people willing to work for the salary $\omega$, or the number $n$ of 
people seeking for a job while employers offer a salary $\omega$ (notice the 
subtle difference in definition, wich will be analyzed in the next section), 
but in order to maximize the $B(n)$ we need the actual relation between 
employement and salary, wich have to be modelled appart. No doubt that offer 
will make the employement a growing function of the salary, but its relation 
to some offer has to be postuled (hopefully derived) appart. Then, replacing 
in $B(n)$ this function and deriving respect to $n$, we find that the 
maximum benefit is \ref{ODE}.

So here is a price not determined by the intersection of an offer and demand 
function. Of course, we cannot ignore the key role that offer and demand 
plays in any trade, but offer-demand curves is not the most general model 
of offer-demmand interplay. We find that if we want to keep as premise the 
maximization of the benefit we have to abandon such description and pass to 
a keynessian approach. What if we reject benefit maximization? Well, if 
there aren't employers and we suppose that 'the society' tries to maximize 
the rent $Y(n)$, then the demand curve would be 
$\omega=\frac{Y(n)}{n}$\footnote{$Y(n)/n$, under the usual asumptions 
$Y'(n)>0$ and $Y''(n)<0$, is qualitatively similar to $Y'(n)$}, 
wich would intersect with an offer curve, and we would recover almost 
the construction. But then it would be strange because no one 'buys' human 
work, or they does it in an indirect way, so this 'salary' would not be 
strictly a 'price'.

\section{employement models and contact with keynessian models}

\label{employement_models}

Let us look at the employement and offer functions. In general, the second
should be somewhat higher than the first. The simplest posibility is that
the employement at equilibrium equals the offer.
But there is no a priori reason for that, and we should analyze it
carefully. Let us look first at the offer function, in a society in wich
most of the population earns its money from a job. 
First we should be clear about what offer are we talking about. The relevant 
offer in order to analyze unemployement is the number of workers seeking for 
a job while employers pay $\omega.$ Depending on prices, cultural 
background, laws, etc, there would be a salary considered as acceptable, say 
$\omega_0$. As at least one person per domestic unit needs a job, there is 
also a minimum demand, say $n_0$. Until the salary reaches $\omega_0$, the 
demand should rise only slightly from $n_0$. Once the offered salary is 
above $\omega$, more people will fill tempted to seek for a job, and the 
demand should increase fastly until it reaches a value close to the total 
adult population. The situation is different of that of a pre-industrial 
society, in wich domestic units could choose between subsistence economy in 
the countryside or employement, so there is no an $n_0$. Let us look now at 
the employement function: it is reasonable that above $\omega_0$ the 
employement almost follows the demand. That is because those people who 
doesn't need the job will eventually be discouraged if they do not find it, 
and in stationary situation the demand would drop close to the actual 
employement. Below $\omega_0$ the situation changes drastically:  people 
will look for jobs with salary $\omega_0$ at least and will be offered a 
lower salary, so the less the salary is, the less people accept the job, 
thus employement fall with $\omega$, and unemployement rises. The 
employement in such a stage thus follow the other thinkable offer function: 
the number of workers willing to work for the offered salary. It might be 
argued that eventually people would accept its poverty and finally if the 
conditions remain long enough both offer functions would equal. But such a 
cultural change may take decades, and is clearly a structural change. Of 
course there is no need for believing that the unemployed are allways the 
same, we could have some kind of dynamical equilibrium of people taking and 
leaving part-time jobs. So unemployement can be an 'equilibrium' solution in 
the usual sense of the expression. Anyway, getting used to poverty is not a 
desirable solution for unemployement: we should get both wealthiness and 
full employement.

This is a 
keynessian-like scenario. Take for instance the case in wich the demand is 
$n_0$ until $\omega_0$, and then becomes rigid, and the employement function 
is completely rigid at $\omega_0$. This is the only case in wich  
we have $Y'(n)=\omega$. This is the simplest keynessian hypothesis, 
so simple that it does not take into account any offer. 

Of course we should 
insert this in a more complete model to see what happen with the prices. 
Werther we stop here in modelling employement, or we try a most complete 
model, is a matter of choice. It looks like, at least in the job market, the 
simplest models are the keynessian ones, because the classic ones are just 
wrong.

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