2.4 functional form

Proportions and Percentages:

Example of quantities reported in the form of percentage: interest rate, inflation
rate, and unemployment rate ect. It is simply obtained by multiplying a proportion
by 100.

If the annual return on Certificate Deposit is 7.6% and you invest $3000 in CD at the
beginning of the year your interest on income is (3000*0.76) = $288. SURPISE!!

In econometrics especially in applied economics we are usually interested in
measuring the changes in various quantities.

Eg:

= initial value and
=subsequent value.

Then the proportional change in x in moving from is simply:

Assuming

It is common to state changes in percentages. And the percentage change in is
going from is simply 100 times the proportional change:

Where is read as percent change in

But what if the variable of interest is itself if percentage?

Suppose;

: Percentage of adults in particular city having a college education where,
=24 (24% have college education)
=30

There are two quantities we can compute to describe how the percentage of college-
educated people has changes:

1. Original Form. The change in ,

. The percentage of people with college education
has changed by 6. “Percentage point change”.
2. Log Form.

Using the equation:
“Percentage Change”

Linear Relationship:

One unit change in results in same change in regardless of the initial value of . It
is the same as saying marginal effect of

Law of Diminishing Marginal Returns: Is not consistent with linear function.

Example Banana.

A non-linear function is characterized by the fact that change in for a given change
in depends on the starting value of

Quadratic Functions:

One way to capture diminishing returns is to add a quadratic term to a linear
relationship:

Where are parameters.

When and the relationship between and are parabolic in shape.

Using calculus it can be shown that the maximum of the function occurs at the point

And

The diminishing marginal effect of is easily seen from the following graph.

Suppose we start with low value of and then increases by some amount say
This has a larger effect on then if we start at a higher value of and increase by
same amount . Thus once an increase in actually decreases

This effect of marginal effect is captured by the slope:

.

Natural Logarithm:

Log-Log Model:Also called log-log, double log, or log linear models.

where:

Thus, it is also called the Constant Elasticity Model.

And:

is the elasticity of with respect to Assuming

Eg: if is the quantity demanded and is the price and these variables are related
by:

Then, the price elasticity of demand is -1.25. Roughly a 1% increase in price leads to
1.25% decrease in quantity demanded.

Exponential Function: Inverse of logs:

Which can alternatively expressed as:

Thus:

Note:

Log-Lin Model or Semi Log Elasticity or Log-Level:

Percentage change in when increases by one unit.

Example:

= 9.4
Interpretation:

Thus it follows that one more years of education increases the wage by
approximately 9.4%. Since the percentage change in wage is the same for each
additional year of education, the change in wage for an extra year of education
increases as education increases; in other words it implies an increasing return to
education.

The semi elasticity is constant and equal to

Lin-Log Model or Level Log:

Where:

In other words is the unit change in when increase by 1 %.

Example:

Or 1% increase in wage increases the weekly hours worked by 0.45 or slightly less
than one-half of an hour. If wage increases by 10% then

However we would not want to use this approximation for much larger percentage
change in wage!!

2.4 functional form

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2.4 functional form