- To motivate the simple linear regression model LRM,
we will discuss a simple economic policy question:
Would an increase in the minimum wage increase unemployment among low-wage workers?
- Let's put this question in the context of demand and supply of labor.
The supply curve of labor is the number of people willing to work as
a function of the market wage. From Econ 110, the location of the curve
depends upon people's taste for work, prices of other goods, etc.
- The demand curve for labor is the number of workers firms are willing
to hire as a function of the market wage. The location of the curve depends
upon technology, prices of other factors of production, etc.
The competitive equilibrium looks like the following:
- The result of the model is an equilibrium wage w* and an equilibrium
employment level N*. People who aren't willing to work at w*, but
are willing to work for some wage, would be unemployed or in
school or just hanging out.
- A minimum wage
acts as a price floor within this simple model.
If the minimum wage is not binding
, then it should have no effect
on employment or unemployment. A binding minimum wage has the following
(Note: the upward sloping line should be labeled Labour Supply.) The
result is a lower level of employment (Nm), or higher rate of unemployment. Of
course, not everyone works at a minimum wage. We need to interpret this as the
market for low-skill (low-wage) workers, such as teenagers who have dropped out of
- With a log-linear demand curve we would write:
where u captures
determinants of labour demand: technology, output prices,
prices of other factors of production, etc.
is the quantity demanded written
as a function of the market wage w. (You should refer to your intermediate
microeconomics textbook or notes to remind yourself that a Cobb-Douglas
production function generates a demand function of the form (D1).)
- The supply curve is irrelevant to the market outcome, because the price
floor means that employment is determined by the supply curve. In other
words, when the price floor (mininum wage) is binding, then the observed
employment rate N is equal to the quantity demanded: N =
the price floor is not binding, then N is found at the intersection of
labour demand and labour supply. Even if they are both simple
log-linear curves, the market outcome N would demand upon four parameters.
With a price floor, we can focus attention upon one equation and
two unknown parameters.
- The slope of the demand curve (graphing
w on the vertical axis) would be
. Since the values of
depend upon unobservable things such as production
functions, these values are unknown constants or
perhaps population parameters.
(Even if we assume that we know the form of the demand curve we
aren't so bold as to assume the slope and intercept.)
- How much
(and through it unemployment)
responds to the minimum wage depends upon the slope of the demand curve:
Recalling the definition of supply elastiticity, we see that
the elasticity of the demand curve. Economic theory predicts that
(downward sloping demand curve). So since
our initial question concerns a change in the
mininum wage we can focus on not two unknown parameters but only
one unknonwn parameter. We therefore can re-form our
initial question into a much easier question:
How big is
in absolute value?
- If we look across labor markets (states, provinces, over time), then we will
get several pairs of minimum wages
and employment rates
is an index the different labor market.
According the demand-curve model, the relationship between the two
values depends upon the other factors operating in the market,
which we might call
- For example,
differences in the local wages commanded by high skilled workers (prices of
other factors of production) would shift the demand curve
for low skilled workers. Data from labour markets that have different
minimum wages might therefore look like the following:
In other words, the slope of the supply curve causes the cloud of
observations to be downward sloping as long as the influence of other factors
is small relative to the slope. But if the variation
is relatively large (or the variation in mininum wages
is relatively small) the cloud of points might appear flat even though
is less than zero.
- To answer even our simplified question,
we would need to answer another question first:
Using sample information, can we isolate the slope
the other factors that disturb the relationship between N and w?
- The change in other factors of demand across observations in the sample
implies that we can't precisely measure the parameters
from data. Instead,
we must estimate the
unknown population paramter
from data on a
sample of employment rates and minimum wages.
- Suppose we have somehow carried out this
estimation. Let the estimate of
, it is simple to form the prediction:
If a politician told you how much the mininum wage was going to change,
you could could plug into this formula and predict for him/her
the percentage change in
. So the easy part of the job is
carrying out the prediction. The hard part is coming up
with an estimate of
in the first place.
- Because your estimate
(it depends on sample information which includes the effect of the
disturbance factor u), your prediction
also a random variable.
Your prediction about the effect of
changing the mininum wage is itself an estimate. You, and
most definitely your client or audience, will ask the further question:
How precise is a prediction based on the estimate
Wanting to know
how precise your estimate is trying to
infer something about the
actual change based on your prediction. You might be asked to form
a confidence interval for the
actual employment response. Or, you might be asked to
test the hypothesis that there is
no employment response,
which is the same as testing the hypothesis that
- We begin by asking a specific policy question about
the effect of raising the mininum wage.
- Using basic economic theory and some
simplifying assumptions (like log-linear demand),
we reduced the question to asking a
specific numerical question about
one theoretical construct, the slope of
the demand curve for low wage workers.
- We then realized that finding this value requires us
to ask a specific statistical question
about estimating a slope from a cloud of points.
- If we are successful in carrying out this step, then
we are lead to ask a specific inferential
question about how precise our estimate is.
- The goal of econometrics
To use statistical techniques to
economic models using economic data.