AP Macroeconomics: Math Geeks Chapter 1
Marginal Analysis
Throughout the text you will see references to the word "marginal," meaning "change in." This idea
has its mathematical roots in the concept of the slope. For example, suppose that the benefit (B) you
derive from your study of economics is proportional to the time (T) you spend studying: B = a x T.
Then the "marginal" benefit of your time is given by the change in B for a given change in T, or:
Marginal benefit =
= a, where a is the slope of the line relating B and T.
More generally, suppose that B is any positive function of time spent: B = b(T). In this case, the
marginal benefit is given by the derivative of B with respect to T:
Marginal benefit =
[See note 3-19]
= b’(T). When evaluated at time T, b’(T) is the slope of the line tangent to b(T).
Ceteris Paribus
In mathematics, the "other-things-equal assumption" is embodied in the concept of the partial
derivative. Suppose, for example, that the amount of Pepsi consumed (QP) depends on its price (PP),
the price of Coca-Cola (PC), consumer incomes (I), and preferences (Z) according to the relationship
QP = f(PP, PC, I, Z). If we wish to investigate the effect of a change in the amount of Pepsi brought
about by a change in its price, other things equal, we would compute
the quantity with respect to its price.
, the partial derivative of
Consider the following example from Rao and Miller. They estimate the demand for Ceylonese Tea to
be QC = 1.85 - 2.10PC + 0.20PB + 1.56PI, where QC is the quantity of Ceylonese tea demanded, PC is
its price, and PB and PI are the prices of Brazilian coffee and Indian tea, respectively (all variables are
expressed in logarithms.) If we wish to know what happens to the quantity of Ceylonese tea
demanded when its price changes, with no change in the prices of Brazilian coffee or Indian tea, we
can simply calculate
= - 2.10. This tells us that each unit increase in the (logarithm of the)
price of Ceylonese tea reduces its (logarithm of) consumption by 2.10. Likewise, each increase in the
price of Brazilian coffee is found to increase the quantity of Ceylonese tea, all else constant, by
= 0.20.
Slope of a Nonlinear Curve
Finding the slope of a straight line of the form Y = a + bX is rather simple: the slope is given by b.
What about the more general case of the unspecified equation Y = f(X)—How do we find its slope? If
f(X) is nonlinear, its slope may be different at each point, but as long as the function is continuous and
differentiable, its slope is given by the derivative of f(X) evaluated at that point.
Consider for example the function Y = f(X) = X 2 + 4. Suppose we want to know its slope at the point
X = 3, Y = 13. The derivative of this function is f ’(X) = 2X, which takes on the value 6 when X = 3.
Hence, the slope of this equation is 6 at the point (3, 13).